Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?sptrs

Solves a system of linear equations with a UDU- or LDL-factored symmetric coefficient matrix using packed storage.

Syntax

lapack_int LAPACKE_ssptrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const float * ap , const lapack_int * ipiv , float * b , lapack_int ldb );

lapack_int LAPACKE_dsptrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const double * ap , const lapack_int * ipiv , double * b , lapack_int ldb );

lapack_int LAPACKE_csptrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const lapack_complex_float * ap , const lapack_int * ipiv , lapack_complex_float * b , lapack_int ldb );

lapack_int LAPACKE_zsptrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const lapack_complex_double * ap , const lapack_int * ipiv , lapack_complex_double * b , lapack_int ldb );

Include Files

  • mkl.h

Description

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 = U*D*UT

if uplo='L',

A = L*D*LT,

where 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

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

uplo

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 = U*D*UT. If uplo = 'L', the array ap stores the packed factor L of the factorization A = L*D*LT.

n

The order of matrix A; n 0.

nrhs

The number of right-hand sides; nrhs 0.

ipiv

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

ap

The dimension of array 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).

b

The array b contains the matrix B whose columns are the right-hand sides for the system of equations. The size of b is max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout.

ldb

The leading dimension of b; ldb max(1, n) for column major layout and ldbnrhs for row major layout.

Output Parameters

b

Overwritten by the solution matrix X.

Return Values

This function returns a value info.

If info=0, the execution is successful.

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

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:


Equation

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.