NAG Library Function Document

nag_complex_qr (f01rcc)

+− 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_complex_qr (f01rcc) finds the

Q R

factorization of the complex

m

n

matrix

A

, where

m \geq n

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_complex_qr (Integer m, Integer n, Complex a[], Integer tda, Complex theta[], 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

unitary matrix and

R

is an

n

n

upper triangular matrix with real diagonal elements.

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}),

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

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

γ_{k}

is a scalar for which

{Re γ}_{k} = 1.0

ζ_{k}

is a real scalar and

z_{k}

is an

(m - k)

element vector.

γ_{k}

ζ_{k}

and

z_{k}

are chosen to annihilate the elements below the triangular part of

A

and to make the diagonal elements real.

The scalar

γ_{k}

and the vector

u_{k}

are returned in the

(k - 1)

th element of the array theta and in the

(k - 1)

th column of a, such that

θ_{k}

, given by

θ_{k} = (ζ_{k}, Im γ_{k}),

is in

theta [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})}^{H} .

A good background description to the

Q R

factorization is given in Dongarra et al. (1979).

4 References

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

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

Constraints:

$n \geq 0$ ;
when $n = 0$ then an immediate return is effected.

3: a[ $m \times tda$ ] – ComplexInput/Output

On entry: the leading

m

n

part of the array a must contain the matrix to be factorized.

On exit: the

n

n

upper triangular part of a will contain the upper triangular matrix

R

, with the imaginary parts of the diagonal elements set to zero, and the

m

n

strictly lower triangular part of a will contain details of the factorization as described above.

4: tda – IntegerInput

On entry: the stride separating matrix column elements in the array a.

Constraint:

tda \geq n

5: theta[n] – ComplexOutput

On exit: the scalar

θ_{k}

for the

k

th transformation. If

T_{k} = I

then

theta [k - 1] = 0.0

; if

T_{k} = (\begin{matrix} α & 0 \\ 0 & I \end{matrix}) Re α < 0.0

then

theta [k - 1] = α

; otherwise

theta [k - 1]

contains

theta [k - 1]

as described in Section 3 and Re

(theta [k - 1])

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‖

ε

being 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 real floating-point operations is given by

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

Following the use of this function the operations

B := Q B and B := Q^{H} B

where

B

is an

m

k

matrix, can be performed by calls to nag_complex_apply_q (f01rdc).

The operation

B : = Q B

can be obtained by the call:

and

B : = Q^{H} B

can be obtained by the call:

B

is a one-dimensional array (single column) then the argument

t d b

can be replaced by

1

. See nag_complex_apply_q (f01rdc) for further details.

The first

k

columns of the unitary matrix

Q

can either be obtained by setting

B

to the first

k

columns of the unit matrix and using the first of the above two calls, or by calling nag_complex_form_q (f01rec), which overwrites the

k

columns of

Q

on the first

k

columns of the array a.

Q

is obtained by the call:

k

is larger than

n

, then

A

must have been declared to have at least

k

columns.

10 Example

To obtain the

Q R

factorization of the 5 by 3 matrix

A = (\begin{matrix} 0.5 i & - 0.5 + 1.5 i & - 1.0 + 1.0 i \\ 0.4 + 0.3 i & 0.9 + 1.3 i & 0.2 + 1.4 i \\ 0.4 & - 0.4 + 0.4 i & 1.8 \\ 0.3 - 0.4 i & 0.1 + 0.7 i & 0.0 \\ - 0.3 i & 0.3 + 0.3 i & 2.4 i \end{matrix})

NAG Library Function Documentnag_complex_qr (f01rcc)

+− 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_complex_qr (f01rcc)