NAG Library Routine Document

F02FKF

Note: this routine uses optional parameters to define choices in the problem specification. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings this must be done by calling the option setting routine F12FDF from the user-supplied subroutine OPTION. Please refer to Section 11 for a detailed description of the specification of the optional parameters.

▸▿ 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

▸▿ 11 Optional Parameters

11.1 Description of the Optional Parameters

1 Purpose

F02FKF computes selected eigenvalues and eigenvectors of a real sparse symmetric matrix.

2 Specification

SUBROUTINE F02FKF (

N, NNZ, A, IROW, ICOL, NEV, NCV, SIGMA, MONIT, OPTION, NCONV, W, V, LDV, RESID, IUSER, RUSER, IFAIL)

INTEGER	N, NNZ, IROW(NNZ), ICOL(NNZ), NEV, NCV, NCONV, LDV, IUSER(*), IFAIL
REAL (KIND=nag_wp)	A(NNZ), SIGMA, W(NCV), V(LDV,), RESID(NEV), RUSER()
EXTERNAL	MONIT, OPTION

3 Description

F02FKF computes selected eigenvalues and the corresponding right eigenvectors of a real sparse symmetric matrix

A

A v_{i} = λ_{i} v_{i} .

A specified number,

n_{e v}

, of eigenvalues

λ_{i}

, or the shifted inverses

μ_{i} = 1 / (λ_{i} - σ)

, may be selected either by largest or smallest modulus, largest or smallest value, or, largest and smallest values (both ends). Convergence is generally faster when selecting larger eigenvalues, smaller eigenvalues can always be selected by choosing a zero inverse shift (

σ = 0.0

). When eigenvalues closest to a given value are required then the shifted inverses of largest magnitude should be selected with shift equal to the required value.

The sparse matrix

A

is stored in symmetric coordinate storage (SCS) format. See Section 2.1.2 in the F11 Chapter Introduction.

F02FKF uses an implicitly restarted Arnoldi (Lanczos) iterative method to converge approximations to a set of required eigenvalues and corresponding eigenvectors. Further algorithmic information is given in Section 9 while a fuller discussion is provided in the F12 Chapter Introduction. If shifts are to be performed then operations using shifted inverse matrices are performed using a direct sparse solver.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia

5 Parameters

1: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N > 0

2: $NNZ$ – INTEGERInput

On entry: the dimension of the array A as declared in the (sub)program from which F02FKF is called.The number of nonzero elements in the lower triangular part of the matrix

A

Constraint:

1 \leq NNZ \leq N \times (N + 1) / 2

3: $A (NNZ)$ – REAL (KIND=nag_wp) arrayInput

On entry: the array of nonzero elements of the lower triangular part of the

n

n

symmetric matrix

A

4: $IROW (NNZ)$ – INTEGER arrayInput

5: $ICOL (NNZ)$ – INTEGER arrayInput

On entry: the row and column indices of the elements supplied in array A.

IROW (k) = i

and

ICOL (k) = j

then

A_{i j}

is stored in

A (k)

. IROW does not need to be ordered, an internal call to F11ZBF forces the correct ordering.

Constraint:

IROW and ICOL must satisfy these constraints:

1 \leq IROW (i) \leq N

and

1 \leq ICOL (i) \leq IROW (i)

, for

i = 1, 2, \dots, NNZ

6: $NEV$ – INTEGERInput

On entry: the number of eigenvalues to be computed.

Constraint:

0 < NEV < N - 1

7: $NCV$ – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which F02FKF is called. The number of Arnoldi basis vectors to use during the computation.

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

NCV \geq 2 \times NEV + 1

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

Constraint:

NEV < NCV \leq N

8: $SIGMA$ – REAL (KIND=nag_wp)Input

On entry: if the Shifted Inverse mode has been selected then SIGMA contains the real shift used; otherwise SIGMA is not referenced. This mode can be selected by setting the appropriate options in the user-supplied subroutine OPTION.

