999 lines
31 KiB
Fortran
999 lines
31 KiB
Fortran
*> \brief \b CLATBS solves a triangular banded system of equations.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download CLATBS + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clatbs.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clatbs.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clatbs.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE CLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
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* SCALE, CNORM, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER DIAG, NORMIN, TRANS, UPLO
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* INTEGER INFO, KD, LDAB, N
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* REAL SCALE
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* ..
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* .. Array Arguments ..
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* REAL CNORM( * )
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* COMPLEX AB( LDAB, * ), X( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> CLATBS solves one of the triangular systems
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*>
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*> A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
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*>
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*> with scaling to prevent overflow, where A is an upper or lower
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*> triangular band matrix. Here A**T denotes the transpose of A, x and b
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*> are n-element vectors, and s is a scaling factor, usually less than
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*> or equal to 1, chosen so that the components of x will be less than
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*> the overflow threshold. If the unscaled problem will not cause
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*> overflow, the Level 2 BLAS routine CTBSV is called. If the matrix A
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*> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
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*> non-trivial solution to A*x = 0 is returned.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> Specifies whether the matrix A is upper or lower triangular.
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*> = 'U': Upper triangular
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*> = 'L': Lower triangular
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*> \endverbatim
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*>
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*> \param[in] TRANS
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*> \verbatim
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*> TRANS is CHARACTER*1
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*> Specifies the operation applied to A.
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*> = 'N': Solve A * x = s*b (No transpose)
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*> = 'T': Solve A**T * x = s*b (Transpose)
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*> = 'C': Solve A**H * x = s*b (Conjugate transpose)
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*> \endverbatim
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*>
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*> \param[in] DIAG
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*> \verbatim
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*> DIAG is CHARACTER*1
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*> Specifies whether or not the matrix A is unit triangular.
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*> = 'N': Non-unit triangular
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*> = 'U': Unit triangular
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*> \endverbatim
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*>
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*> \param[in] NORMIN
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*> \verbatim
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*> NORMIN is CHARACTER*1
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*> Specifies whether CNORM has been set or not.
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*> = 'Y': CNORM contains the column norms on entry
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*> = 'N': CNORM is not set on entry. On exit, the norms will
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*> be computed and stored in CNORM.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] KD
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*> \verbatim
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*> KD is INTEGER
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*> The number of subdiagonals or superdiagonals in the
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*> triangular matrix A. KD >= 0.
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*> \endverbatim
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*>
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*> \param[in] AB
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*> \verbatim
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*> AB is COMPLEX array, dimension (LDAB,N)
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*> The upper or lower triangular band matrix A, stored in the
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*> first KD+1 rows of the array. The j-th column of A is stored
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*> in the j-th column of the array AB as follows:
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*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
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*> \endverbatim
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*>
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*> \param[in] LDAB
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*> \verbatim
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*> LDAB is INTEGER
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*> The leading dimension of the array AB. LDAB >= KD+1.
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*> \endverbatim
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*>
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*> \param[in,out] X
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*> \verbatim
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*> X is COMPLEX array, dimension (N)
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*> On entry, the right hand side b of the triangular system.
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*> On exit, X is overwritten by the solution vector x.
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*> \endverbatim
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*>
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*> \param[out] SCALE
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*> \verbatim
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*> SCALE is REAL
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*> The scaling factor s for the triangular system
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*> A * x = s*b, A**T * x = s*b, or A**H * x = s*b.
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*> If SCALE = 0, the matrix A is singular or badly scaled, and
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*> the vector x is an exact or approximate solution to A*x = 0.
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*> \endverbatim
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*>
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*> \param[in,out] CNORM
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*> \verbatim
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*> CNORM is REAL array, dimension (N)
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*>
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*> If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
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*> contains the norm of the off-diagonal part of the j-th column
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*> of A. If TRANS = 'N', CNORM(j) must be greater than or equal
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*> to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
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*> must be greater than or equal to the 1-norm.
