NAG CL Interface
f02ekc (real_gen_sparse_arnoldi)
Note: this function 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 function f12adc from the usersupplied function option. Please refer to Section 11 for a detailed description of the specification of the optional parameters.
1
Purpose
f02ekc computes selected eigenvalues and eigenvectors of a real sparse general matrix.
2
Specification
void 
f02ekc (Integer n,
Integer nnz,
double a[],
const Integer icolzp[],
const Integer irowix[],
Integer nev,
Integer ncv,
double sigma,
Integer *nconv,
Complex w[],
double v[],
Integer pdv,
double resid[],
Nag_Comm *comm,
NagError *fail) 

The function may be called by the names: f02ekc or nag_eigen_real_gen_sparse_arnoldi.
3
Description
f02ekc computes selected eigenvalues and the corresponding right eigenvectors of a real sparse general matrix
$A$:
A specified number, ${n}_{ev}$, of eigenvalues ${\lambda}_{i}$, or the shifted inverses ${\mu}_{i}=1/\left({\lambda}_{i}\sigma \right)$, 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 ($\sigma =0.0$). 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, ${\lambda}_{i}$ and ${w}_{i}$ may be complex. If ${w}_{i}$ is an eigenvector corresponding to a complex eigenvalue ${\lambda}_{i}$, then the complex conjugate vector ${\overline{w}}_{i}$ is the eigenvector corresponding to the complex conjugate eigenvalue ${\overline{\lambda}}_{i}$. The eigenvalues in a complex conjugate pair ${\lambda}_{i}$ and ${\overline{\lambda}}_{i}$ 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.
f02ekc 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 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; 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 Largescale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
5
Arguments

1:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

2:
$\mathbf{nnz}$ – Integer
Input

On entry: the dimension of the array
a and The number of nonzero elements of the matrix
$A$ and, if a nonzero shifted inverse is to be applied, all diagonal elements. Each nonzero is counted once in the latter case.
Constraint:
$0\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.

3:
$\mathbf{a}\left[{\mathbf{nnz}}\right]$ – double
Input/Output

On entry: the array of nonzero elements (and diagonal elements if a nonzero inverse shift is to be applied) of the $n$ by $n$ 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:
$\mathbf{icolzp}\left[{\mathbf{n}}+1\right]$ – const Integer
Input

On entry:
${\mathbf{icolzp}}\left[i1\right]$ contains the index in
a of the start of column
$\mathit{i}$, for
$\mathit{i}=1,2,\dots ,n$;
${\mathbf{icolzp}}\left[{\mathbf{n}}\right]$ must contain the value
${\mathbf{nnz}}+1$. Thus the number of nonzero elements in column
$\mathit{i}$ of
$A$ is
${\mathbf{icolzp}}\left[i\right]{\mathbf{icolzp}}\left[i1\right]$; when shifts are applied this includes diagonal elements irrespective of value. See
Section 2.1.3 in the
F11 Chapter Introduction.

5:
$\mathbf{irowix}\left[{\mathbf{nnz}}\right]$ – const Integer
Input

On entry:
${\mathbf{irowix}}\left[i1\right]$ contains the row index for each entry in
a. See
Section 2.1.3 in the
F11 Chapter Introduction.

6:
$\mathbf{nev}$ – Integer
Input

On entry: the number of eigenvalues to be computed.
Constraint:
$0<{\mathbf{nev}}<{\mathbf{n}}1$.

7:
$\mathbf{ncv}$ – Integer
Input

On entry: the dimension of the array
w.
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
${\mathbf{ncv}}\ge 2\times {\mathbf{nev}}+1$. 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 matrixvector operations but it also increases the work and storage required to maintain the orthogonal basis vectors. The optimal ‘crossover’ with respect to CPU time is problem dependent and must be determined empirically.
Constraint:
${\mathbf{nev}}+1<{\mathbf{ncv}}\le {\mathbf{n}}$.

8:
$\mathbf{sigma}$ – double
Input

On entry: if the
${\mathbf{Shifted\; Inverse\; Real}}$ 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 usersupplied function
option.

