# ?hptrd

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

## Syntax

FORTRAN 77:

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

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

Fortran 95:

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

## Description

This 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, DIMENSION at least max(1, n(n+1)/2). Contains either upper or lower triangle of A (as specified by uplo) in packed form.

## Output Parameters

ap

  Overwritten by the tridiagonal matrix T and details of the orthogonal 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 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 chptrd

  DOUBLE COMPLEX for zhptrd.

  Arrays, DIMENSION at least max(1, n-1). Contains further details of the orthogonal matrix Q.

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 hptrd interface are the following:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size (n*(n+1)/2).

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

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