393 lines
11 KiB
Fortran
393 lines
11 KiB
Fortran
*> \brief \b SPTRFS
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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 SPTRFS + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sptrfs.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/sptrfs.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/sptrfs.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 SPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
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* BERR, WORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDB, LDX, N, NRHS
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* ..
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* .. Array Arguments ..
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* REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
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* $ E( * ), EF( * ), FERR( * ), WORK( * ),
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* $ X( LDX, * )
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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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*> SPTRFS improves the computed solution to a system of linear
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*> equations when the coefficient matrix is symmetric positive definite
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*> and tridiagonal, and provides error bounds and backward error
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*> estimates for the solution.
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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] 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] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of right hand sides, i.e., the number of columns
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*> of the matrix B. NRHS >= 0.
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*> \endverbatim
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*>
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*> \param[in] D
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*> \verbatim
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*> D is REAL array, dimension (N)
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*> The n diagonal elements of the tridiagonal matrix A.
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*> \endverbatim
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*>
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*> \param[in] E
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*> \verbatim
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*> E is REAL array, dimension (N-1)
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*> The (n-1) subdiagonal elements of the tridiagonal matrix A.
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*> \endverbatim
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*>
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*> \param[in] DF
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*> \verbatim
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*> DF is REAL array, dimension (N)
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*> The n diagonal elements of the diagonal matrix D from the
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*> factorization computed by SPTTRF.
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*> \endverbatim
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*>
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*> \param[in] EF
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*> \verbatim
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*> EF is REAL array, dimension (N-1)
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*> The (n-1) subdiagonal elements of the unit bidiagonal factor
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*> L from the factorization computed by SPTTRF.
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*> \endverbatim
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*>
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*> \param[in] B
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*> \verbatim
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*> B is REAL array, dimension (LDB,NRHS)
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*> The right hand side matrix B.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= 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 REAL array, dimension (LDX,NRHS)
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*> On entry, the solution matrix X, as computed by SPTTRS.
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*> On exit, the improved solution matrix X.
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*> \endverbatim
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*>
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*> \param[in] LDX
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*> \verbatim
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*> LDX is INTEGER
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*> The leading dimension of the array X. LDX >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] FERR
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*> \verbatim
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*> FERR is REAL array, dimension (NRHS)
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*> The forward error bound for each solution vector
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*> X(j) (the j-th column of the solution matrix X).
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*> If XTRUE is the true solution corresponding to X(j), FERR(j)
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*> is an estimated upper bound for the magnitude of the largest
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*> element in (X(j) - XTRUE) divided by the magnitude of the
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*> largest element in X(j).
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*> \endverbatim
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*>
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*> \param[out] BERR
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*> \verbatim
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*> BERR is REAL array, dimension (NRHS)
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*> The componentwise relative backward error of each solution
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*> vector X(j) (i.e., the smallest relative change in
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*> any element of A or B that makes X(j) an exact solution).
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array, dimension (2*N)
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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 = -i, the i-th argument had an illegal value
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*> \endverbatim
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*
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*> \par Internal Parameters:
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* =========================
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*>
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*> \verbatim
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*> ITMAX is the maximum number of steps of iterative refinement.
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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 realPTcomputational
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*
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* =====================================================================
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SUBROUTINE SPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
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$ BERR, WORK, INFO )
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*
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* -- LAPACK computational routine --
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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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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDB, LDX, N, NRHS
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* ..
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* .. Array Arguments ..
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REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
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$ E( * ), EF( * ), FERR( * ), WORK( * ),
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$ X( LDX, * )
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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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INTEGER ITMAX
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PARAMETER ( ITMAX = 5 )
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REAL ZERO
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PARAMETER ( ZERO = 0.0E+0 )
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REAL ONE
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PARAMETER ( ONE = 1.0E+0 )
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REAL TWO
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PARAMETER ( TWO = 2.0E+0 )
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REAL THREE
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PARAMETER ( THREE = 3.0E+0 )
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* ..
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* .. Local Scalars ..
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INTEGER COUNT, I, IX, J, NZ
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REAL BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
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$ SAFMIN
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* ..
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* .. External Subroutines ..
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EXTERNAL SAXPY, SPTTRS, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX
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* ..
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* .. External Functions ..