9:
$\mathbf{monit}$ – function, supplied by the user
External Function

monit is used to monitor the progress of
f02ekc.
monit may be specified as
NULLFN
if no monitoring is actually required.
monit is called after the solution of each eigenvalue subproblem and also just prior to return from
f02ekc.
The specification of
monit is:

1:
$\mathbf{ncv}$ – Integer
Input

On entry: the dimension of the arrays
w and
rzest. The number of Arnoldi basis vectors used during the computation.

2:
$\mathbf{niter}$ – Integer
Input

On entry: the number of the current Arnoldi iteration.

3:
$\mathbf{nconv}$ – Integer
Input

On entry: the number of converged eigenvalues so far.

4:
$\mathbf{w}\left[{\mathbf{ncv}}\right]$ – const Complex
Input

On entry: the first
nconv elements of
w contain the converged approximate eigenvalues.

5:
$\mathbf{rzest}\left[{\mathbf{ncv}}\right]$ – const double
Input

On entry: the first
nconv elements of
rzest contain the Ritz estimates (error bounds) on the converged approximate eigenvalues.

6:
$\mathbf{istat}$ – Integer *
Input/Output

On entry: set to zero.
On exit: if set to a nonzero value
f02ekc returns immediately with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

7:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
monit.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
f02ekc you may allocate memory and initialize these pointers with various quantities for use by
monit when called from
f02ekc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: monit should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
f02ekc. If your code inadvertently
does return any NaNs or infinities,
f02ekc is likely to produce unexpected results.

10:
$\mathbf{option}$ – function, supplied by the user
External Function

You can supply nondefault options to the Arnoldi eigensolver by repeated calls to
f12adc from within
option. (Please note that it is only necessary to call
f12adc; no call to
f12aac is required from within
option.) For example, you can set the mode to
${\mathbf{Shifted\; Inverse\; Real}}$, you can increase the
${\mathbf{Iteration\; Limit}}$ beyond its default and you can print varying levels of detail on the iterative process using
${\mathbf{Print\; Level}}$.
If only the default options (including that the eigenvalues of largest magnitude are sought) are to be used then
option may be may be specified as
NULLFN. See
Section 10 for an example of using
option to set some nondefault options.
The specification of
option is:

1:
$\mathbf{icom}\left[140\right]$ – Integer
Communication Array

On entry: contains details of the default option set. This array must be passed as argument
icomm in any call to
f12adc.
On exit: contains data on the current options set which may be altered from the default set via calls to
f12adc.

2:
$\mathbf{com}\left[60\right]$ – double
Communication Array

On entry: contains details of the default option set. This array must be passed as argument
comm in any call to
f12adc.
On exit: contains data on the current options set which may be altered from the default set via calls to
f12adc.

3:
$\mathbf{istat}$ – Integer *
Input/Output

On entry: set to zero.
On exit: if set to a nonzero value
f02ekc returns immediately with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

4:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
option.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
f02ekc you may allocate memory and initialize these pointers with various quantities for use by
option when called from
f02ekc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

11:
$\mathbf{nconv}$ – Integer *
Output

On exit: the number of converged approximations to the selected eigenvalues. On successful exit, this will normally be either
nev or
${\mathbf{nev}}+1$ depending on the number of complex conjugate pairs of eigenvalues returned.

12:
$\mathbf{w}\left[{\mathbf{ncv}}\right]$ – Complex
Output

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
f02ekc to return one more eigenvalue than requested.

13:
$\mathbf{v}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
v
must be at least
${\mathbf{pdv}}\times {\mathbf{ncv}}$.
On exit: contains the eigenvectors associated with the eigenvalue
${\lambda}_{\mathit{i}}$, for
$\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$ (stored in
w). For a real eigenvalue,
${\lambda}_{j}$, the corresponding eigenvector is real and is stored in
${\mathbf{v}}\left[\left(j1\right)\times {\mathbf{pdv}}+\mathit{i}1\right]$, for
$\mathit{i}=1,2,\dots ,n$. For complex conjugate pairs of eigenvalues,
${w}_{j+1}=\overline{{w}_{j}}$, the real and imaginary parts of the corresponding eigenvectors are stored, respectively, in
${\mathbf{v}}\left[\left(j1\right)\times {\mathbf{pdv}}+\mathit{i}1\right]$ and
${\mathbf{v}}\left[j\times {\mathbf{pdv}}+\mathit{i}1\right]$, for
$\mathit{i}=1,2,\dots ,n$. The imaginary parts stored are for the first of the conjugate pair of eigenvectors; the other eigenvector in the pair is obtained by negating these imaginary parts.

