Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 11/07/2023
Public

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?potrs

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

Syntax

call spotrs( uplo, n, nrhs, a, lda, b, ldb, info )

call dpotrs( uplo, n, nrhs, a, lda, b, ldb, info )

call cpotrs( uplo, n, nrhs, a, lda, b, ldb, info )

call zpotrs( uplo, n, nrhs, a, lda, b, ldb, info )

call potrs( a, b [,uplo] [, info] )

Include Files

  • mkl.fi, lapack.f90

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

uplo

CHARACTER*1. 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

INTEGER. The order of matrix A; n 0.

nrhs

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

a, b

REAL for spotrs

DOUBLE PRECISION for dpotrs

COMPLEX for cpotrs

DOUBLE COMPLEX for zpotrs.

Arrays: a(lda,*), b(ldb,*).

The array a contains the factor U or L (see uplo) as returned by potrf. The second dimension of a must be at least max(1,n).

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

lda

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

ldb

INTEGER. The leading 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.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine potrs interface are as follows:

a

Holds the matrix A of size (n, n).

b

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

uplo

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

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.