*> \brief \b ZLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLA_GBRCOND_X + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * DOUBLE PRECISION FUNCTION ZLA_GBRCOND_X( TRANS, N, KL, KU, AB, * LDAB, AFB, LDAFB, IPIV, * X, INFO, WORK, RWORK ) * * .. Scalar Arguments .. * CHARACTER TRANS * INTEGER N, KL, KU, KD, KE, LDAB, LDAFB, INFO * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ), * $ X( * ) * DOUBLE PRECISION RWORK( * ) * * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLA_GBRCOND_X Computes the infinity norm condition number of *> op(A) * diag(X) where X is a COMPLEX*16 vector. *> \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 = 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] 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) *> \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 the factorization A = P*L*U *> as computed by ZGBTRF; row i of the matrix was interchanged *> with row IPIV(i). *> \endverbatim *> *> \param[in] X *> \verbatim *> X is COMPLEX*16 array, dimension (N) *> The vector X in the formula op(A) * diag(X). *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: Successful exit. *> i > 0: The ith argument is invalid. *> \endverbatim *> *> \param[in] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (2*N). *> Workspace. *> \endverbatim *> *> \param[in] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION array, dimension (N). *> Workspace. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16GBcomputational * * ===================================================================== DOUBLE PRECISION FUNCTION ZLA_GBRCOND_X( TRANS, N, KL, KU, AB, $ LDAB, AFB, LDAFB, IPIV, $ X, INFO, WORK, RWORK ) * * -- 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 N, KL, KU, KD, KE, LDAB, LDAFB, INFO * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ), $ X( * ) DOUBLE PRECISION RWORK( * ) * * * ===================================================================== * * .. Local Scalars .. LOGICAL NOTRANS INTEGER KASE, I, J DOUBLE PRECISION AINVNM, ANORM, TMP COMPLEX*16 ZDUM * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL ZLACN2, ZGBTRS, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function Definitions .. CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) * .. * .. Executable Statements .. * ZLA_GBRCOND_X = 0.0D+0 * INFO = 0 NOTRANS = LSAME( TRANS, 'N' ) IF ( .NOT. NOTRANS .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 .OR. KL.GT.N-1 ) THEN INFO = -3 ELSE IF( KU.LT.0 .OR. KU.GT.N-1 ) THEN INFO = -4 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -6 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZLA_GBRCOND_X', -INFO ) RETURN END IF * * Compute norm of op(A)*op2(C). * KD = KU + 1 KE = KL + 1 ANORM = 0.0D+0 IF ( NOTRANS ) THEN DO I = 1, N TMP = 0.0D+0 DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + CABS1( AB( KD+I-J, J) * X( J ) ) END DO RWORK( I ) = TMP ANORM = MAX( ANORM, TMP ) END DO ELSE DO I = 1, N TMP = 0.0D+0 DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + CABS1( AB( KE-I+J, I ) * X( J ) ) END DO RWORK( I ) = TMP ANORM = MAX( ANORM, TMP ) END DO END IF * * Quick return if possible. * IF( N.EQ.0 ) THEN ZLA_GBRCOND_X = 1.0D+0 RETURN ELSE IF( ANORM .EQ. 0.0D+0 ) THEN RETURN END IF * * Estimate the norm of inv(op(A)). * AINVNM = 0.0D+0 * KASE = 0 10 CONTINUE CALL ZLACN2( N, WORK( N+1 ), WORK, AINVNM, KASE, ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.2 ) THEN * * Multiply by R. * DO I = 1, N WORK( I ) = WORK( I ) * RWORK( I ) END DO * IF ( NOTRANS ) THEN CALL ZGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB, $ IPIV, WORK, N, INFO ) ELSE CALL ZGBTRS( 'Conjugate transpose', N, KL, KU, 1, AFB, $ LDAFB, IPIV, WORK, N, INFO ) ENDIF * * Multiply by inv(X). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) END DO ELSE * * Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) END DO * IF ( NOTRANS ) THEN CALL ZGBTRS( 'Conjugate transpose', N, KL, KU, 1, AFB, $ LDAFB, IPIV, WORK, N, INFO ) ELSE CALL ZGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB, $ IPIV, WORK, N, INFO ) END IF * * Multiply by R. * DO I = 1, N WORK( I ) = WORK( I ) * RWORK( I ) END DO END IF GO TO 10 END IF * * Compute the estimate of the reciprocal condition number. * IF( AINVNM .NE. 0.0D+0 ) $ ZLA_GBRCOND_X = 1.0D+0 / AINVNM * RETURN * END