Computes the inverse of a real or complex symmetric matrix.

lapack_int LAPACKE_ssytri_3 (int matrix_layout, char uplo, lapack_int n, float * A, lapack_int lda, const float * e, const lapack_int * ipiv);

lapack_int LAPACKE_dsytri_3 (int matrix_layout, char uplo, lapack_int n, double * A, lapack_int lda, const double * e, const lapack_int * ipiv);

lapack_int LAPACKE_csytri_3 (int matrix_layout, char uplo, lapack_int n, lapack_complex_float * A, lapack_int lda, const lapack_complex_float * e, const lapack_int * ipiv);

lapack_int LAPACKE_zsytri_3 (int matrix_layout, char uplo, lapack_int n, lapack_complex_double * A, lapack_int lda, const lapack_complex_double * e, const lapack_int * ipiv);

## Description

?sytri_3 computes the inverse of a real or complex symmetric matrix A using the factorization computed by ?sytrf_rk: A = P*U*D*(UT)*(PT) or A = P*L*D*(LT)*(PT), where U (or L) is a unit upper (or lower) triangular matrix, UT (or LT) is the transpose of U (or L), P is a permutation matrix, PT is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

?sytri_3 sets the leading dimension of the workspace before calling ?sytri_3x, which actually computes the inverse. This is the blocked version of the algorithm, calling Level-3 BLAS.

## Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

uplo

Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.

• = 'U': The upper triangle of A is stored.
• = 'L': The lower triangle of A is stored.
n

The order of the matrix A. n ≥ 0.

A

Array of size max(1, lda*n). On entry, diagonal of the block diagonal matrix D and factors U or L as computed by ?sytrf_rk:

• Only diagonal elements of the symmetric block diagonal matrix D on the diagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal) elements of D should be provided on entry in array e.

—and—

• If uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L', factor L in the subdiagonal part of A.

lda

The leading dimension of the array A.

e

Array of size n. On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is not referenced. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is not referenced.

### Note

For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the element e[k-1] is not referenced in both the uplo = 'U' and uplo = 'L' cases.
ipiv

Array of size n. Details of the interchanges and the block structure of D as determined by ?sytrf_rk.

## Output Parameters

A

On exit, if info = 0, the symmetric inverse of the original matrix. If uplo = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced. If uplo = 'L', the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.

## Return Values

This function returns a value info.

= 0: Successful exit.

< 0: If info = -i, the ith argument had an illegal value.

> 0: If info = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.

For more complete information about compiler optimizations, see our Optimization Notice.