?pptri

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?pptri

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

Syntax

call spptri( uplo, n, ap, info )

call dpptri( uplo, n, ap, info )

call cpptri( uplo, n, ap, info )

call zpptri( uplo, n, ap, info )

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

Include Files

mkl.fi, lapack.f90

Description

The routine computes the inverse inv(A) of a symmetric positive definite or, for complex flavors, Hermitian positive-definite matrix A in packed form. Before calling this routine, call ?pptrf to factorize A.

Input Parameters

uplo

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

Indicates whether the upper or lower triangular factor is stored in ap:

If uplo = 'U', then the upper triangular factor is stored.

If uplo = 'L', then the lower triangular factor is stored.

INTEGER. The order of the matrix A; n≥ 0.

REAL for spptri

DOUBLE PRECISION for dpptri

COMPLEX for cpptri

DOUBLE COMPLEX for zpptri.

Array, size at least max(1, n(n+1)/2).

Contains the factorization of the packed matrix A, as returned by ?pptrf.

The dimension ap must be at least max(1,n(n+1)/2).

Output Parameters

Overwritten by the packed n-by-n matrix inv(A).

info

INTEGER.

If info = 0, the execution is successful.

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

If info = i, the i-th diagonal element of the Cholesky factor (and therefore the factor itself) is zero, and the inversion could not be completed.

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 pptri 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

The computed inverse X satisfies the following error bounds:

||XA - I||₂ ≤ c(n)εκ₂(A), ||AX - I||₂ ≤ c(n)εκ₂(A),

where c(n) is a modest linear function of n, and ε is the machine precision; I denotes the identity matrix.

The 2-norm ||A||₂ of a matrix A is defined by ||A||₂ =max_x·x=1(Ax·Ax)^1/2, and the condition number κ₂(A) is defined by κ₂(A) = ||A||₂ ||A^-1||₂ .

The total number of floating-point operations is approximately (2/3)n³ for real flavors and (8/3)n³ for complex flavors.

Parent topic: Matrix Inversion: LAPACK Computational Routines

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?pptri

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