*> \brief \b DLA_GBRCOND estimates the Skeel condition number for a general banded matrix. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLA_GBRCOND + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * DOUBLE PRECISION FUNCTION DLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB, * AFB, LDAFB, IPIV, CMODE, C, * INFO, WORK, IWORK ) * * .. Scalar Arguments .. * CHARACTER TRANS * INTEGER N, LDAB, LDAFB, INFO, KL, KU, CMODE * .. * .. Array Arguments .. * INTEGER IWORK( * ), IPIV( * ) * DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ), * $ C( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLA_GBRCOND Estimates the Skeel condition number of op(A) * op2(C) *> where op2 is determined by CMODE as follows *> CMODE = 1 op2(C) = C *> CMODE = 0 op2(C) = I *> CMODE = -1 op2(C) = inv(C) *> The Skeel condition number cond(A) = norminf( |inv(A)||A| ) *> is computed by computing scaling factors R such that *> diag(R)*A*op2(C) is row equilibrated and computing the standard *> infinity-norm condition number. *> \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 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) *> \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 DOUBLE PRECISION array, dimension (LDAFB,N) *> 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. *> \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 DGBTRF; row i of the matrix was interchanged *> with row IPIV(i). *> \endverbatim *> *> \param[in] CMODE *> \verbatim *> CMODE is INTEGER *> Determines op2(C) in the formula op(A) * op2(C) as follows: *> CMODE = 1 op2(C) = C *> CMODE = 0 op2(C) = I *> CMODE = -1 op2(C) = inv(C) *> \endverbatim *> *> \param[in] C *> \verbatim *> C is DOUBLE PRECISION array, dimension (N) *> The vector C in the formula op(A) * op2(C). *> \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 DOUBLE PRECISION array, dimension (5*N). *> Workspace. *> \endverbatim *> *> \param[in] IWORK *> \verbatim *> IWORK is INTEGER 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 doubleGBcomputational * * ===================================================================== DOUBLE PRECISION FUNCTION DLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB, $ AFB, LDAFB, IPIV, CMODE, C, $ INFO, WORK, IWORK ) * * -- 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, LDAB, LDAFB, INFO, KL, KU, CMODE * .. * .. Array Arguments .. INTEGER IWORK( * ), IPIV( * ) DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ), $ C( * ) * .. * * ===================================================================== * * .. Local Scalars .. LOGICAL NOTRANS INTEGER KASE, I, J, KD, KE DOUBLE PRECISION AINVNM, TMP * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DLACN2, DGBTRS, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX * .. * .. Executable Statements .. * DLA_GBRCOND = 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( 'DLA_GBRCOND', -INFO ) RETURN END IF IF( N.EQ.0 ) THEN DLA_GBRCOND = 1.0D+0 RETURN END IF * * Compute the equilibration matrix R such that * inv(R)*A*C has unit 1-norm. * KD = KU + 1 KE = KL + 1 IF ( NOTRANS ) THEN DO I = 1, N TMP = 0.0D+0 IF ( CMODE .EQ. 1 ) THEN DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + ABS( AB( KD+I-J, J ) * C( J ) ) END DO ELSE IF ( CMODE .EQ. 0 ) THEN DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + ABS( AB( KD+I-J, J ) ) END DO ELSE DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + ABS( AB( KD+I-J, J ) / C( J ) ) END DO END IF WORK( 2*N+I ) = TMP END DO ELSE DO I = 1, N TMP = 0.0D+0 IF ( CMODE .EQ. 1 ) THEN DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + ABS( AB( KE-I+J, I ) * C( J ) ) END DO ELSE IF ( CMODE .EQ. 0 ) THEN DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + ABS( AB( KE-I+J, I ) ) END DO ELSE DO J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + ABS( AB( KE-I+J, I ) / C( J ) ) END DO END IF WORK( 2*N+I ) = TMP END DO END IF * * Estimate the norm of inv(op(A)). * AINVNM = 0.0D+0 KASE = 0 10 CONTINUE CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.2 ) THEN * * Multiply by R. * DO I = 1, N WORK( I ) = WORK( I ) * WORK( 2*N+I ) END DO IF ( NOTRANS ) THEN CALL DGBTRS( 'No transpose', N, KL, KU, 1, AFB, LDAFB, $ IPIV, WORK, N, INFO ) ELSE CALL DGBTRS( 'Transpose', N, KL, KU, 1, AFB, LDAFB, IPIV, $ WORK, N, INFO ) END IF * * Multiply by inv(C). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N WORK( I ) = WORK( I ) / C( I ) END DO ELSE IF ( CMODE .EQ. -1 ) THEN DO I = 1, N WORK( I ) = WORK( I ) * C( I ) END DO END IF ELSE * * Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N WORK( I ) = WORK( I ) / C( I ) END DO ELSE IF ( CMODE .EQ. -1 ) THEN DO I = 1, N WORK( I ) = WORK( I ) * C( I ) END DO END IF IF ( NOTRANS ) THEN CALL DGBTRS( 'Transpose', N, KL, KU, 1, AFB, LDAFB, IPIV, $ WORK, N, INFO ) ELSE CALL DGBTRS( '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 ) * WORK( 2*N+I ) END DO END IF GO TO 10 END IF * * Compute the estimate of the reciprocal condition number. * IF( AINVNM .NE. 0.0D+0 ) $ DLA_GBRCOND = ( 1.0D+0 / AINVNM ) * RETURN * END