NAG Library Routine Document

F08CHF (DGERQF)

+− 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

F08CHF (DGERQF) computes an RQ factorization of a real

m

n

matrix

A

2 Specification

SUBROUTINE F08CHF (

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 dgerqf.

3 Description

F08CHF (DGERQF) forms the

R Q

factorization of an arbitrary rectangular real

m

n

matrix. If

m \leq n

, the factorization is given by

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

where

R

is an

m

m

lower triangular matrix and

Q

is an

n

n

orthogonal matrix. If

m > n

the factorization is given by

A = R Q,

where

R

is an

m

n

upper trapezoidal matrix and

Q

is again an

n

n

orthogonal matrix. In the case where

m < n

the factorization can be expressed as

A = (\begin{matrix} 0 & R \end{matrix}) (\begin{matrix} Q_{1} \\ Q_{2} \end{matrix}) = R Q_{2},

where

Q_{1}

consists of the first

(n - m)

rows of

Q

and

Q_{2}

the remaining

m

rows.

The matrix

Q

is not formed explicitly, but is represented as a product of

\min (m, n)

elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with

Q

in this representation (see Section 8).

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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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 $m$ by $n$ matrix $A$ .

On exit: if $m \leq n$ , the upper triangle of the subarray $A (1 : m, n - m + 1 : n)$ contains the $m$ by $m$ upper triangular matrix $R$ .
If $m \geq n$ , the elements on and above the $(m - n)$ th subdiagonal contain the $m$ by $n$ upper trapezoidal matrix $R$ ; the remaining elements, with the array TAU, represent the orthogonal matrix $Q$ as a product of $\min (m, n)$ 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 F08CHF (DGERQF) 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 $\max (1, \min (M, N))$ .
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 F08CHF (DGERQF) 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

\frac{2}{3} m^{2} (3 n - m)

m \leq n

, or

\frac{2}{3} n^{2} (3 m - n)

m > n

To form the orthogonal matrix

Q

F08CHF (DGERQF) may be followed by a call to F08CJF (DORGRQ):

CALL DORGRQ(N,N,MIN(M,N),A,LDA,TAU,WORK,LWORK,INFO)

but note that the first dimension of the array A must be at least N, which may be larger than was required by F08CHF (DGERQF). When

m \leq n

, it is often only the first

m

rows of

Q

that are required and they may be formed by the call:

CALL DORGRQ(M,N,M,A,LDA,TAU,WORK,LWORK,INFO)

To apply

Q

to an arbitrary real rectangular matrix

C

, F08CHF (DGERQF) may be followed by a call to F08CKF (DORMRQ). For example:

CALL DORMRQ('Left','Transpose',N,P,MIN(M,N),A,LDA,TAU,C,LDC, &
              WORK,LWORK,INFO)

forms

C = Q^{T} C

, where

C

n

p

The complex analogue of this routine is F08CVF (ZGERQF).

9 Example

This example finds the minimum norm solution to the underdetermined equations

A x = b

where

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array}) and b = (\begin{array}{r} - 2.87 \\ 1.63 \\ - 3.52 \\ 0.45 \end{array}) .

The solution is obtained by first obtaining an

R Q

factorization of the matrix

A

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 DocumentF08CHF (DGERQF)

+− 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

F08CHF (DGERQF)