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*>
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*> If NORMIN = 'N', CNORM is an output argument and CNORM(j)
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*> returns the 1-norm of the offdiagonal part of the j-th column
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*> of A.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -k, the k-th argument had an illegal value
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date December 2016
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*
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*> \ingroup complexOTHERauxiliary
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> A rough bound on x is computed; if that is less than overflow, CTBSV
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*> is called, otherwise, specific code is used which checks for possible
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*> overflow or divide-by-zero at every operation.
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*>
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*> A columnwise scheme is used for solving A*x = b. The basic algorithm
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*> if A is lower triangular is
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*>
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*> x[1:n] := b[1:n]
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*> for j = 1, ..., n
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*> x(j) := x(j) / A(j,j)
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*> x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
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*> end
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*>
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*> Define bounds on the components of x after j iterations of the loop:
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*> M(j) = bound on x[1:j]
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*> G(j) = bound on x[j+1:n]
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*> Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
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*>
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*> Then for iteration j+1 we have
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*> M(j+1) <= G(j) / | A(j+1,j+1) |
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*> G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
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*> <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
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*>
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*> where CNORM(j+1) is greater than or equal to the infinity-norm of
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*> column j+1 of A, not counting the diagonal. Hence
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*>
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*> G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
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*> 1<=i<=j
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*> and
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*>
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*> |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
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*> 1<=i< j
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*>
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*> Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTBSV if the
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*> reciprocal of the largest M(j), j=1,..,n, is larger than
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*> max(underflow, 1/overflow).
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*>
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*> The bound on x(j) is also used to determine when a step in the
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*> columnwise method can be performed without fear of overflow. If
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*> the computed bound is greater than a large constant, x is scaled to
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*> prevent overflow, but if the bound overflows, x is set to 0, x(j) to
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*> 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
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*>
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*> Similarly, a row-wise scheme is used to solve A**T *x = b or
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*> A**H *x = b. The basic algorithm for A upper triangular is
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*>
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*> for j = 1, ..., n
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*> x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
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*> end
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*>
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*> We simultaneously compute two bounds
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*> G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
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*> M(j) = bound on x(i), 1<=i<=j
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*>
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*> The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
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*> add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
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*> Then the bound on x(j) is
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*>
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*> M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
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*>
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*> <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
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*> 1<=i<=j
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*>
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*> and we can safely call CTBSV if 1/M(n) and 1/G(n) are both greater
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*> than max(underflow, 1/overflow).
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE CLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
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$ SCALE, CNORM, INFO )
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*
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* -- LAPACK auxiliary routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* December 2016
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*
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* .. Scalar Arguments ..
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CHARACTER DIAG, NORMIN, TRANS, UPLO
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INTEGER INFO, KD, LDAB, N
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REAL SCALE
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* ..
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* .. Array Arguments ..
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REAL CNORM( * )
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COMPLEX AB( LDAB, * ), X( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, HALF, ONE, TWO
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PARAMETER ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0,
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$ TWO = 2.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRAN, NOUNIT, UPPER
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INTEGER I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND
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REAL BIGNUM, GROW, REC, SMLNUM, TJJ, TMAX, TSCAL,
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$ XBND, XJ, XMAX
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COMPLEX CSUMJ, TJJS, USCAL, ZDUM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ICAMAX, ISAMAX
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REAL SCASUM, SLAMCH
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COMPLEX CDOTC, CDOTU, CLADIV
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EXTERNAL LSAME, ICAMAX, ISAMAX, SCASUM, SLAMCH, CDOTC,
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$ CDOTU, CLADIV
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* ..
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* .. External Subroutines ..
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EXTERNAL CAXPY, CSSCAL, CTBSV, SLABAD, SSCAL, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, MIN, REAL
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* ..
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* .. Statement Functions ..
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REAL CABS1, CABS2
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* ..
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* .. Statement Function definitions ..
