NAG Library Routine Document

F02EKF

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 9 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 F12ADF from the user-supplied subroutine OPTION. Please refer to Section 10 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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

+− 10 Optional Parameters

10.1 Description of the Optional Parameters

1 Purpose

F02EKF computes selected eigenvalues and eigenvectors of a real sparse general matrix.

2 Specification

SUBROUTINE F02EKF (

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

INTEGER	N, NNZ, ICOLZP(N+1), IROWIX(NNZ), NEV, NCV, NCONV, LDV, IUSER(*), IFAIL
REAL (KIND=nag_wp)	A(NNZ), SIGMA, V(LDV,), RESID(NEV+1), RUSER()
COMPLEX (KIND=nag_wp)	W(NCV)
EXTERNAL	MONIT, OPTION

3 Description

F02EKF computes selected eigenvalues and the corresponding right eigenvectors of a real sparse general matrix

A

A w_{i} = λ_{i} w_{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 real part, or, largest or smallest imaginary part. 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 real value are required then the shifted inverses of largest magnitude should be selected with shift equal to the required real value.

Note that even though

A

is real,

λ_{i}

and

w_{i}

may be complex. If

w_{i}

is an eigenvector corresponding to a complex eigenvalue

λ_{i}

, then the complex conjugate vector

{\bar{w}}_{i}

is the eigenvector corresponding to the complex conjugate eigenvalue

{\bar{λ}}_{i}

. The eigenvalues in a complex conjugate pair

λ_{i}

and

{\bar{λ}}_{i}

are either both selected or both not selected.

The sparse matrix

A

is stored in compressed column storage (CCS) format. See Section 2.1.3 in the F11 Chapter Introduction.

F02EKF uses an implicitly restarted Arnoldi iterative method to converge approximations to a set of required eigenvalues and corresponding eigenvectors. Further algorithmic information is given in Section 8 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; further information on the solver used is provided in the F11 Chapter Introduction.

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 \geq 0

2: NNZ – INTEGERInput

On entry: the dimension of the array A as declared in the (sub)program from which F02EKF is called. The number of nonzero elements of the matrix

A

and, if a nonzero shifted inverse is to be applied, all diagonal elements. A nonzero diagonal element is counted once in the latter case.

Constraint:

0 \leq NNZ \leq N^{2}

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

On entry: the array of nonzero elements (and diagonal elements if a nonzero inverse shift is to be applied) of the

n

n

general matrix

A

On exit: if a nonzero shifted inverse is to be applied then the diagonal elements of

A

have the shift value, as supplied in SIGMA, subtracted.

4: ICOLZP( $N + 1$ ) – INTEGER arrayInput

On entry:

ICOLZP (i)

contains the index in A of the start of column

i

, for

i = 1, 2, \dots, n

;

ICOLZP (N + 1)

must contain the value

NNZ + 1

. Thus the number of nonzero elements in column

i

A

ICOLZP (i + 1) - ICOLZP (i)

; when shifts are applied this includes diagonal elements irrespective of value. See Section 2.1.3 in the F11 Chapter Introduction.

5: IROWIX( $NNZ$ ) – INTEGER arrayInput

On entry:

IROWIX (i)

contains the row index for each entry in A. See Section 2.1.3 in the F11 Chapter Introduction.

6: NEV – INTEGERInput

On entry: the dimension of the array RESID as declared in the (sub)program from which F02EKF is called. 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 F02EKF 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 CPU time is problem dependent and must be determined empirically.

Constraint:

NEV + 1 < NCV \leq N

8: SIGMA – REAL (KIND=nag_wp)Input

On entry: if the Shifted Inverse Real 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 F02EKF. MONIT may be the dummy subroutine F02EKZ if no monitoring is actually required. (F02EKZ is included in the NAG Library.) MONIT is called after the solution of each eigenvalue sub-problem and also just prior to return from F02EKF.

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)	RZEST(NCV), RUSER(*)
COMPLEX (KIND=nag_wp)	W(NCV)

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) – COMPLEX (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 F02EKF 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 F02EKF. 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 F02EKF 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 F12ADF from within OPTION. For example, you can set the mode to Shifted Inverse Real, 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 9 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 F12ADF.

On exit: contains data on the current options set which may be altered from the default set via calls to F12ADF.
2: COMM( $*$ ) – REAL (KIND=nag_wp) arrayCommunication Array: On entry: contains details of the default option set. This array must be passed as parameter ICOMM in any call to F12ADF.

On exit: contains data on the current options set which may be altered from the default set via calls to F12ADF.
3: ISTAT – INTEGERInput/Output: On entry: set to zero.

