NAG Library Routine Document

F02WGF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     M – INTEGERInput

On entry: $m$, the number of rows of the matrix $A$.

Constraint: $\mathbf{M}\ge 0$.

If $\mathbf{M}=0$, an immediate return is effected.

2:     N – INTEGERInput

On entry: $n$, the number of columns of the matrix $A$.

Constraint: $\mathbf{N}\ge 0$.

If $\mathbf{N}=0$, an immediate return is effected.

3:     K – INTEGERInput

On entry: $k$, the number of singular values to be computed.

Constraint: $0<\mathbf{K}<\min \left(\mathbf{M},\mathbf{N}\right)-1$.

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 $A^{\mathrm{T}}A$ ($m\ge n$) or $A{A}^{\mathrm{T}}$ ($m<n$).

At present there is no a priori analysis to guide the selection of NCV relative to K. However, it is recommended that $\mathbf{NCV}\ge 2\times \mathbf{K}+1$. 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: $\mathbf{K}<\mathbf{NCV}\le \min \left(\mathbf{M},\mathbf{N}\right)$.

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

AV must return the vector result of the matrix-vector product $Ax$ or $A^{\mathrm{T}}x$, 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 $\mathbf{IFLAG}=1$, AX must return the $m$-vector result of the matrix-vector product $Ax$.

If $\mathbf{IFLAG}=2$, AX must return the $n$-vector result of the matrix-vector product $A^{\mathrm{T}}x$.

On exit: may be used as a flag to indicate a failure in the computation of $Ax$ or $A^{\mathrm{T}}x$. 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 $A^{\mathrm{T}}$.

5:     AX($*$) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{IFLAG}=1$, contains the $m$-vector result of the matrix-vector product $Ax$.

If $\mathbf{IFLAG}=2$, contains the $n$-vector result of the matrix-vector product $A^{\mathrm{T}}x$.

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 $\mathit{i}$th element of the $\mathit{j}$th left singular vector $u_{\mathit{j}}$ is stored in $\mathbf{U}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,\mathbf{NCONV}$.

9:     LDU – INTEGERInput

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

Constraint: $\mathbf{LDU}\ge \max \left(1,\mathbf{M}\right)$.

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

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

The $\mathit{i}$th element of the $\mathit{j}$th right singular vector $v_{\mathit{j}}$ is stored in $\mathbf{V}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,\mathbf{NCONV}$.

11:   LDV – INTEGERInput

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

Constraint: $\mathbf{LDV}\ge \max \left(1,\mathbf{N}\right)$.

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

On exit: the residual $‖A{v}_j-{\sigma }_j{u}_j‖$, for $m\ge n$, or $‖A^{\mathrm{T}}{u}_j-{\sigma }_j{v}_j‖$, for $m<n$, for each of the converged singular values ${\sigma }_j$ 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 $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ 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 $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

F02WGF returns with $\mathbf{IFAIL}=\mathbf{0}$ if at least $k$ singular values have converged and the corresponding left and right singular vectors have been computed.

6  Error Indicators and Warnings

$\mathbf{IFAIL}<0$

IFLAG was set to a negative value following a call to AV.

$\mathbf{IFAIL}=1$

On entry,

$\mathbf{M}<0$.

$\mathbf{IFAIL}=2$

On entry,

$\mathbf{N}<0$.

$\mathbf{IFAIL}=3$

On entry,

$\mathbf{K}\le 0$.

$\mathbf{IFAIL}=4$

On entry,	$\mathbf{NCV}\le \mathbf{K}+1$,
or	$\mathbf{NCV}>\min \left(\mathbf{M},\mathbf{N}\right)$.

$\mathbf{IFAIL}=5$

On entry,

$\mathbf{LDU}<\mathbf{M}$.

$\mathbf{IFAIL}=6$

On entry,

$\mathbf{LDV}<\mathbf{N}$.

$\mathbf{IFAIL}=7$

No converged singular values were found to sufficient accuracy.

$\mathbf{IFAIL}=8$

The internal maximum number of iterations has been reached. Some singular values may have converged.

$\mathbf{IFAIL}=9$

$\mathbf{IFAIL}=10$

$\mathbf{IFAIL}=20$

An error occurred during an internal call. Consider increasing the size of NCV relative to K.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f02wgfe.f90)

9.2  Program Data

Program Data (f02wgfe.d)

9.3  Program Results

Program Results (f02wgfe.r)