NAG Library Function Document

nag_dgeqpf (f08bec)

+− 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

nag_dgeqpf (f08bec) computes the

Q R

factorization, with column pivoting, of a real

m

n

matrix.

2 Specification

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_dgeqpf (f08bec) 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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd 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: 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

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

6: jpvt[ $\dim$ ] – IntegerInput/Output

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

\max (1, n)

On entry: if

jpvt [i - 1] \neq 0

, then the

i

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

i

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 [i - 1] = k

, then the

k

th column of

A

is moved to become the

i

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[ $\min (m, n)$ ] – doubleOutput

On exit: further details of the orthogonal matrix

Q

8: 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: $n \geq 0$ .
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.

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_dgeqpf (f08bec) is not threaded by NAG in any implementation.

nag_dgeqpf (f08bec) 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 form the orthogonal matrix

Q

nag_dgeqpf (f08bec) may be followed by a call to nag_dorgqr (f08afc):

nag_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 nag_dgeqpf (f08bec).

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

To apply

Q

to an arbitrary real rectangular matrix

C

, nag_dgeqpf (f08bec) may be followed by a call to nag_dormqr (f08agc). For example,

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

forms

C = Q^{T} C

, where

C

m

p

To compute a

Q R

factorization without column pivoting, use nag_dgeqrf (f08aec).

The complex analogue of this function is nag_zgeqpf (f08bsc).

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} - 0.09 & 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} - 0.01 & - 0.04 \\ 0.04 & - 0.03 \\ 0.05 & 0.01 \\ - 0.03 & - 0.02 \\ 0.02 & 0.05 \\ - 0.06 & 0.07 \end{array}) .

Here

A

is approximately rank-deficient, and hence it is preferable to use nag_dgeqpf (f08bec) rather than nag_dgeqrf (f08aec).

NAG Library Function Documentnag_dgeqpf (f08bec)

+− 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_dgeqpf (f08bec)