?hetrd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?hetrd

Reduces a complex Hermitian matrix to tridiagonal form.

Syntax

lapack_int LAPACKE_chetrd( int matrix_layout, char uplo, lapack_int n, lapack_complex_float* a, lapack_int lda, float* d, float* e, lapack_complex_float* tau );

lapack_int LAPACKE_zhetrd( int matrix_layout, char uplo, lapack_int n, lapack_complex_double* a, lapack_int lda, double* d, double* e, lapack_complex_double* tau );

Include Files

mkl.h

Description

The routine reduces a complex Hermitian matrix A to symmetric tridiagonal form T by a unitary similarity transformation: A = Q*T*Q^H. The unitary matrix Q is not formed explicitly but is represented as a product of n-1 elementary reflectors. Routines are provided to work with Q in this representation. (They are described later in this topic.)

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'.

If uplo = 'U', a stores the upper triangular part of A.

If uplo = 'L', a stores the lower triangular part of A.

n

The order of the matrix A (n≥ 0).

a

a (size max(1, lda*n)) is an array containing either upper or lower triangular part of the matrix A, as specified by uplo. If uplo = 'U', the leading n-by-n upper triangular part of a contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo = 'L', the leading n-by-n lower triangular part of a contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

lda

The leading dimension of a; at least max(1, n).

Output Parameters

a

On exit,

if uplo = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors;

if uplo = 'L', the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors.

d, e

Arrays:

d contains the diagonal elements of the matrix T.

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

e contains the off-diagonal elements of T.

The dimension of e must be at least max(1, n-1).

tau

Array, size at least max(1, n-1). Stores (n-1) scalars that define elementary reflectors in decomposition of the unitary matrix Q in a product of n-1 elementary reflectors.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Application Notes

The computed matrix T is exactly similar to a matrix A + E, where ||E||₂ = c(n)*ε*||A||₂, c(n) is a modestly increasing function of n, and ε is the machine precision.

The approximate number of floating-point operations is (16/3)n³.

After calling this routine, you can call the following:

ungtr: to form the computed matrix Q explicitly
unmtr: to multiply a complex matrix by Q.

The real counterpart of this routine is ?sytrd.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?hetrd