14:
$\mathbf{pdv}$ – Integer
Input

On entry: the stride separating, in the array
v, real and imaginary parts of elements of a conjugate pair of eigenvectors, or separating the elements of a real eigenvector from the corresponding real parts of the next eigenvector.
Constraint:
${\mathbf{pdv}}\ge {\mathbf{n}}$.

15:
$\mathbf{resid}\left[{\mathbf{nev}}+1\right]$ – double
Output

On exit: the residual ${\Vert A{w}_{\mathit{i}}{\lambda}_{\mathit{i}}{w}_{\mathit{i}}\Vert}_{2}$ for the estimates to the eigenpair ${\lambda}_{\mathit{i}}$ and ${w}_{\mathit{i}}$ is returned in ${\mathbf{resid}}\left[\mathit{i}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$.

16:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

17:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_DIAG_ELEMENTS

On entry, in shifted inverse mode, the $j$th diagonal element of $A$ is not defined, for $j=\u2329\mathit{\text{value}}\u232a$.
 NE_EIGENVALUES

The number of eigenvalues found to sufficient accuracy is zero.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{nev}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nev}}>0$.
On entry, ${\mathbf{nnz}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nnz}}>0$.
 NE_INT_2

On entry, ${\mathbf{ncv}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ncv}}\le {\mathbf{n}}$.
On entry, ${\mathbf{ncv}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{nev}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ncv}}>{\mathbf{nev}}+1$.
On entry, ${\mathbf{nnz}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nnz}}\le {\mathbf{n}}\times {\mathbf{n}}$.
On entry, ${\mathbf{pdv}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pdv}}\ge {\mathbf{n}}$.
 NE_INTERNAL_EIGVAL_FAIL

Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix.
Please contact
NAG.
 NE_INTERNAL_EIGVEC_FAIL

In calculating eigenvectors, an internal call returned with an error.
Please contact
NAG.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
An unexpected internal error occurred when postprocessing an eigenproblem.
This error should not occur. Please contact
NAG.
An unexpected internal error occurred when solving an eigenproblem.
This error should not occur. Please contact
NAG.
Internal inconsistency in the number of converged Ritz values. Number counted $\text{}=\u2329\mathit{\text{value}}\u232a$, number expected $\text{}=\u2329\mathit{\text{value}}\u232a$.
 NE_INVALID_OPTION

The maximum number of iterations $\text{}\le 0$, the optional parameter ${\mathbf{Iteration\; Limit}}$ has been set to $\u2329\mathit{\text{value}}\u232a$.
 NE_NO_ARNOLDI_FAC

Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $\text{}=\u2329\mathit{\text{value}}\u232a$.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_SHIFTS_APPLIED

No shifts could be applied during a cycle of the implicitly restarted Arnoldi iteration.
 NE_SCHUR_EIG_FAIL

During calculation of a real Schur form, there was a failure to compute $\u2329\mathit{\text{value}}\u232a$ eigenvalues in a total of $\u2329\mathit{\text{value}}\u232a$ iterations.
 NE_SCHUR_REORDER

The computed Schur form could not be reordered by an internal call.
This routine returned with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=\u2329\mathit{\text{value}}\u232a$.
Please contact
NAG.
 NE_SINGULAR

On entry, the matrix $A\sigma \times I$ is nearly numerically singular and could not be inverted. Try perturbing the value of $\sigma $. Norm of matrix $\text{}=\u2329\mathit{\text{value}}\u232a$, Reciprocal condition number $\text{}=\u2329\mathit{\text{value}}\u232a$.
On entry, the matrix $A\sigma \times I$ is numerically singular and could not be inverted. Try perturbing the value of $\sigma $.
 NE_SPARSE_COL

