version 1.7, 2010/12/21 13:53:34
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version 1.20, 2023/08/07 08:39:00
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*> \brief \b DLATRS solves a triangular system of equations with the scale factor set to prevent overflow. |
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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 DLATRS + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrs.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/dlatrs.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/dlatrs.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 DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, |
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* 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, LDA, N |
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* DOUBLE PRECISION SCALE |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), CNORM( * ), 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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*> DLATRS solves one of the triangular systems |
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*> |
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*> A *x = s*b or A**T *x = s*b |
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*> |
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*> with scaling to prevent overflow. Here A is an upper or lower |
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*> triangular matrix, A**T denotes the transpose of A, x and b are |
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*> 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 DTRSV 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**T* x = s*b (Conjugate transpose = 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] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimension (LDA,N) |
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*> The triangular matrix A. If UPLO = 'U', the leading n by n |
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*> upper triangular part of the array A contains the upper |
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*> triangular matrix, and the strictly lower triangular part of |
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*> A is not referenced. If UPLO = 'L', the leading n by n lower |
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*> triangular part of the array A contains the lower triangular |
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*> matrix, and the strictly upper triangular part of A is not |
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*> referenced. If DIAG = 'U', the diagonal elements of A are |
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*> also not referenced and are assumed to be 1. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max (1,N). |
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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 DOUBLE PRECISION 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 DOUBLE PRECISION |
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*> The scaling factor s for the triangular system |
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*> A * x = s*b or A**T* 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 DOUBLE PRECISION 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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*> \ingroup doubleOTHERauxiliary |
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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, DTRSV |
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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 DTRSV 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. The basic |
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*> 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]**T * 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]**T * 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 DTRSV 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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* ===================================================================== |
SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, |
SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, |
$ CNORM, INFO ) |
$ CNORM, INFO ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER DIAG, NORMIN, TRANS, UPLO |
CHARACTER DIAG, NORMIN, TRANS, UPLO |
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DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * ) |
DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
|
* |
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* DLATRS solves one of the triangular systems |
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* |
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* A *x = s*b or A'*x = s*b |
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* |
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* with scaling to prevent overflow. Here A is an upper or lower |
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* triangular matrix, A' denotes the transpose of A, x and b are |
|
* n-element vectors, and s is a scaling factor, usually less than |
|
* or equal to 1, chosen so that the components of x will be less than |
|
* the overflow threshold. If the unscaled problem will not cause |
|
* overflow, the Level 2 BLAS routine DTRSV 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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* |
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* Arguments |
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* ========= |
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* |
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* UPLO (input) 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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* |
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* TRANS (input) 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'* x = s*b (Transpose) |
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* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) |
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* |
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* DIAG (input) 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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* |
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* NORMIN (input) 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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* |
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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* A (input) DOUBLE PRECISION array, dimension (LDA,N) |
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* The triangular matrix A. If UPLO = 'U', the leading n by n |
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* upper triangular part of the array A contains the upper |
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* triangular matrix, and the strictly lower triangular part of |
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* A is not referenced. If UPLO = 'L', the leading n by n lower |
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* triangular part of the array A contains the lower triangular |
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* matrix, and the strictly upper triangular part of A is not |
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* referenced. If DIAG = 'U', the diagonal elements of A are |
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* also not referenced and are assumed to be 1. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max (1,N). |
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* |
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* X (input/output) DOUBLE PRECISION 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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* |
