NAG Library Function Document

nag_dgeqrt (f08abc)

It differs from nag_dgeqrf (f08aec) 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). Functions 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: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: m – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: n – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: nb – IntegerInput

On entry: the explicitly chosen block size to be used in computing the

Q R

factorization. See Section 9 for details.

Constraints:

$nb \geq 1$ ;
if $\min (m, n) > 0$ , $nb \leq \min (m, n)$ .

5: a[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

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

n

upper triangular matrix

R

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

n

upper trapezoidal matrix

R

6: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

7: t[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array t must be at least

$\max (1, pdt \times \min (m, n))$ when $order = Nag_ColMajor$ ;
$\max (1, nb \times pdt)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

T

is stored in

$t [(j - 1) \times pdt + i - 1]$ when $order = Nag_ColMajor$ ;
$t [(i - 1) \times pdt + j - 1]$ when $order = Nag_RowMajor$ .

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

n

matrix

T

T = [T_{1} | T_{2} | \dots | T_{b}]

8: pdt – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array t.

Constraints:

if $order = Nag_ColMajor$ , $pdt \geq nb$ ;
if $order = Nag_RowMajor$ , $pdt \geq \max (1, \min (m, n))$ .

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pda \geq \max (1, m)$ .
On entry, $pda = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pda \geq \max (1, n)$ .
On entry, $pdt = ⟨value⟩$ and $nb = ⟨value⟩$ .
Constraint: $pdt \geq nb$ .
NE_INT_3: On entry, $nb = ⟨value⟩$ , $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $nb \geq 1$ and
if $\min (m, n) > 0$ , $nb \leq \min (m, n)$ .
On entry, $pdt = ⟨value⟩$ , $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdt \geq \max (1, \min (m, n))$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

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

nag_dgeqrt (f08abc) is not threaded by NAG in any implementation.

nag_dgeqrt (f08abc) 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 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

, nag_dgeqrt (f08abc) may be followed by a call to nag_dgemqrt (f08acc). For example,

nag_dgemqrt(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),nb,a,pda,t,pdt, 
c,pdc,&fail)

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 nag_dgemqrt (f08acc) as above.

The block size, nb, used by nag_dgeqrt (f08abc) 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 nag_dtpqrt (f08bbc) or nag_dgeqpf (f08bec).

The complex analogue of this function is nag_zgeqrt (f08apc).

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 Function Documentnag_dgeqrt (f08abc)

+− 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

NAG Library Function Document

nag_dgeqrt (f08abc)