*> \brief \b DLA_GBRCOND estimates the Skeel condition number for a general banded matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
* 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[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (5*N).
*> Workspace.
*> \endverbatim
*>
*> \param[out] 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.
*
*> \ingroup la_gbrcond
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLA_GBRCOND( TRANS, N, KL, KU, AB,
$ LDAB,
$ AFB, LDAFB, IPIV, CMODE, C,
$ INFO, WORK, IWORK )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. 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 of DLA_GBRCOND
*
END