On entry, for $i=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{icolzp}}\left[i1\right]=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{icolzp}}\left[i\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{icolzp}}\left[i1\right]<{\mathbf{icolzp}}\left[i\right]$.
On entry, ${\mathbf{icolzp}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{icolzp}}\left[0\right]=1$.
On entry, ${\mathbf{icolzp}}\left[{\mathbf{n}}\right]=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{nnz}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{icolzp}}\left[{\mathbf{n}}\right]={\mathbf{nnz}}+1$.
 NE_SPARSE_ROW

On entry, in specification of column $\u2329\mathit{\text{value}}\u232a$, and for $j=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{irowix}}\left[j1\right]=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{irowix}}\left[j\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irowix}}\left[j1\right]<{\mathbf{irowix}}\left[j\right]$.
 NE_TOO_MANY_ITER

The maximum number of iterations has been reached.
The maximum number of iterations $\text{}=\u2329\mathit{\text{value}}\u232a$.
The number of converged eigenvalues $\text{}=\u2329\mathit{\text{value}}\u232a$.
See the function document for further details.
 NE_USER_STOP

User requested termination in
monit,
${\mathbf{istat}}=\u2329\mathit{\text{value}}\u232a$.
User requested termination in
option,
${\mathbf{istat}}=\u2329\mathit{\text{value}}\u232a$.
 NE_ZERO_RESID

An unexpected internal error occurred when solving an eigenproblem.
This error should not occur. Please contact
NAG.
7
Accuracy
The relative accuracy of a Ritz value (eigenvalue approximation),
$\lambda $, is considered acceptable if its Ritz estimate
$\le {\mathbf{Tolerance}}\times \lambda $. The default value for
${\mathbf{Tolerance}}$ is the
machine precision given by
X02AJC. The Ritz estimates are available via the
monit function at each iteration in the Arnoldi process, or can be printed by setting option
${\mathbf{Print\; Level}}$ to a positive value.
8
Parallelism and Performance
f02ekc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f02ekc 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 function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
f02ekc calls functions based on the ARPACK suite in
Chapter F12. These functions 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 ${\mathbf{Regular}}$ mode, only matrixvector multiplications are performed using the sparse matrix $A$ during the Arnoldi process. Each iteration is therefore cheap computationally, relative to the alternative, ${\mathbf{Shifted\; Inverse\; Real}}$, 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 ${\mathbf{Smallest\; Magnitude}}$ 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, ${\mathbf{Regular}}$ mode may still prove to be the most efficient since no inversions are required. Where smallest eigenvalues are sought and ${\mathbf{Regular}}$ mode is not suitable, or eigenvalues close to a given real value are sought, the ${\mathbf{Shifted\; Inverse\; Real}}$ mode should be used.
If the
${\mathbf{Shifted\; Inverse\; Real}}$ mode is used (via a call to
f12adc in
option) then the matrix
$A\sigma I$ is used in linear system solves by the Arnoldi process. This is first factorized internally using the direct
$LU$ factorization function
f11mec. The condition number of
$A\sigma I$ is then calculated by a call to
f11mgc. If the condition number is too big then the matrix is considered to be nearly singular, i.e.,
$\sigma $ is an approximate eigenvalue of
$A$, and the function exits with an error. In this situation it is normally sufficient to perturb
$\sigma $ by a small amount and call
f02ekc again. After successful factorization, subsequent solves are performed by calls to
f11mfc.
Finally, f02ekc transforms the eigenvectors. Each eigenvector $w$ (real or complex) is normalized so that ${\Vert w\Vert}_{2}=1$, and the element of largest absolute value is real.
The monitoring function
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
${\mathbf{Print\; Level}}$. If this is set to a positive value then information will be printed, by default, to standard output. The
${\mathbf{Monitoring}}$ option may be used to select a monitoring
file by setting the option to a file identification (unit) number associated with
${\mathbf{Monitoring}}$ (see
x04acc).
10
Example
This example computes the four eigenvalues of the matrix
$A$ which lie closest to the value
$\sigma =5.5$ on the real line, and their corresponding eigenvectors, where
$A$ is the tridiagonal matrix with elements
10.1
Program Text
10.2
Program Data
10.3
Program Results
11
Optional Parameters
Internally
f02ekc calls functions from the suite
f12aac,
f12abc,
f12acc,
f12adc and
f12aec. Several optional parameters for these computational functions define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of
f02ekc 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 usersupplied function
option in the call to
f02ekc.
option must be coded such that one call to
f12adc 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 $\epsilon $ is a generic notation for machine precision (see X02AJC).
Keywords and character values are case and white space insensitive.
Optional parameters used to specify files (e.g.,
${\mathbf{Advisory}}$ and
${\mathbf{Monitoring}}$) have type Integer. This Integer value corresponds with the Nag_FileID as returned by
x04acc. See
Section 10 for an example of the use of this facility.
Advisory  $i$  Default $\text{}=0$

