# ?hetrd

Reduces a complex Hermitian matrix to tridiagonal form.

## Syntax

FORTRAN 77:

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

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

Fortran 95:

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

## Description

This routine reduces a complex Hermitian matrix A to symmetric tridiagonal form T by a unitary similarity transformation: `A = Q*T*QH`. 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.)

This routine calls ?latrd to reduce a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation.

## 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.

  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 first 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

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

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 chetrd DOUBLE COMPLEX for zhetrd.

  Array, DIMENSION at least max(1, n-1). Stores further details of the unitary matrix Q.

`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.

## Fortran 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 Fortran 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).

d

  Holds the vector of length (n).

e

  Holds the vector of length (n-1).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## 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