# ?unghr

Generates the complex unitary matrix Q determined by ?gehrd.

## Syntax

FORTRAN 77:

call cunghr(n, ilo, ihi, a, lda, tau, work, lwork, info)

call zunghr(n, ilo, ihi, a, lda, tau, work, lwork, info)

Fortran 95:

call unghr(a, tau [,ilo] [,ihi] [,info])

## Description

This routine is intended to be used following a call to cgehrd/zgehrd, which reduces a complex matrix A to upper Hessenberg form H by a unitary similarity transformation: `A = Q*H*QH`. ?gehrd represents the matrix Q as a product of ihi-ilo elementary reflectors. Here ilo and ihi are values determined by cgebal/zgebal when balancing the matrix; if the matrix has not been balanced, `ilo = 1` and `ihi = n`.

Use the routine ?unghr to generate Q explicitly as a square matrix. The matrix Q has the structure:

where Q22 occupies rows and columns ilo to ihi.

## Input Parameters

n

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

ilo, ihi

  INTEGER. These must be the same parameters ilo and ihi, respectively, as supplied to ?gehrd . (If `n > 0`, then `1 ≤ ilo ≤ ihi ≤ n`. If n = 0, then `ilo = 1` and `ihi = 0`.)

a, tau, work

  COMPLEX for cunghr

  DOUBLE COMPLEX for zunghr.

  Arrays:

  a(lda,*) contains details of the vectors which define the elementary reflectors, as returned by ?gehrd.

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

  tau(*) contains further details of the elementary reflectors, as returned by ?gehrd .

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

  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 ≥ max(1, ihi-ilo).

  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 n-by-n 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 unghr interface are the following:

a

  Holds the matrix A of size (n,n).

tau

  Holds the vector of length (n-1).

ilo

  Default value for this argument is `ilo = 1`.

ihi

  Default value for this argument is `ihi = n`.

## Application Notes

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

If it is not clear how much workspace to supply, use a generous value of lwork for the first run, or set `lwork = -1`.

In first case 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 `lwork = -1`, then 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 lwork is less than the minimal required value and is not equal to -1, then the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The computed matrix Q differs from the exact result by a matrix E such that `||E||2 = O(ε)`, where ε is the machine precision.

The approximate number of real floating-point operations is `(16/3)(ihi-ilo)3`.

The real counterpart of this routine is ?orghr.

Parent topic: Nonsymmetric Eigenvalue Problems