# ?gehrd

Reduces a general matrix to upper Hessenberg form.

## Syntax

FORTRAN 77:

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

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

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

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

Fortran 95:

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

## Description

The routine reduces a general matrix A to upper Hessenberg form H by an orthogonal or unitary similarity transformation `A = Q*H*QH`. Here H has real subdiagonal elements.

The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of elementary reflectors. Routines are provided to work with Q in this representation.

## Input Parameters

n

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

ilo, ihi

  INTEGER. If A is an output by ?gebal, then ilo and ihi must contain the values returned by that routine. Otherwise `ilo = 1` and `ihi = n`. (If `n > 0`, then `1 ≤ ilo ≤ ihi ≤ n`; if `n = 0`, `ilo = 1` and `ihi = 0`.)

a, work

  REAL for sgehrd

  DOUBLE PRECISION for dgehrd

  COMPLEX for cgehrd

  DOUBLE COMPLEX for zgehrd.

  Arrays:

  a (lda,*) contains the matrix A.

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

  work (lwork) is a workspace array.

lda

  INTEGER. The first dimension of a; at least max(1, n).

lwork

  INTEGER. The size of the work array; at least max(1, 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 upper Hessenberg matrix H and details of the matrix Q. The subdiagonal elements of H are real.

tau

  REAL for sgehrd

  DOUBLE PRECISION for dgehrd

  COMPLEX for cgehrd

  DOUBLE COMPLEX for zgehrd.

  Array, DIMENSION at least max (1, n-1).

  Contains additional information on the 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 gehrd 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 = 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 Hessenberg matrix H is exactly similar to a nearby 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 for real flavors is `(2/3)*(ihi - ilo)2(2ihi + 2ilo + 3n)`; for complex flavors it is 4 times greater.

Parent topic: Nonsymmetric Eigenvalue Problems