version 1.3, 2010/08/06 15:28:46
|
version 1.9, 2011/11/21 20:43:02
|
Line 1
|
Line 1
|
|
*> \brief <b> DPOSVX computes the solution to system of linear equations A * X = B for PO matrices</b> |
|
* |
|
* =========== DOCUMENTATION =========== |
|
* |
|
* Online html documentation available at |
|
* http://www.netlib.org/lapack/explore-html/ |
|
* |
|
*> \htmlonly |
|
*> Download DPOSVX + dependencies |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposvx.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposvx.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposvx.f"> |
|
*> [TXT]</a> |
|
*> \endhtmlonly |
|
* |
|
* Definition: |
|
* =========== |
|
* |
|
* SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, |
|
* S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, |
|
* IWORK, INFO ) |
|
* |
|
* .. Scalar Arguments .. |
|
* CHARACTER EQUED, FACT, UPLO |
|
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS |
|
* DOUBLE PRECISION RCOND |
|
* .. |
|
* .. Array Arguments .. |
|
* INTEGER IWORK( * ) |
|
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), |
|
* $ BERR( * ), FERR( * ), S( * ), WORK( * ), |
|
* $ X( LDX, * ) |
|
* .. |
|
* |
|
* |
|
*> \par Purpose: |
|
* ============= |
|
*> |
|
*> \verbatim |
|
*> |
|
*> DPOSVX uses the Cholesky factorization A = U**T*U or A = L*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 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 = 'E', real scaling factors are computed to equilibrate |
|
*> the system: |
|
*> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B |
|
*> Whether or not the system will be equilibrated depends on the |
|
*> scaling of the matrix A, but if equilibration is used, A is |
|
*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B. |
|
*> |
|
*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to |
|
*> factor the matrix A (after equilibration if FACT = 'E') as |
|
*> A = U**T* U, if UPLO = 'U', or |
|
*> A = L * L**T, if UPLO = 'L', |
|
*> where U is an upper triangular matrix and L is a lower triangular |
|
*> matrix. |
|
*> |
|
*> 3. 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. |
|
*> |
|
*> 4. The system of equations is solved for X using the factored form |
|
*> of A. |
|
*> |
|
*> 5. Iterative refinement is applied to improve the computed solution |
|
*> matrix and calculate error bounds and backward error estimates |
|
*> for it. |
|
*> |
|
*> 6. If equilibration was used, the matrix X is premultiplied by |
|
*> diag(S) so that it solves the original system before |
|
*> equilibration. |
|
*> \endverbatim |
|
* |
|
* Arguments: |
|
* ========== |
|
* |
|
*> \param[in] FACT |
|
*> \verbatim |
|
*> FACT is CHARACTER*1 |
|
*> Specifies whether or not the factored form of the matrix A is |
|
*> supplied on entry, and if not, whether the matrix A should be |
|
*> equilibrated before it is factored. |
|
*> = 'F': On entry, AF contains the factored form of A. |
|
*> If EQUED = 'Y', the matrix A has been equilibrated |
|
*> with scaling factors given by S. A and AF will not |
|
*> be modified. |
|
*> = 'N': The matrix A will be copied to AF and factored. |
|
*> = 'E': The matrix A will be equilibrated if necessary, then |
|
*> copied to AF and factored. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] UPLO |
|
*> \verbatim |
|
*> UPLO is CHARACTER*1 |
|
*> = 'U': Upper triangle of A is stored; |
|
*> = 'L': Lower triangle of A is stored. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] N |
|
*> \verbatim |
|
*> N is INTEGER |
|
*> The number of linear equations, i.e., 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,out] A |
|
*> \verbatim |
|
*> A is DOUBLE PRECISION array, dimension (LDA,N) |
|
*> On entry, the symmetric matrix A, except if FACT = 'F' and |
|
*> EQUED = 'Y', then A must contain the equilibrated matrix |
|
*> diag(S)*A*diag(S). If UPLO = 'U', the leading |
|
*> N-by-N upper triangular part of A contains the upper |
|
*> triangular part of the matrix A, and the strictly lower |
|
*> triangular part of A is not referenced. If UPLO = 'L', the |
|
*> leading N-by-N lower triangular part of A contains the lower |
|
*> triangular part of the matrix A, and the strictly upper |
|
*> triangular part of A is not referenced. A is not modified if |
|
*> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. |
|
*> |
|
*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by |
|
*> diag(S)*A*diag(S). |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDA |
|
*> \verbatim |
|
*> LDA is INTEGER |
|
*> The leading dimension of the array A. LDA >= max(1,N). |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] AF |
|
*> \verbatim |
|
*> AF is or output) DOUBLE PRECISION array, dimension (LDAF,N) |
|
*> If FACT = 'F', then AF is an input argument and on entry |
|
*> contains the triangular factor U or L from the Cholesky |
|