9: $MONIT$ – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

MONIT is used to monitor the progress of F02FKF. MONIT may be the dummy subroutine F02FKZ if no monitoring is actually required. (F02FKZ is included in the NAG Library.) MONIT is called after the solution of each eigenvalue sub-problem and also just prior to return from F02FKF.

The specification of MONIT is:

SUBROUTINE MONIT (

NCV, NITER, NCONV, W, RZEST, ISTAT, IUSER, RUSER)

INTEGER	NCV, NITER, NCONV, ISTAT, IUSER(*)
REAL (KIND=nag_wp)	W(NCV), RZEST(NCV), RUSER(*)

1: $NCV$ – INTEGERInput: On entry: the dimension of the arrays W and RZEST. The number of Arnoldi basis vectors used during the computation.
2: $NITER$ – INTEGERInput: On entry: the number of the current Arnoldi iteration.
3: $NCONV$ – INTEGERInput: On entry: the number of converged eigenvalues so far.
4: $W (NCV)$ – REAL (KIND=nag_wp) arrayInput: On entry: the first NCONV elements of W contain the converged approximate eigenvalues.
5: $RZEST (NCV)$ – REAL (KIND=nag_wp) arrayInput: On entry: the first NCONV elements of RZEST contain the Ritz estimates (error bounds) on the converged approximate eigenvalues.
6: $ISTAT$ – INTEGERInput/Output: On entry: set to zero.

On exit: if set to a nonzero value F02FKF returns immediately with $IFAIL = 9$ .
7: $IUSER (*)$ – INTEGER arrayUser Workspace
8: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace: MONIT is called with the parameters IUSER and RUSER as supplied to F02FKF. You are free to use the arrays IUSER and RUSER to supply information to MONIT as an alternative to using COMMON global variables.

MONIT must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F02FKF is called. Parameters denoted as Input must not be changed by this procedure.

10: $OPTION$ – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

You can supply non-default options to the Arnoldi eigensolver by repeated calls to F12FDF from within OPTION. (Please note that it is only necessary to call F12FDF; no call to F12FAF is required from within OPTION.) For example, you can set the mode to Shifted Inverse, you can increase the Iteration Limit beyond its default and you can print varying levels of detail on the iterative process using Print Level.

If only the default options (including that the eigenvalues of largest magnitude are sought) are to be used then OPTION may be the dummy subroutine F02EKY (F02EKY is included in the NAG Library). See Section 10 for an example of using OPTION to set some non-default options.

The specification of OPTION is:

SUBROUTINE OPTION (

ICOMM, COMM, ISTAT, IUSER, RUSER)

INTEGER	ICOMM(), ISTAT, IUSER()
REAL (KIND=nag_wp)	COMM(), RUSER()

1: $ICOMM (*)$ – INTEGER arrayCommunication Array: On entry: contains details of the default option set. This array must be passed as parameter ICOMM in any call to F12FDF.

On exit: contains data on the current options set which may be altered from the default set via calls to F12FDF.
2: $COMM (*)$ – REAL (KIND=nag_wp) arrayCommunication Array: On entry: contains details of the default option set. This array must be passed as parameter COMM in any call to F12FDF.

On exit: contains data on the current options set which may be altered from the default set via calls to F12FDF.
3: $ISTAT$ – INTEGERInput/Output: On entry: set to zero.

On exit: if set to a nonzero value F02FKF returns immediately with $IFAIL = 10$ .
4: $IUSER (*)$ – INTEGER arrayUser Workspace
5: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace: OPTION is called with the parameters IUSER and RUSER as supplied to F02FKF. You are free to use the arrays IUSER and RUSER to supply information to OPTION as an alternative to using COMMON global variables.

OPTION must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F02FKF is called.

11: $NCONV$ – INTEGEROutput

On exit: the number of converged approximations to the selected eigenvalues. On successful exit, this will normally be NEV.