On exit: if set to a nonzero value F02EKF 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 F02EKF. 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 F02EKF is called.

11: NCONV – INTEGEROutput

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

NEV + 1

depending on the number of complex conjugate pairs of eigenvalues returned.

12: W(NCV) – COMPLEX (KIND=nag_wp) arrayOutput

On exit: the first NCONV elements contain the converged approximations to the selected eigenvalues. Since complex conjugate pairs of eigenvalues appear together, it is possible (given an odd number of converged real eigenvalues) for F02EKF to return one more eigenvalue than requested.

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

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

NEV + 1

On exit: contains the eigenvectors associated with the eigenvalue

λ_{i}

, for

i = 1, 2, \dots, NCONV

(stored in W). For complex conjugate pairs of eigenvalues,

w_{j + 1} = \bar{w_{j}}

, the real and imaginary parts of the corresponding eigenvectors are stored, respectively, in

V (i, j)

and

V (i, j + 1)

, for

i = 1, 2, \dots, n

; the second vector stored is for the first of the conjugate pair which must be negated for the second of the pair.

14: LDV – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F02EKF is called.

Constraint:

LDV \geq \max (1, N)

15: RESID( $NEV + 1$ ) – 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 F02EKF, 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 F02EKF, 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: $NNZ \leq N \times N$ .

$IFAIL = 3$: On entry, in shifted inverse mode, the $j$ th diagonal element of $A$ is not defined, for $j = ⟨value⟩$ .

$IFAIL = 4$: On entry, for $i = ⟨value⟩$ , $ICOLZP (i) = ⟨value⟩$ and $ICOLZP (I + 1) = ⟨value⟩$ .
Constraint: $ICOLZP (I) < ICOLZP (I + 1)$ .

On entry, $ICOLZP (1) = ⟨value⟩$ .
Constraint: $ICOLZP (1) = 1$ .

On entry, $ICOLZP (N + 1) = ⟨value⟩$ and $NNZ = ⟨value⟩$ .
Constraint: $ICOLZP (N + 1) = NNZ + 1$ .

$IFAIL = 5$: On entry, in specification of column $⟨value⟩$ , and for $j = ⟨value⟩$ , $IROWIX (j) = ⟨value⟩$ and $IROWIX (j + 1) = ⟨value⟩$ .
Constraint: $IROWIX (j) < IROWIX (j + 1)$ .

$IFAIL = 6$: On entry, $NEV = ⟨value⟩$ .
Constraint: $NEV > 0$ .

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

On entry, $NCV = ⟨value⟩$ and $NEV = ⟨value⟩$ .
Constraint: $NCV > NEV + 1$ .

$IFAIL = 8$: On entry, the matrix $A - σ \times I$ is nearly numerically singular and could not be inverted. Try perturbing the value of sigma. Norm of matrix $= ⟨value⟩$ , Reciprocal condition number $= ⟨value⟩$ .

On entry, the matrix $A - σ \times I$ 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: $LDV \geq N$ .

$IFAIL = 21$: The maximum number of iterations $\leq 0$ , the optional parameter Iteration Limit has been set to $⟨value⟩$ .

$IFAIL = 22$: An internal call to F12ABF returned with $IFAIL = 2$ .
This error should not occur. Please contact NAG.

$IFAIL = 23$: An internal call to F12ABF returned with $IFAIL = 3$ .

$IFAIL = 24$: 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.

$IFAIL = 25$: No shifts could be applied during a cycle of the implicitly restarted Arnoldi iteration.

$IFAIL = 26$: Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $= ⟨value⟩$ .

$IFAIL = 27$: Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix.
Please contact NAG.

$IFAIL = 32$: An internal call to F12ACF returned with $IFAIL = 2$ .

$IFAIL = 33$: The number of eigenvalues found to sufficient accuracy, is zero.

$IFAIL = 34$: Internal inconsistency in the number of converged Ritz values number counted $= ⟨value⟩$ , number expected $= ⟨value⟩$ .

$IFAIL = 35$: During calculation of a real Schur form, there was a failure to compute $⟨value⟩$ eigenvalues in a total of $⟨value⟩$ iterations.

$IFAIL = 36$: The computed Schur form could not be reordered by an internal call.
This routine returned with $IFAIL = ⟨value⟩$ .
Please contact NAG.

$IFAIL = 37$: In calculating eigenvectors, an internal call returned with an error.
Please contact NAG.

$IFAIL = - 999$: Dynamic memory allocation failed.

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, or can be printed by setting option Print Level to a positive value.

