444 lines
13 KiB
Fortran
444 lines
13 KiB
Fortran
*> \brief \b SPBRFS
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download SPBRFS + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spbrfs.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/spbrfs.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spbrfs.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
|
|
* LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* CHARACTER UPLO
|
|
* INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* INTEGER IWORK( * )
|
|
* REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
|
* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> SPBRFS improves the computed solution to a system of linear
|
|
*> equations when the coefficient matrix is symmetric positive definite
|
|
*> and banded, and provides error bounds and backward error estimates
|
|
*> for the solution.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \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 order of the matrix A. N >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] KD
|
|
*> \verbatim
|
|
*> KD is INTEGER
|
|
*> The number of superdiagonals of the matrix A if UPLO = 'U',
|
|
*> or the number of subdiagonals if UPLO = 'L'. KD >= 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] AB
|
|
*> \verbatim
|
|
*> AB is REAL array, dimension (LDAB,N)
|
|
*> The upper or lower triangle of the symmetric band matrix A,
|
|
*> stored in the first KD+1 rows of the array. The j-th column
|
|
*> of A is stored in the j-th column of the array AB as follows:
|
|
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
|
|
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDAB
|
|
*> \verbatim
|
|
*> LDAB is INTEGER
|
|
*> The leading dimension of the array AB. LDAB >= KD+1.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] AFB
|
|
*> \verbatim
|
|
*> AFB is REAL array, dimension (LDAFB,N)
|
|
*> The triangular factor U or L from the Cholesky factorization
|
|
*> A = U**T*U or A = L*L**T of the band matrix A as computed by
|
|
*> SPBTRF, in the same storage format as A (see AB).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDAFB
|
|
*> \verbatim
|
|
*> LDAFB is INTEGER
|
|
*> The leading dimension of the array AFB. LDAFB >= KD+1.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] B
|
|
*> \verbatim
|
|
*> B is REAL array, dimension (LDB,NRHS)
|
|
*> The right hand side matrix B.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDB
|
|
*> \verbatim
|
|
*> LDB is INTEGER
|
|
*> The leading dimension of the array B. LDB >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] X
|
|
*> \verbatim
|
|
*> X is REAL array, dimension (LDX,NRHS)
|
|
*> On entry, the solution matrix X, as computed by SPBTRS.
|
|
*> On exit, the improved solution matrix X.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDX
|
|
*> \verbatim
|
|
*> LDX is INTEGER
|
|
*> The leading dimension of the array X. LDX >= max(1,N).
|
|
*> \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
|
|
*> \endverbatim
|
|
*
|
|
*> \par Internal Parameters:
|
|
* =========================
|
|
*>
|
|
*> \verbatim
|
|
*> ITMAX is the maximum number of steps of iterative refinement.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \date December 2016
|
|
*
|
|
*> \ingroup realOTHERcomputational
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
|
|
$ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
|
|
*
|
|
* -- LAPACK computational 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..--
|
|
* December 2016
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER UPLO
|
|
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER IWORK( * )
|
|
REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
|
|
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
INTEGER ITMAX
|
|
PARAMETER ( ITMAX = 5 )
|
|
REAL ZERO
|
|
PARAMETER ( ZERO = 0.0E+0 )
|
|
REAL ONE
|
|
PARAMETER ( ONE = 1.0E+0 )
|
|
REAL TWO
|
|
PARAMETER ( TWO = 2.0E+0 )
|
|
REAL THREE
|
|
PARAMETER ( THREE = 3.0E+0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL UPPER
|
|
INTEGER COUNT, I, J, K, KASE, L, NZ
|
|
REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
|
|
* ..
|
|
* .. Local Arrays ..
|
|
INTEGER ISAVE( 3 )
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL SAXPY, SCOPY, SLACN2, SPBTRS, SSBMV, XERBLA
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC ABS, MAX, MIN
|
|
* ..
|
|
* .. External Functions ..
|
|
LOGICAL LSAME
|
|
REAL SLAMCH
|
|
EXTERNAL LSAME, SLAMCH
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input parameters.
|
|
*
|
|
INFO = 0
|
|
UPPER = LSAME( UPLO, 'U' )
|
|
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
|
|
INFO = -1
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( KD.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -4
|
|
ELSE IF( LDAB.LT.KD+1 ) THEN
|
|
INFO = -6
|
|
ELSE IF( LDAFB.LT.KD+1 ) THEN
|
|
INFO = -8
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -10
|
|
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
|
INFO = -12
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'SPBRFS', -INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
|
|
DO 10 J = 1, NRHS
|
|
FERR( J ) = ZERO
|
|
BERR( J ) = ZERO
|
|
10 CONTINUE
|
|
RETURN
|
|
END IF
|
|
*
|
|
* NZ = maximum number of nonzero elements in each row of A, plus 1
|
|
*
|
|
NZ = MIN( N+1, 2*KD+2 )
|
|
EPS = SLAMCH( 'Epsilon' )
|
|
SAFMIN = SLAMCH( 'Safe minimum' )
|
|
SAFE1 = NZ*SAFMIN
|
|
SAFE2 = SAFE1 / EPS
|
|
*
|
|
* Do for each right hand side
|
|
*
|
|
DO 140 J = 1, NRHS
|
|
*
|
|
COUNT = 1
|
|
LSTRES = THREE
|
|
20 CONTINUE
|
|
*
|
|
* Loop until stopping criterion is satisfied.