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CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
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CABS2( ZDUM ) = ABS( REAL( ZDUM ) / 2. ) +
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$ ABS( AIMAG( ZDUM ) / 2. )
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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UPPER = LSAME( UPLO, 'U' )
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NOTRAN = LSAME( TRANS, 'N' )
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NOUNIT = LSAME( DIAG, 'N' )
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*
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* Test the input parameters.
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*
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IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
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INFO = -1
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ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
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$ LSAME( TRANS, 'C' ) ) THEN
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INFO = -2
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ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
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INFO = -3
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ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
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$ LSAME( NORMIN, 'N' ) ) THEN
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INFO = -4
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ELSE IF( N.LT.0 ) THEN
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INFO = -5
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ELSE IF( KD.LT.0 ) THEN
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INFO = -6
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ELSE IF( LDAB.LT.KD+1 ) THEN
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INFO = -8
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CLATBS', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Determine machine dependent parameters to control overflow.
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*
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SMLNUM = SLAMCH( 'Safe minimum' )
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BIGNUM = ONE / SMLNUM
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CALL SLABAD( SMLNUM, BIGNUM )
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SMLNUM = SMLNUM / SLAMCH( 'Precision' )
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BIGNUM = ONE / SMLNUM
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SCALE = ONE
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*
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IF( LSAME( NORMIN, 'N' ) ) THEN
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*
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* Compute the 1-norm of each column, not including the diagonal.
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*
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IF( UPPER ) THEN
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*
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* A is upper triangular.
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*
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DO 10 J = 1, N
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JLEN = MIN( KD, J-1 )
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CNORM( J ) = SCASUM( JLEN, AB( KD+1-JLEN, J ), 1 )
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10 CONTINUE
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ELSE
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*
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* A is lower triangular.
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*
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DO 20 J = 1, N
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JLEN = MIN( KD, N-J )
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IF( JLEN.GT.0 ) THEN
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CNORM( J ) = SCASUM( JLEN, AB( 2, J ), 1 )
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ELSE
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CNORM( J ) = ZERO
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END IF
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20 CONTINUE
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END IF
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END IF
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*
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* Scale the column norms by TSCAL if the maximum element in CNORM is
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* greater than BIGNUM/2.
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*
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IMAX = ISAMAX( N, CNORM, 1 )
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TMAX = CNORM( IMAX )
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IF( TMAX.LE.BIGNUM*HALF ) THEN
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TSCAL = ONE
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ELSE
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TSCAL = HALF / ( SMLNUM*TMAX )
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CALL SSCAL( N, TSCAL, CNORM, 1 )
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END IF
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*
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* Compute a bound on the computed solution vector to see if the
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* Level 2 BLAS routine CTBSV can be used.
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*
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XMAX = ZERO
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DO 30 J = 1, N
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XMAX = MAX( XMAX, CABS2( X( J ) ) )
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30 CONTINUE
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XBND = XMAX
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IF( NOTRAN ) THEN
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*
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* Compute the growth in A * x = b.
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*
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IF( UPPER ) THEN
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JFIRST = N
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JLAST = 1
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JINC = -1
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MAIND = KD + 1
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ELSE
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JFIRST = 1
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JLAST = N
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JINC = 1
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MAIND = 1
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END IF
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*
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IF( TSCAL.NE.ONE ) THEN
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GROW = ZERO
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GO TO 60
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END IF
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*
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IF( NOUNIT ) THEN
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*
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* A is non-unit triangular.
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*
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* Compute GROW = 1/G(j) and XBND = 1/M(j).
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* Initially, G(0) = max{x(i), i=1,...,n}.
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*
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GROW = HALF / MAX( XBND, SMLNUM )
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XBND = GROW
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DO 40 J = JFIRST, JLAST, JINC
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*
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* Exit the loop if the growth factor is too small.
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*
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IF( GROW.LE.SMLNUM )
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$ GO TO 60
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*
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TJJS = AB( MAIND, J )
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TJJ = CABS1( TJJS )
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*
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IF( TJJ.GE.SMLNUM ) THEN
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*
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* M(j) = G(j-1) / abs(A(j,j))
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*
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XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
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ELSE
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*
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* M(j) could overflow, set XBND to 0.