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* SCALE (output) DOUBLE PRECISION |
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* The scaling factor s for the triangular system |
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* A * x = s*b or A'* 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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* |
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* CNORM (input or output) DOUBLE PRECISION 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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* |
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* INFO (output) 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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* |
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* Further Details |
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* ======= ======= |
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* |
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* A rough bound on x is computed; if that is less than overflow, DTRSV |
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* is called, otherwise, specific code is used which checks for possible |
|
* overflow or divide-by-zero at every operation. |
|
* |
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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 DTRSV 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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* 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 |
|
* 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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* Similarly, a row-wise scheme is used to solve A'*x = b. The basic |
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* algorithm for A upper triangular is |
|
* |
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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. |
|
* Then the bound on x(j) is |
|
* |
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* M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | |
|
* |
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* <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) |
|
* 1<=i<=j |
|
* |
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* and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater |
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* than max(underflow, 1/overflow). |
|
* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* .. External Functions .. |
* .. External Functions .. |
LOGICAL LSAME |
LOGICAL LSAME |
INTEGER IDAMAX |
INTEGER IDAMAX |
DOUBLE PRECISION DASUM, DDOT, DLAMCH |
DOUBLE PRECISION DASUM, DDOT, DLAMCH, DLANGE |
EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH |
EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH, DLANGE |
* .. |
* .. |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DAXPY, DSCAL, DTRSV, XERBLA |
EXTERNAL DAXPY, DSCAL, DTRSV, XERBLA |
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* |
* |
* Quick return if possible |
* Quick return if possible |
* |
* |
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SCALE = ONE |
IF( N.EQ.0 ) |
IF( N.EQ.0 ) |
$ RETURN |
$ RETURN |
* |
* |
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* |
* |
SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) |
SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) |
BIGNUM = ONE / SMLNUM |
BIGNUM = ONE / SMLNUM |
SCALE = ONE |
|
* |
* |
IF( LSAME( NORMIN, 'N' ) ) THEN |
IF( LSAME( NORMIN, 'N' ) ) THEN |
* |
* |
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IF( TMAX.LE.BIGNUM ) THEN |
IF( TMAX.LE.BIGNUM ) THEN |
TSCAL = ONE |
TSCAL = ONE |
ELSE |
ELSE |
TSCAL = ONE / ( SMLNUM*TMAX ) |
* |
CALL DSCAL( N, TSCAL, CNORM, 1 ) |
* Avoid NaN generation if entries in CNORM exceed the |
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* overflow threshold |
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* |
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IF( TMAX.LE.DLAMCH('Overflow') ) THEN |
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* Case 1: All entries in CNORM are valid floating-point numbers |
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TSCAL = ONE / ( SMLNUM*TMAX ) |
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CALL DSCAL( N, TSCAL, CNORM, 1 ) |
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ELSE |
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* Case 2: At least one column norm of A cannot be represented |
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* as floating-point number. Find the offdiagonal entry A( I, J ) |
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* with the largest absolute value. If this entry is not +/- Infinity, |
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* use this value as TSCAL. |
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TMAX = ZERO |
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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 J = 2, N |
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TMAX = MAX( DLANGE( 'M', J-1, 1, A( 1, J ), 1, SUMJ ), |
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$ TMAX ) |
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END DO |
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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 J = 1, N - 1 |
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TMAX = MAX( DLANGE( 'M', N-J, 1, A( J+1, J ), 1, |
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$ SUMJ ), TMAX ) |
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END DO |
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END IF |
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* |
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IF( TMAX.LE.DLAMCH('Overflow') ) THEN |
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TSCAL = ONE / ( SMLNUM*TMAX ) |
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DO J = 1, N |
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IF( CNORM( J ).LE.DLAMCH('Overflow') ) THEN |
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CNORM( J ) = CNORM( J )*TSCAL |
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ELSE |
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* Recompute the 1-norm without introducing Infinity |
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* in the summation |
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CNORM( J ) = ZERO |
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IF( UPPER ) THEN |
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DO I = 1, J - 1 |
|
CNORM( J ) = CNORM( J ) + |
|
$ TSCAL * ABS( A( I, J ) ) |
|
END DO |
|
ELSE |
|
DO I = J + 1, N |
|
CNORM( J ) = CNORM( J ) + |
|
$ TSCAL * ABS( A( I, J ) ) |
|
END DO |
|
END IF |
|
END IF |
|
END DO |
|
ELSE |
|
* At least one entry of A is not a valid floating-point entry. |
|
* Rely on TRSV to propagate Inf and NaN. |
|
CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 ) |
|
RETURN |
|
END IF |
|
END IF |
END IF |
END IF |
* |
* |
* Compute a bound on the computed solution vector to see if the |
* Compute a bound on the computed solution vector to see if the |
Line 346
|
Line 487
|
* |
* |
ELSE |
ELSE |
* |
* |
* Compute the growth in A' * x = b. |
* Compute the growth in A**T * x = b. |
* |
* |
IF( UPPER ) THEN |
IF( UPPER ) THEN |
JFIRST = 1 |
JFIRST = 1 |
Line 554
|
Line 695
|
* |
* |
ELSE |
ELSE |
* |
* |
* Solve A' * x = b |
* Solve A**T * x = b |
* |
* |
DO 160 J = JFIRST, JLAST, JINC |
DO 160 J = JFIRST, JLAST, JINC |
* |
* |
Line 666
|
Line 807
|
ELSE |
ELSE |
* |
* |
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and |
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and |
* scale = 0, and compute a solution to A'*x = 0. |
* scale = 0, and compute a solution to A**T*x = 0. |
* |
* |
DO 140 I = 1, N |
DO 140 I = 1, N |
X( I ) = ZERO |
X( I ) = ZERO |