495 lines
16 KiB
Fortran
495 lines
16 KiB
Fortran
*> \brief <b> SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download SPPSVX + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sppsvx.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sppsvx.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sppsvx.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
|
|
* X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* CHARACTER EQUED, FACT, UPLO
|
|
* INTEGER INFO, LDB, LDX, N, NRHS
|
|
* REAL RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* INTEGER IWORK( * )
|
|
* REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
|
|
* $ FERR( * ), S( * ), WORK( * ), X( LDX, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
|
|
*> compute the solution to a real system of linear equations
|
|
*> A * X = B,
|
|
*> where A is an N-by-N symmetric positive definite matrix stored in
|
|
*> packed format 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:
|
|
*> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*B.
|
|
*>
|
|
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
|
|
*> factor the matrix A (after equilibration if FACT = 'E') as
|
|
*> A = U**T* U, if UPLO = 'U', or
|
|
*> A = L * L**T, if UPLO = 'L',
|
|
*> where U is an upper triangular matrix and L is a lower triangular
|
|
*> matrix.
|
|
*>
|
|
*> 3. If the leading i-by-i principal minor is not positive definite,
|
|
*> 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(S) 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, AFP contains the factored form of A.
|
|
*> If EQUED = 'Y', the matrix A has been equilibrated
|
|
*> with scaling factors given by S. AP and AFP will not
|
|
*> be modified.
|
|
*> = 'N': The matrix A will be copied to AFP and factored.
|
|
*> = 'E': The matrix A will be equilibrated if necessary, then
|
|
*> copied to AFP and factored.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] UPLO
|
|
*> \verbatim
|
|
*> UPLO is CHARACTER*1
|
|
*> = 'U': Upper triangle of A is stored;
|
|
*> = 'L': Lower triangle of A is stored.
|
|
*> \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] AP
|
|
*> \verbatim
|
|
*> AP is REAL array, dimension (N*(N+1)/2)
|
|
*> On entry, the upper or lower triangle of the symmetric matrix
|
|
*> A, packed columnwise in a linear array, except if FACT = 'F'
|
|
*> and EQUED = 'Y', then A must contain the equilibrated matrix
|
|
*> diag(S)*A*diag(S). The j-th column of A is stored in the
|
|
*> array AP as follows:
|
|
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
|
|
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
|
|
*> See below for further details. A is not modified if
|
|
*> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
|
|
*>
|
|
*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
|
|
*> diag(S)*A*diag(S).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] AFP
|
|
*> \verbatim
|
|
*> AFP is REAL array, dimension
|
|
*> (N*(N+1)/2)
|
|
*> If FACT = 'F', then AFP is an input argument and on entry
|
|
*> contains the triangular factor U or L from the Cholesky
|
|
*> factorization A = U**T*U or A = L*L**T, in the same storage
|
|
*> format as A. If EQUED .ne. 'N', then AFP is the factored
|
|
*> form of the equilibrated matrix A.
|
|
*>
|
|
*> If FACT = 'N', then AFP is an output argument and on exit
|
|
*> returns the triangular factor U or L from the Cholesky
|
|
*> factorization A = U**T * U or A = L * L**T of the original
|
|
*> matrix A.
|
|
*>
|
|
*> If FACT = 'E', then AFP is an output argument and on exit
|
|
*> returns the triangular factor U or L from the Cholesky
|
|
*> factorization A = U**T * U or A = L * L**T of the equilibrated
|
|
*> matrix A (see the description of AP for the form of the
|
|
*> equilibrated matrix).
|
|
*> \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').
|
|
*> = 'Y': Equilibration was done, i.e., A has been replaced by
|
|
*> diag(S) * A * diag(S).
|
|
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
|
|
*> output argument.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] S
|
|
*> \verbatim
|
|
*> S is REAL array, dimension (N)
|
|
*> The scale factors for A; not accessed if EQUED = 'N'. S is
|
|
*> an input argument if FACT = 'F'; otherwise, S is an output
|
|
*> argument. If FACT = 'F' and EQUED = 'Y', each element of S
|
|
*> 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 EQUED = 'Y',
|
|
*> B is overwritten by diag(S) * 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 if EQUED = 'Y',
|
|
*> A and B are modified on exit, and the solution to the
|
|
*> equilibrated system is inv(diag(S))*X.
|
|
*> \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 (3*N)
|
|
*> \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: the leading minor of order i of A is
|
|
*> not positive definite, so the factorization
|
|
*> could not be completed, and the solution has not
|
|
*> been 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.