*> factorization A = U**T*U or A = L*L**T, in the same storage |
|
*> format as A. If EQUED .ne. 'N', then AF is the factored form |
|
*> of the equilibrated matrix diag(S)*A*diag(S). |
|
*> |
|
*> If FACT = 'N', then AF is an output argument and on exit |
|
*> returns the triangular factor U or L from the Cholesky |
|
*> factorization A = U**T*U or A = L*L**T of the original |
|
*> matrix A. |
|
*> |
|
*> If FACT = 'E', then AF is an output argument and on exit |
|
*> returns the triangular factor U or L from the Cholesky |
|
*> factorization A = U**T*U or A = L*L**T of the equilibrated |
|
*> matrix A (see the description of A for the form of the |
|
*> equilibrated matrix). |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDAF |
|
*> \verbatim |
|
*> LDAF is INTEGER |
|
*> The leading dimension of the array AF. LDAF >= max(1,N). |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] EQUED |
|
*> \verbatim |
|
*> EQUED is or output) CHARACTER*1 |
|
*> Specifies the form of equilibration that was done. |
|
*> = 'N': No equilibration (always true if FACT = 'N'). |
|
*> = 'Y': Equilibration was done, i.e., A has been replaced by |
|
*> diag(S) * A * diag(S). |
|
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an |
|
*> output argument. |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] S |
|
*> \verbatim |
|
*> S is or output) DOUBLE PRECISION array, dimension (N) |
|
*> The scale factors for A; not accessed if EQUED = 'N'. S is |
|
*> an input argument if FACT = 'F'; otherwise, S is an output |
|
*> argument. If FACT = 'F' and EQUED = 'Y', each element of S |
|
*> must be positive. |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] B |
|
*> \verbatim |
|
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) |
|
*> On entry, the N-by-NRHS right hand side matrix B. |
|
*> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', |
|
*> B is overwritten by diag(S) * 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 or INFO = N+1, the N-by-NRHS solution matrix X to |
|
*> the original system of equations. Note that if EQUED = 'Y', |
|
*> A and B are modified on exit, and the solution to the |
|
*> equilibrated system is inv(diag(S))*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 estimate of the reciprocal condition number of the matrix |
|
*> A after equilibration (if done). 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 estimated 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). The estimate is as reliable as |
|
*> the estimate for RCOND, and is almost always a slight |
|
*> overestimate of the true error. |
|
*> \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 (3*N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] IWORK |
|
*> \verbatim |
|
*> IWORK is INTEGER array, dimension (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 doublePOsolve |
|
* |
|
* ===================================================================== |
SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, |
SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, |
$ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, |
$ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, |
$ IWORK, INFO ) |
$ IWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine (version 3.4.0) -- |
* -- 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 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER EQUED, FACT, UPLO |
CHARACTER EQUED, FACT, UPLO |
Line 19
|
Line 324
|
$ X( LDX, * ) |
$ X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* DPOSVX uses the Cholesky factorization A = U**T*U or A = L*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 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 = 'E', real scaling factors are computed to equilibrate |
|
* the system: |
|
* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B |
|
* Whether or not the system will be equilibrated depends on the |
|
* scaling of the matrix A, but if equilibration is used, A is |
|
* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. |
|
* |
|
* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to |
|
* factor the matrix A (after equilibration if FACT = 'E') as |
|
* A = U**T* U, if UPLO = 'U', or |
|
* A = L * L**T, if UPLO = 'L', |
|
* where U is an upper triangular matrix and L is a lower triangular |
|
* matrix. |
|
* |
|
* 3. 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. |
|
* |
|
* 4. The system of equations is solved for X using the factored form |
|
* of A. |
|
* |
|
* 5. Iterative refinement is applied to improve the computed solution |
|
* matrix and calculate error bounds and backward error estimates |
|
* for it. |
|
* |
|
* 6. If equilibration was used, the matrix X is premultiplied by |
|
* diag(S) so that it solves the original system before |
|
* equilibration. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* FACT (input) CHARACTER*1 |
