?pptri

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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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 =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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