Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?potrs

Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix.

Syntax

lapack_int LAPACKE_spotrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const float * a , lapack_int lda , float * b , lapack_int ldb );

lapack_int LAPACKE_dpotrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const double * a , lapack_int lda , double * b , lapack_int ldb );

lapack_int LAPACKE_cpotrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const lapack_complex_float * a , lapack_int lda , lapack_complex_float * b , lapack_int ldb );

lapack_int LAPACKE_zpotrs (int matrix_layout , char uplo , lapack_int n , lapack_int nrhs , const lapack_complex_double * a , lapack_int lda , 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 positive-definite or, for complex data, Hermitian positive-definite matrix A, given the Cholesky factorization of A:

A = UT*U for real data, A = UH*U for complex data if uplo='U'
A = L*LT for real data, A = L*LH for complex data if uplo='L'

where L is a lower triangular matrix and U is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix B.

Before calling this routine, you must call ?potrf to compute the Cholesky factorization of A.

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', U is stored, whereA = UT*U for real data, A = UH*U for complex data.

If uplo = 'L', L is stored, whereA = L*LT for real data, A = L*LH for complex data.

n

The order of matrix A; n 0.

nrhs

The number of right-hand sides (nrhs 0).

a

Array A of size at least max(1, lda*n)

The array a contains the factor U or L (see uplo) as returned by potrf. .

lda

The leading dimension of a. lda max(1, n).

b

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. The size of b must be at least 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

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

|E| 
					c(n)ε |U
					
						H
					||U|

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

A similar estimate holds for uplo = 'L'. 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 approximate number of floating-point operations for one right-hand side vector b is 2n2 for real flavors and 8n2 for complex flavors.

To estimate the condition number κ(A), call ?pocon.

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