The routine may be called by the names f08bhf, nagf_lapackeig_dtzrzf or its LAPACK name dtzrzf.
The () real upper trapezoidal matrix given by
where is an upper triangular matrix and is an matrix, is factorized as
where is also an upper triangular matrix and is an orthogonal matrix.
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
1: – IntegerInput
On entry: , the number of rows of the matrix .
2: – IntegerInput
On entry: , the number of columns of the matrix .
3: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the leading upper trapezoidal part of the array a must contain the matrix to be factorized.
On exit: the leading upper triangular part of a contains the upper triangular matrix , and elements to n of the first rows of a, with the array tau, represent the orthogonal matrix as a product of elementary reflectors (see Section 3.3.6 in the F08 Chapter Introduction).
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08bhf is called.
5: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau
must be at least
On exit: the scalar factors of the elementary reflectors.
6: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
7: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08bhf is called.
If , 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.
for optimal performance, , where is the optimal block size.
8: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed factorization is the exact factorization of a nearby matrix , where
and is the machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08bhf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is approximately .
This example solves the linear least squares problems
for the minimum norm solutions and , where is the th column of the matrix ,
The solution is obtained by first obtaining a factorization with column pivoting of the matrix , and then the factorization of the leading part of is computed, where is the estimated rank of . A tolerance of is used to estimate the rank of from the upper triangular factor, .
Note that the block size (NB) of assumed in this example is not realistic for such a small problem, but should be suitable for large problems.