--- rpl/lapack/lapack/dptsvx.f 2011/07/22 07:38:10 1.8 +++ rpl/lapack/lapack/dptsvx.f 2011/11/21 20:43:02 1.9 @@ -1,10 +1,237 @@ +*> \brief \b DPTSVX +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DPTSVX + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, +* RCOND, FERR, BERR, WORK, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER FACT +* INTEGER INFO, LDB, LDX, N, NRHS +* DOUBLE PRECISION RCOND +* .. +* .. Array Arguments .. +* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), +* $ E( * ), EF( * ), FERR( * ), WORK( * ), +* $ X( LDX, * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DPTSVX uses the factorization A = L*D*L**T to compute the solution +*> to a real system of linear equations A*X = B, where A is an N-by-N +*> symmetric positive definite tridiagonal matrix and X and B are +*> N-by-NRHS matrices. +*> +*> Error bounds on the solution and a condition estimate are also +*> provided. +*> \endverbatim +* +*> \par Description: +* ================= +*> +*> \verbatim +*> +*> The following steps are performed: +*> +*> 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L +*> is a unit lower bidiagonal matrix and D is diagonal. The +*> factorization can also be regarded as having the form +*> A = U**T*D*U. +*> +*> 2. If the leading i-by-i principal minor is not positive definite, +*> then the routine returns with INFO = i. Otherwise, the factored +*> form of A is used to estimate the condition number of the matrix +*> A. If the reciprocal of the condition number is less than machine +*> precision, INFO = N+1 is returned as a warning, but the routine +*> still goes on to solve for X and compute error bounds as +*> described below. +*> +*> 3. The system of equations is solved for X using the factored form +*> of A. +*> +*> 4. Iterative refinement is applied to improve the computed solution +*> matrix and calculate error bounds and backward error estimates +*> for it. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] FACT +*> \verbatim +*> FACT is CHARACTER*1 +*> Specifies whether or not the factored form of A has been +*> supplied on entry. +*> = 'F': On entry, DF and EF contain the factored form of A. +*> D, E, DF, and EF will not be modified. +*> = 'N': The matrix A will be copied to DF and EF and +*> factored. +*> \endverbatim +*> +*> \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 matrices B and X. 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,out] DF +*> \verbatim +*> DF is or output) DOUBLE PRECISION array, dimension (N) +*> If FACT = 'F', then DF is an input argument and on entry +*> contains the n diagonal elements of the diagonal matrix D +*> from the L*D*L**T factorization of A. +*> If FACT = 'N', then DF is an output argument and on exit +*> contains the n diagonal elements of the diagonal matrix D +*> from the L*D*L**T factorization of A. +*> \endverbatim +*> +*> \param[in,out] EF +*> \verbatim +*> EF is or output) DOUBLE PRECISION array, dimension (N-1) +*> If FACT = 'F', then EF is an input argument and on entry +*> contains the (n-1) subdiagonal elements of the unit +*> bidiagonal factor L from the L*D*L**T factorization of A. +*> If FACT = 'N', then EF is an output argument and on exit +*> contains the (n-1) subdiagonal elements of the unit +*> bidiagonal factor L from the L*D*L**T factorization of A. +*> \endverbatim +*> +*> \param[in] B +*> \verbatim +*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) +*> The N-by-NRHS 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[out] X +*> \verbatim +*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) +*> If INFO = 0 of INFO = N+1, the N-by-NRHS 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] RCOND +*> \verbatim +*> RCOND is DOUBLE PRECISION +*> The reciprocal condition number of the matrix A. If RCOND +*> is less than the machine precision (in particular, if +*> RCOND = 0), the matrix is singular to working precision. +*> This condition is indicated by a return code of INFO > 0. +*> \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 +*> > 0: if INFO = i, and i is +*> <= N: the leading minor of order i of A is +*> not positive definite, so the factorization +*> could not be completed, and the solution has not +*> been computed. RCOND = 0 is returned. +*> = N+1: U is nonsingular, but RCOND is less than machine +*> precision, meaning that the matrix is singular +*> to working precision. Nevertheless, the +*> solution and error bounds are computed because +*> there are a number of situations where the +*> computed solution can be more accurate than the +*> value of RCOND would suggest. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2011 +* +*> \ingroup doubleOTHERcomputational +* +* ===================================================================== SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, $ RCOND, FERR, BERR, WORK, INFO ) * -* -- LAPACK routine (version 3.3.1) -- +* -- LAPACK computational routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* -- April 2011 -- +* November 2011 * * .. Scalar Arguments .. CHARACTER FACT @@ -17,131 +244,6 @@ $ X( LDX, * ) * .. * -* Purpose -* ======= -* -* DPTSVX uses the factorization A = L*D*L**T to compute the solution -* to a real system of linear equations A*X = B, where A is an N-by-N -* symmetric positive definite tridiagonal matrix and X and B are -* N-by-NRHS matrices. -* -* Error bounds on the solution and a condition estimate are also -* provided. -* -* Description -* =========== -* -* The following steps are performed: -* -* 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L -* is a unit lower bidiagonal matrix and D is diagonal. The -* factorization can also be regarded as having the form -* A = U**T*D*U. -* -* 2. If the leading i-by-i principal minor is not positive definite, -* then the routine returns with INFO = i. Otherwise, the factored -* form of A is used to estimate the condition number of the matrix -* A. If the reciprocal of the condition number is less than machine -* precision, INFO = N+1 is returned as a warning, but the routine -* still goes on to solve for X and compute error bounds as -* described below. -* -* 3. The system of equations is solved for X using the factored form -* of A. -* -* 4. Iterative refinement is applied to improve the computed solution -* matrix and calculate error bounds and backward error estimates -* for it. -* -* Arguments -* ========= -* -* FACT (input) CHARACTER*1 -* Specifies whether or not the factored form of A has been -* supplied on entry. -* = 'F': On entry, DF and EF contain the factored form of A. -* D, E, DF, and EF will not be modified. -* = 'N': The matrix A will be copied to DF and EF and -* factored. -* -* 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 matrices B and X. 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 or output) DOUBLE PRECISION array, dimension (N) -* If FACT = 'F', then DF is an input argument and on entry -* contains the n diagonal elements of the diagonal matrix D -* from the L*D*L**T factorization of A. -* If FACT = 'N', then DF is an output argument and on exit -* contains the n diagonal elements of the diagonal matrix D -* from the L*D*L**T factorization of A. -* -* EF (input or output) DOUBLE PRECISION array, dimension (N-1) -* If FACT = 'F', then EF is an input argument and on entry -* contains the (n-1) subdiagonal elements of the unit -* bidiagonal factor L from the L*D*L**T factorization of A. -* If FACT = 'N', then EF is an output argument and on exit -* contains the (n-1) subdiagonal elements of the unit -* bidiagonal factor L from the L*D*L**T factorization of A. -* -* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) -* The N-by-NRHS right hand side matrix B. -* -* LDB (input) INTEGER -* The leading dimension of the array B. LDB >= max(1,N). -* -* X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) -* If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X. -* -* LDX (input) INTEGER -* The leading dimension of the array X. LDX >= max(1,N). -* -* RCOND (output) DOUBLE PRECISION -* The reciprocal condition number of the matrix A. If RCOND -* is less than the machine precision (in particular, if -* RCOND = 0), the matrix is singular to working precision. -* This condition is indicated by a return code of INFO > 0. -* -* 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 -* > 0: if INFO = i, and i is -* <= N: the leading minor of order i of A is -* not positive definite, so the factorization -* could not be completed, and the solution has not -* been computed. RCOND = 0 is returned. -* = N+1: U is nonsingular, but RCOND is less than machine -* precision, meaning that the matrix is singular -* to working precision. Nevertheless, the -* solution and error bounds are computed because -* there are a number of situations where the -* computed solution can be more accurate than the -* value of RCOND would suggest. -* * ===================================================================== * * .. Parameters ..