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INTEGER ISAMAX
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REAL SLAMCH
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EXTERNAL ISAMAX, SLAMCH
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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IF( N.LT.0 ) THEN
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INFO = -1
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -2
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
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INFO = -10
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SPTRFS', -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 .OR. NRHS.EQ.0 ) THEN
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DO 10 J = 1, NRHS
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FERR( J ) = ZERO
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BERR( J ) = ZERO
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10 CONTINUE
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RETURN
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END IF
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*
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* NZ = maximum number of nonzero elements in each row of A, plus 1
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*
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NZ = 4
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EPS = SLAMCH( 'Epsilon' )
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SAFMIN = SLAMCH( 'Safe minimum' )
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SAFE1 = NZ*SAFMIN
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SAFE2 = SAFE1 / EPS
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*
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* Do for each right hand side
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*
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DO 90 J = 1, NRHS
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*
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COUNT = 1
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LSTRES = THREE
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20 CONTINUE
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*
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* Loop until stopping criterion is satisfied.
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*
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* Compute residual R = B - A * X. Also compute
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* abs(A)*abs(x) + abs(b) for use in the backward error bound.
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*
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IF( N.EQ.1 ) THEN
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BI = B( 1, J )
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DX = D( 1 )*X( 1, J )
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WORK( N+1 ) = BI - DX
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WORK( 1 ) = ABS( BI ) + ABS( DX )
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ELSE
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BI = B( 1, J )
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DX = D( 1 )*X( 1, J )
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EX = E( 1 )*X( 2, J )
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WORK( N+1 ) = BI - DX - EX
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WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
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DO 30 I = 2, N - 1
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BI = B( I, J )
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CX = E( I-1 )*X( I-1, J )
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DX = D( I )*X( I, J )
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EX = E( I )*X( I+1, J )
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WORK( N+I ) = BI - CX - DX - EX
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WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
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30 CONTINUE
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BI = B( N, J )
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CX = E( N-1 )*X( N-1, J )
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DX = D( N )*X( N, J )
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WORK( N+N ) = BI - CX - DX
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WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
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END IF
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*
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* Compute componentwise relative backward error from formula
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*
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* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
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*
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* where abs(Z) is the componentwise absolute value of the matrix
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* or vector Z. If the i-th component of the denominator is less
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* than SAFE2, then SAFE1 is added to the i-th components of the
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* numerator and denominator before dividing.
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*
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S = ZERO
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DO 40 I = 1, N
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IF( WORK( I ).GT.SAFE2 ) THEN
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S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
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ELSE
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S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
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$ ( WORK( I )+SAFE1 ) )
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END IF
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40 CONTINUE
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BERR( J ) = S
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*
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* Test stopping criterion. Continue iterating if
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* 1) The residual BERR(J) is larger than machine epsilon, and
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* 2) BERR(J) decreased by at least a factor of 2 during the
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* last iteration, and
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* 3) At most ITMAX iterations tried.
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*
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IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
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$ COUNT.LE.ITMAX ) THEN
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*
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* Update solution and try again.
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*
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CALL SPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
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CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
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LSTRES = BERR( J )
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COUNT = COUNT + 1
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GO TO 20
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END IF
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*
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* Bound error from formula
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*
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* norm(X - XTRUE) / norm(X) .le. FERR =
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* norm( abs(inv(A))*
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* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
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*
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* where
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* norm(Z) is the magnitude of the largest component of Z
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* inv(A) is the inverse of A
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* abs(Z) is the componentwise absolute value of the matrix or
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* vector Z
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* NZ is the maximum number of nonzeros in any row of A, plus 1
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* EPS is machine epsilon
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*
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* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
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* is incremented by SAFE1 if the i-th component of
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* abs(A)*abs(X) + abs(B) is less than SAFE2.
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*
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DO 50 I = 1, N
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IF( WORK( I ).GT.SAFE2 ) THEN
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WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
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ELSE
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WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
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END IF
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50 CONTINUE
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IX = ISAMAX( N, WORK, 1 )
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FERR( J ) = WORK( IX )
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*
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* Estimate the norm of inv(A).
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*
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* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
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*
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* m(i,j) = abs(A(i,j)), i = j,
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* m(i,j) = -abs(A(i,j)), i .ne. j,
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*
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* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
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*
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* Solve M(L) * x = e.
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*
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WORK( 1 ) = ONE
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DO 60 I = 2, N
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WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
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60 CONTINUE
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*
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* Solve D * M(L)**T * x = b.
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*
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WORK( N ) = WORK( N ) / DF( N )
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DO 70 I = N - 1, 1, -1
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WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
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70 CONTINUE
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*
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* Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
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*
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IX = ISAMAX( N, WORK, 1 )
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FERR( J ) = FERR( J )*ABS( WORK( IX ) )
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*
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* Normalize error.
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*
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LSTRES = ZERO
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DO 80 I = 1, N
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LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
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80 CONTINUE
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IF( LSTRES.NE.ZERO )
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$ FERR( J ) = FERR( J ) / LSTRES
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*
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90 CONTINUE
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*
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RETURN
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*
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* End of SPTRFS
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*
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END
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