Merge pull request #4043 from martin-frbg/lapack809-811-812
Fix typos in LAPACK comments (Reference-LAPACK PRs 809,811,812)
This commit is contained in:
Martin Kroeker
GitHub
commit
617e8bcfe7
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@ -52,10 +52,10 @@
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*> are computed and stored in the arrays U and V, respectively. The diagonal
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*> of [SIGMA] is computed and stored in the array SVA.
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*> \endverbatim
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*>
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*> Arguments:
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*> ==========
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*>
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*
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* Arguments:
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* ==========
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*
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*> \param[in] JOBA
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*> \verbatim
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*> JOBA is CHARACTER*1
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@ -151,7 +151,7 @@
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*> transposed A if A^* seems to be better with respect to convergence.
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*> If the matrix is not square, JOBT is ignored.
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*> The decision is based on two values of entropy over the adjoint
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*> orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
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*> orbit of A^* * A. See the descriptions of RWORK(6) and RWORK(7).
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*> = 'T': transpose if entropy test indicates possibly faster
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*> convergence of Jacobi process if A^* is taken as input. If A is
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*> replaced with A^*, then the row pivoting is included automatically.
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@ -209,11 +209,11 @@
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*> \verbatim
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*> SVA is REAL array, dimension (N)
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*> On exit,
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*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
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*> computation SVA contains Euclidean column norms of the
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*> - For RWORK(1)/RWORK(2) = ONE: The singular values of A. During
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*> the computation SVA contains Euclidean column norms of the
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*> iterated matrices in the array A.
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*> - For WORK(1) .NE. WORK(2): The singular values of A are
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*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
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*> - For RWORK(1) .NE. RWORK(2): The singular values of A are
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*> (RWORK(1)/RWORK(2)) * SVA(1:N). This factored form is used if
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*> sigma_max(A) overflows or if small singular values have been
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*> saved from underflow by scaling the input matrix A.
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*> - If JOBR='R' then some of the singular values may be returned
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@ -104,6 +104,7 @@
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*> \endverbatim
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= MB*M.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -106,6 +106,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= NB*N.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -53,7 +53,7 @@
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*>
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*> S*x = w*P*x, (y**H)*S = w*(y**H)*P,
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*>
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*> where y**H denotes the conjugate tranpose of y.
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*> where y**H denotes the conjugate transpose of y.
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*> The eigenvalues are not input to this routine, but are computed
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*> directly from the diagonal elements of S and P.
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*>
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@ -154,7 +154,7 @@
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*> \verbatim
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*> VR is COMPLEX array, dimension (LDVR,MM)
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*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
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*> contain an N-by-N matrix Q (usually the unitary matrix Z
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*> contain an N-by-N matrix Z (usually the unitary matrix Z
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*> of right Schur vectors returned by CHGEQZ).
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*> On exit, if SIDE = 'R' or 'B', VR contains:
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*> if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
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@ -259,7 +259,7 @@
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EXTERNAL LSAME, SLAMCH, CLADIV
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* ..
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* .. External Subroutines ..
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EXTERNAL CGEMV, SLABAD, XERBLA
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EXTERNAL CGEMV, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, MIN, REAL
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@ -367,7 +367,6 @@
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*
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SAFMIN = SLAMCH( 'Safe minimum' )
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BIG = ONE / SAFMIN
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CALL SLABAD( SAFMIN, BIG )
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ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' )
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SMALL = SAFMIN*N / ULP
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BIG = ONE / SMALL
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@ -212,13 +212,13 @@
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*> LRWORK is INTEGER
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*> The dimension of the array RWORK.
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*>
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*> If LRWORK = -1, then a workspace query is assumed; the routine
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*> If LRWORK=-1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK and RWORK
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*> arrays, returns this value as the first entry of the WORK
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*> and RWORK array, respectively, and no error message related
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*> to LWORK or LRWORK is issued by XERBLA.
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*> \endverbatim
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*
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
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@ -133,6 +133,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= (M+NB)*N.
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*> If LWORK = -1, then a workspace query is assumed.
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*> The routine only calculates the optimal size of the WORK
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@ -104,6 +104,7 @@
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*> \endverbatim
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= MB*M.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -106,6 +106,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= NB*N.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -133,6 +133,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= (M+NB)*N.
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*> If LWORK = -1, then a workspace query is assumed.
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*> The routine only calculates the optimal size of the WORK
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@ -104,6 +104,7 @@
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*> \endverbatim
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= MB * M.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -106,6 +106,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= NB*N.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -133,6 +133,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= (M+NB)*N.
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*> If LWORK = -1, then a workspace query is assumed.
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*> The routine only calculates the optimal size of the WORK
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@ -52,10 +52,10 @@
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*> are computed and stored in the arrays U and V, respectively. The diagonal
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*> of [SIGMA] is computed and stored in the array SVA.
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*> \endverbatim
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*>
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*> Arguments:
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*> ==========
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*>
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*
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* Arguments:
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* ==========
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*
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*> \param[in] JOBA
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*> \verbatim
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*> JOBA is CHARACTER*1
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@ -151,7 +151,7 @@
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*> transposed A if A^* seems to be better with respect to convergence.
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*> If the matrix is not square, JOBT is ignored.
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*> The decision is based on two values of entropy over the adjoint
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*> orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
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*> orbit of A^* * A. See the descriptions of RWORK(6) and RWORK(7).
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*> = 'T': transpose if entropy test indicates possibly faster
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*> convergence of Jacobi process if A^* is taken as input. If A is
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*> replaced with A^*, then the row pivoting is included automatically.
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@ -209,11 +209,11 @@
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*> \verbatim
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*> SVA is DOUBLE PRECISION array, dimension (N)
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*> On exit,
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*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
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*> computation SVA contains Euclidean column norms of the
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*> - For RWORK(1)/RWORK(2) = ONE: The singular values of A. During
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*> the computation SVA contains Euclidean column norms of the
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*> iterated matrices in the array A.
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*> - For WORK(1) .NE. WORK(2): The singular values of A are
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*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
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*> - For RWORK(1) .NE. RWORK(2): The singular values of A are
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*> (RWORK(1)/RWORK(2)) * SVA(1:N). This factored form is used if
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*> sigma_max(A) overflows or if small singular values have been
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*> saved from underflow by scaling the input matrix A.
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*> - If JOBR='R' then some of the singular values may be returned
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@ -104,6 +104,7 @@
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*> \endverbatim
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= MB*M.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -106,6 +106,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= NB*N.
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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@ -53,7 +53,7 @@
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*>
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*> S*x = w*P*x, (y**H)*S = w*(y**H)*P,
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*>
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*> where y**H denotes the conjugate tranpose of y.
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*> where y**H denotes the conjugate transpose of y.
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*> The eigenvalues are not input to this routine, but are computed
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*> directly from the diagonal elements of S and P.
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*>
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@ -154,7 +154,7 @@
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*> \verbatim
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*> VR is COMPLEX*16 array, dimension (LDVR,MM)
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*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
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*> contain an N-by-N matrix Q (usually the unitary matrix Z
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*> contain an N-by-N matrix Z (usually the unitary matrix Z
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*> of right Schur vectors returned by ZHGEQZ).
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*> On exit, if SIDE = 'R' or 'B', VR contains:
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*> if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
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@ -259,7 +259,7 @@
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EXTERNAL LSAME, DLAMCH, ZLADIV
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* ..
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* .. External Subroutines ..
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EXTERNAL DLABAD, XERBLA, ZGEMV
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EXTERNAL XERBLA, ZGEMV
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
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@ -367,7 +367,6 @@
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*
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SAFMIN = DLAMCH( 'Safe minimum' )
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BIG = ONE / SAFMIN
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CALL DLABAD( SAFMIN, BIG )
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ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
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SMALL = SAFMIN*N / ULP
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BIG = ONE / SMALL
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@ -211,13 +211,13 @@
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*> LRWORK is INTEGER
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*> The dimension of the array RWORK.
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*>
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*> If LRWORK = -1, then a workspace query is assumed; the routine
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*> If LRWORK=-1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK and RWORK
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*> arrays, returns this value as the first entry of the WORK
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*> and RWORK array, respectively, and no error message related
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*> to LWORK or LRWORK is issued by XERBLA.
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*> \endverbatim
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*
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
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@ -133,6 +133,7 @@
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= (M+NB)*N.
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*> If LWORK = -1, then a workspace query is assumed.
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*> The routine only calculates the optimal size of the WORK
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@ -0,0 +1,644 @@
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*> \brief <b> CGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b>
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download CGBSVX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgbsvx.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgbsvx.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgbsvx.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE CGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
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* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
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* RCOND, FERR, BERR, WORK, RWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER EQUED, FACT, TRANS
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* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
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* REAL RCOND
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* REAL BERR( * ), C( * ), FERR( * ), R( * ),
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* $ RWORK( * )
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* COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
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* $ WORK( * ), X( LDX, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> CGBSVX uses the LU factorization to compute the solution to a complex
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*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
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*> where A is a band matrix of order N with KL subdiagonals and KU
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*> superdiagonals, and X and B are N-by-NRHS matrices.
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*>
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*> Error bounds on the solution and a condition estimate are also
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*> provided.