|
|
*
|
|
*> \date April 2012
|
|
*
|
|
*> \ingroup realOTHERsolve
|
|
*
|
|
*> \par Further Details:
|
|
* =====================
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> The packed storage scheme is illustrated by the following example
|
|
*> when N = 4, UPLO = 'U':
|
|
*>
|
|
*> Two-dimensional storage of the symmetric matrix A:
|
|
*>
|
|
*> a11 a12 a13 a14
|
|
*> a22 a23 a24
|
|
*> a33 a34 (aij = conjg(aji))
|
|
*> a44
|
|
*>
|
|
*> Packed storage of the upper triangle of A:
|
|
*>
|
|
*> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
|
|
*> \endverbatim
|
|
*>
|
|
* =====================================================================
|
|
SUBROUTINE SPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
|
|
$ X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
|
|
*
|
|
* -- LAPACK driver routine (version 3.7.0) --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
* April 2012
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER EQUED, FACT, UPLO
|
|
INTEGER INFO, LDB, LDX, N, NRHS
|
|
REAL RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER IWORK( * )
|
|
REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
|
|
$ FERR( * ), S( * ), WORK( * ), X( LDX, * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
REAL ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL EQUIL, NOFACT, RCEQU
|
|
INTEGER I, INFEQU, J
|
|
REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
|
|
* ..
|
|
* .. External Functions ..
|
|
LOGICAL LSAME
|
|
REAL SLAMCH, SLANSP
|
|
EXTERNAL LSAME, SLAMCH, SLANSP
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL SCOPY, SLACPY, SLAQSP, SPPCON, SPPEQU, SPPRFS,
|
|
$ SPPTRF, SPPTRS, XERBLA
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, MIN
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
INFO = 0
|
|
NOFACT = LSAME( FACT, 'N' )
|
|
EQUIL = LSAME( FACT, 'E' )
|
|
IF( NOFACT .OR. EQUIL ) THEN
|
|
EQUED = 'N'
|
|
RCEQU = .FALSE.
|
|
ELSE
|
|
RCEQU = LSAME( EQUED, 'Y' )
|
|
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.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
|
|
$ THEN
|
|
INFO = -2
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -4
|
|
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
|
|
$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
|
|
INFO = -7
|
|
ELSE
|
|
IF( RCEQU ) THEN
|
|
SMIN = BIGNUM
|
|
SMAX = ZERO
|
|
DO 10 J = 1, N
|
|
SMIN = MIN( SMIN, S( J ) )
|
|
SMAX = MAX( SMAX, S( J ) )
|
|
10 CONTINUE
|
|
IF( SMIN.LE.ZERO ) THEN
|
|
INFO = -8
|
|
ELSE IF( N.GT.0 ) THEN
|
|
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
|
|
ELSE
|
|
SCOND = ONE
|
|
END IF
|
|
END IF
|
|
IF( INFO.EQ.0 ) THEN
|
|
IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -10
|
|
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
|
INFO = -12
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'SPPSVX', -INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
IF( EQUIL ) THEN
|
|
*
|
|
* Compute row and column scalings to equilibrate the matrix A.
|
|
*
|
|
CALL SPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
|
|
IF( INFEQU.EQ.0 ) THEN
|
|
*
|
|
* Equilibrate the matrix.
|
|
*
|
|
CALL SLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
|
|
RCEQU = LSAME( EQUED, 'Y' )
|
|
END IF
|
|
END IF
|
|
*
|
|
* Scale the right-hand side.
|
|
*
|
|
IF( RCEQU ) THEN
|
|
DO 30 J = 1, NRHS
|
|
DO 20 I = 1, N
|
|
B( I, J ) = S( I )*B( I, J )
|
|
20 CONTINUE
|
|
30 CONTINUE
|
|
END IF
|
|
*
|
|
IF( NOFACT .OR. EQUIL ) THEN
|
|
*
|
|
* Compute the Cholesky factorization A = U**T * U or A = L * L**T.
|
|
*
|
|
CALL SCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
|
|
CALL SPPTRF( UPLO, N, AFP, INFO )
|
|
*
|
|
* Return if INFO is non-zero.
|
|
*
|
|
IF( INFO.GT.0 )THEN
|
|
RCOND = ZERO
|
|
RETURN
|
|
END IF
|
|
END IF
|
|
*
|
|
* Compute the norm of the matrix A.
|
|
*
|
|
ANORM = SLANSP( 'I', UPLO, N, AP, WORK )
|
|
*
|
|
* Compute the reciprocal of the condition number of A.
|
|
*
|
|
CALL SPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, IWORK, INFO )
|
|
*
|
|
* Compute the solution matrix X.
|
|
*
|
|
CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
|
CALL SPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO )
|
|
*
|
|
* Use iterative refinement to improve the computed solution and
|
|
* compute error bounds and backward error estimates for it.
|
|
*
|
|
CALL SPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR,
|
|
$ WORK, IWORK, INFO )
|
|
*
|
|
* Transform the solution matrix X to a solution of the original
|
|
* system.
|
|
*
|
|
IF( RCEQU ) THEN
|
|
DO 50 J = 1, NRHS
|
|
DO 40 I = 1, N
|
|
X( I, J ) = S( I )*X( I, J )
|
|
40 CONTINUE
|
|
50 CONTINUE
|
|
DO 60 J = 1, NRHS
|
|
FERR( J ) = FERR( J ) / SCOND
|
|
60 CONTINUE
|
|
END IF
|
|
*
|
|
* Set INFO = N+1 if the matrix is singular to working precision.
|
|
*
|
|
IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
|
|
$ INFO = N + 1
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SPPSVX
|
|
*
|
|
END
|