NAG Library Routine Document

F02WUF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F02WUF returns all, or part, of the singular value decomposition of a real upper triangular matrix.

2 Specification

SUBROUTINE F02WUF (

N, A, LDA, NCOLB, B, LDB, WANTQ, Q, LDQ, SV, WANTP, WORK, IFAIL)

INTEGER	N, LDA, NCOLB, LDB, LDQ, IFAIL
REAL (KIND=nag_wp)	A(LDA,), B(LDB,), Q(LDQ,), SV(N), WORK()
LOGICAL	WANTQ, WANTP

3 Description

The

n

n

upper triangular matrix

R

is factorized as

R = Q S P^{T},

where

Q

and

P

are

n

n

orthogonal matrices and

S

is an

n

n

diagonal matrix with non-negative diagonal elements,

σ_{1}, σ_{2}, \dots, σ_{n}

, ordered such that

σ_{1} \geq σ_{2} \geq \dots \geq σ_{n} \geq 0 .

The columns of

Q

are the left-hand singular vectors of

R

, the diagonal elements of

S

are the singular values of

R

and the columns of

P

are the right-hand singular vectors of

R

Either or both of

Q

and

P^{T}

may be requested and the matrix

C

given by

C = Q^{T} B,

where

B

is an

n

ncolb

given matrix, may also be requested.

The routine obtains the singular value decomposition by first reducing

R

to bidiagonal form by means of Givens plane rotations and then using the

Q R

algorithm to obtain the singular value decomposition of the bidiagonal form.

Good background descriptions to the singular value decomposition are given in Chan (1982), Dongarra et al. (1979), Golub and Van Loan (1996), Hammarling (1985) and Wilkinson (1978).

Note that if

K

is any orthogonal diagonal matrix so that

K K^{T} = I

(that is the diagonal elements of

K

are

+ 1

- 1

) then

A = (Q K) S {(P K)}^{T}

is also a singular value decomposition of

A

4 References

Chan T F (1982) An improved algorithm for computing the singular value decomposition ACM Trans. Math. Software 8 72–83

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

Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25

Wilkinson J H (1978) Singular Value Decomposition – Basic Aspects Numerical Software – Needs and Availability (ed D A H Jacobs) Academic Press

5 Parameters

1: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

R

N = 0

, an immediate return is effected.

Constraint:

N \geq 0

2: $A (LDA *)$ – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the leading

n

n

upper triangular part of the array A must contain the upper triangular matrix

R

On exit: if

WANTP = .TRUE.

, the

n

n

part of A will contain the

n

n

orthogonal matrix

P^{T}

, otherwise the

n

n

upper triangular part of A is used as internal workspace, but the strictly lower triangular part of A is not referenced.

3: $LDA$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F02WUF is called.

Constraint:

LDA \geq \max (1, N)

4: $NCOLB$ – INTEGERInput

On entry:

ncolb

, the number of columns of the matrix

B

NCOLB = 0

, the array B is not referenced.

Constraint:

NCOLB \geq 0

5: $B (LDB *)$ – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, NCOLB)

On entry: with

NCOLB > 0

, the leading

n

ncolb

part of the array B must contain the matrix to be transformed.

On exit: the leading

n

ncolb

part of the array B is overwritten by the matrix

Q^{T} B

6: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F02WUF is called.

Constraints:

if $NCOLB > 0$ , $LDB \geq \max (1, N)$ ;
otherwise $LDB \geq 1$ .

7: $WANTQ$ – LOGICALInput

On entry: must be .TRUE. if the matrix

Q

is required.

WANTQ = .FALSE.

, the array Q is not referenced.

8: $Q (LDQ *)$ – REAL (KIND=nag_wp) arrayOutput

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

\max (1, N)

WANTQ = .TRUE.

, and at least

1

otherwise.

On exit: with

WANTQ = .TRUE.

, the leading

n

n

part of the array Q will contain the orthogonal matrix

Q

. Otherwise the array Q is not referenced.

9: $LDQ$ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F02WUF is called.

Constraints:

if $WANTQ = .TRUE.$ , $LDQ \geq \max (1, N)$ ;
otherwise $LDQ \geq 1$ .

10: $SV (N)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the array SV will contain the

n

diagonal elements of the matrix

S

11: $WANTP$ – LOGICALInput

On entry: must be .TRUE. if the matrix

P^{T}

is required, in which case

P^{T}

is overwritten on the array A, otherwise WANTP must be .FALSE..

12: $WORK (*)$ – REAL (KIND=nag_wp) arrayOutput

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

\max (1, 2 \times (N - 1))

NCOLB = 0

and

WANTQ = .FALSE.

and

WANTP = .FALSE.

\max (1, 3 \times (N - 1))

if (

NCOLB = 0

and

WANTQ = .FALSE.

and

WANTP = .TRUE.

) or (

WANTP = .FALSE.

and (

NCOLB > 0

WANTQ = .TRUE.

)), and at least

\max (1, 5 \times (N - 1))

otherwise.

On exit:

WORK (N)

contains the total number of iterations taken by the

Q R

algorithm.

The rest of the array is used as internal workspace.

13: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = - 1$

On entry,	$N < 0$ ,
or	$LDA < N$ ,
or	$NCOLB < 0$ ,
or	$LDB < N$ and $NCOLB > 0$ ,
or	$LDQ < N$ and $WANTQ = .TRUE.$ .

$IFAIL > 0$: The $Q R$ algorithm has failed to converge in $50 \times N$ iterations. In this case $SV (1), SV (2), \dots, SV (IFAIL)$ may not have been found correctly and the remaining singular values may not be the smallest. The matrix $R$ will nevertheless have been factorized as $R = Q E P^{T}$ , where $E$ is a bidiagonal matrix with $SV (1), SV (2), \dots, SV (n)$ as the diagonal elements and $WORK (1), WORK (2), \dots, WORK (n - 1)$ as the superdiagonal elements.

This failure is not likely to occur.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The computed factors

Q

S

and

P

satisfy the relation

Q S P^{T} = R + E,

where

‖E‖ \leq c ε ‖A‖,

ε

is the machine precision,

c

is a modest function of

n

and

‖.‖

denotes the spectral (two) norm. Note that

‖A‖ = SV (1)

A similar result holds for the computed matrix

Q^{T} B

The computed matrix

Q

satisfies the relation

Q = T + F,

where

T

is exactly orthogonal and

‖F‖ \leq d ε,

where

d

is a modest function of

n

. A similar result holds for

P

8 Parallelism and Performance

F02WUF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F02WUF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

For given values of NCOLB, WANTQ and WANTP, the number of floating-point operations required is approximately proportional to

n^{3}

Following the use of this routine the rank of

R

may be estimated by a call to the INTEGER FUNCTION F06KLF. The statement

IRANK = F06KLF(N,SV,1,TOL)

returns the value

(k - 1)

in IRANK, where

k

is the smallest integer for which

SV (k) < tol \times SV (1)

, and

tol

is the tolerance supplied in TOL, so that IRANK is an estimate of the rank of

S

and thus also of

R

. If TOL is supplied as negative then the machine precision is used in place of TOL .

10 Example

This example finds the singular value decomposition of the

3

3

upper triangular matrix

A = (\begin{array}{r} - 4 & - 2 & - 3 \\ 0 & - 3 & - 2 \\ 0 & 0 & - 4 \end{array}),

together with the vector

Q^{T} b

for the vector

b = (\begin{array}{r} - 1 \\ - 1 \\ - 1 \end{array}) .

NAG Library Routine DocumentF02WUF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

F02WUF