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*> \endverbatim
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*
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*> \par Description:
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* =================
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*>
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*> \verbatim
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*>
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*> The following steps are performed by this subroutine:
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*>
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*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
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*> the system:
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*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
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*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
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*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
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*> Whether or not the system will be equilibrated depends on the
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*> scaling of the matrix A, but if equilibration is used, A is
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*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
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*> or diag(C)*B (if TRANS = 'T' or 'C').
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*>
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*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
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*> matrix A (after equilibration if FACT = 'E') as
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*> A = L * U,
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*> where L is a product of permutation and unit lower triangular
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*> matrices with KL subdiagonals, and U is upper triangular with
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*> KL+KU superdiagonals.
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*>
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*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
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*> returns with INFO = i. Otherwise, the factored form of A is used
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*> to estimate the condition number of the matrix A. If the
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*> reciprocal of the condition number is less than machine precision,
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*> INFO = N+1 is returned as a warning, but the routine still goes on
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*> to solve for X and compute error bounds as described below.
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*>
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*> 4. The system of equations is solved for X using the factored form
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*> of A.
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*>
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*> 5. Iterative refinement is applied to improve the computed solution
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*> matrix and calculate error bounds and backward error estimates
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*> for it.
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*>
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*> 6. If equilibration was used, the matrix X is premultiplied by
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*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
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*> that it solves the original system before equilibration.
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*> \endverbatim
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*
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* Arguments:
|
||||
* ==========
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||||
*
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*> \param[in] FACT
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*> \verbatim
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*> FACT is CHARACTER*1
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*> Specifies whether or not the factored form of the matrix A is
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*> supplied on entry, and if not, whether the matrix A should be
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*> equilibrated before it is factored.
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*> = 'F': On entry, AFB and IPIV contain the factored form of
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*> A. If EQUED is not 'N', the matrix A has been
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*> equilibrated with scaling factors given by R and C.
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*> AB, AFB, and IPIV are not modified.
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*> = 'N': The matrix A will be copied to AFB and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AFB and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations.
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Conjugate transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KL
|
||||
*> \verbatim
|
||||
*> KL is INTEGER
|
||||
*> The number of subdiagonals within the band of A. KL >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KU
|
||||
*> \verbatim
|
||||
*> KU is INTEGER
|
||||
*> The number of superdiagonals within the band of A. KU >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AB
|
||||
*> \verbatim
|
||||
*> AB is COMPLEX array, dimension (LDAB,N)
|
||||
*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
|
||||
*> The j-th column of A is stored in the j-th column of the
|
||||
*> array AB as follows:
|
||||
*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
|
||||
*>
|
||||
*> If FACT = 'F' and EQUED is not 'N', then A must have been
|
||||
*> equilibrated by the scaling factors in R and/or C. AB is not
|
||||
*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
|
||||
*> EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAB
|
||||
*> \verbatim
|
||||
*> LDAB is INTEGER
|
||||
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AFB
|
||||
*> \verbatim
|
||||
*> AFB is COMPLEX array, dimension (LDAFB,N)
|
||||
*> If FACT = 'F', then AFB is an input argument and on entry
|
||||
*> contains details of the LU factorization of the band matrix
|
||||
*> A, as computed by CGBTRF. U is stored as an upper triangular
|
||||
*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
|
||||
*> and the multipliers used during the factorization are stored
|
||||
*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
|
||||
*> the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of A.
|
||||
*>
|
||||
*> If FACT = 'E', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of the equilibrated
|
||||
*> matrix A (see the description of AB for the form of the
|
||||
*> equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAFB
|
||||
*> \verbatim
|
||||
*> LDAFB is INTEGER
|
||||
*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> as computed by CGBTRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is REAL array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is REAL array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is COMPLEX array, dimension (LDB,NRHS)
|
||||
*> On entry, the right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is COMPLEX array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is REAL
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is REAL array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is REAL array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is COMPLEX array, dimension (2*N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RWORK
|
||||
*> \verbatim
|
||||
*> RWORK is REAL array, dimension (MAX(1,N))
|
||||
*> On exit, RWORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If RWORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> RWORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization
|
||||
*> has been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> value of RCOND would suggest.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup complexGBsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE CGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
||||
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
||||
$ RCOND, FERR, BERR, WORK, RWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
|
||||
REAL RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * )
|
||||
REAL BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ RWORK( * )
|
||||
COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
* Moved setting of INFO = N+1 so INFO does not subsequently get
|
||||
* overwritten. Sven, 17 Mar 05.
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J, J1, J2
|
||||
REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
REAL CLANGB, CLANTB, SLAMCH
|
||||
EXTERNAL LSAME, CLANGB, CLANTB, SLAMCH
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL CCOPY, CGBCON, CGBEQU, CGBRFS, CGBTRF, CGBTRS,
|
||||
$ CLACPY, CLAQGB, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = SLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( KL.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( KU.LT.0 ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
|
||||
INFO = -10
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -12
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -13
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -18
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'CGBSVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL CGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL CLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of the band matrix A.
|
||||
*
|
||||
DO 70 J = 1, N
|
||||
J1 = MAX( J-KU, 1 )
|
||||
J2 = MIN( J+KL, N )
|
||||
CALL CCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
|
||||
$ AFB( KL+KU+1-J+J1, J ), 1 )
|
||||
70 CONTINUE
|
||||
*
|
||||
CALL CGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
ANORM = ZERO
|
||||
DO 90 J = 1, INFO
|
||||
DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
|
||||
ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
|
||||
80 CONTINUE
|
||||
90 CONTINUE
|
||||
RPVGRW = CLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
|
||||
$ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
|
||||
$ RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = ANORM / RPVGRW
|
||||
END IF
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = CLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
|
||||
RPVGRW = CLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = CLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL CGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
|
||||
$ WORK, RWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL CGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
|
||||
$ INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL CGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
|
||||
$ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 140 J = 1, NRHS
|
||||
DO 130 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
130 CONTINUE
|
||||
140 CONTINUE
|
||||
DO 150 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
150 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
*
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RETURN
|
||||
*
|
||||
* End of CGBSVX
|
||||
*
|
||||
END
|
||||
File diff suppressed because it is too large
Load Diff
|
|
@ -0,0 +1,602 @@
|
|||
*> \brief <b> CGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download CGESVX + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgesvx.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgesvx.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgesvx.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE CGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
* EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
* WORK, RWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER EQUED, FACT, TRANS
|
||||
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
* REAL RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* INTEGER IPIV( * )
|
||||
* REAL BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
* $ RWORK( * )
|
||||
* COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
* $ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> CGESVX uses the LU factorization to compute the solution to a complex
|
||||
*> system of linear equations
|
||||
*> A * X = B,
|
||||
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
|
||||
*>
|
||||
*> Error bounds on the solution and a condition estimate are also
|
||||
*> provided.
|
||||
*> \endverbatim
|
||||
*
|
||||
*> \par Description:
|
||||
* =================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> The following steps are performed:
|
||||
*>
|
||||
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
|
||||
*> the system:
|
||||
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
|
||||
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
|
||||
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
|
||||
*> Whether or not the system will be equilibrated depends on the
|
||||
*> scaling of the matrix A, but if equilibration is used, A is
|
||||
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
|
||||
*> or diag(C)*B (if TRANS = 'T' or 'C').
|
||||
*>
|
||||
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
|
||||
*> matrix A (after equilibration if FACT = 'E') as
|
||||
*> A = P * L * U,
|
||||
*> where P is a permutation matrix, L is a unit lower triangular
|
||||
*> matrix, and U is upper triangular.
|
||||
*>
|
||||
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
|
||||
*> returns with INFO = i. Otherwise, the factored form of A is used
|
||||
*> to estimate the condition number of the matrix A. If the
|
||||
*> reciprocal of the condition number is less than machine precision,
|
||||
*> INFO = N+1 is returned as a warning, but the routine still goes on
|
||||
*> to solve for X and compute error bounds as described below.
|
||||
*>
|
||||
*> 4. The system of equations is solved for X using the factored form
|
||||
*> of A.
|
||||
*>
|
||||
*> 5. Iterative refinement is applied to improve the computed solution
|
||||
*> matrix and calculate error bounds and backward error estimates
|
||||
*> for it.
|
||||
*>
|
||||
*> 6. If equilibration was used, the matrix X is premultiplied by
|
||||
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
|
||||
*> that it solves the original system before equilibration.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] FACT
|
||||
*> \verbatim
|
||||
*> FACT is CHARACTER*1
|
||||
*> Specifies whether or not the factored form of the matrix A is
|
||||
*> supplied on entry, and if not, whether the matrix A should be
|
||||
*> equilibrated before it is factored.
|
||||
*> = 'F': On entry, AF and IPIV contain the factored form of A.
|
||||
*> If EQUED is not 'N', the matrix A has been
|
||||
*> equilibrated with scaling factors given by R and C.
|
||||
*> A, AF, and IPIV are not modified.