If the optional parameter
${\mathbf{List}}$ is set then optional parameter specifications are listed in a
${\mathbf{List}}$ file by setting the option to a file identification (unit) number associated with advisory messages (see
x04acc).
This special keyword may be used to reset all optional parameters to their default values.
Iteration Limit  $i$ 
Default $\text{}=300$

The limit on the number of Arnoldi iterations that can be performed before
f02ekc exits with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOO_MANY_ITER.
Largest Magnitude   Default 
The Arnoldi iterative method converges on a number of eigenvalues with given properties. The default is to compute the eigenvalues of largest magnitude using ${\mathbf{Largest\; Magnitude}}$. Alternatively, eigenvalues may be chosen which have ${\mathbf{Largest\; Real}}$ part, ${\mathbf{Largest\; Imaginary}}$ part, ${\mathbf{Smallest\; Magnitude}}$, ${\mathbf{Smallest\; Real}}$ part or ${\mathbf{Smallest\; Imaginary}}$ part.
Note that these options select the eigenvalue properties for eigenvalues of $\mathrm{OP}$ the linear operator determined by the computational mode and problem type.
Optional parameter ${\mathbf{List}}$ enables printing of each optional parameter specification as it is supplied. ${\mathbf{Nolist}}$ suppresses this printing.
Monitoring  $i$  Default $\text{}=1$ 
Unless
${\mathbf{Monitoring}}$ is set to
$1$ (the default), monitoring information is output to the file associated with Nag_FileID
$i$ during the solution of each problem; this may be the same as
${\mathbf{Advisory}}$. The type of information produced is dependent on the value of
${\mathbf{Print\; Level}}$, see the description of the optional parameter
${\mathbf{Print\; Level}}$ in this section for details of the information produced. Please see
x04acc to associate a file with a given Nag_FileID (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Print Level  $i$  Default $\text{}=0$ 
This controls the amount of printing produced by
f02ekc as follows.
$=0$  No output except error messages. 
$>0$  The set of selected options. 
$=2$  Problem and timing statistics when all calls to f12abc have been completed. 
$\ge 5$  A single line of summary output at each Arnoldi iteration. 
$\ge 10$ 
If ${\mathbf{Monitoring}}$ is set to a valid Nag_FileID then at each iteration, the length and additional steps of the current Arnoldi factorization and the number of converged Ritz values; during reorthogonalization, the norm of initial/restarted starting vector. 
$\ge 20$  Problem and timing statistics on final exit from f12abc. If ${\mathbf{Monitoring}}$ is set to a valid Nag_FileID 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 reorthogonalization. 
$\ge 30$  If ${\mathbf{Monitoring}}$ is set to a valid Nag_FileID 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 reorthogonalization, 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. 
$\ge 40$  If ${\mathbf{Monitoring}}$ is set to a valid Nag_FileID 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 $H$. 
$\ge 50$  If ${\mathbf{Monitoring}}$ is set to a valid Nag_FileID 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. 
These options define the computational mode which in turn defines the form of operation $\mathrm{OP}\left(x\right)$ to be performed.
Given a standard eigenvalue problem in the form
$Ax=\lambda x$ then the following modes are available with the appropriate operator
$\mathrm{OP}\left(x\right)$.
${\mathbf{Regular}}$ 
$\mathrm{OP}=A$ 
${\mathbf{Shifted\; Inverse\; Real}}$ 
$\mathrm{OP}={\left(A\sigma I\right)}^{1}$ where $\sigma $ is real 
Tolerance  $r$ 
Default $\text{}=\epsilon $

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