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

On exit: the first NCONV elements contain the converged approximations to the selected eigenvalues.

13: $V (LDV *)$ – REAL (KIND=nag_wp) arrayOutput

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

NCV

On exit: contains the eigenvectors associated with the eigenvalue

λ_{i}

, for

i = 1, 2, \dots, NCONV

(stored in W). For eigenvalue,

λ_{j}

, the corresponding eigenvector is stored in

V (i, j)

, for

i = 1, 2, \dots, n

14: $LDV$ – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F02FKF is called.

Constraint:

LDV \geq N

15: $RESID (NEV)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the residual

{‖A w_{i} - λ_{i} w_{i}‖}_{2}

for the estimates to the eigenpair

λ_{i}

and

w_{i}

is returned in

RESID (i)

, for

i = 1, 2, \dots, NCONV

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

IUSER is not used by F02FKF, but is passed directly to MONIT and OPTION and may be used to pass information to these routines as an alternative to using COMMON global variables.

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

RUSER is not used by F02FKF, but is passed directly to MONIT and OPTION and may be used to pass information to these routines as an alternative to using COMMON global variables.

18: $IFAIL$ – 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, $N = 〈value〉$ .
Constraint: $N > 0$ .

$IFAIL = 2$: On entry, $NNZ = 〈value〉$ .
Constraint: $NNZ > 0$ .

On entry, $NNZ = 〈value〉$ and $N = 〈value〉$ .
Constraint: $NNZ \leq N \times (N + 1) / 2$ .

$IFAIL = 4$: On entry, for $i = 〈value〉$ , $IROW (i) = 〈value〉$ .
Constraint: $1 \leq IROW (i) \leq N$ .

$IFAIL = 5$: On entry, for $i = 〈value〉$ , $ICOL (i) = 〈value〉$ , $IROW (i) = 〈value〉$ .
Constraint: $1 \leq ICOL (i) \leq IROW (i)$ .

$IFAIL = 6$: On entry, $NEV = 〈value〉$ .
Constraint: $NEV > 0$ .

On entry, $NEV = 〈value〉$ and $N = 〈value〉$ .
Constraint: $NEV < (N - 1)$ .

$IFAIL = 7$: On entry, $NCV = 〈value〉$ and $N = 〈value〉$ .
Constraint: $NCV \leq N$ .

On entry, $NCV = 〈value〉$ and $NEV = 〈value〉$ .
Constraint: $NCV > NEV$ .

$IFAIL = 8$: On entry, the matrix $(A - σ I)$ is numerically singular and could not be inverted. Try perturbing the value of $σ$ .

$IFAIL = 9$: User requested termination in MONIT, $ISTAT = 〈value〉$ .

$IFAIL = 10$: User requested termination in OPTION, $ISTAT = 〈value〉$ .

$IFAIL = 14$: On entry, $LDV = 〈value〉$ and $N = 〈value〉$ .
Constraint: $LDV \geq N$ .

$IFAIL = 20$: The maximum number of iterations, through the optional parameter Iteration Limit, has been set to a non-positive value.

$IFAIL = 21$: The option Both Ends has been set but only $1$ eigenvalue is requested.

$IFAIL = 22$: The maximum number of iterations has been reached.
The maximum number of iterations $= 〈value〉$ .
The number of converged eigenvalues $= 〈value〉$ .
See the routine document for further details.

$IFAIL = 30$: A serious error, code $〈value〉$ , has occurred in an internal call to $〈value〉$ . Check all subroutine calls and array sizes. If the call is correct then please contact NAG for assistance.

$IFAIL = - 99$: 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 relative accuracy of a Ritz value (eigenvalue approximation),

λ

, is considered acceptable if its Ritz estimate

\leq Tolerance \times λ

. The default value for Tolerance is the machine precision given by X02AJF. The Ritz estimates are available via the MONIT subroutine at each iteration in the Arnoldi process, or can be printed by setting option Print Level to a positive value.