|
||||
*> = 'N': The matrix A will be copied to AF and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AF and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations:
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Conjugate transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is COMPLEX array, dimension (LDA,N)
|
||||
*> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
|
||||
*> not 'N', then A must have been equilibrated by the scaling
|
||||
*> factors in R and/or C. A is not modified if FACT = 'F' or
|
||||
*> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of the array A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AF
|
||||
*> \verbatim
|
||||
*> AF is COMPLEX array, dimension (LDAF,N)
|
||||
*> If FACT = 'F', then AF is an input argument and on entry
|
||||
*> contains the factors L and U from the factorization
|
||||
*> A = P*L*U as computed by CGETRF. If EQUED .ne. 'N', then
|
||||
*> AF is the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A (see the description of A for
|
||||
*> the form of the equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAF
|
||||
*> \verbatim
|
||||
*> LDAF is INTEGER
|
||||
*> The leading dimension of the array AF. LDAF >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> as computed by CGETRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is REAL array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is REAL array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is COMPLEX array, dimension (LDB,NRHS)
|
||||
*> On entry, the N-by-NRHS right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is COMPLEX array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is REAL
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is REAL array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is REAL array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is COMPLEX array, dimension (2*N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RWORK
|
||||
*> \verbatim
|
||||
*> RWORK is REAL array, dimension (MAX(1,2*N))
|
||||
*> On exit, RWORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If RWORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> RWORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization has
|
||||
*> been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> value of RCOND would suggest.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup complexGEsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE CGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
$ WORK, RWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
REAL RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * )
|
||||
REAL BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ RWORK( * )
|
||||
COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J
|
||||
REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
REAL CLANGE, CLANTR, SLAMCH
|
||||
EXTERNAL LSAME, CLANGE, CLANTR, SLAMCH
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL CGECON, CGEEQU, CGERFS, CGETRF, CGETRS, CLACPY,
|
||||
$ CLAQGE, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = SLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -10
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -11
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'CGESVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL CGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL CLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
|
||||
$ EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of A.
|
||||
*
|
||||
CALL CLACPY( 'Full', N, N, A, LDA, AF, LDAF )
|
||||
CALL CGETRF( N, N, AF, LDAF, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
RPVGRW = CLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
|
||||
$ RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = CLANGE( 'M', N, INFO, A, LDA, RWORK ) /
|
||||
$ RPVGRW
|
||||
END IF
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = CLANGE( NORM, N, N, A, LDA, RWORK )
|
||||
RPVGRW = CLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = CLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL CGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL CGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL CGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
|
||||
$ LDX, FERR, BERR, WORK, RWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 80 J = 1, NRHS
|
||||
DO 70 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
70 CONTINUE
|
||||
80 CONTINUE
|
||||
DO 90 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
90 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
*
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RETURN
|
||||
*
|
||||
* End of CGESVX
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,639 @@
|
|||
*> \brief <b> DGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download DGBSVX + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsvx.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsvx.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsvx.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
||||
* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
||||
* RCOND, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER EQUED, FACT, TRANS
|
||||
* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
|
||||
* DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* INTEGER IPIV( * ), IWORK( * )
|
||||
* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
||||
* $ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
* $ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> DGBSVX uses the LU factorization to compute the solution to a real
|
||||
*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
|
||||
*> where A is a band matrix of order N with KL subdiagonals and KU
|
||||
*> superdiagonals, and X and B are N-by-NRHS matrices.
|
||||
*>
|
||||
*> Error bounds on the solution and a condition estimate are also
|
||||
*> provided.
|
||||
*> \endverbatim
|
||||
*
|
||||
*> \par Description:
|
||||
* =================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> The following steps are performed by this subroutine:
|
||||
*>
|
||||
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
|
||||
*> the system:
|
||||
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
|
||||
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
|
||||
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
|
||||
*> Whether or not the system will be equilibrated depends on the
|
||||
*> scaling of the matrix A, but if equilibration is used, A is
|
||||
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
|
||||
*> or diag(C)*B (if TRANS = 'T' or 'C').
|
||||
*>
|
||||
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
|
||||
*> matrix A (after equilibration if FACT = 'E') as
|
||||
*> A = L * U,
|
||||
*> where L is a product of permutation and unit lower triangular
|
||||
*> matrices with KL subdiagonals, and U is upper triangular with
|
||||
*> KL+KU superdiagonals.
|
||||
*>
|
||||
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
|
||||
*> returns with INFO = i. Otherwise, the factored form of A is used
|
||||
*> to estimate the condition number of the matrix A. If the
|
||||
*> reciprocal of the condition number is less than machine precision,
|
||||
*> INFO = N+1 is returned as a warning, but the routine still goes on
|
||||
*> to solve for X and compute error bounds as described below.
|
||||
*>
|
||||
*> 4. The system of equations is solved for X using the factored form
|
||||
*> of A.
|
||||
*>
|
||||
*> 5. Iterative refinement is applied to improve the computed solution
|
||||
*> matrix and calculate error bounds and backward error estimates
|
||||
*> for it.
|
||||
*>
|
||||
*> 6. If equilibration was used, the matrix X is premultiplied by
|
||||
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
|
||||
*> that it solves the original system before equilibration.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] FACT
|
||||
*> \verbatim
|
||||
*> FACT is CHARACTER*1
|
||||
*> Specifies whether or not the factored form of the matrix A is
|
||||
*> supplied on entry, and if not, whether the matrix A should be
|
||||
*> equilibrated before it is factored.
|
||||
*> = 'F': On entry, AFB and IPIV contain the factored form of
|
||||
*> A. If EQUED is not 'N', the matrix A has been
|
||||
*> equilibrated with scaling factors given by R and C.
|
||||
*> AB, AFB, and IPIV are not modified.
|
||||
*> = 'N': The matrix A will be copied to AFB and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AFB and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations.
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KL
|
||||
*> \verbatim
|
||||
*> KL is INTEGER
|
||||
*> The number of subdiagonals within the band of A. KL >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KU
|
||||
*> \verbatim
|
||||
*> KU is INTEGER
|
||||
*> The number of superdiagonals within the band of A. KU >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AB
|
||||
*> \verbatim
|
||||
*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
|
||||
*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
|
||||
*> The j-th column of A is stored in the j-th column of the
|
||||
*> array AB as follows:
|
||||
*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
|
||||
*>
|
||||
*> If FACT = 'F' and EQUED is not 'N', then A must have been
|
||||
*> equilibrated by the scaling factors in R and/or C. AB is not
|
||||
*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
|
||||
*> EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAB
|
||||
*> \verbatim
|
||||
*> LDAB is INTEGER
|
||||
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AFB
|
||||
*> \verbatim
|
||||
*> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
|
||||
*> If FACT = 'F', then AFB is an input argument and on entry
|
||||
*> contains details of the LU factorization of the band matrix
|
||||
*> A, as computed by DGBTRF. U is stored as an upper triangular
|
||||
*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
|
||||
*> and the multipliers used during the factorization are stored
|
||||
*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
|
||||
*> the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of A.
|
||||
*>
|
||||
*> If FACT = 'E', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of the equilibrated
|
||||
*> matrix A (see the description of AB for the form of the
|
||||
*> equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAFB
|
||||
*> \verbatim
|
||||
*> LDAFB is INTEGER
|
||||
*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> as computed by DGBTRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is DOUBLE PRECISION array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is DOUBLE PRECISION array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
|
||||
*> On entry, the right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is DOUBLE PRECISION
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,3*N))
|
||||
*> On exit, WORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If WORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> WORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] IWORK
|
||||
*> \verbatim
|
||||
*> IWORK is INTEGER array, dimension (N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization
|
||||
*> has been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> value of RCOND would suggest.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup doubleGBsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
||||
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
||||
$ RCOND, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
|
||||
DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * ), IWORK( * )
|
||||
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
||||
$ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J, J1, J2
|
||||
DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
DOUBLE PRECISION DLAMCH, DLANGB, DLANTB
|
||||
EXTERNAL LSAME, DLAMCH, DLANGB, DLANTB
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF, DGBTRS,
|
||||
$ DLACPY, DLAQGB, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = DLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( KL.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( KU.LT.0 ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
|
||||
INFO = -10
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -12
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -13
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -18
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'DGBSVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of the band matrix A.
|
||||
*
|
||||
DO 70 J = 1, N
|
||||
J1 = MAX( J-KU, 1 )
|
||||
J2 = MIN( J+KL, N )
|
||||
CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
|
||||
$ AFB( KL+KU+1-J+J1, J ), 1 )
|
||||
70 CONTINUE
|
||||
*
|
||||
CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
ANORM = ZERO
|
||||
DO 90 J = 1, INFO
|
||||
DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
|
||||
ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
|
||||
80 CONTINUE
|
||||
90 CONTINUE
|
||||
RPVGRW = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
|
||||
$ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
|
||||
$ WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = ANORM / RPVGRW
|
||||
END IF
|
||||
WORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
|
||||
RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
|
||||
$ WORK, IWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
|
||||
$ INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
|
||||
$ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 140 J = 1, NRHS
|
||||
DO 130 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
130 CONTINUE
|
||||
140 CONTINUE
|
||||
DO 150 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
150 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
*
|
||||
WORK( 1 ) = RPVGRW
|
||||
RETURN
|
||||
*
|
||||
* End of DGBSVX
|
||||
*
|
||||
END
|
||||
File diff suppressed because it is too large
Load Diff
|
|
@ -0,0 +1,599 @@
|
|||
*> \brief <b> DGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download DGESVX + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvx.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvx.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvx.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
* EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
* WORK, IWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER EQUED, FACT, TRANS
|
||||
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
* DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* INTEGER IPIV( * ), IWORK( * )
|
||||
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
* $ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
* $ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> DGESVX uses the LU factorization to compute the solution to a real
|
||||
*> system of linear equations
|
||||
*> A * X = B,
|
||||
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
|
||||
*>
|
||||
*> Error bounds on the solution and a condition estimate are also
|
||||
*> provided.
