void	nag_dtpqrt (Nag_OrderType order, Integer m, Integer n, Integer l, Integer nb, double a[], Integer pda, double b[], Integer pdb, double t[], Integer pdt, NagError *fail)

3 Description

nag_dtpqrt (f08bbc) forms the

Q R

factorization of a real

(m + n)

n

triangular-pentagonal matrix

C

C = (\begin{matrix} A \\ B \end{matrix})

where

A

is an upper triangular

n

n

matrix and

B

is an

m

n

pentagonal matrix consisting of an

(m - l)

n

rectangular matrix

B_{1}

on top of an

l

n

upper trapezoidal matrix

B_{2}

B = (\begin{matrix} B_{1} \\ B_{2} \end{matrix}) .

The upper trapezoidal matrix

B_{2}

consists of the first

l

rows of an

n

n

upper triangular matrix, where

0 \leq l \leq \min (m, n)

. If

l = 0

B

m

n

rectangular; if

l = n

and

m = n

B

is upper triangular.

A recursive, explicitly blocked,

Q R

factorization (see nag_dgeqrt (f08abc)) is performed on the matrix

C

. The upper triangular matrix

R

, details of the orthogonal matrix

Q

, and further details (the block reflector factors) of

Q

are returned.

Typically the matrix

A

B_{2}

contains the matrix

R

from the

Q R

factorization of a subproblem and nag_dtpqrt (f08bbc) performs the

Q R

update operation from the inclusion of matrix

B_{1}

For example, consider the

Q R

factorization of an

l

n

matrix

\hat{B}

with

l < n

\hat{B} = \hat{Q} \hat{R}

\hat{R} = (\begin{matrix} \hat{R_{1}} & \hat{R_{2}} \end{matrix})

, where

\hat{R_{1}}

l

l

upper triangular and

\hat{R_{2}}

(n - l)

n

rectangular (this can be performed by nag_dgeqrt (f08abc)). Given an initial least-squares problem

\hat{B} \hat{X} = \hat{Y}

where

X

and

Y

are

l

nrhs

matrices, we have

\hat{R} \hat{X} = {\hat{Q}}^{T} \hat{Y}

Now, adding an additional

m - l

rows to the original system gives the augmented least squares problem

B X = Y

where

B

is an

m

n

matrix formed by adding

m - l

rows on top of

\hat{R}

and

Y

is an

m

nrhs

matrix formed by adding

m - l

rows on top of

{\hat{Q}}^{T} \hat{Y}

nag_dtpqrt (f08bbc) can then be used to perform the

Q R

factorization of the pentagonal matrix

B

; the

n

n

matrix

A

will be zero on input and contain

R

on output.

In the case where

\hat{B}

r

n

r \geq n

\hat{R}

n

n

upper triangular (forming

A

) on top of

r - n

rows of zeros (forming first

r - n

rows of

B

). Augmentation is then performed by adding rows to the bottom of

B

with

l = 0

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 2.3.1.3 in How to Use the NAG Library and its Documentation 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

B

Constraint:

m \geq 0

3: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

B

and the order of the upper triangular matrix

A

Constraint:

n \geq 0

4: $l$ – IntegerInput

On entry:

l

, the number of rows of the trapezoidal part of

B

(i.e.,

B_{2}

Constraint:

0 \leq l \leq \min (m, n)

5: $nb$ – IntegerInput

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

Q R

factorization. See Section 9 for details.

Constraints:

$nb \geq 1$ ;
if $n > 0$ , $nb \leq n$ .

6: $a [\dim]$ – doubleInput/Output

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

\max (1, pda \times n)

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

n

n

upper triangular matrix

A

On exit: the upper triangle is overwritten by the corresponding elements of the

n

n

upper triangular matrix

R

7: $pda$ – IntegerInput

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

Constraint:

pda \geq \max (1, n)

8: $b [\dim]$ – doubleInput/Output

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

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

The

(i, j)

th element of the matrix

B

is stored in

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

On entry: the

m

n

pentagonal matrix

B

composed of an

(m - l)

n

rectangular matrix

B_{1}

above an

l

n

upper trapezoidal matrix

B_{2}

On exit: details of the orthogonal matrix

Q

9: $pdb$ – IntegerInput

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

Constraints:

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

10: $t [\dim]$ – doubleOutput

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

$\max (1, pdt \times n)$ 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}]

11: $pdt$ – 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 $order = Nag_RowMajor$ , $pdt \geq n$ .

12: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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, $nb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $nb \geq 1$ and
if $n > 0$ , $nb \leq n$ .

On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdb \geq \max (1, m)$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdt = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdt \geq n$ .

On entry, $pdt = 〈value〉$ and $nb = 〈value〉$ .
Constraint: $pdt \geq nb$ .
NE_INT_3: On entry, $l = 〈value〉$ , $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: $0 \leq l \leq \min (m, n)$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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_dtpqrt (f08bbc) 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 function. 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

The block size, nb, used by nag_dtpqrt (f08bbc) 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 apply

Q

to an arbitrary real rectangular matrix

C

, nag_dtpqrt (f08bbc) may be followed by a call to nag_dtpmqrt (f08bcc). For example,

nag_dtpmqrt(Nag_ColMajor,Nag_LeftSide,Nag_Trans,m,p,n,l,nb,b,pdb,
t,pdt,c,pdc,&c(n+1,1),ldc,&fail)

forms

C = Q^{T} C

, where

C

(m + n)

p

To form the orthogonal matrix

Q

explicitly set

p = m + n

, initialize

C

to the identity matrix and make a call to nag_dtpmqrt (f08bcc) as above.

10 Example

This example finds the basic solutions for 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}) .

Q R

factorization is performed on the first

4

rows of

A

using nag_dgeqrt (f08abc) after which the first

4

rows of

B

are updated by applying

Q^{T}

using nag_dgemqrt (f08acc). The remaining row is added by performing a

Q R

update using nag_dtpqrt (f08bbc);

B

is updated by applying the new

Q^{T}

using nag_dtpmqrt (f08bcc); the solution is finally obtained by triangular solve using

R

from the updated

Q R

NAG Library Function Documentnag_dtpqrt (f08bbc)

▸▿ 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_dtpqrt (f08bbc)