?hptrd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?hptrd

Reduces a complex Hermitian matrix to tridiagonal form using packed storage.

Syntax

call chptrd(uplo, n, ap, d, e, tau, info)

call zhptrd(uplo, n, ap, d, e, tau, info)

call hptrd(ap, tau [,uplo] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine reduces a packed 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 for working with Q in this representation (see Application Notes below).

Input Parameters

uplo

CHARACTER*1. Must be 'U' or 'L'.

If uplo = 'U', ap stores the packed upper triangle of A.

If uplo = 'L', ap stores the packed lower triangle of A.

n

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

ap

COMPLEX for chptrd

DOUBLE COMPLEX for zhptrd.

Array, size at least max(1, n(n+1)/2). Contains either upper or lower triangle of A (as specified by uplo) in the packed form described in "Matrix Storage Schemes.

Output Parameters

ap

Overwritten by the tridiagonal matrix T and details of the unitary matrix Q, as specified by uplo.

d, e

REAL for chptrd

DOUBLE PRECISION for zhptrd.

Arrays:

d(*) contains the diagonal elements of the matrix T.

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

e(*) contains the off-diagonal elements of T.

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

tau

COMPLEX for chptrd

DOUBLE COMPLEX for zhptrd.

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 reflectors.

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 hptrd interface are the following:

ap: Holds the array A of size (n*(n+1)/2).
tau: Holds the vector with the number of elements 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

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:

upgtr: to form the computed matrix Q explicitly
upmtr: to multiply a complex matrix by Q.

The real counterpart of this routine is sptrd.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?hptrd