?sptrs

Syntax

FORTRAN 77:

call ssptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

call dsptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

call csptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

call zsptrs( uplo, n, nrhs, ap, ipiv, b, ldb, info )

Fortran 95:

call sptrs( ap, b, ipiv [, uplo] [,info] )

Description

This routine is declared in mkl_lapack.fi for FORTRAN 77 interface, in mkl_lapack.f90 for Fortran 95 interface, and in mkl_lapack.h for C interface.

The routine solves for X the system of linear equations A*X = B with a symmetric matrix A, given the Bunch-Kaufman factorization of A:

if uplo='U',

A = PUDU^TP^T

if uplo='L',

A = PLDL^TP^T,

where P is a permutation matrix, U and L are upper and lower packed triangular matrices with unit diagonal, and D is a symmetric block-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix B. You must supply the factor U (or L) and the array ipiv returned by the factorization routine ?sptrf.

Input Parameters

uplo

CHARACTER*1. Must be 'U' or 'L'.

Indicates how the input matrix A has been factored:

If uplo = 'U', the array ap stores the packed factor U of the factorization A = P*U*D*U^T*P^T. If uplo = 'L', the array ap stores the packed factor L of the factorization A = P*L*D*L^T*P^T.

n

INTEGER. The order of matrix A; n ≥ 0.

nrhs

INTEGER. The number of right-hand sides; nrhs ≥ 0.

ipiv

INTEGER.

Array, DIMENSION at least max(1, n). The ipiv array, as returned by ?sptrf.

ap, b

REAL for ssptrs

DOUBLE PRECISION for dsptrs

COMPLEX for csptrs

DOUBLE COMPLEX for zsptrs.

Arrays: ap(*), b(ldb,*).

The dimension of ap must be at least max(1,n(n+1)/2). The array ap contains the factor U or L, as specified by uplo, in packed storage (see Matrix Storage Schemes).

The array b contains the matrix B whose columns are the right-hand sides for the system of equations. The second dimension of b must be at least max(1, nrhs).

ldb

INTEGER. The first dimension of b; ldb ≥ max(1, n).

Output Parameters

b

Overwritten by the solution matrix X.

info

INTEGER. If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine sptrs interface are as follows:

ap

Holds the array A of size (n*(n+1)/2).

b

Holds the matrix B of size (n, nrhs).

ipiv

Holds the vector of length n.

uplo

Must be 'U' or 'L'. The default value is 'U'.

Application Notes

For each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A + E)x = b, where

|E| ≤ c(n)ε P|U||D||UT|PT or |E| ≤ c(n)ε P|L||D||LT|PT

c(n) is a modest linear function of n, and ε is the machine precision.

If x₀ is the true solution, the computed solution x satisfies this error bound:

where cond(A,x)= || |A^-1||A| |x| ||_∞ / ||x||_∞ ≤ ||A^-1||_∞ ||A||_∞ = κ_∞(A).

Note that cond(A,x) can be much smaller than κ_∞(A).

The total number of floating-point operations for one right-hand side vector is approximately 2n² for real flavors or 8n² for complex flavors.

To estimate the condition number κ_∞(A), call ?spcon.

To refine the solution and estimate the error, call ?sprfs.