F02WGF (PDF version)

F02 Chapter Contents

F02 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

F02WGF

▸▿ 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

F02WGF returns leading terms in the singular value decomposition (SVD) of a real general matrix and computes the corresponding left and right singular vectors.

2 Specification

SUBROUTINE F02WGF (

M, N, K, NCV, AV, NCONV, SIGMA, U, LDU, V, LDV, RESID, IUSER, RUSER, IFAIL)

INTEGER	M, N, K, NCV, NCONV, LDU, LDV, IUSER(*), IFAIL
REAL (KIND=nag_wp)	SIGMA(NCV), U(LDU,NCV), V(LDV,NCV), RESID(NCV), RUSER(*)
EXTERNAL	AV

3 Description

F02WGF computes a few,

k

, of the largest singular values and corresponding vectors of an

m

by

n

matrix

A

. The value of

k

should be small relative to

m

and

n

, for example

k \sim O (\min (m, n))

. The full singular value decomposition (SVD) of an

m

by

n

matrix

A

is given by

A = U Σ V^{T},

where

U

and

V

are orthogonal and

Σ

is an

m

by

n

diagonal matrix with real diagonal elements,

σ_{i}

, such that

σ_{1} \geq σ_{2} \geq \dots \geq σ_{\min (m, n)} \geq 0 .

The

σ_{i}

are the singular values of

A

and the first

\min (m, n)

columns of

U

and

V

are the left and right singular vectors of

A

.

If

U_{k}

,

V_{k}

denote the leading

k

columns of

U

and

V

respectively, and if

Σ_{k}

denotes the leading principal submatrix of

Σ

, then

A_{k} \equiv U_{k} Σ_{k} {V^{T}}_{k}

is the best rank-

k

approximation to

A

in both the

2

-norm and the Frobenius norm.

The singular values and singular vectors satisfy

A v_{i} = σ_{i} u_{i} and A^{T} u_{i} = σ_{i} v_{i} so that A^{T} A ν_{i} = σ_{i}^{2} ν_{i} ​ and ​ A A^{T} u_{i} = σ_{i}^{2} u_{i},

where

u_{i}

and

v_{i}

are the

i

th columns of

U_{k}

and

V_{k}

respectively.

Thus, for

m \geq n

, the largest singular values and corresponding right singular vectors are computed by finding eigenvalues and eigenvectors for the symmetric matrix

A^{T} A

. For

m < n

, the largest singular values and corresponding left singular vectors are computed by finding eigenvalues and eigenvectors for the symmetric matrix

A A^{T}

. These eigenvalues and eigenvectors are found using routines from Chapter F12. You should read the F12 Chapter Introduction for full details of the method used here.

The real matrix

A

is not explicitly supplied to F02WGF. Instead, you are required to supply a routine, AV, that must calculate one of the requested matrix-vector products

A x

or

A^{T} x

for a given real vector

x

(of length

n

or

m

respectively).

4 References

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

5 Parameters

1: $M$ – INTEGERInput

On entry:

m

, the number of rows of the matrix

A

.

Constraint:

M \geq 0

.

If

M = 0

, an immediate return is effected.

2: $N$ – INTEGERInput

On entry:

n

, the number of columns of the matrix

A

.

Constraint:

N \geq 0

.

If

N = 0

, an immediate return is effected.

3: $K$ – INTEGERInput

On entry:

k

, the number of singular values to be computed.

Constraint:

0 < K < \min (M, N) - 1

.

4: $NCV$ – INTEGERInput

On entry: the dimension of the arrays SIGMA and RESID and the second dimension of the arrays U and V as declared in the (sub)program from which F02WGF is called. This is the number of Lanczos basis vectors to use during the computation of the largest eigenvalues of

A^{T} A

(

m \geq n

) or

A A^{T}

(

m < n

).

At present there is no a priori analysis to guide the selection of NCV relative to K. However, it is recommended that

NCV \geq 2 \times K + 1

. If many problems of the same type are to be solved, you should experiment with varying NCV while keeping K fixed for a given test problem. This will usually decrease the required number of matrix-vector operations but it also increases the internal storage required to maintain the orthogonal basis vectors. The optimal ‘cross-over’ with respect to CPU time is problem dependent and must be determined empirically.

Constraint:

K < NCV \leq \min (M, N)

.

5: $AV$ – SUBROUTINE, supplied by the user.External Procedure

AV must return the vector result of the matrix-vector product

A x

or

A^{T} x

, as indicated by the input value of IFLAG, for the given vector

x

.

AV is called from F02WGF with the parameter IUSER and RUSER as supplied to F02WGF. You are free to use these arrays to supply information to AV.

The specification of AV is:

SUBROUTINE AV (

IFLAG, M, N, X, AX, IUSER, RUSER)

INTEGER	IFLAG, M, N, IUSER(*)
REAL (KIND=nag_wp)	X(), AX(), RUSER(*)

1: $IFLAG$ – INTEGERInput/Output: On entry: if $IFLAG = 1$ , AX must return the $m$ -vector result of the matrix-vector product $A x$ .
If $IFLAG = 2$ , AX must return the $n$ -vector result of the matrix-vector product $A^{T} x$ .