8 Further Comments

F02EKF calls routines based on the ARPACK suite in Chapter F12. These routines use an implicitly restarted Arnoldi 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 Arnoldi process. Each iteration is therefore cheap computationally, relative to the alternative, Shifted Inverse Real, 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 Real mode should be used.

If the Shifted Inverse Real mode is used (via a call to F12ADF in OPTION) then the matrix

A - σ I

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

L U

factorization routine F11MEF. The condition number of

A - σ I

is then calculated by a call to F11MGF. If the condition number is too big then the matrix is considered to be nearly singular, i.e.,

σ

is an approximate eigenvalue of

A

, and the routine exits with an error. In this situation it is normally sufficient to perturb

σ

by a small amount and call F02EKF again. After successful factorization, subsequent solves are performed by calls to F11MFF.

Finally, F02EKF transforms the eigenvectors. Each eigenvector

w

(real or complex) is normalized so that

{‖w‖}_{2} = 1

, and the element of largest absolute value is real and positive.

The monitoring routine MONIT provides some basic information on the convergence of the Arnoldi iterations. Much greater levels of detail on the Arnoldi 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 Monitoring option may be used to select a channel number for the printing of monitoring information and X04ACF may be used to associate that unit number with a file.

9 Example

This example computes the four eigenvalues of the matrix

A

which lie closest to the value

σ = 5.5

on the real line, and their corresponding eigenvectors, where

A

is the tridiagonal matrix with elements

a_{i j} = \{\begin{cases} 2 + i, & j = i \\ 3, & j = i - 1 \\ - 1 + ρ / (2 n + 2), & j = i + 1 ​ with ​ ρ = 10.0 . \end{cases}

10 Optional Parameters

Internally F02EKF calls routines from the suite F12AAF, F12ABF, F12ACF, F12ADF and F12AEF. 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 F02EKF 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 F02EKF. OPTION must be coded such that one call to F12ADF 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 10.1.

10.1 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 printed on the advisory message channel number

i

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 Arnoldi iterations that can be performed before F02EKF exits with

IFAIL \neq 0

Largest Magnitude

Default

Largest Imaginary

Largest Real

Smallest Imaginary

Smallest Magnitude

Smallest Real

The Arnoldi iterative method converges on a number of eigenvalues with given properties. The default is to compute the eigenvalues of largest magnitude using Largest Magnitude. Alternatively, eigenvalues may be chosen which have Largest Real part, Largest Imaginary part, Smallest Magnitude, Smallest Real part or Smallest Imaginary part.

Note that these options select the eigenvalue properties for eigenvalues of

OP

the linear operator determined by the computational mode and problem type.

Nolist

Default

List

Normally each optional parameter specification is not printed to the advisory channel as it is supplied. Optional parameter List may be used to enable printing and optional parameter Nolist may be used to suppress the printing.

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 F02EKF as follows.

$= 0$	No output except error messages.
$> 0$	The set of selected options.
$= 2$	Problem and timing statistics when all calls to F12ABF have been completed.
$\geq 5$	A single line of summary output at each Arnoldi iteration.
$\geq 10$	If $Monitoring > 0$ , then at each iteration, the length and additional steps of the current Arnoldi factorization and the number of converged Ritz values; during re-orthogonalization, the norm of initial/restarted starting vector.
$\geq 20$	Problem and timing statistics on final exit from F12ABF. If $Monitoring > 0$ , then at each iteration, the number of shifts being applied, the eigenvalues and estimates of the Hessenberg matrix $H$ , the size of the Arnoldi basis, the wanted Ritz values and associated Ritz estimates and the shifts applied; vector norms prior to and following re-orthogonalization.
$\geq 30$	If $Monitoring > 0$ , 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 compressed upper Hessenberg matrix $H$ ; during re-orthogonalization, the initial/restarted starting vector; during the Arnoldi iteration loop, a restart is flagged and the number of the residual requiring iterative refinement; while applying shifts, the indices of the shifts being applied.
$\geq 40$	If $Monitoring > 0$ , then during the Arnoldi iteration loop, the Arnoldi 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 $Monitoring > 0$ , then during Arnoldi iteration loop: norms of key components and the active column of $H$ , norms of residuals during iterative refinement, the final upper Hessenberg matrix $H$ ; while applying shifts: number of shifts, shift values, block indices, updated matrix $H$ ; while computing eigenvalues of $H$ : the matrix $H$ , the computed eigenvalues and Ritz estimates.

Regular

Default

Shifted Inverse Real

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

OP (x)

to be performed.

Given a standard eigenvalue problem in the form

A x = λ x

then the following modes are available with the appropriate operator

OP (x)

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

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.

F02EKF (PDF version)

F02 Chapter Contents

F02 Chapter Introduction

NAG Library Manual