|
||||
*> \endverbatim
|
||||
*
|
||||
*> \par Description:
|
||||
* =================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> The following steps are performed:
|
||||
*>
|
||||
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
|
||||
*> the system:
|
||||
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
|
||||
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
|
||||
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
|
||||
*> Whether or not the system will be equilibrated depends on the
|
||||
*> scaling of the matrix A, but if equilibration is used, A is
|
||||
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
|
||||
*> or diag(C)*B (if TRANS = 'T' or 'C').
|
||||
*>
|
||||
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
|
||||
*> matrix A (after equilibration if FACT = 'E') as
|
||||
*> A = P * L * U,
|
||||
*> where P is a permutation matrix, L is a unit lower triangular
|
||||
*> matrix, and U is upper triangular.
|
||||
*>
|
||||
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
|
||||
*> returns with INFO = i. Otherwise, the factored form of A is used
|
||||
*> to estimate the condition number of the matrix A. If the
|
||||
*> reciprocal of the condition number is less than machine precision,
|
||||
*> INFO = N+1 is returned as a warning, but the routine still goes on
|
||||
*> to solve for X and compute error bounds as described below.
|
||||
*>
|
||||
*> 4. The system of equations is solved for X using the factored form
|
||||
*> of A.
|
||||
*>
|
||||
*> 5. Iterative refinement is applied to improve the computed solution
|
||||
*> matrix and calculate error bounds and backward error estimates
|
||||
*> for it.
|
||||
*>
|
||||
*> 6. If equilibration was used, the matrix X is premultiplied by
|
||||
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
|
||||
*> that it solves the original system before equilibration.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] FACT
|
||||
*> \verbatim
|
||||
*> FACT is CHARACTER*1
|
||||
*> Specifies whether or not the factored form of the matrix A is
|
||||
*> supplied on entry, and if not, whether the matrix A should be
|
||||
*> equilibrated before it is factored.
|
||||
*> = 'F': On entry, AF and IPIV contain the factored form of A.
|
||||
*> If EQUED is not 'N', the matrix A has been
|
||||
*> equilibrated with scaling factors given by R and C.
|
||||
*> A, AF, and IPIV are not modified.
|
||||
*> = 'N': The matrix A will be copied to AF and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AF and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations:
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is DOUBLE PRECISION array, dimension (LDA,N)
|
||||
*> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
|
||||
*> not 'N', then A must have been equilibrated by the scaling
|
||||
*> factors in R and/or C. A is not modified if FACT = 'F' or
|
||||
*> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of the array A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AF
|
||||
*> \verbatim
|
||||
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
|
||||
*> If FACT = 'F', then AF is an input argument and on entry
|
||||
*> contains the factors L and U from the factorization
|
||||
*> A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
|
||||
*> AF is the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A (see the description of A for
|
||||
*> the form of the equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAF
|
||||
*> \verbatim
|
||||
*> LDAF is INTEGER
|
||||
*> The leading dimension of the array AF. LDAF >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> as computed by DGETRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is DOUBLE PRECISION array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is DOUBLE PRECISION array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
|
||||
*> On entry, the N-by-NRHS right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is DOUBLE PRECISION
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,4*N))
|
||||
*> On exit, WORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If WORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> WORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] IWORK
|
||||
*> \verbatim
|
||||
*> IWORK is INTEGER array, dimension (N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization has
|
||||
*> been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> value of RCOND would suggest.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup doubleGEsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
$ WORK, IWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * ), IWORK( * )
|
||||
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
$ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J
|
||||
DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
DOUBLE PRECISION DLAMCH, DLANGE, DLANTR
|
||||
EXTERNAL LSAME, DLAMCH, DLANGE, DLANTR
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL DGECON, DGEEQU, DGERFS, DGETRF, DGETRS, DLACPY,
|
||||
$ DLAQGE, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = DLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -10
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -11
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'DGESVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
|
||||
$ EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of A.
|
||||
*
|
||||
CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF )
|
||||
CALL DGETRF( N, N, AF, LDAF, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
|
||||
$ WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = DLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW
|
||||
END IF
|
||||
WORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = DLANGE( NORM, N, N, A, LDA, WORK )
|
||||
RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = DLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
|
||||
$ LDX, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 80 J = 1, NRHS
|
||||
DO 70 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
70 CONTINUE
|
||||
80 CONTINUE
|
||||
DO 90 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
90 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
*
|
||||
WORK( 1 ) = RPVGRW
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
RETURN
|
||||
*
|
||||
* End of DGESVX
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,641 @@
|
|||
*> \brief <b> SGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download SGBSVX + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgbsvx.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgbsvx.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgbsvx.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
||||
* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
||||
* RCOND, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER EQUED, FACT, TRANS
|
||||
* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
|
||||
* REAL RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* INTEGER IPIV( * ), IWORK( * )
|
||||
* REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
||||
* $ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
* $ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> SGBSVX uses the LU factorization to compute the solution to a real
|
||||
*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
|
||||
*> where A is a band matrix of order N with KL subdiagonals and KU
|
||||
*> superdiagonals, and X and B are N-by-NRHS matrices.
|
||||
*>
|
||||
*> Error bounds on the solution and a condition estimate are also
|
||||
*> provided.
|
||||
*> \endverbatim
|
||||
*
|
||||
*> \par Description:
|
||||
* =================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> The following steps are performed by this subroutine:
|
||||
*>
|
||||
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
|
||||
*> the system:
|
||||
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
|
||||
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
|
||||
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
|
||||
*> Whether or not the system will be equilibrated depends on the
|
||||
*> scaling of the matrix A, but if equilibration is used, A is
|
||||
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
|
||||
*> or diag(C)*B (if TRANS = 'T' or 'C').
|
||||
*>
|
||||
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
|
||||
*> matrix A (after equilibration if FACT = 'E') as
|
||||
*> A = L * U,
|
||||
*> where L is a product of permutation and unit lower triangular
|
||||
*> matrices with KL subdiagonals, and U is upper triangular with
|
||||
*> KL+KU superdiagonals.
|
||||
*>
|
||||
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
|
||||
*> returns with INFO = i. Otherwise, the factored form of A is used
|
||||
*> to estimate the condition number of the matrix A. If the
|
||||
*> reciprocal of the condition number is less than machine precision,
|
||||
*> INFO = N+1 is returned as a warning, but the routine still goes on
|
||||
*> to solve for X and compute error bounds as described below.
|
||||
*>
|
||||
*> 4. The system of equations is solved for X using the factored form
|
||||
*> of A.
|
||||
*>
|
||||
*> 5. Iterative refinement is applied to improve the computed solution
|
||||
*> matrix and calculate error bounds and backward error estimates
|
||||
*> for it.
|
||||
*>
|
||||
*> 6. If equilibration was used, the matrix X is premultiplied by
|
||||
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
|
||||
*> that it solves the original system before equilibration.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] FACT
|
||||
*> \verbatim
|
||||
*> FACT is CHARACTER*1
|
||||
*> Specifies whether or not the factored form of the matrix A is
|
||||
*> supplied on entry, and if not, whether the matrix A should be
|
||||
*> equilibrated before it is factored.
|
||||
*> = 'F': On entry, AFB and IPIV contain the factored form of
|
||||
*> A. If EQUED is not 'N', the matrix A has been
|
||||
*> equilibrated with scaling factors given by R and C.
|
||||
*> AB, AFB, and IPIV are not modified.
|
||||
*> = 'N': The matrix A will be copied to AFB and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AFB and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations.
