--- rpl/lapack/lapack/dptrfs.f 2010/08/07 13:22:24 1.5 +++ rpl/lapack/lapack/dptrfs.f 2023/08/07 08:39:05 1.19 @@ -1,10 +1,169 @@ +*> \brief \b DPTRFS +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DPTRFS + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, +* BERR, WORK, INFO ) +* +* .. Scalar Arguments .. +* INTEGER INFO, LDB, LDX, N, NRHS +* .. +* .. Array Arguments .. +* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), +* $ E( * ), EF( * ), FERR( * ), WORK( * ), +* $ X( LDX, * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DPTRFS improves the computed solution to a system of linear +*> equations when the coefficient matrix is symmetric positive definite +*> and tridiagonal, and provides error bounds and backward error +*> estimates for the solution. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of right hand sides, i.e., the number of columns +*> of the matrix B. NRHS >= 0. +*> \endverbatim +*> +*> \param[in] D +*> \verbatim +*> D is DOUBLE PRECISION array, dimension (N) +*> The n diagonal elements of the tridiagonal matrix A. +*> \endverbatim +*> +*> \param[in] E +*> \verbatim +*> E is DOUBLE PRECISION array, dimension (N-1) +*> The (n-1) subdiagonal elements of the tridiagonal matrix A. +*> \endverbatim +*> +*> \param[in] DF +*> \verbatim +*> DF is DOUBLE PRECISION array, dimension (N) +*> The n diagonal elements of the diagonal matrix D from the +*> factorization computed by DPTTRF. +*> \endverbatim +*> +*> \param[in] EF +*> \verbatim +*> EF is DOUBLE PRECISION array, dimension (N-1) +*> The (n-1) subdiagonal elements of the unit bidiagonal factor +*> L from the factorization computed by DPTTRF. +*> \endverbatim +*> +*> \param[in] B +*> \verbatim +*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) +*> The right hand side matrix B. +*> \endverbatim +*> +*> \param[in] LDB +*> \verbatim +*> LDB is INTEGER +*> The leading dimension of the array B. LDB >= max(1,N). +*> \endverbatim +*> +*> \param[in,out] X +*> \verbatim +*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) +*> On entry, the solution matrix X, as computed by DPTTRS. +*> On exit, the improved solution matrix X. +*> \endverbatim +*> +*> \param[in] LDX +*> \verbatim +*> LDX is INTEGER +*> The leading dimension of the array X. LDX >= max(1,N). +*> \endverbatim +*> +*> \param[out] FERR +*> \verbatim +*> FERR is DOUBLE PRECISION array, dimension (NRHS) +*> The forward error bound for each solution vector +*> X(j) (the j-th column of the solution matrix X). +*> If XTRUE is the true solution corresponding to X(j), FERR(j) +*> is an estimated upper bound for the magnitude of the largest +*> element in (X(j) - XTRUE) divided by the magnitude of the +*> largest element in X(j). +*> \endverbatim +*> +*> \param[out] BERR +*> \verbatim +*> BERR is DOUBLE PRECISION array, dimension (NRHS) +*> The componentwise relative backward error of each solution +*> vector X(j) (i.e., the smallest relative change in +*> any element of A or B that makes X(j) an exact solution). +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is DOUBLE PRECISION array, dimension (2*N) +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -i, the i-th argument had an illegal value +*> \endverbatim +* +*> \par Internal Parameters: +* ========================= +*> +*> \verbatim +*> ITMAX is the maximum number of steps of iterative refinement. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \ingroup doublePTcomputational +* +* ===================================================================== SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, $ BERR, WORK, INFO ) * -* -- LAPACK routine (version 3.2) -- +* -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 * * .. Scalar Arguments .. INTEGER INFO, LDB, LDX, N, NRHS @@ -15,75 +174,6 @@ $ X( LDX, * ) * .. * -* Purpose -* ======= -* -* DPTRFS improves the computed solution to a system of linear -* equations when the coefficient matrix is symmetric positive definite -* and tridiagonal, and provides error bounds and backward error -* estimates for the solution. -* -* Arguments -* ========= -* -* N (input) INTEGER -* The order of the matrix A. N >= 0. -* -* NRHS (input) INTEGER -* The number of right hand sides, i.e., the number of columns -* of the matrix B. NRHS >= 0. -* -* D (input) DOUBLE PRECISION array, dimension (N) -* The n diagonal elements of the tridiagonal matrix A. -* -* E (input) DOUBLE PRECISION array, dimension (N-1) -* The (n-1) subdiagonal elements of the tridiagonal matrix A. -* -* DF (input) DOUBLE PRECISION array, dimension (N) -* The n diagonal elements of the diagonal matrix D from the -* factorization computed by DPTTRF. -* -* EF (input) DOUBLE PRECISION array, dimension (N-1) -* The (n-1) subdiagonal elements of the unit bidiagonal factor -* L from the factorization computed by DPTTRF. -* -* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) -* The right hand side matrix B. -* -* LDB (input) INTEGER -* The leading dimension of the array B. LDB >= max(1,N). -* -* X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) -* On entry, the solution matrix X, as computed by DPTTRS. -* On exit, the improved solution matrix X. -* -* LDX (input) INTEGER -* The leading dimension of the array X. LDX >= max(1,N). -* -* FERR (output) DOUBLE PRECISION array, dimension (NRHS) -* The forward error bound for each solution vector -* X(j) (the j-th column of the solution matrix X). -* If XTRUE is the true solution corresponding to X(j), FERR(j) -* is an estimated upper bound for the magnitude of the largest -* element in (X(j) - XTRUE) divided by the magnitude of the -* largest element in X(j). -* -* BERR (output) DOUBLE PRECISION array, dimension (NRHS) -* The componentwise relative backward error of each solution -* vector X(j) (i.e., the smallest relative change in -* any element of A or B that makes X(j) an exact solution). -* -* WORK (workspace) DOUBLE PRECISION array, dimension (2*N) -* -* INFO (output) INTEGER -* = 0: successful exit -* < 0: if INFO = -i, the i-th argument had an illegal value -* -* Internal Parameters -* =================== -* -* ITMAX is the maximum number of steps of iterative refinement. -* * ===================================================================== * * .. Parameters .. @@ -263,7 +353,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T. * * Solve M(L) * x = e. * @@ -272,7 +362,7 @@ WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) ) 60 CONTINUE * -* Solve D * M(L)' * x = b. +* Solve D * M(L)**T * x = b. * WORK( N ) = WORK( N ) / DF( N ) DO 70 I = N - 1, 1, -1