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])
C:
lapack_int LAPACKE_<?>tzrzf( int matrix_order, lapack_int m, lapack_int n, <datatype>* a, lapack_int lda, <datatype>* tau );
Include Files
The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.
Description
The 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.
The ?tzrzf routine replaces the deprecated ?tzrqf routine.
Input Parameters
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
- 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 leading 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
The factorization is obtained by Householder's method. The k-th transformation matrix, z(k), which is used to introduce zeros into the (m - k + 1)-th row of A, is given in the form
where for real flavors
and for complex flavors
tau is a scalar and z(k) is an l-element vector. tau and z(k) are chosen to annihilate the elements of the k-th row of X.
The scalar tau is returned in the k-th element of tau and the vector u(k) in the k-th row of A, such that the elements of z(k) are in a(k, m+1), ..., a(k, n).
The elements of r are returned in the upper triangular part of A.
Z is given by
Z = Z(1)*Z(2)*...*Z(m).
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: