?hetrd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?hetrd

Reduces a complex Hermitian matrix to tridiagonal form.

Syntax

call chetrd(uplo, n, a, lda, d, e, tau, work, lwork, info)

call zhetrd(uplo, n, a, lda, d, e, tau, work, lwork, info)

call hetrd(a, tau [,uplo] [,info])

Include Files

mkl.fi, lapack.f90

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

uplo

CHARACTER*1. 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

INTEGER. The order of the matrix A (n≥ 0).

a, work

COMPLEX for chetrd

DOUBLE COMPLEX for zhetrd.

a(lda,*) 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.

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

work is a workspace array, its dimension max(1, lwork).

lda

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

lwork

INTEGER. The size of the work array (lwork≥n).

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

See Application Notes for the suggested value of lwork.

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

REAL for chetrd

DOUBLE PRECISION for zhetrd.

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

COMPLEX for chetrdDOUBLE COMPLEX for zhetrd.

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.

work(1)

If info = 0, on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

INTEGER.

If info = 0, the execution is successful.

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

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 restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine hetrd interface are the following:

a: Holds the matrix A of size (n,n).
tau: Holds the vector of length (n-1).
uplo: Must be 'U' or 'L'. The default value is 'U'.

Note that diagonal (d) and off-diagonal (e) elements of the matrix T are omitted because they are kept in the matrix A on exit.

Application Notes

For better performance, try using lwork =n*blocksize, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

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 Fortran

?hetrd