f06qsf : NAG Library, Mark 26

f06qsf performs a

Q R

R Q

factorization (as a sequence of plane rotations) of a real upper spiked matrix.

Specification

Fortran Interface

Subroutine f06qsf (

side, n, k1, k2, c, s, a, lda)

Integer, Intent (In)	::	n, k1, k2, lda
Real (Kind=nag_wp), Intent (Inout)	::	s(), a(lda,)
Real (Kind=nag_wp), Intent (Out)	::	c(k2-1)
Character (1), Intent (In)	::	side

C Header Interface

#include nagmk26.h

void	f06qsf_ ( const char side, const Integer n, const Integer k1, const Integer k2, double c[], double s[], double a[], const Integer *lda, const Charlen length_side)

Description

f06qsf transforms an

n

n

real upper spiked matrix

H

to upper triangular form

R

by applying a real orthogonal matrix

P

from the left or the right.

P

is formed as a sequence of plane rotations in planes

k_{1}

k_{2}

side ='L'

H

is assumed to have a row spike, with nonzero elements

h_{k_{2}, k}

, for

k = k_{1}, \dots, k_{2} - 1

. The rotations are applied from the left:

P H = R,

where

P = P_{k_{2} - 1} \dots P_{k_{1} + 1} P_{k_{1}}

and

P_{k}

is a rotation in the

(k, k_{2})

plane.

side ='R'

H

is assumed to have a column spike, with nonzero elements

h_{k + 1, k_{1}}

, for

k = k_{1}, \dots, k_{2} - 1

. The rotations are applied from the right:

H P^{T} = R,

where

P = P_{k_{1}} P_{k_{1} + 1} \dots P_{k_{2} - 1}

and

P_{k}

is a rotation in the

(k_{1}, k + 1)

plane.

The

2

2

plane rotation part of

P_{k}

has the form

(\begin{array}{r} c_{k} & s_{k} \\ - s_{k} & c_{k} \end{array}) .

Arguments

side

– Character(1)Input

On entry: specifies whether

H

is operated on from the left or the right.

$side ='L'$: $H$ is pre-multiplied from the left.
$side ='R'$: $H$ is post-multiplied from the right.

Constraint:

side ='L'

'R'

n

– IntegerInput

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

k1

– IntegerInput

k2

– IntegerInput

On entry: the values

k_{1}

and

k_{2}

k1 < 1

k2 \leq k1

k2 > n

, an immediate return is effected.

c (k2 - 1)

– Real (Kind=nag_wp) arrayOutput

On exit:

c (k)

holds

c_{k}

, the cosine of the rotation

P_{k}

, for

k = k_{1}, \dots, k_{2} - 1

s (*)

– Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array s must be at least

k2 - 1

On entry: the nonzero elements of the spike of

H

s (k)

must hold

h_{k_{2}, k}

side ='L'

, and

h_{k + 1, k_{1}}

side ='R'

, for

k = k_{1}, \dots, k_{2} - 1

On exit:

s (k)

holds

s_{k}

, the sine of the rotation

P_{k}

, for

k = k_{1}, \dots, k_{2} - 1

a (lda *)

– Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least

n

On entry: the upper triangular part of the

n

n

upper spiked matrix

H

On exit: the upper triangular matrix

R

lda

– IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f06qsf is called.

Constraint:

lda \geq \max (1, n)

NAG Library Routine Document

f06qsf (dusqr)

▸▿ 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

NAG Library Routine Document

f06qsf (dusqr)

▸▿ 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

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