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KL
|
||||
*> \verbatim
|
||||
*> KL is INTEGER
|
||||
*> The number of subdiagonals within the band of A. KL >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KU
|
||||
*> \verbatim
|
||||
*> KU is INTEGER
|
||||
*> The number of superdiagonals within the band of A. KU >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AB
|
||||
*> \verbatim
|
||||
*> AB is REAL array, dimension (LDAB,N)
|
||||
*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
|
||||
*> The j-th column of A is stored in the j-th column of the
|
||||
*> array AB as follows:
|
||||
*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
|
||||
*>
|
||||
*> If FACT = 'F' and EQUED is not 'N', then A must have been
|
||||
*> equilibrated by the scaling factors in R and/or C. AB is not
|
||||
*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
|
||||
*> EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAB
|
||||
*> \verbatim
|
||||
*> LDAB is INTEGER
|
||||
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AFB
|
||||
*> \verbatim
|
||||
*> AFB is REAL array, dimension (LDAFB,N)
|
||||
*> If FACT = 'F', then AFB is an input argument and on entry
|
||||
*> contains details of the LU factorization of the band matrix
|
||||
*> A, as computed by SGBTRF. U is stored as an upper triangular
|
||||
*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
|
||||
*> and the multipliers used during the factorization are stored
|
||||
*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
|
||||
*> the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of A.
|
||||
*>
|
||||
*> If FACT = 'E', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of the equilibrated
|
||||
*> matrix A (see the description of AB for the form of the
|
||||
*> equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAFB
|
||||
*> \verbatim
|
||||
*> LDAFB is INTEGER
|
||||
*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> as computed by SGBTRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is REAL array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is REAL array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is REAL array, dimension (LDB,NRHS)
|
||||
*> On entry, the right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is REAL array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is REAL
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is REAL array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is REAL array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is REAL array, dimension (MAX(1,3*N))
|
||||
*> On exit, WORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If WORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> WORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] IWORK
|
||||
*> \verbatim
|
||||
*> IWORK is INTEGER array, dimension (N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization
|
||||
*> has been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup realGBsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
||||
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
||||
$ RCOND, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
|
||||
REAL RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * ), IWORK( * )
|
||||
REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
||||
$ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
* Moved setting of INFO = N+1 so INFO does not subsequently get
|
||||
* overwritten. Sven, 17 Mar 05.
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J, J1, J2
|
||||
REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
REAL SLAMCH, SLANGB, SLANTB
|
||||
EXTERNAL LSAME, SLAMCH, SLANGB, SLANTB
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL SCOPY, SGBCON, SGBEQU, SGBRFS, SGBTRF, SGBTRS,
|
||||
$ SLACPY, SLAQGB, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = SLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( KL.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( KU.LT.0 ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
|
||||
INFO = -10
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -12
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -13
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -18
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'SGBSVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL SGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL SLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of the band matrix A.
|
||||
*
|
||||
DO 70 J = 1, N
|
||||
J1 = MAX( J-KU, 1 )
|
||||
J2 = MIN( J+KL, N )
|
||||
CALL SCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
|
||||
$ AFB( KL+KU+1-J+J1, J ), 1 )
|
||||
70 CONTINUE
|
||||
*
|
||||
CALL SGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
ANORM = ZERO
|
||||
DO 90 J = 1, INFO
|
||||
DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
|
||||
ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
|
||||
80 CONTINUE
|
||||
90 CONTINUE
|
||||
RPVGRW = SLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
|
||||
$ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
|
||||
$ WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = ANORM / RPVGRW
|
||||
END IF
|
||||
WORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = SLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
|
||||
RPVGRW = SLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = SLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL SGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
|
||||
$ WORK, IWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL SGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
|
||||
$ INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL SGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
|
||||
$ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 140 J = 1, NRHS
|
||||
DO 130 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
130 CONTINUE
|
||||
140 CONTINUE
|
||||
DO 150 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
150 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
*
|
||||
WORK( 1 ) = RPVGRW
|
||||
RETURN
|
||||
*
|
||||
* End of SGBSVX
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,599 @@
|
|||
*> \brief <b> SGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download SGESVX + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgesvx.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgesvx.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgesvx.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE SGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
* EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
* WORK, IWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER EQUED, FACT, TRANS
|
||||
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
* REAL RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* INTEGER IPIV( * ), IWORK( * )
|
||||
* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
* $ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
* $ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> SGESVX uses the LU factorization to compute the solution to a real
|
||||
*> system of linear equations
|
||||
*> A * X = B,
|
||||
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
|
||||
*>
|
||||
*> Error bounds on the solution and a condition estimate are also
|
||||
*> provided.
|
||||
*> \endverbatim
|
||||
*
|
||||
*> \par Description:
|
||||
* =================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> The following steps are performed:
|
||||
*>
|
||||
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
|
||||
*> the system:
|
||||
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
|
||||
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
|
||||
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
|
||||
*> Whether or not the system will be equilibrated depends on the
|
||||
*> scaling of the matrix A, but if equilibration is used, A is
|
||||
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
|
||||
*> or diag(C)*B (if TRANS = 'T' or 'C').
|
||||
*>
|
||||
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
|
||||
*> matrix A (after equilibration if FACT = 'E') as
|
||||
*> A = P * L * U,
|
||||
*> where P is a permutation matrix, L is a unit lower triangular
|
||||
*> matrix, and U is upper triangular.
|
||||
*>
|
||||
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
|
||||
*> returns with INFO = i. Otherwise, the factored form of A is used
|
||||
*> to estimate the condition number of the matrix A. If the
|
||||
*> reciprocal of the condition number is less than machine precision,
|
||||
*> INFO = N+1 is returned as a warning, but the routine still goes on
|
||||
*> to solve for X and compute error bounds as described below.
|
||||
*>
|
||||
*> 4. The system of equations is solved for X using the factored form
|
||||
*> of A.
|
||||
*>
|
||||
*> 5. Iterative refinement is applied to improve the computed solution
|
||||
*> matrix and calculate error bounds and backward error estimates
|
||||
*> for it.
|
||||
*>
|
||||
*> 6. If equilibration was used, the matrix X is premultiplied by
|
||||
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
|
||||
*> that it solves the original system before equilibration.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] FACT
|
||||
*> \verbatim
|
||||
*> FACT is CHARACTER*1
|
||||
*> Specifies whether or not the factored form of the matrix A is
|
||||
*> supplied on entry, and if not, whether the matrix A should be
|
||||
*> equilibrated before it is factored.
|
||||
*> = 'F': On entry, AF and IPIV contain the factored form of A.
|
||||
*> If EQUED is not 'N', the matrix A has been
|
||||
*> equilibrated with scaling factors given by R and C.
|
||||
*> A, AF, and IPIV are not modified.
|
||||
*> = 'N': The matrix A will be copied to AF and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AF and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations:
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is REAL array, dimension (LDA,N)
|
||||
*> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
|
||||
*> not 'N', then A must have been equilibrated by the scaling
|
||||
*> factors in R and/or C. A is not modified if FACT = 'F' or
|
||||
*> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of the array A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AF
|
||||
*> \verbatim
|
||||
*> AF is REAL array, dimension (LDAF,N)
|
||||
*> If FACT = 'F', then AF is an input argument and on entry
|
||||
*> contains the factors L and U from the factorization
|
||||
*> A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then
|
||||
*> AF is the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A (see the description of A for
|
||||
*> the form of the equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAF
|
||||
*> \verbatim
|
||||
*> LDAF is INTEGER
|
||||
*> The leading dimension of the array AF. LDAF >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> as computed by SGETRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is REAL array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is REAL array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is REAL array, dimension (LDB,NRHS)
|
||||
*> On entry, the N-by-NRHS right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is REAL array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is REAL
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is REAL array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is REAL array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is REAL array, dimension (MAX(1,4*N))
|
||||
*> On exit, WORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If WORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> WORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] IWORK
|
||||
*> \verbatim
|
||||
*> IWORK is INTEGER array, dimension (N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization has
|
||||
*> been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> value of RCOND would suggest.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup realGEsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE SGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
$ WORK, IWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
REAL RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * ), IWORK( * )
|
||||
REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
$ BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
REAL ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J
|
||||
REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
REAL SLAMCH, SLANGE, SLANTR
|
||||
EXTERNAL LSAME, SLAMCH, SLANGE, SLANTR
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL SGECON, SGEEQU, SGERFS, SGETRF, SGETRS, SLACPY,
|
||||
$ SLAQGE, XERBLA
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = SLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -10
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -11
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'SGESVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL SGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL SLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
|
||||
$ EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of A.
|
||||
*
|
||||
CALL SLACPY( 'Full', N, N, A, LDA, AF, LDAF )
|
||||
CALL SGETRF( N, N, AF, LDAF, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
RPVGRW = SLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
|
||||
$ WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = SLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW
|
||||
END IF
|
||||
WORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = SLANGE( NORM, N, N, A, LDA, WORK )
|
||||
RPVGRW = SLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = SLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL SGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL SGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL SGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
|
||||
$ LDX, FERR, BERR, WORK, IWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 80 J = 1, NRHS
|
||||
DO 70 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
70 CONTINUE
|
||||
80 CONTINUE
|
||||
DO 90 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
90 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
*
|
||||
WORK( 1 ) = RPVGRW
|
||||
RETURN
|
||||
*
|
||||
* End of SGESVX
|
||||
*
|
||||
END
|
||||
|
|
@ -0,0 +1,644 @@
|
|||
*> \brief <b> ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download ZGBSVX + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbsvx.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbsvx.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbsvx.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
||||
* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
||||
* RCOND, FERR, BERR, WORK, RWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER EQUED, FACT, TRANS
|
||||
* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
|
||||
* DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* INTEGER IPIV( * )
|
||||
* DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
* $ RWORK( * )
|
||||
* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
||||
* $ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> ZGBSVX uses the LU factorization to compute the solution to a complex
|
||||
*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
|
||||
*> where A is a band matrix of order N with KL subdiagonals and KU
|
||||
*> superdiagonals, and X and B are N-by-NRHS matrices.
