f08bf: FL CL CPP AD

NAG CL Interface
f08bfc (dgeqp3)

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NAG Library Manual, Mark 27

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08bf: FL CL CPP AD

▸▿ 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

f08bfc computes the

Q R

factorization, with column pivoting, of a real

m

n

matrix.

2 Specification

#include <nag.h>

void	f08bfc (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, Integer jpvt[], double tau[], NagError *fail)

The function may be called by the names: f08bfc, nag_lapackeig_dgeqp3 or nag_dgeqp3.

3 Description

f08bfc forms the

Q R

factorization, with column pivoting, of an arbitrary rectangular real

m

n

matrix.

m \geq n

, the factorization is given by:

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

where

R

is an

n

n

upper triangular matrix,

Q

is an

m

m

orthogonal matrix and

P

is an

n

n

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

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

which reduces to

A P = 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 trapezoidal, and the factorization can be written

A P = 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 in the first

k

columns of the array a represents a

Q R

factorization of the first

k

columns of the permuted matrix

A P

The function allows specified columns of

A

to be moved to the leading columns of

A P

at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the

i

th stage the pivot column is chosen to be the column which maximizes the

2

-norm of elements

i

m

over columns

i

n

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 https://www.netlib.org/lapack/lug

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

5 Arguments

1: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: $a [\dim]$ – double Input/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

5: $pda$ – Integer Input

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 $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

6: $jpvt [\dim]$ – Integer Input/Output

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

\max (1, n)

On entry: if

jpvt [j - 1] \neq 0

, the

j

th column of

A

is moved to the beginning of

A P

before the decomposition is computed and is fixed in place during the computation. Otherwise, the

j

th column of

A

is a free column (i.e., one which may be interchanged during the computation with any other free column).

On exit: details of the permutation matrix

P

. More precisely, if

jpvt [j - 1] = k

, the

k

th column of

A

is moved to become the

j

th column of

A P

; in other words, the columns of

A P

are the columns of

A

in the order

jpvt [0], jpvt [1], \dots, jpvt [n - 1]

7: $tau [\dim]$ – double Output

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

\max (1, \min (m, n))

On exit: the scalar factors of the elementary reflectors.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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: $n \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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

f08bfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08bfc 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

To form the orthogonal matrix

Q

f08bfc may be followed by a call to f08afc :

nag_lapackeig_dorgqr(order,m,m,MIN(m,n),&a,pda,tau,&fail)

but note that the second dimension of the array a must be at least m, which may be larger than was required by f08bfc.

When

m \geq n

, it is often only the first

n

columns of

Q

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

nag_lapackeig_dorgqr(order,m,n,n,&a,pda,tau,&fail)

To apply

Q

to an arbitrary

m

p

real rectangular matrix

C

, f08bfc may be followed by a call to f08agc . For example,

nag_lapackeig_dormqr(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,tau,
&c,pdc,&fail)

forms the matrix product

C = Q^{T} C

To compute a

Q R

factorization without column pivoting, use f08aec.

The complex analogue of this function is f08btc.

10 Example

This example solves the linear least squares problems

\min_{x} {‖b_{j} - A x_{j}‖}_{2}, j = 1, 2

for the basic solutions

x_{1}

and

x_{2}

, where

A = (\begin{array}{r} - 0.09 & 0.14 & - 0.46 & 0.68 & 1.29 \\ - 1.56 & 0.20 & 0.29 & 1.09 & 0.51 \\ - 1.48 & - 0.43 & 0.89 & - 0.71 & - 0.96 \\ - 1.09 & 0.84 & 0.77 & 2.11 & - 1.27 \\ 0.08 & 0.55 & - 1.13 & 0.14 & 1.74 \\ - 1.59 & - 0.72 & 1.06 & 1.24 & 0.34 \end{array}) and B = (\begin{array}{r} 7.4 & 2.7 \\ 4.2 & - 3.0 \\ - 8.3 & - 9.6 \\ 1.8 & 1.1 \\ 8.6 & 4.0 \\ 2.1 & - 5.7 \end{array})

and

b_{j}

is the

j

th column of the matrix

B

. The solution is obtained by first obtaining a

Q R

factorization with column pivoting of the matrix

A

. A tolerance of

0.01

is used to estimate the rank of

A

from the upper triangular factor,

R

f08bf: FL CL CPP AD

NAG CL Interfacef08bfc (dgeqp3)

▸▿ 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 CL Interface
f08bfc (dgeqp3)