?sytri

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/22/2024

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?sytri_3

Computes the inverse of a real or complex symmetric matrix.

call ssytri_3(uplo, n, A, lda, e, ipiv, work, lwork, info)

call dsytri_3(uplo, n, A, lda, e, ipiv, work, lwork, info)

call csytri_3(uplo, n, A, lda, e, ipiv, work, lwork, info)

call zsytri_3(uplo, n, A, lda, e, ipiv, work, lwork, info)

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*(U^T)*(P^T) or A = P*L*D*(L^T)*(P^T), where U (or L) is a unit upper (or lower) triangular matrix, U^T (or L^T) is the transpose of U (or L), P is a permutation matrix, P^T 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

uplo

CHARACTER*1

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

INTEGER

The order of the matrix A. n ≥ 0.

A

REAL for ssytri_3

DOUBLE PRECISION for dsytri_3

COMPLEX for csytri_3

COMPLEX*16 for zsytri_3

Array, dimension (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

INTEGER

The leading dimension of the array A.lda ≥ max(1, n).

e

REAL for ssytri_3

DOUBLE PRECISION for dsytri_3

COMPLEX for csytri_3

COMPLEX*16 for zsytri_3

Array, dimension (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) is not referenced in both the uplo = 'U' and uplo = 'L' cases.

ipiv

INTEGER

Array, dimension (n). Details of the interchanges and the block structure of D as determined by ?sytrf_rk.

lwork

INTEGER

The length of the array work.

If LDWORK = -1, a workspace query is assumed; the routine calculates only the optimal size of the optimal size of the work array and returns this value as the first entry of the work array, and no error message related to lwork is issued by XERBLA.

Output Parameters

A

REAL for ssytri_3

DOUBLE PRECISION for dsytri_3

COMPLEX for csytri_3

COMPLEX*16 for zsytri_3

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.

work

REAL for ssytri_3

DOUBLE PRECISION for dsytri_3

COMPLEX for csytri_3

COMPLEX*16 for zsytri_3

Array, dimension (n+NB+1)*(NB+3). On exit, if info = 0, work(1) returns the optimal lwork.

info

INTEGER

= 0: successful exit.
< 0: If info = -i, the i^th argument had an illegal value.
> 0: If info = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.

Parent topic: Matrix Inversion: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?sytri_3

Description

Input Parameters

Output Parameters