?potrs

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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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 = U^TU` for real data, `A = U^HU` for complex data	if `uplo='U'`
`A = LL^T` for real data, `A = LL^H` 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	CHARACTER1. Must be 'U' or 'L'. Indicates how the input matrix `A` has been factored: If `uplo = 'U'`, `U` is stored, where`A` = `U`^T`U` for real data, `A` = `U`^H`U` for complex data. If `uplo = 'L'`, `L` is stored, where`A` = `L``L`^T for real data, `A` = `L`*`L`^H 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

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:

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.

Parent topic: Solving Systems of Linear Equations: LAPACK Computational Routines

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