8 Parallelism and Performance

F02FKF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F02FKF 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

F02FKF calls routines based on the ARPACK suite in Chapter F12. These routines use an implicitly restarted Lanczos iterative method to converge to approximations to a set of required eigenvalues (see the F12 Chapter Introduction).

In the default Regular mode, only matrix-vector multiplications are performed using the sparse matrix

A

during the Lanczos process; F11XEF can be used to perform this task. Each iteration is therefore cheap computationally, relative to the alternative, Shifted Inverse, mode described below. It is most efficient (i.e., the total number of iterations required is small) when the eigenvalues of largest magnitude are sought and these are distinct.

Although there is an option for returning the smallest eigenvalues using this mode (see Smallest Magnitude option), the number of iterations required for convergence will be far greater or the method may not converge at all. However, where convergence is achieved, Regular mode may still prove to be the most efficient since no inversions are required. Where smallest eigenvalues are sought and Regular mode is not suitable, or eigenvalues close to a given real value are sought, the Shifted Inverse mode should be used.

If the Shifted Inverse mode is used (via a call to F12FDF in OPTION) then the matrix

A - σ I

is used in linear system solves by the Lanczos process. This is first factorized internally using a direct sparse

L D L^{T}

factorization under the assumption that the matrix is indefinite. If the factorization determines that the matrix is numerically singular then the routine exits with an error. In this situation it is normally sufficient to perturb

σ

by a small amount and call F02FKF again. After successful factorization, subsequent solves are performed by backsubstitution using the sparse factorization.

Finally, F02FKF transforms the eigenvectors. Each eigenvector

w

is normalized so that

{‖w‖}_{2} = 1

The monitoring routine MONIT provides some basic information on the convergence of the Lanczos iterations. Much greater levels of detail on the Lanczos process are available via option Print Level. If this is set to a positive value then information will be printed, by default, to standard output. The destination of monitoring information can be changed using the Monitoring option.

10 Example

This example solves

A x = λ x

in Shifted Inverse mode, where

A

is obtained from the standard central difference discretization of the one-dimensional Laplacian operator

\frac{\partial^{2} u}{\partial x^{2}}

[0, 1]

, with zero Dirichlet boundary conditions.

11 Optional Parameters

Internally F02FKF calls routines from the suite F12FAF, F12FBF, F12FCF, F12FDF and F12FEF. Several optional parameters for these computational routines define choices in the problem specification or the algorithm logic. In order to reduce the number of formal parameters of F02FKF these optional parameters are also used here and have associated default values that are usually appropriate. Therefore, you need only specify those optional parameters whose values are to be different from their default values.

Optional parameters may be specified via the user-supplied subroutine OPTION in the call to F02FKF. OPTION must be coded such that one call to F12FDF is necessary to set each optional parameter. All optional parameters you do not specify are set to their default values.

The remainder of this section can be skipped if you wish to use the default values for all optional parameters.

The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 11.1.

11.1 Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

the keywords, where the minimum abbreviation of each keyword is underlined;
a parameter value, where the letters $a$ , $i$ and $r$ denote options that take character, integer and real values respectively;
the default value, where the symbol $ε$ is a generic notation for machine precision (see X02AJF).

Keywords and character values are case and white space insensitive.

Advisory

i

Default

=

the value returned by X04ABF

If the optional parameter List is set then optional parameter specifications are listed in a List file by setting the option to a file identification (unit) number associated with Advisory messages (see X04ABF and X04ACF).

Defaults

This special keyword may be used to reset all optional parameters to their default values.

Iteration Limit

i

Default

= 300

The limit on the number of Lanczos iterations that can be performed before F12FBF exits. If not all requested eigenvalues have converged to within Tolerance and the number of Lanczos iterations has reached this limit then F12FBF exits with an error; F12FCF can still be called subsequently to return the number of converged eigenvalues, the converged eigenvalues and, if requested, the corresponding eigenvectors.