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*
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XBND = ZERO
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END IF
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*
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IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
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*
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* G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
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*
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GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
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ELSE
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*
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* G(j) could overflow, set GROW to 0.
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*
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GROW = ZERO
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END IF
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40 CONTINUE
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GROW = XBND
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ELSE
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*
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* A is unit triangular.
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*
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* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
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*
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GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
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DO 50 J = JFIRST, JLAST, JINC
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*
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* Exit the loop if the growth factor is too small.
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*
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IF( GROW.LE.SMLNUM )
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$ GO TO 60
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*
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* G(j) = G(j-1)*( 1 + CNORM(j) )
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*
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GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
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50 CONTINUE
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END IF
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60 CONTINUE
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*
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ELSE
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*
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|
* Compute the growth in A**T * x = b or A**H * x = b.
|
|
*
|
|
IF( UPPER ) THEN
|
|
JFIRST = 1
|
|
JLAST = N
|
|
JINC = 1
|
|
MAIND = KD + 1
|
|
ELSE
|
|
JFIRST = N
|
|
JLAST = 1
|
|
JINC = -1
|
|
MAIND = 1
|
|
END IF
|
|
*
|
|
IF( TSCAL.NE.ONE ) THEN
|
|
GROW = ZERO
|
|
GO TO 90
|
|
END IF
|
|
*
|
|
IF( NOUNIT ) THEN
|
|
*
|
|
* A is non-unit triangular.
|
|
*
|
|
* Compute GROW = 1/G(j) and XBND = 1/M(j).
|
|
* Initially, M(0) = max{x(i), i=1,...,n}.
|
|
*
|
|
GROW = HALF / MAX( XBND, SMLNUM )
|
|
XBND = GROW
|
|
DO 70 J = JFIRST, JLAST, JINC
|
|
*
|
|
* Exit the loop if the growth factor is too small.
|
|
*
|
|
IF( GROW.LE.SMLNUM )
|
|
$ GO TO 90
|
|
*
|
|
* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
|
|
*
|
|
XJ = ONE + CNORM( J )
|
|
GROW = MIN( GROW, XBND / XJ )
|
|
*
|
|
TJJS = AB( MAIND, J )
|
|
TJJ = CABS1( TJJS )
|
|
*
|
|
IF( TJJ.GE.SMLNUM ) THEN
|
|
*
|
|
* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
|
|
*
|
|
IF( XJ.GT.TJJ )
|
|
$ XBND = XBND*( TJJ / XJ )
|
|
ELSE
|
|
*
|
|
* M(j) could overflow, set XBND to 0.
|
|
*
|
|
XBND = ZERO
|
|
END IF
|
|
70 CONTINUE
|
|
GROW = MIN( GROW, XBND )
|
|
ELSE
|
|
*
|
|
* A is unit triangular.
|
|
*
|
|
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
|
|
*
|
|
GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
|
|
DO 80 J = JFIRST, JLAST, JINC
|
|
*
|
|
* Exit the loop if the growth factor is too small.
|
|
*
|
|
IF( GROW.LE.SMLNUM )
|
|
$ GO TO 90
|
|
*
|
|
* G(j) = ( 1 + CNORM(j) )*G(j-1)
|
|
*
|
|
XJ = ONE + CNORM( J )
|
|
GROW = GROW / XJ
|
|
80 CONTINUE
|
|
END IF
|
|
90 CONTINUE
|
|
END IF
|
|
*
|
|
IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
|
|
*
|
|
* Use the Level 2 BLAS solve if the reciprocal of the bound on
|
|
* elements of X is not too small.
|
|
*
|
|
CALL CTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, X, 1 )
|
|
ELSE
|
|
*
|
|
* Use a Level 1 BLAS solve, scaling intermediate results.