|
* Specifies whether or not the factored form of the matrix A is |
|
* supplied on entry, and if not, whether the matrix A should be |
|
* equilibrated before it is factored. |
|
* = 'F': On entry, AF contains the factored form of A. |
|
* If EQUED = 'Y', the matrix A has been equilibrated |
|
* with scaling factors given by S. A and AF will not |
|
* be modified. |
|
* = 'N': The matrix A will be copied to AF and factored. |
|
* = 'E': The matrix A will be equilibrated if necessary, then |
|
* copied to AF and factored. |
|
* |
|
* UPLO (input) CHARACTER*1 |
|
* = 'U': Upper triangle of A is stored; |
|
* = 'L': Lower triangle of A is stored. |
|
* |
|
* N (input) INTEGER |
|
* The number of linear equations, i.e., 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. |
|
* |
|
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
|
* On entry, the symmetric matrix A, except if FACT = 'F' and |
|
* EQUED = 'Y', then A must contain the equilibrated matrix |
|
* diag(S)*A*diag(S). If UPLO = 'U', the leading |
|
* N-by-N upper triangular part of A contains the upper |
|
* triangular part of the matrix A, and the strictly lower |
|
* triangular part of A is not referenced. If UPLO = 'L', the |
|
* leading N-by-N lower triangular part of A contains the lower |
|
* triangular part of the matrix A, and the strictly upper |
|
* triangular part of A is not referenced. A is not modified if |
|
* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. |
|
* |
|
* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by |
|
* diag(S)*A*diag(S). |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,N). |
|
* |
|
* AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N) |
|
* If FACT = 'F', then AF is an input argument and on entry |
|
* contains the triangular factor U or L from the Cholesky |
|
* factorization A = U**T*U or A = L*L**T, in the same storage |
|
* format as A. If EQUED .ne. 'N', then AF is the factored form |
|
* of the equilibrated matrix diag(S)*A*diag(S). |
|
* |
|
* If FACT = 'N', then AF is an output argument and on exit |
|
* returns the triangular factor U or L from the Cholesky |
|
* factorization A = U**T*U or A = L*L**T of the original |
|
* matrix A. |
|
* |
|
* If FACT = 'E', then AF is an output argument and on exit |
|
* returns the triangular factor U or L from the Cholesky |
|
* factorization A = U**T*U or A = L*L**T of the equilibrated |
|
* matrix A (see the description of A for the form of the |
|
* equilibrated matrix). |
|
* |
|
* LDAF (input) INTEGER |
|
* The leading dimension of the array AF. LDAF >= max(1,N). |
|
* |
|
* EQUED (input or output) CHARACTER*1 |
|
* Specifies the form of equilibration that was done. |
|
* = 'N': No equilibration (always true if FACT = 'N'). |
|
* = 'Y': Equilibration was done, i.e., A has been replaced by |
|
* diag(S) * A * diag(S). |
|
* EQUED is an input argument if FACT = 'F'; otherwise, it is an |
|
* output argument. |
|
* |
|
* S (input or output) DOUBLE PRECISION array, dimension (N) |
|
* The scale factors for A; not accessed if EQUED = 'N'. S is |
|
* an input argument if FACT = 'F'; otherwise, S is an output |
|
* argument. If FACT = 'F' and EQUED = 'Y', each element of S |
|
* must be positive. |
|
* |
|
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) |
|
* On entry, the N-by-NRHS right hand side matrix B. |
|
* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', |
|
* B is overwritten by diag(S) * 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 or INFO = N+1, the N-by-NRHS solution matrix X to |
|
* the original system of equations. Note that if EQUED = 'Y', |
|
* A and B are modified on exit, and the solution to the |
|
* equilibrated system is inv(diag(S))*X. |
|
* |
|
* LDX (input) INTEGER |
|
* The leading dimension of the array X. LDX >= max(1,N). |
|
* |
|
* RCOND (output) DOUBLE PRECISION |
|
* The estimate of the reciprocal condition number of the matrix |
|
* A after equilibration (if done). 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 estimated 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). The estimate is as reliable as |
|
* the estimate for RCOND, and is almost always a slight |
|
* overestimate of the true error. |
|
* |
|
* 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 (3*N) |
|
* |
|
* IWORK (workspace) INTEGER array, dimension (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 .. |
* .. Parameters .. |
Line 320
|
Line 436
|
* |
* |
IF( NOFACT .OR. EQUIL ) THEN |
IF( NOFACT .OR. EQUIL ) THEN |
* |
* |
* Compute the Cholesky factorization A = U'*U or A = L*L'. |
* Compute the Cholesky factorization A = U**T *U or A = L*L**T. |
* |
* |
CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF ) |
CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF ) |
CALL DPOTRF( UPLO, N, AF, LDAF, INFO ) |
CALL DPOTRF( UPLO, N, AF, LDAF, INFO ) |