*> \brief \b ZGBRFS * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGBRFS + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, * IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, * INFO ) * * .. Scalar Arguments .. * CHARACTER TRANS * INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS * .. * .. Array Arguments .. * INTEGER IPIV( * ) * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), * $ WORK( * ), X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGBRFS improves the computed solution to a system of linear *> equations when the coefficient matrix is banded, and provides *> error bounds and backward error estimates for the solution. *> \endverbatim * * Arguments: * ========== * *> \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 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] AB *> \verbatim *> AB is COMPLEX*16 array, dimension (LDAB,N) *> The original band matrix A, stored 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). *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KL+KU+1. *> \endverbatim *> *> \param[in] AFB *> \verbatim *> AFB is COMPLEX*16 array, dimension (LDAFB,N) *> 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. *> \endverbatim *> *> \param[in] LDAFB *> \verbatim *> LDAFB is INTEGER *> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. *> \endverbatim *> *> \param[in] IPIV *> \verbatim *> IPIV is INTEGER array, dimension (N) *> The pivot indices from ZGBTRF; for 1<=i<=N, row i of the *> matrix was interchanged with row IPIV(i). *> \endverbatim *> *> \param[in] B *> \verbatim *> B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS) *> On entry, the solution matrix X, as computed by ZGBTRS. *> 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 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 (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 complex16GBcomputational * * ===================================================================== SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, $ 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 TRANS INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS * .. * .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), $ WORK( * ), X( LDX, * ) * .. * * ===================================================================== * * .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) * .. * .. Local Scalars .. LOGICAL NOTRAN CHARACTER TRANSN, TRANST INTEGER COUNT, I, J, K, KASE, KK, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK COMPLEX*16 ZDUM * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Subroutines .. EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGBMV, ZGBTRS, ZLACN2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DIMAG, MAX, MIN * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. $ LSAME( TRANS, 'C' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -7 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -9 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -12 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGBRFS', -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 * IF( NOTRAN ) THEN TRANSN = 'N' TRANST = 'C' ELSE TRANSN = 'C' TRANST = 'N' END IF * * NZ = maximum number of nonzero elements in each row of A, plus 1 * NZ = MIN( KL+KU+2, N+1 ) EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( '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 - op(A) * X, * where op(A) = A, A**T, or A**H, depending on TRANS. * CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) CALL ZGBMV( TRANS, N, N, KL, KU, -CONE, AB, LDAB, X( 1, J ), 1, $ CONE, WORK, 1 ) * * Compute componentwise relative backward error from formula * * max(i) ( abs(R(i)) / ( abs(op(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 RWORK( I ) = CABS1( B( I, J ) ) 30 CONTINUE * * Compute abs(op(A))*abs(X) + abs(B). * IF( NOTRAN ) THEN DO 50 K = 1, N KK = KU + 1 - K XK = CABS1( X( K, J ) ) DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL ) RWORK( I ) = RWORK( I ) + CABS1( AB( KK+I, K ) )*XK 40 CONTINUE 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO KK = KU + 1 - K DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL ) S = S + CABS1( AB( KK+I, K ) )*CABS1( X( I, J ) ) 60 CONTINUE RWORK( K ) = RWORK( K ) + S 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( RWORK( I ).GT.SAFE2 ) THEN S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) ELSE S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / $ ( RWORK( 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 ZGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, WORK, N, $ INFO ) CALL ZAXPY( N, CONE, WORK, 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(op(A)))* * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) * * where * norm(Z) is the magnitude of the largest component of Z * inv(op(A)) is the inverse of op(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(op(A))*abs(X)+abs(B)) * is incremented by SAFE1 if the i-th component of * abs(op(A))*abs(X) + abs(B) is less than SAFE2. * * Use ZLACN2 to estimate the infinity-norm of the matrix * inv(op(A)) * diag(W), * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) * DO 90 I = 1, N IF( RWORK( I ).GT.SAFE2 ) THEN RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) ELSE RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + $ SAFE1 END IF 90 CONTINUE * KASE = 0 100 CONTINUE CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Multiply by diag(W)*inv(op(A)**H). * CALL ZGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV, $ WORK, N, INFO ) DO 110 I = 1, N WORK( I ) = RWORK( I )*WORK( I ) 110 CONTINUE ELSE * * Multiply by inv(op(A))*diag(W). * DO 120 I = 1, N WORK( I ) = RWORK( I )*WORK( I ) 120 CONTINUE CALL ZGBTRS( TRANSN, N, KL, KU, 1, AFB, LDAFB, IPIV, $ WORK, N, INFO ) END IF GO TO 100 END IF * * Normalize error. * LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) $ FERR( J ) = FERR( J ) / LSTRES * 140 CONTINUE * RETURN * * End of ZGBRFS * END