NAG Library Routine Document

F08BHF (DTZRZF)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

F08BHF (DTZRZF) reduces the

m

n

(

m \leq n

) real upper trapezoidal matrix

A

to upper triangular form by means of orthogonal transformations.

2 Specification

SUBROUTINE F08BHF (

M, N, A, LDA, TAU, WORK, LWORK, INFO)

INTEGER	M, N, LDA, LWORK, INFO
REAL (KIND=nag_wp)	A(LDA,), TAU(), WORK(max(1,LWORK))

The routine may be called by its LAPACK name dtzrzf.

3 Description

The

m

n

(

m \leq n

) real upper trapezoidal matrix

A

given by

A = (\begin{matrix} R_{1} & R_{2} \end{matrix}),

where

R_{1}

is an

m

m

upper triangular matrix and

R_{2}

is an

m

(n - m)

matrix, is factorized as

A = (\begin{matrix} R & 0 \end{matrix}) Z,

where

R

is also an

m

m

upper triangular matrix and

Z

is an

n

n

orthogonal matrix.

4 References

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 http://www.netlib.org/lapack/lug

5 Parameters

1: $M$ – INTEGERInput: On entry: $m$ , the number of rows of the matrix $A$ .

Constraint: $M \geq 0$ .
2: $N$ – INTEGERInput: On entry: $n$ , the number of columns of the matrix $A$ .

Constraint: $N \geq 0$ .
3: $A (LDA *)$ – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .

On entry: the leading $m$ by $n$ upper trapezoidal part of the array A must contain the matrix to be factorized.

On exit: 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 (see Section 3.3.6 in the F08 Chapter Introduction).
4: $LDA$ – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F08BHF (DTZRZF) is called.

Constraint: $LDA \geq \max (1, M)$ .
5: $TAU (*)$ – REAL (KIND=nag_wp) arrayOutput: Note: the dimension of the array TAU must be at least $\max (1, M)$ .

On exit: the scalar factors of the elementary reflectors.
6: $WORK (\max (1, LWORK))$ – REAL (KIND=nag_wp) arrayWorkspace: On exit: if $INFO = 0$ , $WORK (1)$ contains the minimum value of LWORK required for optimal performance.
7: $LWORK$ – INTEGERInput: On entry: the dimension of the array WORK as declared in the (sub)program from which F08BHF (DTZRZF) is called.
If $LWORK = - 1$ , 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.

Suggested value: for optimal performance, $LWORK \geq M \times nb$ , where $nb$ is the optimal block size.

Constraint: $LWORK \geq \max (1, M)$ or $LWORK = - 1$ .
8: $INFO$ – INTEGEROutput: On exit: $INFO = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$INFO < 0$: If $INFO = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The computed factorization is the exact factorization of a nearby matrix

A + E

, where

{‖E‖}_{2} = O ε {‖A‖}_{2}

and

ε

is the machine precision.

8 Parallelism and Performance

F08BHF (DTZRZF) is not threaded by NAG in any implementation.

F08BHF (DTZRZF) 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.

9 Further Comments

The total number of floating-point operations is approximately

4 m^{2} (n - m)

The complex analogue of this routine is F08BVF (ZTZRZF).

10 Example

This example solves the linear least squares problems

\min_{x} {‖b_{j} - A x_{j}‖}_{2}, j = 1, 2

for the minimum norm solutions

x_{1}

and

x_{2}

, where

b_{j}

is the

j

th column of the matrix

B

A = (\begin{array}{r} - 0.09 & 0.14 & - 0.46 & 0.68 & 1.29 \\ - 1.56 & 0.20 & 0.29 & 1.09 & 0.51 \\ - 1.48 & - 0.43 & 0.89 & - 0.71 & - 0.96 \\ - 1.09 & 0.84 & 0.77 & 2.11 & - 1.27 \\ 0.08 & 0.55 & - 1.13 & 0.14 & 1.74 \\ - 1.59 & - 0.72 & 1.06 & 1.24 & 0.34 \end{array}) and B = (\begin{array}{r} 7.4 & 2.7 \\ 4.2 & - 3.0 \\ - 8.3 & - 9.6 \\ 1.8 & 1.1 \\ 8.6 & 4.0 \\ 2.1 & - 5.7 \end{array}) .

The solution is obtained by first obtaining a

Q R

factorization with column pivoting of the matrix

A

, and then the

R Z

factorization of the leading

k

k

part of

R

is computed, where

k

is the estimated rank of

A

. A tolerance of

0.01

is used to estimate the rank of

A

from the upper triangular factor,

R

Note that the block size (NB) of

64

assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

NAG Library Routine DocumentF08BHF (DTZRZF)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

F08BHF (DTZRZF)