NAG Library Function Document

nag_dgeqrf (f08aec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_dgeqrf (f08aec) computes the

Q R

factorization of a real

m

n

matrix.

2 Specification

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_dgeqrf (f08aec) forms the

Q R

factorization of an arbitrary rectangular real

m

n

matrix. No pivoting is performed.

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 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 8).

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 original matrix

A

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

or 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: tau[ $\dim$ ] – doubleOutput

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

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

On exit: further details of the orthogonal matrix

Q

7: 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 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_dgeqrf (f08aec) 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_dgeqrf (f08aec).

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_dgeqrf (f08aec) 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 with column pivoting, use nag_dgeqpf (f08bec).

The complex analogue of this function is nag_zgeqrf (f08asc).

9 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} - 3.15 & 2.19 \\ - 0.11 & - 3.64 \\ 1.99 & 0.57 \\ - 2.70 & 8.23 \\ 0.26 & - 6.35 \\ 4.50 & - 1.48 \end{array}) .

NAG Library Function Documentnag_dgeqrf (f08aec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_dgeqrf (f08aec)