# ?tzrzf

Reduces the upper trapezoidal matrix A to upper triangular form.

## Syntax

FORTRAN 77:

call stzrzf(m, n, a, lda, tau, work, lwork, info)

call dtzrzf(m, n, a, lda, tau, work, lwork, info)

call ctzrzf(m, n, a, lda, tau, work, lwork, info)

call ztzrzf(m, n, a, lda, tau, work, lwork, info)

Fortran 95:

call tzrzf(a [, tau] [,info])

## Description

This routine reduces the m-by-n (`m ≤ n`) real/complex upper trapezoidal matrix A to upper triangular form by means of orthogonal/unitary transformations. The upper trapezoidal matrix A is factored as

`A = (R 0)*Z`,

where Z is an n-by-n orthogonal/unitary matrix and R is an m-by-m upper triangular matrix.

See ?larz that applies an elementary reflector returned by ?tzrzf to a general matrix.

## Input Parameters

m

  INTEGER. The number of rows in the matrix A (`m ≥ 0`).

n

  INTEGER. The number of columns in A (n ≥ m).

a, work

  REAL for stzrzf

  DOUBLE PRECISION for dtzrzf

  COMPLEX for ctzrzf

  DOUBLE COMPLEX for ztzrzf.

  Arrays: a(lda,*), work(lwork).The leading m-by-n upper trapezoidal part of the array a contains the matrix A to be factorized.

  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, m).

lwork

  INTEGER. The size of the work array;

  `lwork ≥ max(1, m)`.

  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 on exit by the factorization data as follows:

  the leading m-by-m upper triangular part of a contains the upper triangular matrix R, and elements m +1 to n of the first m rows of a, with the array tau, represent the orthogonal matrix Z as a product of m elementary reflectors.

tau

  REAL for stzrzf

  DOUBLE PRECISION for dtzrzf

  COMPLEX for ctzrzf

  DOUBLE COMPLEX for ztzrzf.

  Array, DIMENSION at least max (1, m). Contains scalar factors of the elementary reflectors for the matrix Z.

`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 tzrzf interface are the following:

a

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

tau

  Holds the vector of length (m).

## Application Notes

For better performance, try using `lwork =m*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.

Related routines include:

?ormrz

  to apply matrix Q (for real matrices)

?unmrz

  to apply matrix Q (for complex matrices).

Parent topic: Orthogonal Factorizations