NAG Library Routine Document

Integer, Intent (In)	::	m, n, nb, lda, ldt
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), t(ldt,)
Real (Kind=nag_wp), Intent (Out)	::	work(nb*n)

C Header Interface

#include <nagmk26.h>

void	f08abf_ (const Integer m, const Integer n, const Integer nb, double a[], const Integer lda, double t[], const Integer ldt, double work[], Integer info)

The routine may be called by its LAPACK name dgeqrt.

3

Description

f08abf (dgeqrt) forms the

Q R

factorization of an arbitrary rectangular real

m

n

matrix. No pivoting is performed.

It differs from f08aef (dgeqrf) in that it: requires an explicit block size; stores reflector factors that are upper triangular matrices of the chosen block size (rather than scalars); and recursively computes the

Q R

factorization based on the algorithm of Elmroth and Gustavson (2000).

m \geq n

, the factorization is given by:

A = Q (\begin{array}{r} R \\ 0 \end{array}),

where

R

is an

n

n

upper triangular matrix and

Q

is an

m

m

orthogonal matrix. It is sometimes more convenient to write the factorization as

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

which reduces to

A = Q_{1} R,

where

Q_{1}

consists of the first

n

columns of

Q

, and

Q_{2}

the remaining

m - n

columns.

m < n

R

is upper trapezoidal, and the factorization can be written

A = Q (\begin{array}{r} R_{1} & R_{2} \end{array}),

where

R_{1}

is upper triangular and

R_{2}

is rectangular.

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 9).

Note also that for any

k < n

, the information returned represents a

Q R

factorization of the first

k

columns of the original matrix

A

4

References

Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel

Q R

Factorization Leads to Better Performance IBM Journal of Research and Development. (Volume 44) 4 605–624

Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

5

Arguments

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: $nb$ – IntegerInput: On entry: the explicitly chosen block size to be used in computing the $Q R$ factorization. See Section 9 for details.

Constraint: if $\min (m, n) > 0$ , $1 \leq nb \leq \min (m, n)$ .
4: $a (lda *)$ – 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 $m \geq n$ , the elements below the diagonal are overwritten by details of the orthogonal matrix $Q$ and the upper triangle is overwritten by the corresponding elements of the $n$ by $n$ upper triangular matrix $R$ .
If $m < n$ , the strictly lower triangular part is overwritten by details of the orthogonal matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m$ by $n$ upper trapezoidal matrix $R$ .
5: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f08abf (dgeqrt) is called.

Constraint: $lda \geq \max (1, m)$ .
6: $t (ldt *)$ – Real (Kind=nag_wp) arrayOutput: Note: the second dimension of the array t must be at least $\max (1, \min (m, n))$ .

On exit: further details of the orthogonal matrix $Q$ . The number of blocks is $b = ⌈\frac{k}{nb}⌉$ , where $k = \min (m, n)$ and each block is of order nb except for the last block, which is of order $k - (b - 1) \times nb$ . For each of the blocks, an upper triangular block reflector factor is computed: $T_{1}, T_{2}, \dots, T_{b}$ . These are stored in the $nb$ by $n$ matrix $T$ as $T = [T_{1} | T_{2} | \dots | T_{b}]$ .
7: $ldt$ – IntegerInput: On entry: the first dimension of the array t as declared in the (sub)program from which f08abf (dgeqrt) is called.

Constraint: $ldt \geq nb$ .
8: $work (nb \times n)$ – Real (Kind=nag_wp) arrayWorkspace
9: $info$ – 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

f08abf (dgeqrt) 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

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

m \geq n

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

m < n

To apply

Q

to an arbitrary real rectangular matrix

C

, f08abf (dgeqrt) may be followed by a call to f08acf (dgemqrt). For example,

Call dgemqrt('Left','Transpose',m,p,min(m,n),nb,a,lda,t,ldt,c,ldc, &
              work,info)

forms

C = Q^{T} C

, where

C

m

p

To form the orthogonal matrix

Q

explicitly, simply initialize the

m

m

matrix

C

to the identity matrix and form

C = Q C

using f08acf (dgemqrt) as above.

The block size, nb, used by f08abf (dgeqrt) is supplied explicitly through the interface. For moderate and large sizes of matrix, the block size can have a marked effect on the efficiency of the algorithm with the optimal value being dependent on problem size and platform. A value of

nb = 64 ≪ \min (m, n)

is likely to achieve good efficiency and it is unlikely that an optimal value would exceed

340

To compute a

Q R

factorization with column pivoting, use f08bbf (dtpqrt) or f08bef (dgeqpf).

The complex analogue of this routine is f08apf (zgeqrt).

10

Example

This example solves the linear least squares problems

minimize {‖A x_{i} - b_{i}‖}_{2}, i = 1, 2

where

b_{1}

and

b_{2}

are the columns of the matrix

B

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 & 0.25 \\ - 1.93 & 1.08 & - 0.31 & - 2.14 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ - 1.93 & 0.64 & - 0.66 & 0.08 \\ 0.15 & 0.30 & 0.15 & - 2.13 \\ - 0.02 & 1.03 & - 1.43 & 0.50 \end{array}) and B = (\begin{array}{r} - 2.67 & 0.41 \\ - 0.55 & - 3.10 \\ 3.34 & - 4.01 \\ - 0.77 & 2.76 \\ 0.48 & - 6.17 \\ 4.10 & 0.21 \end{array}) .

NAG Library Routine Document

f08abf (dgeqrt)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

2

Specification

3

Description

4

References

5

Arguments

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