?pptri

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 4/28/2026

Version

Public

?pptri

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

Syntax

lapack_intLAPACKE_spptri ( intmatrix_layout , charuplo , lapack_intn , float*ap );

lapack_intLAPACKE_dpptri ( intmatrix_layout , charuplo , lapack_intn , double*ap );

lapack_intLAPACKE_cpptri ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_float*ap );

lapack_intLAPACKE_zpptri ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_double*ap );

Include Files

mkl.h

Description

spptri dpptri cpptri zpptri pptri

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

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.

LAPACK 95 Interface Notes

There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.

Application Notes

The computed inverse \(X\) satisfies the following error bounds:

\[||XA - I||_{2} \leq c(n)\varepsilon\kappa_{2}(A), ||AX - I||_{2} \leq c(n)\varepsilon\kappa_{2}(A),\]

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

The 2-norm \(||A||_{2}\) of a matrix \(A\) is defined by \(||A||_{2} =max_{x \cdot x=1}(Ax \cdot Ax)^{1/2}\) , and the condition number \(\kappa_{2}(A)\) is defined by \(\kappa_{2}(A) = ||A||_{2} ||A^{-1}||_{2}\) .

The total number of floating-point operations is approximately \((2/3)n^{3}\) for real flavors and \((8/3)n^{3}\) for complex flavors.

NOTE:

Matrix Storage Schemes

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?pptri

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

LAPACK 95 Interface Notes

Application Notes