|
|
*
|
|
* Compute residual R = B - A * X
|
|
*
|
|
CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
|
|
CALL SSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
|
|
$ WORK( N+1 ), 1 )
|
|
*
|
|
* Compute componentwise relative backward error from formula
|
|
*
|
|
* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
|
|
*
|
|
* where abs(Z) is the componentwise absolute value of the matrix
|
|
* or vector Z. If the i-th component of the denominator is less
|
|
* than SAFE2, then SAFE1 is added to the i-th components of the
|
|
* numerator and denominator before dividing.
|
|
*
|
|
DO 30 I = 1, N
|
|
WORK( I ) = ABS( B( I, J ) )
|
|
30 CONTINUE
|
|
*
|
|
* Compute abs(A)*abs(X) + abs(B).
|
|
*
|
|
IF( UPPER ) THEN
|
|
DO 50 K = 1, N
|
|
S = ZERO
|
|
XK = ABS( X( K, J ) )
|
|
L = KD + 1 - K
|
|
DO 40 I = MAX( 1, K-KD ), K - 1
|
|
WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
|
|
S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
|
|
40 CONTINUE
|
|
WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
|
|
50 CONTINUE
|
|
ELSE
|
|
DO 70 K = 1, N
|
|
S = ZERO
|
|
XK = ABS( X( K, J ) )
|
|
WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
|
|
L = 1 - K
|
|
DO 60 I = K + 1, MIN( N, K+KD )
|
|
WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
|
|
S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
|
|
60 CONTINUE
|
|
WORK( K ) = WORK( K ) + S
|
|
70 CONTINUE
|
|
END IF
|
|
S = ZERO
|
|
DO 80 I = 1, N
|
|
IF( WORK( I ).GT.SAFE2 ) THEN
|
|
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
|
|
ELSE
|
|
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
|
|
$ ( WORK( I )+SAFE1 ) )
|
|
END IF
|
|
80 CONTINUE
|
|
BERR( J ) = S
|
|
*
|
|
* Test stopping criterion. Continue iterating if
|
|
* 1) The residual BERR(J) is larger than machine epsilon, and
|
|
* 2) BERR(J) decreased by at least a factor of 2 during the
|
|
* last iteration, and
|
|
* 3) At most ITMAX iterations tried.
|
|
*
|
|
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
|
|
$ COUNT.LE.ITMAX ) THEN
|
|
*
|
|
* Update solution and try again.
|
|
*
|
|
CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
|
|
$ INFO )
|
|
CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
|
|
LSTRES = BERR( J )
|
|
COUNT = COUNT + 1
|
|
GO TO 20
|
|
END IF
|
|
*
|
|
* Bound error from formula
|
|
*
|
|
* norm(X - XTRUE) / norm(X) .le. FERR =
|
|
* norm( abs(inv(A))*
|
|
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
|
|
*
|
|
* where
|
|
* norm(Z) is the magnitude of the largest component of Z
|
|
* inv(A) is the inverse of A
|
|
* abs(Z) is the componentwise absolute value of the matrix or
|
|
* vector Z
|
|
* NZ is the maximum number of nonzeros in any row of A, plus 1
|
|
* EPS is machine epsilon
|
|
*
|
|
* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
|
|
* is incremented by SAFE1 if the i-th component of
|
|
* abs(A)*abs(X) + abs(B) is less than SAFE2.
|
|
*
|
|
* Use SLACN2 to estimate the infinity-norm of the matrix
|
|
* inv(A) * diag(W),
|
|
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
|
|
*
|
|
DO 90 I = 1, N
|
|
IF( WORK( I ).GT.SAFE2 ) THEN
|
|
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
|
|
ELSE
|
|
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
|
|
END IF
|
|
90 CONTINUE
|
|
*
|
|
KASE = 0
|
|
100 CONTINUE
|
|
CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
|
|
$ KASE, ISAVE )
|
|
IF( KASE.NE.0 ) THEN
|
|
IF( KASE.EQ.1 ) THEN
|
|
*
|
|
* Multiply by diag(W)*inv(A**T).
|
|
*
|
|
CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
|
|
$ INFO )
|
|
DO 110 I = 1, N
|
|
WORK( N+I ) = WORK( N+I )*WORK( I )
|
|
110 CONTINUE
|
|
ELSE IF( KASE.EQ.2 ) THEN
|
|
*
|
|
* Multiply by inv(A)*diag(W).
|
|
*
|
|
DO 120 I = 1, N
|
|
WORK( N+I ) = WORK( N+I )*WORK( I )
|
|
120 CONTINUE
|
|
CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
|
|
$ INFO )
|
|
END IF
|
|
GO TO 100
|
|
END IF
|
|
*
|
|
* Normalize error.
|
|
*
|
|
LSTRES = ZERO
|
|
DO 130 I = 1, N
|
|
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
|
|
130 CONTINUE
|
|
IF( LSTRES.NE.ZERO )
|
|
$ FERR( J ) = FERR( J ) / LSTRES
|
|
*
|
|
140 CONTINUE
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SPBRFS
|
|
*
|
|
END
|