|
|
*
|
|
IF( XMAX.GT.BIGNUM*HALF ) THEN
|
|
*
|
|
* Scale X so that its components are less than or equal to
|
|
* BIGNUM in absolute value.
|
|
*
|
|
SCALE = ( BIGNUM*HALF ) / XMAX
|
|
CALL CSSCAL( N, SCALE, X, 1 )
|
|
XMAX = BIGNUM
|
|
ELSE
|
|
XMAX = XMAX*TWO
|
|
END IF
|
|
*
|
|
IF( NOTRAN ) THEN
|
|
*
|
|
* Solve A * x = b
|
|
*
|
|
DO 110 J = JFIRST, JLAST, JINC
|
|
*
|
|
* Compute x(j) = b(j) / A(j,j), scaling x if necessary.
|
|
*
|
|
XJ = CABS1( X( J ) )
|
|
IF( NOUNIT ) THEN
|
|
TJJS = AB( MAIND, J )*TSCAL
|
|
ELSE
|
|
TJJS = TSCAL
|
|
IF( TSCAL.EQ.ONE )
|
|
$ GO TO 105
|
|
END IF
|
|
TJJ = CABS1( TJJS )
|
|
IF( TJJ.GT.SMLNUM ) THEN
|
|
*
|
|
* abs(A(j,j)) > SMLNUM:
|
|
*
|
|
IF( TJJ.LT.ONE ) THEN
|
|
IF( XJ.GT.TJJ*BIGNUM ) THEN
|
|
*
|
|
* Scale x by 1/b(j).
|
|
*
|
|
REC = ONE / XJ
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
END IF
|
|
X( J ) = CLADIV( X( J ), TJJS )
|
|
XJ = CABS1( X( J ) )
|
|
ELSE IF( TJJ.GT.ZERO ) THEN
|
|
*
|
|
* 0 < abs(A(j,j)) <= SMLNUM:
|
|
*
|
|
IF( XJ.GT.TJJ*BIGNUM ) THEN
|
|
*
|
|
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
|
|
* to avoid overflow when dividing by A(j,j).
|
|
*
|
|
REC = ( TJJ*BIGNUM ) / XJ
|
|
IF( CNORM( J ).GT.ONE ) THEN
|
|
*
|
|
* Scale by 1/CNORM(j) to avoid overflow when
|
|
* multiplying x(j) times column j.
|
|
*
|
|
REC = REC / CNORM( J )
|
|
END IF
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
X( J ) = CLADIV( X( J ), TJJS )
|
|
XJ = CABS1( X( J ) )
|
|
ELSE
|
|
*
|
|
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
|
|
* scale = 0, and compute a solution to A*x = 0.
|
|
*
|
|
DO 100 I = 1, N
|
|
X( I ) = ZERO
|
|
100 CONTINUE
|
|
X( J ) = ONE
|
|
XJ = ONE
|
|
SCALE = ZERO
|
|
XMAX = ZERO
|
|
END IF
|
|
105 CONTINUE
|
|
*
|
|
* Scale x if necessary to avoid overflow when adding a
|
|
* multiple of column j of A.
|
|
*
|
|
IF( XJ.GT.ONE ) THEN
|
|
REC = ONE / XJ
|
|
IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
|
|
*
|
|
* Scale x by 1/(2*abs(x(j))).
|
|
*
|
|
REC = REC*HALF
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
END IF
|
|
ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
|
|
*
|
|
* Scale x by 1/2.
