F06TPF : NAG Library, Mark 24

NAG Library Routine Document

F06TPF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

1 Purpose

F06TPF performs a

Q R

factorization (as a sequence of plane rotations) of a complex upper triangular matrix that has been modified by a rank-1 update.

2 Specification

3 Description

F06TPF performs a

Q R

factorization of an upper triangular matrix which has been modified by a rank-1 update:

α x y^{T} + U = Q R

where

U

and

R

are

n

n

complex upper triangular matrices with real diagonal elements,

x

and

y

are

n

-element complex vectors,

α

is a complex scalar, and

Q

is an

n

n

complex unitary matrix.

Q

is formed as the product of two sequences of plane rotations and a unitary diagonal matrix

D

Q^{H} = D Q_{n - 1} \dots Q_{2} Q_{1} P_{1} P_{2} \dots P_{n - 1}

where

$P_{k}$ is a rotation in the $(k, n)$ plane, chosen to annihilate $x_{k}$ : thus $P x = β e_{n}$ , where $P = P_{1} P_{2} \dots P_{n - 1}$ and $e_{n}$ is the last column of the unit matrix;
$Q_{k}$ is a rotation in the $(k, n)$ plane, chosen to annihilate the $(n, k)$ element of $(α β e_{n} y^{T} + P U)$ , and thus restore it to upper triangular form;
$D = diag (1, \dots, 1, d_{n})$ , with $d_{n}$ chosen to make $r_{n n}$ real; $|d_{n}| = 1$ .

The

2

2

plane rotation part of

P_{k}

Q_{k}

has the form

(\begin{array}{r} c_{k} & {\bar{s}}_{k} \\ - s_{k} & c_{k} \end{array})

with

c_{k}

real. The tangents of the rotations

P_{k}

are returned in the array X; the cosines and sines of these rotations can be recovered by calling F06BCF. The cosines and sines of the rotations

Q_{k}

are returned directly in the arrays C and S.

5 Parameters

1: N – INTEGERInput

On entry:

n

, the order of the matrices

U

and

R

Constraint:

N \geq 0

2: ALPHA – COMPLEX (KIND=nag_wp)Input

On entry: the scalar

α

3: X(

*

) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, 1 + (N - 1) \times INCX)

On entry: the

n

-element vector

x

x_{i}

must be stored in

X (1 + (i - 1) \times INCX)

, for

i = 1, 2, \dots, N

Intermediate elements of X are not referenced.

On exit: the referenced elements are overwritten by details of the sequence of plane rotations.

4: INCX – INTEGERInput

On entry: the increment in the subscripts of X between successive elements of

x

Constraint:

INCX > 0

5: Y(

*

) – COMPLEX (KIND=nag_wp) arrayInput

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

\max (1, 1 + (N - 1) \times INCY)

On entry: the

n

-element vector

y

y_{i}

must be stored in

Y (1 + (i - 1) \times INCY)

, for

i = 1, 2, \dots, N

Intermediate elements of Y are not referenced.

6: INCY – INTEGERInput

On entry: the increment in the subscripts of Y between successive elements of

y

Constraint:

INCY > 0

7: A(LDA,

*

) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

N

On entry: the

n

n

upper triangular matrix

U

. The imaginary parts of the diagonal elements must be zero.

On exit: the upper triangular matrix

R

. The imaginary parts of the diagonal elements must be zero.

8: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F06TPF is called.

Constraint:

LDA \geq \max (1, N)

9: C(

N - 1

) – REAL (KIND=nag_wp) arrayOutput

On exit: the cosines of the rotations

Q_{k}

, for

k = 1, 2, \dots, n - 1

10: S(N) – COMPLEX (KIND=nag_wp) arrayOutput

On exit: the sines of the rotations

Q_{k}

, for

k = 1, 2, \dots, n - 1

;

S (n)

holds

d_{n}

, the

n

th diagonal element of

D

NAG Library Routine Document

F06TPF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine DocumentF06TPF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine Document

F06TPF