NAG Library Function Document

nag_real_qr (f01qcc)

+− 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_real_qr (f01qcc) finds the

Q R

factorization of the real

m

n

matrix

A

, where

m \geq n

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_real_qr (Integer m, Integer n, double a[], Integer tda, double zeta[], NagError *fail)

3 Description

The

m

n

matrix

A

is factorized as

\begin{matrix} A = Q (\begin{matrix} R \\ 0 \end{matrix}) & when ​ m > n, \\ A = Q R & when ​ m = n, \end{matrix}

where

Q

is an

m

m

orthogonal matrix and

R

is an

n

n

upper triangular matrix. The factorization is obtained by Householder's method. The

k

th transformation matrix,

Q_{k}

, which is used to introduce zeros into the

k

th column of

A

is given in the form

Q_{k} = (\begin{matrix} I & 0 \\ 0 & T_{k} \end{matrix})

where

T_{k} = I - u_{k} u_{k}^{T},

u_{k} = (\begin{matrix} ζ_{k} \\ z_{k} \end{matrix}),

ζ_{k}

is a scalar and

z_{k}

is an

(m - k)

element vector.

ζ_{k}

and

z_{k}

are chosen to annihilate the elements below the triangular part of

A

The vector

u_{k}

is returned in the

(k - 1)

th element of the array zeta and in the

(k - 1)

th column of a, such that

ζ_{k}

is in

zeta [k - 1]

and the elements of

z_{k}

are in

a [(k) \times tda + k - 1], \dots, a [(m - 1) \times tda + k - 1]

. The elements of

R

are returned in the upper triangular part of a.

Q

is given by

Q = {(Q_{n} Q_{n - 1} \dots Q_{1})}^{T} .

Good background descriptions to the

Q R

factorization are given in Dongarra et al. (1979) and Golub and Van Loan (1996).

4 References

Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979) LINPACK Users' Guide SIAM, Philadelphia

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

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

5 Arguments

1: m – IntegerInput: On entry: $m$ , the number of rows of $A$ .
Constraint: $m \geq n$ .
2: n – IntegerInput: On entry: $n$ , the number of columns of $A$ .
When $n = 0$ then an immediate return is effected.

Constraint: $n \geq 0$ .
3: a[ $m \times tda$ ] – doubleInput/Output: On entry: the leading $m$ by $n$ part of the array a must contain the matrix to be factorized.

On exit: the $n$ by $n$ upper triangular part of a will contain the upper triangular matrix $R$ and the $m$ by $n$ strictly lower triangular part of a will contain details of the factorization as described in Section 3
4: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
5: zeta[n] – doubleOutput: On exit: $zeta [k - 1]$ contains the scalar $ζ_{k}$ for the $k$ th transformation. If $T_{k} = I$ then zeta $(k - 1) = 0.0$ , otherwise $zeta [k - 1]$ contains $ζ_{k}$ as described in Section 3 and $ζ_{k}$ is always in the range $(1.0, \sqrt{2.0})$ .
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $m = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $m \geq n$ .
On entry, $tda = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $tda \geq n$ .
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .

7 Accuracy

The computed factors

Q

and

R

satisfy the relation

Q (\begin{matrix} R \\ 0 \end{matrix}) = A + E

where

‖E‖ \leq c ε ‖A‖

, and

ε

is the machine precision,

c

is a modest function of

m

and

n

and

.

denotes the spectral (two) norm.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The approximate number of floating-point operations is given by

2 n^{2} (3 m - n) / 3

10 Example

To obtain the

Q R

factorization of the 5 by 3 matrix

A = (\begin{matrix} 2.0 & 2.5 & 2.5 \\ 2.0 & 2.5 & 2.5 \\ 1.6 & - 0.4 & 2.8 \\ 2.0 & - 0.5 & 0.5 \\ 1.2 & - 0.3 & - 2.9 \end{matrix}) .

NAG Library Function Documentnag_real_qr (f01qcc)

+− 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_real_qr (f01qcc)