|
|
*
|
|
CALL CSSCAL( N, HALF, X, 1 )
|
|
SCALE = SCALE*HALF
|
|
END IF
|
|
*
|
|
IF( UPPER ) THEN
|
|
IF( J.GT.1 ) THEN
|
|
*
|
|
* Compute the update
|
|
* x(max(1,j-kd):j-1) := x(max(1,j-kd):j-1) -
|
|
* x(j)* A(max(1,j-kd):j-1,j)
|
|
*
|
|
JLEN = MIN( KD, J-1 )
|
|
CALL CAXPY( JLEN, -X( J )*TSCAL,
|
|
$ AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 )
|
|
I = ICAMAX( J-1, X, 1 )
|
|
XMAX = CABS1( X( I ) )
|
|
END IF
|
|
ELSE IF( J.LT.N ) THEN
|
|
*
|
|
* Compute the update
|
|
* x(j+1:min(j+kd,n)) := x(j+1:min(j+kd,n)) -
|
|
* x(j) * A(j+1:min(j+kd,n),j)
|
|
*
|
|
JLEN = MIN( KD, N-J )
|
|
IF( JLEN.GT.0 )
|
|
$ CALL CAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1,
|
|
$ X( J+1 ), 1 )
|
|
I = J + ICAMAX( N-J, X( J+1 ), 1 )
|
|
XMAX = CABS1( X( I ) )
|
|
END IF
|
|
110 CONTINUE
|
|
*
|
|
ELSE IF( LSAME( TRANS, 'T' ) ) THEN
|
|
*
|
|
* Solve A**T * x = b
|
|
*
|
|
DO 150 J = JFIRST, JLAST, JINC
|
|
*
|
|
* Compute x(j) = b(j) - sum A(k,j)*x(k).
|
|
* k<>j
|
|
*
|
|
XJ = CABS1( X( J ) )
|
|
USCAL = TSCAL
|
|
REC = ONE / MAX( XMAX, ONE )
|
|
IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
|
|
*
|
|
* If x(j) could overflow, scale x by 1/(2*XMAX).
|
|
*
|
|
REC = REC*HALF
|
|
IF( NOUNIT ) THEN
|
|
TJJS = AB( MAIND, J )*TSCAL
|
|
ELSE
|
|
TJJS = TSCAL
|
|
END IF
|
|
TJJ = CABS1( TJJS )
|
|
IF( TJJ.GT.ONE ) THEN
|
|
*
|
|
* Divide by A(j,j) when scaling x if A(j,j) > 1.
|
|
*
|
|
REC = MIN( ONE, REC*TJJ )
|
|
USCAL = CLADIV( USCAL, TJJS )
|
|
END IF
|
|
IF( REC.LT.ONE ) THEN
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
END IF
|
|
*
|
|
CSUMJ = ZERO
|
|
IF( USCAL.EQ.CMPLX( ONE ) ) THEN
|
|
*
|
|
* If the scaling needed for A in the dot product is 1,
|
|
* call CDOTU to perform the dot product.
|
|
*
|
|
IF( UPPER ) THEN
|
|
JLEN = MIN( KD, J-1 )
|
|
CSUMJ = CDOTU( JLEN, AB( KD+1-JLEN, J ), 1,
|
|
$ X( J-JLEN ), 1 )
|
|
ELSE
|
|
JLEN = MIN( KD, N-J )
|
|
IF( JLEN.GT.1 )
|
|
$ CSUMJ = CDOTU( JLEN, AB( 2, J ), 1, X( J+1 ),
|
|
$ 1 )
|
|
END IF
|
|
ELSE
|
|
*
|
|
* Otherwise, use in-line code for the dot product.
|
|
*
|
|
IF( UPPER ) THEN
|
|
JLEN = MIN( KD, J-1 )
|
|
DO 120 I = 1, JLEN
|
|
CSUMJ = CSUMJ + ( AB( KD+I-JLEN, J )*USCAL )*
|
|
$ X( J-JLEN-1+I )
|
|
120 CONTINUE
|
|
ELSE
|
|
JLEN = MIN( KD, N-J )
|
|
DO 130 I = 1, JLEN
|
|
CSUMJ = CSUMJ + ( AB( I+1, J )*USCAL )*X( J+I )
|
|
130 CONTINUE
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( USCAL.EQ.CMPLX( TSCAL ) ) THEN
|
|
*
|
|
* Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
|
|
* was not used to scale the dotproduct.
