NAG Library Routine Document

F08BVF (ZTZRZF)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08BVF (ZTZRZF) reduces the

m

n

(

m \leq n

) complex upper trapezoidal matrix

A

to upper triangular form by means of unitary transformations.

2 Specification

SUBROUTINE F08BVF (

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

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

The routine may be called by its LAPACK name ztzrzf.

3 Description

The

m

n

(

m \leq n

) complex 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

unitary 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, $*$ ) – COMPLEX (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 unitary 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 F08BVF (ZTZRZF) is called.
Constraint: $LDA \geq \max (1, M)$ .
5: TAU( $*$ ) – COMPLEX (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( $\max (1, LWORK)$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace: On exit: if $INFO = 0$ , the real part of $WORK (1)$ 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 F08BVF (ZTZRZF) 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

Errors or warnings detected by the routine:

$INFO < 0$: 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 Further Comments

The total number of floating point operations is approximately

16 m^{2} (n - m)

The real analogue of this routine is F08BHF (DTZRZF).

9 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} 0.47 - 0.34 i & - 0.40 + 0.54 i & 0.60 + 0.01 i & 0.80 - 1.02 i \\ - 0.32 - 0.23 i & - 0.05 + 0.20 i & - 0.26 - 0.44 i & - 0.43 + 0.17 i \\ 0.35 - 0.60 i & - 0.52 - 0.34 i & 0.87 - 0.11 i & - 0.34 - 0.09 i \\ 0.89 + 0.71 i & - 0.45 - 0.45 i & - 0.02 - 0.57 i & 1.14 - 0.78 i \\ - 0.19 + 0.06 i & 0.11 - 0.85 i & 1.44 + 0.80 i & 0.07 + 1.14 i \end{array})

and

B = (\begin{array}{r} - 1.08 - 2.59 i & 2.22 + 2.35 i \\ - 2.61 - 1.49 i & 1.62 - 1.48 i \\ 3.13 - 3.61 i & 1.65 + 3.43 i \\ 7.33 - 8.01 i & - 0.98 + 3.08 i \\ 9.12 + 7.63 i & - 2.84 + 2.78 i \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 DocumentF08BVF (ZTZRZF)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F08BVF (ZTZRZF)