?sytri

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?sytri

Computes the inverse of a symmetric matrix using U*D*U^T or L*D*L^T Bunch-Kaufman factorization.

Syntax

call ssytri( uplo, n, a, lda, ipiv, work, info )

call dsytri( uplo, n, a, lda, ipiv, work, info )

call csytri( uplo, n, a, lda, ipiv, work, info )

call zsytri( uplo, n, a, lda, ipiv, work, info )

call sytri( a, ipiv [,uplo] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine computes the inverse inv(A) of a symmetric matrix A. Before calling this routine, call ?sytrf to factorize A.

Input Parameters

uplo	CHARACTER1. Must be 'U' or 'L'. Indicates how the input matrix `A` has been factored: If `uplo = 'U'`, the array `a` stores the Bunch-Kaufman factorization `A = UDU^T`. If `uplo = 'L'`, the array `a` stores the Bunch-Kaufman factorization `A = LD*L^T`.
n	INTEGER. The order of the matrix `A`; `n≥ 0`.
a, work	REAL for ssytri DOUBLE PRECISION for dsytri COMPLEX for csytri DOUBLE COMPLEX for zsytri. Arrays: `a`(size `lda` by ) contains the factorization of the matrix `A`, as returned by ?sytrf. The second dimension of `a` must be at least `max(1,n)`. `work()` is a workspace array. The dimension of `work` must be at least `max(1,2*n)`.
lda	INTEGER. The leading dimension of `a`; `lda≥ max(1, n)`.
ipiv	INTEGER. Array, size at least `max(1, n)`. The `ipiv` array, as returned by ?sytrf.

Output Parameters

Overwritten by the 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 D is zero, D is singular, 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 sytri interface are as follows:

a	Holds the matrix `A` of size (`n,n`).
ipiv	Holds the vector of length `n`.
uplo	Must be 'U' or 'L'. The default value is 'U'.

Application Notes

The computed inverse X satisfies the following error bounds:

|D*U^T*P^T*X*P*U - I| ≤ c(n)ε(|D||U^T|P^T|X|P|U| + |D||D^-1|)

for uplo = 'U', and

|D*L^T*P^T*X*P*L - I| ≤ c(n)ε(|D||L^T|P^T|X|P|L| + |D||D^-1|)

for uplo = 'L'. Here c(n) is a modest linear function of n, and ε is the machine precision; I denotes the identity matrix.

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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?sytri

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