?potri

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 4/28/2026

Version

Public

?potri

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

Syntax

call spotri ( uplo , n , a , lda , info )

call dpotri ( uplo , n , a , lda , info )

call cpotri ( uplo , n , a , lda , info )

call zpotri ( uplo , n , a , lda , info )

call potri ( a [ , uplo ] [ , info ] )

Include Files

mkl.fi , mkl_lapack.f90

Description

spotri dpotri cpotri zpotri potri

The routine computes the inverse inv(A) of a symmetric positive definite or, for complex flavors, Hermitian positive-definite matrix A . Before calling this routine, call ?potrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix.) to factorize A .

Input Parameters

uplo

CHARACTER*1 . Must be ‘U’ or ‘L’ .

Indicates how the input matrix A has been factored:

If uplo = 'U' , the upper triangle of A is stored.

If uplo = 'L' , the lower triangle of A is stored.

n

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

a

REAL for spotri

DOUBLE PRECISION for dpotri

COMPLEX for cpotri

DOUBLE COMPLEX for zpotri .

Array a (size lda by *) Contains the factorization of the matrix A , as returned by GUID-15A9E907-2600-430B-BF11-DFA0A375711B.xml#GUID-15A9E907-2600-430B-BF11-DFA0A375711B .

The second dimension of a must be at least max(1, n) .

lda

INTEGER . The leading dimension of a . lda≥ max(1, n).

Output Parameters

a

Overwritten by the upper or lower triangle of the inverse of 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.

Return Values

No return value, info is an Output Parameter.

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 potri interface are as follows:

a

Holds the matrix A of size ( n,n ).

uplo

Must be ‘U’ or ‘L’ . The default value is ‘U’ .

Application Notes

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

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

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 Fortran

?potri

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

LAPACK 95 Interface Notes

Application Notes