Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?pptri

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

Syntax

lapack_int LAPACKE_spptri (int matrix_layout , char uplo , lapack_int n , float * ap );

lapack_int LAPACKE_dpptri (int matrix_layout , char uplo , lapack_int n , double * ap );

lapack_int LAPACKE_cpptri (int matrix_layout , char uplo , lapack_int n , lapack_complex_float * ap );

lapack_int LAPACKE_zpptri (int matrix_layout , char uplo , lapack_int n , lapack_complex_double * ap );

Include Files

  • mkl.h

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

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 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.

n

The order of the matrix A; n 0.

ap

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

ap

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

Return Values

This function returns a value info.

If info = 0, the execution is successful.

If info = -i, parameter i 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.

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 =maxx·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.