|
||||
*>
|
||||
*> Error bounds on the solution and a condition estimate are also
|
||||
*> provided.
|
||||
*> \endverbatim
|
||||
*
|
||||
*> \par Description:
|
||||
* =================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> The following steps are performed by this subroutine:
|
||||
*>
|
||||
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
|
||||
*> the system:
|
||||
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
|
||||
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
|
||||
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
|
||||
*> Whether or not the system will be equilibrated depends on the
|
||||
*> scaling of the matrix A, but if equilibration is used, A is
|
||||
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
|
||||
*> or diag(C)*B (if TRANS = 'T' or 'C').
|
||||
*>
|
||||
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
|
||||
*> matrix A (after equilibration if FACT = 'E') as
|
||||
*> A = L * U,
|
||||
*> where L is a product of permutation and unit lower triangular
|
||||
*> matrices with KL subdiagonals, and U is upper triangular with
|
||||
*> KL+KU superdiagonals.
|
||||
*>
|
||||
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
|
||||
*> returns with INFO = i. Otherwise, the factored form of A is used
|
||||
*> to estimate the condition number of the matrix A. If the
|
||||
*> reciprocal of the condition number is less than machine precision,
|
||||
*> INFO = N+1 is returned as a warning, but the routine still goes on
|
||||
*> to solve for X and compute error bounds as described below.
|
||||
*>
|
||||
*> 4. The system of equations is solved for X using the factored form
|
||||
*> of A.
|
||||
*>
|
||||
*> 5. Iterative refinement is applied to improve the computed solution
|
||||
*> matrix and calculate error bounds and backward error estimates
|
||||
*> for it.
|
||||
*>
|
||||
*> 6. If equilibration was used, the matrix X is premultiplied by
|
||||
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
|
||||
*> that it solves the original system before equilibration.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] FACT
|
||||
*> \verbatim
|
||||
*> FACT is CHARACTER*1
|
||||
*> Specifies whether or not the factored form of the matrix A is
|
||||
*> supplied on entry, and if not, whether the matrix A should be
|
||||
*> equilibrated before it is factored.
|
||||
*> = 'F': On entry, AFB and IPIV contain the factored form of
|
||||
*> A. If EQUED is not 'N', the matrix A has been
|
||||
*> equilibrated with scaling factors given by R and C.
|
||||
*> AB, AFB, and IPIV are not modified.
|
||||
*> = 'N': The matrix A will be copied to AFB and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AFB and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations.
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Conjugate transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KL
|
||||
*> \verbatim
|
||||
*> KL is INTEGER
|
||||
*> The number of subdiagonals within the band of A. KL >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] KU
|
||||
*> \verbatim
|
||||
*> KU is INTEGER
|
||||
*> The number of superdiagonals within the band of A. KU >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AB
|
||||
*> \verbatim
|
||||
*> AB is COMPLEX*16 array, dimension (LDAB,N)
|
||||
*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
|
||||
*> The j-th column of A is stored in the j-th column of the
|
||||
*> array AB as follows:
|
||||
*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
|
||||
*>
|
||||
*> If FACT = 'F' and EQUED is not 'N', then A must have been
|
||||
*> equilibrated by the scaling factors in R and/or C. AB is not
|
||||
*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
|
||||
*> EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAB
|
||||
*> \verbatim
|
||||
*> LDAB is INTEGER
|
||||
*> The leading dimension of the array AB. LDAB >= KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AFB
|
||||
*> \verbatim
|
||||
*> AFB is COMPLEX*16 array, dimension (LDAFB,N)
|
||||
*> If FACT = 'F', then AFB is an input argument and on entry
|
||||
*> contains details of the LU factorization of the band matrix
|
||||
*> A, as computed by ZGBTRF. U is stored as an upper triangular
|
||||
*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
|
||||
*> and the multipliers used during the factorization are stored
|
||||
*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
|
||||
*> the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of A.
|
||||
*>
|
||||
*> If FACT = 'E', then AFB is an output argument and on exit
|
||||
*> returns details of the LU factorization of the equilibrated
|
||||
*> matrix A (see the description of AB for the form of the
|
||||
*> equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAFB
|
||||
*> \verbatim
|
||||
*> LDAFB is INTEGER
|
||||
*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> as computed by ZGBTRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is DOUBLE PRECISION array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is DOUBLE PRECISION array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is COMPLEX*16 array, dimension (LDB,NRHS)
|
||||
*> On entry, the right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is COMPLEX*16 array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is DOUBLE PRECISION
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is COMPLEX*16 array, dimension (2*N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RWORK
|
||||
*> \verbatim
|
||||
*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,N))
|
||||
*> On exit, RWORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If RWORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> RWORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization
|
||||
*> has been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> value of RCOND would suggest.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup complex16GBsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
|
||||
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
|
||||
$ RCOND, FERR, BERR, WORK, RWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
|
||||
DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * )
|
||||
DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ RWORK( * )
|
||||
COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
* Moved setting of INFO = N+1 so INFO does not subsequently get
|
||||
* overwritten. Sven, 17 Mar 05.
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J, J1, J2
|
||||
DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB
|
||||
EXTERNAL LSAME, DLAMCH, ZLANGB, ZLANTB
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL XERBLA, ZCOPY, ZGBCON, ZGBEQU, ZGBRFS, ZGBTRF,
|
||||
$ ZGBTRS, ZLACPY, ZLAQGB
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC ABS, MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = DLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( KL.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( KU.LT.0 ) THEN
|
||||
INFO = -5
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
|
||||
INFO = -10
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -12
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -13
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -18
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'ZGBSVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL ZGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
|
||||
$ AMAX, EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of the band matrix A.
|
||||
*
|
||||
DO 70 J = 1, N
|
||||
J1 = MAX( J-KU, 1 )
|
||||
J2 = MIN( J+KL, N )
|
||||
CALL ZCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
|
||||
$ AFB( KL+KU+1-J+J1, J ), 1 )
|
||||
70 CONTINUE
|
||||
*
|
||||
CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
ANORM = ZERO
|
||||
DO 90 J = 1, INFO
|
||||
DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
|
||||
ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
|
||||
80 CONTINUE
|
||||
90 CONTINUE
|
||||
RPVGRW = ZLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
|
||||
$ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
|
||||
$ RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = ANORM / RPVGRW
|
||||
END IF
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
|
||||
RPVGRW = ZLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = ZLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
|
||||
$ WORK, RWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
|
||||
$ INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
|
||||
$ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 140 J = 1, NRHS
|
||||
DO 130 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
130 CONTINUE
|
||||
140 CONTINUE
|
||||
DO 150 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
150 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
*
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RETURN
|
||||
*
|
||||
* End of ZGBSVX
|
||||
*
|
||||
END
|
||||
File diff suppressed because it is too large
Load Diff
|
|
@ -0,0 +1,602 @@
|
|||
*> \brief <b> ZGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
|
||||
*
|
||||
* =========== DOCUMENTATION ===========
|
||||
*
|
||||
* Online html documentation available at
|
||||
* http://www.netlib.org/lapack/explore-html/
|
||||
*
|
||||
*> \htmlonly
|
||||
*> Download ZGESVX + dependencies
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgesvx.f">
|
||||
*> [TGZ]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgesvx.f">
|
||||
*> [ZIP]</a>
|
||||
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgesvx.f">
|
||||
*> [TXT]</a>
|
||||
*> \endhtmlonly
|
||||
*
|
||||
* Definition:
|
||||
* ===========
|
||||
*
|
||||
* SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
* EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
* WORK, RWORK, INFO )
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
* CHARACTER EQUED, FACT, TRANS
|
||||
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
* DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
* INTEGER IPIV( * )
|
||||
* DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
* $ RWORK( * )
|
||||
* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
* $ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
*
|
||||
*> \par Purpose:
|
||||
* =============
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> ZGESVX uses the LU factorization to compute the solution to a complex
|
||||
*> system of linear equations
|
||||
*> A * X = B,
|
||||
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
|
||||
*>
|
||||
*> Error bounds on the solution and a condition estimate are also
|
||||
*> provided.
|
||||
*> \endverbatim
|
||||
*
|
||||
*> \par Description:
|
||||
* =================
|
||||
*>
|
||||
*> \verbatim
|
||||
*>
|
||||
*> The following steps are performed:
|
||||
*>
|
||||
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
|
||||
*> the system:
|
||||
*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
|
||||
*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
|
||||
*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
|
||||
*> Whether or not the system will be equilibrated depends on the
|
||||
*> scaling of the matrix A, but if equilibration is used, A is
|
||||
*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
|
||||
*> or diag(C)*B (if TRANS = 'T' or 'C').
|
||||
*>
|
||||
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
|
||||
*> matrix A (after equilibration if FACT = 'E') as
|
||||
*> A = P * L * U,
|
||||
*> where P is a permutation matrix, L is a unit lower triangular
|
||||
*> matrix, and U is upper triangular.
|
||||
*>
|
||||
*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
|
||||
*> returns with INFO = i. Otherwise, the factored form of A is used
|
||||
*> to estimate the condition number of the matrix A. If the
|
||||
*> reciprocal of the condition number is less than machine precision,
|
||||
*> INFO = N+1 is returned as a warning, but the routine still goes on
|
||||
*> to solve for X and compute error bounds as described below.