|
|
*
|
|
X( J ) = X( J ) - CSUMJ
|
|
XJ = CABS1( X( J ) )
|
|
IF( NOUNIT ) THEN
|
|
*
|
|
* Compute x(j) = x(j) / A(j,j), scaling if necessary.
|
|
*
|
|
TJJS = AB( MAIND, J )*TSCAL
|
|
ELSE
|
|
TJJS = TSCAL
|
|
IF( TSCAL.EQ.ONE )
|
|
$ GO TO 145
|
|
END IF
|
|
TJJ = CABS1( TJJS )
|
|
IF( TJJ.GT.SMLNUM ) THEN
|
|
*
|
|
* abs(A(j,j)) > SMLNUM:
|
|
*
|
|
IF( TJJ.LT.ONE ) THEN
|
|
IF( XJ.GT.TJJ*BIGNUM ) THEN
|
|
*
|
|
* Scale X by 1/abs(x(j)).
|
|
*
|
|
REC = ONE / XJ
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
END IF
|
|
X( J ) = CLADIV( X( J ), TJJS )
|
|
ELSE IF( TJJ.GT.ZERO ) THEN
|
|
*
|
|
* 0 < abs(A(j,j)) <= SMLNUM:
|
|
*
|
|
IF( XJ.GT.TJJ*BIGNUM ) THEN
|
|
*
|
|
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
|
|
*
|
|
REC = ( TJJ*BIGNUM ) / XJ
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
X( J ) = CLADIV( X( J ), TJJS )
|
|
ELSE
|
|
*
|
|
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
|
|
* scale = 0 and compute a solution to A**T *x = 0.
|
|
*
|
|
DO 140 I = 1, N
|
|
X( I ) = ZERO
|
|
140 CONTINUE
|
|
X( J ) = ONE
|
|
SCALE = ZERO
|
|
XMAX = ZERO
|
|
END IF
|
|
145 CONTINUE
|
|
ELSE
|
|
*
|
|
* Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
|
|
* product has already been divided by 1/A(j,j).
|
|
*
|
|
X( J ) = CLADIV( X( J ), TJJS ) - CSUMJ
|
|
END IF
|
|
XMAX = MAX( XMAX, CABS1( X( J ) ) )
|
|
150 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Solve A**H * x = b
|
|
*
|
|
DO 190 J = JFIRST, JLAST, JINC
|
|
*
|
|
* Compute x(j) = b(j) - sum A(k,j)*x(k).
|
|
* k<>j
|
|
*
|
|
XJ = CABS1( X( J ) )
|
|
USCAL = TSCAL
|
|
REC = ONE / MAX( XMAX, ONE )
|
|
IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
|
|
*
|
|
* If x(j) could overflow, scale x by 1/(2*XMAX).
|
|
*
|
|
REC = REC*HALF
|
|
IF( NOUNIT ) THEN
|
|
TJJS = CONJG( AB( MAIND, J ) )*TSCAL
|
|
ELSE
|
|
TJJS = TSCAL
|
|
END IF
|
|
TJJ = CABS1( TJJS )
|
|
IF( TJJ.GT.ONE ) THEN
|
|
*
|
|
* Divide by A(j,j) when scaling x if A(j,j) > 1.
|
|
*
|
|
REC = MIN( ONE, REC*TJJ )
|
|
USCAL = CLADIV( USCAL, TJJS )
|
|
END IF
|
|
IF( REC.LT.ONE ) THEN
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
END IF
|
|
*
|
|
CSUMJ = ZERO
|
|
IF( USCAL.EQ.CMPLX( ONE ) ) THEN
|
|
*
|
|
* If the scaling needed for A in the dot product is 1,
|
|
* call CDOTC to perform the dot product.
|
|
*
|
|
IF( UPPER ) THEN
|
|
JLEN = MIN( KD, J-1 )
|
|
CSUMJ = CDOTC( JLEN, AB( KD+1-JLEN, J ), 1,
|
|
$ X( J-JLEN ), 1 )
|
|
ELSE
|
|
JLEN = MIN( KD, N-J )
|
|
IF( JLEN.GT.1 )
|
|
$ CSUMJ = CDOTC( JLEN, AB( 2, J ), 1, X( J+1 ),
|
|
$ 1 )
|
|
END IF
|
|
ELSE
|
|
*
|
|
* Otherwise, use in-line code for the dot product.
