?pptrf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?pptrf

Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.

Syntax

call spptrf( uplo, n, ap, info )

call dpptrf( uplo, n, ap, info )

call cpptrf( uplo, n, ap, info )

call zpptrf( uplo, n, ap, info )

call pptrf( ap [, uplo] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian positive-definite packed matrix 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.

NOTE:

This routine supports the Progress Routine feature. See Progress Function for details.

Input Parameters

uplo

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

Indicates whether the upper or lower triangular part of A is packed in the array ap, and how A is factored:

If uplo = 'U', the array ap stores the upper triangular part of the matrix A, and A is factored as U^H*U.

If uplo = 'L', the array ap stores the lower triangular part of the matrix A; A is factored as L*L^H.

INTEGER. The order of matrix A; n≥ 0.

REAL for spptrf

DOUBLE PRECISION for dpptrf

COMPLEX for cpptrf

DOUBLE COMPLEX for zpptrf.

Array, size at least max(1, n(n+1)/2). The array ap contains either the upper or the lower triangular part of the matrix A (as specified by uplo) in packed storage (see Matrix Storage Schemes).

Output Parameters

Overwritten by the Cholesky factor U or L, as specified by uplo.

info

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

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

If info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the factorization could not be completed. This may indicate an error in forming the matrix A.

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 pptrf interface are as follows:

ap	Holds the array `A` of size `(n*(n+1)/2)`.
uplo	Must be 'U' or 'L'. The default value is 'U'.

Application Notes

If uplo = 'U', the computed factor U is the exact factor of a perturbed matrix A + E, where

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

A similar estimate holds for uplo = 'L'.

The total number of floating-point operations is approximately (1/3)n³ for real flavors and (4/3)n³ for complex flavors.

After calling this routine, you can call the following routines:

?pptrs	to solve `A*X = B`
?ppcon	to estimate the condition number of `A`
?pptri	to compute the inverse of `A`.

Parent topic: Matrix Factorization: LAPACK Computational Routines

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