Largest Magnitude

Default

Both Ends

Largest Algebraic

Smallest Algebraic

Smallest Magnitude

The Lanczos iterative method converges on a number of eigenvalues with given properties. The default is for F12FBF to compute the eigenvalues of largest magnitude using Largest Magnitude. Alternatively, eigenvalues may be chosen which have Largest Algebraic part, Smallest Magnitude, or Smallest Algebraic part; or eigenvalues which are from Both Ends of the algebraic spectrum.

Nolist

Default

List

Normally each optional parameter specification is not listed as it is supplied. This behaviour can be changed using the List and Nolist options.

Monitoring

i

Default

= - 1

i > 0

, monitoring information is output to channel number

i

during the solution of each problem; this may be the same as the Advisory channel number. The type of information produced is dependent on the value of Print Level, see the description of the optional parameter Print Level for details of the information produced. Please see X04ACF to associate a file with a given channel number.

Print Level

i

Default

= 0

This controls the amount of printing produced by F02FKF as follows.

$= 0$	No output except error messages. If you want to suppress all output, set $Print Level = 0$ .
$> 0$	The set of selected options.
$= 2$	Problem and timing statistics on final exit from F12FBF.
$\geq 5$	A single line of summary output at each Lanczos iteration.
$\geq 10$	If Monitoring is set, then at each iteration, the length and additional steps of the current Lanczos factorization and the number of converged Ritz values; during re-orthogonalization, the norm of initial/restarted starting vector; on a final Lanczos iteration, the number of update iterations taken, the number of converged eigenvalues, the converged eigenvalues and their Ritz estimates.
$\geq 20$	Problem and timing statistics on final exit from F12FBF. If $Monitoring > 0$ , Monitoring is set, then at each iteration, the number of shifts being applied, the eigenvalues and estimates of the symmetric tridiagonal matrix $H$ , the size of the Lanczos basis, the wanted Ritz values and associated Ritz estimates and the shifts applied; vector norms prior to and following re-orthogonalization.
$\geq 30$	If $Monitoring > 0$ , Monitoring is set, then on final iteration, the norm of the residual; when computing the Schur form, the eigenvalues and Ritz estimates both before and after sorting; for each iteration, the norm of residual for compressed factorization and the symmetric tridiagonal matrix $H$ ; during re-orthogonalization, the initial/restarted starting vector; during the Lanczos iteration loop, a restart is flagged and the number of the residual requiring iterative refinement; while applying shifts, some indices.
$\geq 40$	If $Monitoring > 0$ , Monitoring is set, then during the Lanczos iteration loop, the Lanczos vector number and norm of the current residual; while applying shifts, key measures of progress and the order of $H$ ; while computing eigenvalues of $H$ , the last rows of the Schur and eigenvector matrices; when computing implicit shifts, the eigenvalues and Ritz estimates of $H$ .
$\geq 50$	If Monitoring is set, then during Lanczos iteration loop: norms of key components and the active column of $H$ , norms of residuals during iterative refinement, the final symmetric tridiagonal matrix $H$ ; while applying shifts: number of shifts, shift values, block indices, updated tridiagonal matrix $H$ ; while computing eigenvalues of $H$ : the diagonals of $H$ , the computed eigenvalues and Ritz estimates.

Note that setting

Print Level \geq 30

can result in very lengthy Monitoring output.

Regular

Default

Regular Inverse

Shifted Inverse

These options define the computational mode which in turn defines the form of operation

OP (x)

to be performed.

Regular	$OP = A$
Shifted Inverse	$OP = {(A - σ I)}^{- 1}$ where $σ$ is real
Regular Inverse	$OP = A^{- 1}$

Tolerance

r

Default

= ε

An approximate eigenvalue has deemed to have converged when the corresponding Ritz estimate is within Tolerance relative to the magnitude of the eigenvalue.

F02FKF (PDF version)

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F02 Chapter Introduction

NAG Library Manual