# ?orghr

Generates the real orthogonal matrix Q determined by ?gehrd.

## Syntax

FORTRAN 77:

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

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

Fortran 95:

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

## Description

This routine explicitly generates the orthogonal matrix Q that has been determined by a preceding call to sgehrd/dgehrd. (The routine ?gehrd reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation, `A = Q*H*QT`, and represents the matrix Q as a product of `ihi-ilo` elementary reflectors. Here ilo and ihi are values determined by sgebal/dgebal when balancing the matrix; if the matrix has not been balanced, `ilo = 1` and `ihi = n`.)

The matrix Q generated by ?orghr 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`, `ilo = 1` and `ihi = 0`.)

a, tau, work

  REAL for sorghr

  DOUBLE PRECISION for dorghr

  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 orthogonal 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 orghr 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 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 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 floating-point operations is `(4/3)(ihi-ilo)3`.

The complex counterpart of this routine is ?unghr.

Parent topic: Nonsymmetric Eigenvalue Problems