On exit: may be used as a flag to indicate a failure in the computation of $A x$ or $A^{T} x$ . If IFLAG is negative on exit from AV, F02WGF will exit immediately with IFAIL set to IFLAG.
2: $M$ – INTEGERInput: On entry: the number of rows of the matrix $A$ .
3: $N$ – INTEGERInput: On entry: the number of columns of the matrix $A$ .
4: $X (*)$ – REAL (KIND=nag_wp) arrayInput: On entry: the vector to be pre-multiplied by the matrix $A$ or $A^{T}$ .
5: $AX (*)$ – REAL (KIND=nag_wp) arrayOutput: On exit: if $IFLAG = 1$ , contains the $m$ -vector result of the matrix-vector product $A x$ .
If $IFLAG = 2$ , contains the $n$ -vector result of the matrix-vector product $A^{T} x$ .
6: $IUSER (*)$ – INTEGER arrayUser Workspace
7: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace: AV is called with the parameters IUSER and RUSER as supplied to F02WGF. You are free to use the arrays IUSER and RUSER to supply information to AV as an alternative to using COMMON global variables.

AV must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F02WGF is called. Parameters denoted as Input must not be changed by this procedure.

6: $NCONV$ – INTEGEROutput

On exit: the number of converged singular values found.

7: $SIGMA (NCV)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the NCONV converged singular values are stored in the first NCONV elements of SIGMA.

8: $U (LDU, NCV)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the left singular vectors corresponding to the singular values stored in SIGMA.

The

i

th element of the

j

th left singular vector

u_{j}

is stored in

U (i, j)

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, NCONV

.

9: $LDU$ – INTEGERInput

On entry: the first dimension of the array U as declared in the (sub)program from which F02WGF is called.

Constraint:

LDU \geq \max (1, M)

.

10: $V (LDV, NCV)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the right singular vectors corresponding to the singular values stored in SIGMA.

The

i

th element of the

j

th right singular vector

v_{j}

is stored in

V (i, j)

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, NCONV

.

11: $LDV$ – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F02WGF is called.

Constraint:

LDV \geq \max (1, N)

.

12: $RESID (NCV)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the residual

‖A v_{j} - σ_{j} u_{j}‖

, for

m \geq n

, or

‖A^{T} u_{j} - σ_{j} v_{j}‖

, for

m < n

, for each of the converged singular values

σ_{j}

and corresponding left and right singular vectors

u_{j}

and

v_{j}

.

13: $IUSER (*)$ – INTEGER arrayUser Workspace

14: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by F02WGF, but are passed directly to AV and may be used to pass information to this routine as an alternative to using COMMON global variables.

15: $IFAIL$ – 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).

F02WGF returns with

IFAIL = 0

if at least

k

singular values have converged and the corresponding left and right singular vectors have been computed.

6 Error Indicators and Warnings

If on entry

IFAIL = 0

or

- 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, $M = 〈value〉$ .
Constraint: $M \geq 0$ .

$IFAIL = 2$: On entry, $N = 〈value〉$ .
Constraint: $N \geq 0$ .

$IFAIL = 3$: On entry, $K = 〈value〉$ .
Constraint: $K > 0$ .

$IFAIL = 4$: On entry, $K = 〈value〉$ , $NCV = 〈value〉$ , $M = 〈value〉$ and $N = 〈value〉$ .
Constraint: $K < NCV \leq \min (M, N)$ .

$IFAIL = 5$: On entry, $LDU = 〈value〉$ and $M = 〈value〉$ .
Constraint: $LDU \geq M$ .

$IFAIL = 6$: On entry, $LDV = 〈value〉$ and $N = 〈value〉$ .
Constraint: $LDV \geq N$ .

$IFAIL = 8$: The maximum number of iterations has been reached. The maximum number of iterations $= 〈value〉$ . The number of converged eigenvalues $= 〈value〉$ .

$IFAIL = 9$: No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.

$IFAIL = 10$: Could not build a full Lanczos factorization.

$IFAIL = 11$: The number of eigenvalues found to sufficient accuracy is zero.

$IFAIL = 20$: An error occurred during an internal call. Consider increasing the size of NCV relative to K.

$IFAIL < 0$: On output from user-defined routine AV, IFLAG was set to a negative value, $IFLAG = 〈value〉$ .

$IFAIL = - 99$: 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

See Section 2.14.2 in the F08 Chapter Introduction.

8 Parallelism and Performance

F02WGF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F02WGF 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

None.

10 Example

This example finds the four largest singular values (

σ

) and corresponding right and left singular vectors for the matrix

A

, where

A

is the

m

by

n

real matrix derived from the simplest finite difference discretization of the two-dimensional kernal

k (s, t) d t

where

k (s, t) = \{\begin{matrix} s (t - 1) & if ​ 0 \leq s \leq t \leq 1 \\ t (s - 1) & if ​ 0 \leq t < s \leq 1 \end{matrix} .

10.1 Program Text

Program Text (f02wgfe.f90)

10.2 Program Data

Program Data (f02wgfe.d)

10.3 Program Results

Program Results (f02wgfe.r)

F02WGF (PDF version)

F02 Chapter Contents

F02 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015