table of contents
/home/abuild/rpmbuild/BUILD/lapack-3.12.0/SRC/sppsvx.f(3) | Library Functions Manual | /home/abuild/rpmbuild/BUILD/lapack-3.12.0/SRC/sppsvx.f(3) |
NAME¶
/home/abuild/rpmbuild/BUILD/lapack-3.12.0/SRC/sppsvx.f
SYNOPSIS¶
Functions/Subroutines¶
subroutine SPPSVX (fact, uplo, n, nrhs, ap, afp, equed, s,
b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SPPSVX computes the solution to system of linear equations A * X = B for
OTHER matrices
Function/Subroutine Documentation¶
subroutine SPPSVX (character fact, character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( * ) afp, character equed, real, dimension( * ) s, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)¶
SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
Purpose:
!> !> SPPSVX 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 stored in !> packed format 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 principal minor of order i is not positive, !> 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. !>
Parameters
FACT
!> 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, AFP contains the factored form of A. !> If EQUED = 'Y', the matrix A has been equilibrated !> with scaling factors given by S. AP and AFP will not !> be modified. !> = 'N': The matrix A will be copied to AFP and factored. !> = 'E': The matrix A will be equilibrated if necessary, then !> copied to AFP and factored. !>
UPLO
!> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !>
N
!> N is INTEGER !> The number of linear equations, i.e., the order of the !> matrix A. N >= 0. !>
NRHS
!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !>
AP
!> AP is REAL array, dimension (N*(N+1)/2) !> On entry, the upper or lower triangle of the symmetric matrix !> A, packed columnwise in a linear array, except if FACT = 'F' !> and EQUED = 'Y', then A must contain the equilibrated matrix !> diag(S)*A*diag(S). The j-th column of A is stored in the !> array AP as follows: !> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; !> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. !> See below for further details. 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). !>
AFP
!> AFP is REAL array, dimension (N*(N+1)/2) !> If FACT = 'F', then AFP 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 AFP is the factored !> form of the equilibrated matrix A. !> !> If FACT = 'N', then AFP 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 AFP 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 AP for the form of the !> equilibrated matrix). !>
EQUED
!> EQUED is 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
!> S is REAL 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
!> B is REAL 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
!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
X
!> X is REAL 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
!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>
RCOND
!> RCOND is REAL !> 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
!> FERR is REAL 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
!> BERR is REAL 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
!> WORK is REAL array, dimension (3*N) !>
IWORK
!> IWORK is INTEGER array, dimension (N) !>
INFO
!> 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 principal minor of order i of A !> is not positive, 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. !>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
!> !> The packed storage scheme is illustrated by the following example !> when N = 4, UPLO = 'U': !> !> Two-dimensional storage of the symmetric matrix A: !> !> a11 a12 a13 a14 !> a22 a23 a24 !> a33 a34 (aij = conjg(aji)) !> a44 !> !> Packed storage of the upper triangle of A: !> !> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] !>
Definition at line 309 of file sppsvx.f.
Author¶
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