|
|
*
|
|
IF( UPPER ) THEN
|
|
JLEN = MIN( KD, J-1 )
|
|
DO 160 I = 1, JLEN
|
|
CSUMJ = CSUMJ + ( CONJG( AB( KD+I-JLEN, J ) )*
|
|
$ USCAL )*X( J-JLEN-1+I )
|
|
160 CONTINUE
|
|
ELSE
|
|
JLEN = MIN( KD, N-J )
|
|
DO 170 I = 1, JLEN
|
|
CSUMJ = CSUMJ + ( CONJG( AB( I+1, J ) )*USCAL )*
|
|
$ X( J+I )
|
|
170 CONTINUE
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( USCAL.EQ.CMPLX( TSCAL ) ) THEN
|
|
*
|
|
* Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
|
|
* was not used to scale the dotproduct.
|
|
*
|
|
X( J ) = X( J ) - CSUMJ
|
|
XJ = CABS1( X( J ) )
|
|
IF( NOUNIT ) THEN
|
|
*
|
|
* Compute x(j) = x(j) / A(j,j), scaling if necessary.
|
|
*
|
|
TJJS = CONJG( AB( MAIND, J ) )*TSCAL
|
|
ELSE
|
|
TJJS = TSCAL
|
|
IF( TSCAL.EQ.ONE )
|
|
$ GO TO 185
|
|
END IF
|
|
TJJ = CABS1( TJJS )
|
|
IF( TJJ.GT.SMLNUM ) THEN
|
|
*
|
|
* abs(A(j,j)) > SMLNUM:
|
|
*
|
|
IF( TJJ.LT.ONE ) THEN
|
|
IF( XJ.GT.TJJ*BIGNUM ) THEN
|
|
*
|
|
* Scale X by 1/abs(x(j)).
|
|
*
|
|
REC = ONE / XJ
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
END IF
|
|
X( J ) = CLADIV( X( J ), TJJS )
|
|
ELSE IF( TJJ.GT.ZERO ) THEN
|
|
*
|
|
* 0 < abs(A(j,j)) <= SMLNUM:
|
|
*
|
|
IF( XJ.GT.TJJ*BIGNUM ) THEN
|
|
*
|
|
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
|
|
*
|
|
REC = ( TJJ*BIGNUM ) / XJ
|
|
CALL CSSCAL( N, REC, X, 1 )
|
|
SCALE = SCALE*REC
|
|
XMAX = XMAX*REC
|
|
END IF
|
|
X( J ) = CLADIV( X( J ), TJJS )
|
|
ELSE
|
|
*
|
|
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
|
|
* scale = 0 and compute a solution to A**H *x = 0.
|
|
*
|
|
DO 180 I = 1, N
|
|
X( I ) = ZERO
|
|
180 CONTINUE
|
|
X( J ) = ONE
|
|
SCALE = ZERO
|
|
XMAX = ZERO
|
|
END IF
|
|
185 CONTINUE
|
|
ELSE
|
|
*
|
|
* Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
|
|
* product has already been divided by 1/A(j,j).
|
|
*
|
|
X( J ) = CLADIV( X( J ), TJJS ) - CSUMJ
|
|
END IF
|
|
XMAX = MAX( XMAX, CABS1( X( J ) ) )
|
|
190 CONTINUE
|
|
END IF
|
|
SCALE = SCALE / TSCAL
|
|
END IF
|
|
*
|
|
* Scale the column norms by 1/TSCAL for return.
|
|
*
|
|
IF( TSCAL.NE.ONE ) THEN
|
|
CALL SSCAL( N, ONE / TSCAL, CNORM, 1 )
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of CLATBS
|
|
*
|
|
END
|