Integer, Intent (In)	::	n, lda, ncolb, ldb, ldq
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	rwork(*)
Real (Kind=nag_wp), Intent (Out)	::	sv(n)
Complex (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,), q(ldq,*)
Complex (Kind=nag_wp), Intent (Out)	::	cwork(max(1,n-1))
Logical, Intent (In)	::	wantq, wantp

C Header Interface

#include nagmk26.h

void	f02xuf_ ( const Integer n, Complex a[], const Integer lda, const Integer ncolb, Complex b[], const Integer ldb, const logical wantq, Complex q[], const Integer ldq, double sv[], const logical wantp, double rwork[], Complex cwork[], Integer ifail)

3

Description

The

n

n

upper triangular matrix

R

is factorized as

R = Q S P^{H},

where

Q

and

P

are

n

n

unitary matrices and

S

is an

n

n

diagonal matrix with real non-negative diagonal elements,

s v_{1}, s v_{2}, \dots, s v_{n}

, ordered such that

s v_{1} \geq s v_{2} \geq \dots \geq s v_{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^{H}

may be requested and the matrix

C

given by

C = Q^{H} B,

where

B

is an

n

ncolb

given matrix, may also be requested.

f02xuf 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 Dongarra et al. (1979), Hammarling (1985) and Wilkinson (1978).

Note that if

K

is any unitary diagonal matrix so that

K K^{H} = I,

then

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

is also a singular value decomposition of

A

4

References

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

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

Arguments

1: $n$ – IntegerInput

On entry:

n

, the order of the matrix

R

n = 0

, an immediate return is effected.

Constraint:

n \geq 0

2: $a (lda *)$ – Complex (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

unitary matrix

P^{H}

, 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 f02xuf 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: $b (ldb *)$ – Complex (Kind=nag_wp) arrayInput/Output

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

\max (1, ncolb)

On entry: if

ncolb > 0

, the leading

n

ncolb

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

On exit: is overwritten by the

n

ncolb

matrix

Q^{H} B

6: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f02xuf 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.

then the array q is not referenced.

8: $q (ldq *)$ – Complex (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: if

wantq = .TRUE.

, the leading

n

n

part of the array q will contain the unitary 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 f02xuf 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

n

diagonal elements of the matrix

S

11: $wantp$ – LogicalInput

On entry: must be .TRUE. if the matrix

P^{H}

is required, in which case

P^{H}

is returned in the array a, otherwise wantp must be .FALSE..

12: $rwork (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array rwork 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 (

ncolb > 0

and

wantp = .FALSE.

) or (

wantq = .TRUE.

and

wantp = .FALSE.

), and at least

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

otherwise.

On exit: rwork(n) contains the total number of iterations taken by the

Q R

algorithm.

The rest of the array is used as workspace.

13: $cwork (\max (1, n - 1))$ – Complex (Kind=nag_wp) arrayWorkspace

14: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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 $R = Q E P^{H}$ , where $E$ is a bidiagonal matrix with $sv (1), sv (2), \dots, sv (n)$ as the diagonal elements and $rwork (1), rwork (2), \dots, rwork (n - 1)$ 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 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computed factors

Q

S

and

P

satisfy the relation

Q S P^{H} = A + 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‖ = s v_{1}

8

Parallelism and Performance

f02xuf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f02xuf 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

s v (k) < tol \times s v (1)

, where

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} 1 & 1 + i & 1 + i \\ 0 & - 2 & - 1 - i \\ 0 & 0 & - 3 \end{array})

together with the vector

Q^{H} b

for the vector

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

NAG Library Routine Document

f02xuf (complex_triang_svd)

▸▿ 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

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