|
||||
*>
|
||||
*> 4. The system of equations is solved for X using the factored form
|
||||
*> of A.
|
||||
*>
|
||||
*> 5. Iterative refinement is applied to improve the computed solution
|
||||
*> matrix and calculate error bounds and backward error estimates
|
||||
*> for it.
|
||||
*>
|
||||
*> 6. If equilibration was used, the matrix X is premultiplied by
|
||||
*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
|
||||
*> that it solves the original system before equilibration.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Arguments:
|
||||
* ==========
|
||||
*
|
||||
*> \param[in] FACT
|
||||
*> \verbatim
|
||||
*> FACT is CHARACTER*1
|
||||
*> Specifies whether or not the factored form of the matrix A is
|
||||
*> supplied on entry, and if not, whether the matrix A should be
|
||||
*> equilibrated before it is factored.
|
||||
*> = 'F': On entry, AF and IPIV contain the factored form of A.
|
||||
*> If EQUED is not 'N', the matrix A has been
|
||||
*> equilibrated with scaling factors given by R and C.
|
||||
*> A, AF, and IPIV are not modified.
|
||||
*> = 'N': The matrix A will be copied to AF and factored.
|
||||
*> = 'E': The matrix A will be equilibrated if necessary, then
|
||||
*> copied to AF and factored.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] TRANS
|
||||
*> \verbatim
|
||||
*> TRANS is CHARACTER*1
|
||||
*> Specifies the form of the system of equations:
|
||||
*> = 'N': A * X = B (No transpose)
|
||||
*> = 'T': A**T * X = B (Transpose)
|
||||
*> = 'C': A**H * X = B (Conjugate transpose)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] N
|
||||
*> \verbatim
|
||||
*> N is INTEGER
|
||||
*> The number of linear equations, i.e., the order of the
|
||||
*> matrix A. N >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] NRHS
|
||||
*> \verbatim
|
||||
*> NRHS is INTEGER
|
||||
*> The number of right hand sides, i.e., the number of columns
|
||||
*> of the matrices B and X. NRHS >= 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] A
|
||||
*> \verbatim
|
||||
*> A is COMPLEX*16 array, dimension (LDA,N)
|
||||
*> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
|
||||
*> not 'N', then A must have been equilibrated by the scaling
|
||||
*> factors in R and/or C. A is not modified if FACT = 'F' or
|
||||
*> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
|
||||
*>
|
||||
*> On exit, if EQUED .ne. 'N', A is scaled as follows:
|
||||
*> EQUED = 'R': A := diag(R) * A
|
||||
*> EQUED = 'C': A := A * diag(C)
|
||||
*> EQUED = 'B': A := diag(R) * A * diag(C).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDA
|
||||
*> \verbatim
|
||||
*> LDA is INTEGER
|
||||
*> The leading dimension of the array A. LDA >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] AF
|
||||
*> \verbatim
|
||||
*> AF is COMPLEX*16 array, dimension (LDAF,N)
|
||||
*> If FACT = 'F', then AF is an input argument and on entry
|
||||
*> contains the factors L and U from the factorization
|
||||
*> A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
|
||||
*> AF is the factored form of the equilibrated matrix A.
|
||||
*>
|
||||
*> If FACT = 'N', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then AF is an output argument and on exit
|
||||
*> returns the factors L and U from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A (see the description of A for
|
||||
*> the form of the equilibrated matrix).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDAF
|
||||
*> \verbatim
|
||||
*> LDAF is INTEGER
|
||||
*> The leading dimension of the array AF. LDAF >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] IPIV
|
||||
*> \verbatim
|
||||
*> IPIV is INTEGER array, dimension (N)
|
||||
*> If FACT = 'F', then IPIV is an input argument and on entry
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> as computed by ZGETRF; row i of the matrix was interchanged
|
||||
*> with row IPIV(i).
|
||||
*>
|
||||
*> If FACT = 'N', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the original matrix A.
|
||||
*>
|
||||
*> If FACT = 'E', then IPIV is an output argument and on exit
|
||||
*> contains the pivot indices from the factorization A = P*L*U
|
||||
*> of the equilibrated matrix A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] EQUED
|
||||
*> \verbatim
|
||||
*> EQUED is CHARACTER*1
|
||||
*> Specifies the form of equilibration that was done.
|
||||
*> = 'N': No equilibration (always true if FACT = 'N').
|
||||
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
||||
*> diag(R).
|
||||
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
||||
*> by diag(C).
|
||||
*> = 'B': Both row and column equilibration, i.e., A has been
|
||||
*> replaced by diag(R) * A * diag(C).
|
||||
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
||||
*> output argument.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] R
|
||||
*> \verbatim
|
||||
*> R is DOUBLE PRECISION array, dimension (N)
|
||||
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
||||
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
||||
*> is not accessed. R is an input argument if FACT = 'F';
|
||||
*> otherwise, R is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'R' or 'B', each element of R must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] C
|
||||
*> \verbatim
|
||||
*> C is DOUBLE PRECISION array, dimension (N)
|
||||
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
||||
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
||||
*> is not accessed. C is an input argument if FACT = 'F';
|
||||
*> otherwise, C is an output argument. If FACT = 'F' and
|
||||
*> EQUED = 'C' or 'B', each element of C must be positive.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in,out] B
|
||||
*> \verbatim
|
||||
*> B is COMPLEX*16 array, dimension (LDB,NRHS)
|
||||
*> On entry, the N-by-NRHS right hand side matrix B.
|
||||
*> On exit,
|
||||
*> if EQUED = 'N', B is not modified;
|
||||
*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
|
||||
*> diag(R)*B;
|
||||
*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
|
||||
*> overwritten by diag(C)*B.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDB
|
||||
*> \verbatim
|
||||
*> LDB is INTEGER
|
||||
*> The leading dimension of the array B. LDB >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] X
|
||||
*> \verbatim
|
||||
*> X is COMPLEX*16 array, dimension (LDX,NRHS)
|
||||
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
|
||||
*> to the original system of equations. Note that A and B are
|
||||
*> modified on exit if EQUED .ne. 'N', and the solution to the
|
||||
*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
|
||||
*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
|
||||
*> and EQUED = 'R' or 'B'.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[in] LDX
|
||||
*> \verbatim
|
||||
*> LDX is INTEGER
|
||||
*> The leading dimension of the array X. LDX >= max(1,N).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RCOND
|
||||
*> \verbatim
|
||||
*> RCOND is DOUBLE PRECISION
|
||||
*> The estimate of the reciprocal condition number of the matrix
|
||||
*> A after equilibration (if done). If RCOND is less than the
|
||||
*> machine precision (in particular, if RCOND = 0), the matrix
|
||||
*> is singular to working precision. This condition is
|
||||
*> indicated by a return code of INFO > 0.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] FERR
|
||||
*> \verbatim
|
||||
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The estimated forward error bound for each solution vector
|
||||
*> X(j) (the j-th column of the solution matrix X).
|
||||
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
|
||||
*> is an estimated upper bound for the magnitude of the largest
|
||||
*> element in (X(j) - XTRUE) divided by the magnitude of the
|
||||
*> largest element in X(j). The estimate is as reliable as
|
||||
*> the estimate for RCOND, and is almost always a slight
|
||||
*> overestimate of the true error.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] BERR
|
||||
*> \verbatim
|
||||
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
|
||||
*> The componentwise relative backward error of each solution
|
||||
*> vector X(j) (i.e., the smallest relative change in
|
||||
*> any element of A or B that makes X(j) an exact solution).
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] WORK
|
||||
*> \verbatim
|
||||
*> WORK is COMPLEX*16 array, dimension (2*N)
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] RWORK
|
||||
*> \verbatim
|
||||
*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,2*N))
|
||||
*> On exit, RWORK(1) contains the reciprocal pivot growth
|
||||
*> factor norm(A)/norm(U). The "max absolute element" norm is
|
||||
*> used. If RWORK(1) is much less than 1, then the stability
|
||||
*> of the LU factorization of the (equilibrated) matrix A
|
||||
*> could be poor. This also means that the solution X, condition
|
||||
*> estimator RCOND, and forward error bound FERR could be
|
||||
*> unreliable. If factorization fails with 0<INFO<=N, then
|
||||
*> RWORK(1) contains the reciprocal pivot growth factor for the
|
||||
*> leading INFO columns of A.
|
||||
*> \endverbatim
|
||||
*>
|
||||
*> \param[out] INFO
|
||||
*> \verbatim
|
||||
*> INFO is INTEGER
|
||||
*> = 0: successful exit
|
||||
*> < 0: if INFO = -i, the i-th argument had an illegal value
|
||||
*> > 0: if INFO = i, and i is
|
||||
*> <= N: U(i,i) is exactly zero. The factorization has
|
||||
*> been completed, but the factor U is exactly
|
||||
*> singular, so the solution and error bounds
|
||||
*> could not be computed. RCOND = 0 is returned.
|
||||
*> = N+1: U is nonsingular, but RCOND is less than machine
|
||||
*> precision, meaning that the matrix is singular
|
||||
*> to working precision. Nevertheless, the
|
||||
*> solution and error bounds are computed because
|
||||
*> there are a number of situations where the
|
||||
*> computed solution can be more accurate than the
|
||||
*> value of RCOND would suggest.
|
||||
*> \endverbatim
|
||||
*
|
||||
* Authors:
|
||||
* ========
|
||||
*
|
||||
*> \author Univ. of Tennessee
|
||||
*> \author Univ. of California Berkeley
|
||||
*> \author Univ. of Colorado Denver
|
||||
*> \author NAG Ltd.
|
||||
*
|
||||
*> \ingroup complex16GEsolve
|
||||
*
|
||||
* =====================================================================
|
||||
SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
||||
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
|
||||
$ WORK, RWORK, INFO )
|
||||
*
|
||||
* -- LAPACK driver routine --
|
||||
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||||
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||||
*
|
||||
* .. Scalar Arguments ..
|
||||
CHARACTER EQUED, FACT, TRANS
|
||||
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
|
||||
DOUBLE PRECISION RCOND
|
||||
* ..
|
||||
* .. Array Arguments ..
|
||||
INTEGER IPIV( * )
|
||||
DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
|
||||
$ RWORK( * )
|
||||
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
||||
$ WORK( * ), X( LDX, * )
|
||||
* ..
|
||||
*
|
||||
* =====================================================================
|
||||
*
|
||||
* .. Parameters ..
|
||||
DOUBLE PRECISION ZERO, ONE
|
||||
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
|
||||
* ..
|
||||
* .. Local Scalars ..
|
||||
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
|
||||
CHARACTER NORM
|
||||
INTEGER I, INFEQU, J
|
||||
DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
|
||||
$ ROWCND, RPVGRW, SMLNUM
|
||||
* ..
|
||||
* .. External Functions ..
|
||||
LOGICAL LSAME
|
||||
DOUBLE PRECISION DLAMCH, ZLANGE, ZLANTR
|
||||
EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANTR
|
||||
* ..
|
||||
* .. External Subroutines ..
|
||||
EXTERNAL XERBLA, ZGECON, ZGEEQU, ZGERFS, ZGETRF, ZGETRS,
|
||||
$ ZLACPY, ZLAQGE
|
||||
* ..
|
||||
* .. Intrinsic Functions ..
|
||||
INTRINSIC MAX, MIN
|
||||
* ..
|
||||
* .. Executable Statements ..
|
||||
*
|
||||
INFO = 0
|
||||
NOFACT = LSAME( FACT, 'N' )
|
||||
EQUIL = LSAME( FACT, 'E' )
|
||||
NOTRAN = LSAME( TRANS, 'N' )
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
EQUED = 'N'
|
||||
ROWEQU = .FALSE.
|
||||
COLEQU = .FALSE.
|
||||
ELSE
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
SMLNUM = DLAMCH( 'Safe minimum' )
|
||||
BIGNUM = ONE / SMLNUM
|
||||
END IF
|
||||
*
|
||||
* Test the input parameters.
|
||||
*
|
||||
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
|
||||
$ THEN
|
||||
INFO = -1
|
||||
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
|
||||
$ LSAME( TRANS, 'C' ) ) THEN
|
||||
INFO = -2
|
||||
ELSE IF( N.LT.0 ) THEN
|
||||
INFO = -3
|
||||
ELSE IF( NRHS.LT.0 ) THEN
|
||||
INFO = -4
|
||||
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -6
|
||||
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -8
|
||||
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
||||
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
||||
INFO = -10
|
||||
ELSE
|
||||
IF( ROWEQU ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 10 J = 1, N
|
||||
RCMIN = MIN( RCMIN, R( J ) )
|
||||
RCMAX = MAX( RCMAX, R( J ) )
|
||||
10 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -11
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
ROWCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( COLEQU .AND. INFO.EQ.0 ) THEN
|
||||
RCMIN = BIGNUM
|
||||
RCMAX = ZERO
|
||||
DO 20 J = 1, N
|
||||
RCMIN = MIN( RCMIN, C( J ) )
|
||||
RCMAX = MAX( RCMAX, C( J ) )
|
||||
20 CONTINUE
|
||||
IF( RCMIN.LE.ZERO ) THEN
|
||||
INFO = -12
|
||||
ELSE IF( N.GT.0 ) THEN
|
||||
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
|
||||
ELSE
|
||||
COLCND = ONE
|
||||
END IF
|
||||
END IF
|
||||
IF( INFO.EQ.0 ) THEN
|
||||
IF( LDB.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -14
|
||||
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
||||
INFO = -16
|
||||
END IF
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
IF( INFO.NE.0 ) THEN
|
||||
CALL XERBLA( 'ZGESVX', -INFO )
|
||||
RETURN
|
||||
END IF
|
||||
*
|
||||
IF( EQUIL ) THEN
|
||||
*
|
||||
* Compute row and column scalings to equilibrate the matrix A.
|
||||
*
|
||||
CALL ZGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
|
||||
IF( INFEQU.EQ.0 ) THEN
|
||||
*
|
||||
* Equilibrate the matrix.
|
||||
*
|
||||
CALL ZLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
|
||||
$ EQUED )
|
||||
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
||||
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Scale the right hand side.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( ROWEQU ) THEN
|
||||
DO 40 J = 1, NRHS
|
||||
DO 30 I = 1, N
|
||||
B( I, J ) = R( I )*B( I, J )
|
||||
30 CONTINUE
|
||||
40 CONTINUE
|
||||
END IF
|
||||
ELSE IF( COLEQU ) THEN
|
||||
DO 60 J = 1, NRHS
|
||||
DO 50 I = 1, N
|
||||
B( I, J ) = C( I )*B( I, J )
|
||||
50 CONTINUE
|
||||
60 CONTINUE
|
||||
END IF
|
||||
*
|
||||
IF( NOFACT .OR. EQUIL ) THEN
|
||||
*
|
||||
* Compute the LU factorization of A.
|
||||
*
|
||||
CALL ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
|
||||
CALL ZGETRF( N, N, AF, LDAF, IPIV, INFO )
|
||||
*
|
||||
* Return if INFO is non-zero.
|
||||
*
|
||||
IF( INFO.GT.0 ) THEN
|
||||
*
|
||||
* Compute the reciprocal pivot growth factor of the
|
||||
* leading rank-deficient INFO columns of A.
|
||||
*
|
||||
RPVGRW = ZLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
|
||||
$ RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = ZLANGE( 'M', N, INFO, A, LDA, RWORK ) /
|
||||
$ RPVGRW
|
||||
END IF
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RCOND = ZERO
|
||||
RETURN
|
||||
END IF
|
||||
END IF
|
||||
*
|
||||
* Compute the norm of the matrix A and the
|
||||
* reciprocal pivot growth factor RPVGRW.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
NORM = '1'
|
||||
ELSE
|
||||
NORM = 'I'
|
||||
END IF
|
||||
ANORM = ZLANGE( NORM, N, N, A, LDA, RWORK )
|
||||
RPVGRW = ZLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK )
|
||||
IF( RPVGRW.EQ.ZERO ) THEN
|
||||
RPVGRW = ONE
|
||||
ELSE
|
||||
RPVGRW = ZLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW
|
||||
END IF
|
||||
*
|
||||
* Compute the reciprocal of the condition number of A.
|
||||
*
|
||||
CALL ZGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
|
||||
*
|
||||
* Compute the solution matrix X.
|
||||
*
|
||||
CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
||||
CALL ZGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
|
||||
*
|
||||
* Use iterative refinement to improve the computed solution and
|
||||
* compute error bounds and backward error estimates for it.
|
||||
*
|
||||
CALL ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
|
||||
$ LDX, FERR, BERR, WORK, RWORK, INFO )
|
||||
*
|
||||
* Transform the solution matrix X to a solution of the original
|
||||
* system.
|
||||
*
|
||||
IF( NOTRAN ) THEN
|
||||
IF( COLEQU ) THEN
|
||||
DO 80 J = 1, NRHS
|
||||
DO 70 I = 1, N
|
||||
X( I, J ) = C( I )*X( I, J )
|
||||
70 CONTINUE
|
||||
80 CONTINUE
|
||||
DO 90 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / COLCND
|
||||
90 CONTINUE
|
||||
END IF
|
||||
ELSE IF( ROWEQU ) THEN
|
||||
DO 110 J = 1, NRHS
|
||||
DO 100 I = 1, N
|
||||
X( I, J ) = R( I )*X( I, J )
|
||||
100 CONTINUE
|
||||
110 CONTINUE
|
||||
DO 120 J = 1, NRHS
|
||||
FERR( J ) = FERR( J ) / ROWCND
|
||||
120 CONTINUE
|
||||
END IF
|
||||
*
|
||||
* Set INFO = N+1 if the matrix is singular to working precision.
|
||||
*
|
||||
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
|
||||
$ INFO = N + 1
|
||||
*
|
||||
RWORK( 1 ) = RPVGRW
|
||||
RETURN
|
||||
*
|
||||
* End of ZGESVX
|
||||
*
|
||||
END
|
||||
Loading…
Reference in New Issue