# NAG FL Interfacef02ekf (real_​gen_​sparse_​arnoldi)

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 f12adf from the user-supplied subroutine option. Please refer to Section 11 for a detailed description of the specification of the optional parameters.

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## 1Purpose

f02ekf computes selected eigenvalues and eigenvectors of a real sparse general matrix.

## 2Specification

Fortran Interface
 Subroutine f02ekf ( n, nnz, a, nev, ncv, w, v, ldv,
 Integer, Intent (In) :: n, nnz, icolzp(n+1), irowix(nnz), nev, ncv, ldv Integer, Intent (Inout) :: iuser(*), ifail Integer, Intent (Out) :: nconv Real (Kind=nag_wp), Intent (In) :: sigma Real (Kind=nag_wp), Intent (Inout) :: a(nnz), v(ldv,*), ruser(*) Real (Kind=nag_wp), Intent (Out) :: resid(nev+1) Complex (Kind=nag_wp), Intent (Out) :: w(ncv) External :: monit, option
#include <nag.h>
 void f02ekf_ (const Integer *n, const Integer *nnz, double a[], const Integer icolzp[], const Integer irowix[], const Integer *nev, const Integer *ncv, const double *sigma, void (NAG_CALL *monit)(const Integer *ncv, const Integer *niter, const Integer *nconv, const Complex w[], const double rzest[], Integer *istat, Integer iuser[], double ruser[]),void (NAG_CALL *option)(Integer icomm[], double comm[], Integer *istat, Integer iuser[], double ruser[]),Integer *nconv, Complex w[], double v[], const Integer *ldv, double resid[], Integer iuser[], double ruser[], Integer *ifail)
The routine may be called by the names f02ekf or nagf_eigen_real_gen_sparse_arnoldi.

## 3Description

f02ekf computes selected eigenvalues and the corresponding right eigenvectors of a real sparse general matrix $A$:
 $Awi = λi wi .$
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.
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 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.

## 4References

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, Philadelphia

## 5Arguments

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: $\mathbf{a}\left({\mathbf{nnz}}\right)$Real (Kind=nag_wp) array Input/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: $\mathbf{icolzp}\left({\mathbf{n}}+1\right)$Integer array Input
On entry: ${\mathbf{icolzp}}\left(i\right)$ contains the index in a of the start of column $\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$; ${\mathbf{icolzp}}\left({\mathbf{n}}+1\right)$ must contain the value ${\mathbf{nnz}}+1$. Thus the number of nonzero elements in column $\mathit{i}$ of $A$ is ${\mathbf{icolzp}}\left(i+1\right)-{\mathbf{icolzp}}\left(i\right)$; when shifts are applied this includes diagonal elements irrespective of value. See Section 2.1.3 in the F11 Chapter Introduction.
5: $\mathbf{irowix}\left({\mathbf{nnz}}\right)$Integer array Input
On entry: ${\mathbf{irowix}}\left(i\right)$ 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 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 ${\mathbf{ncv}}\ge 2×{\mathbf{nev}}+1$. 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 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: ${\mathbf{nev}}+1<{\mathbf{ncv}}\le {\mathbf{n}}$.
8: $\mathbf{sigma}$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: $\mathbf{monit}$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:
Fortran Interface
 Subroutine monit ( ncv, w,
 Integer, Intent (In) :: ncv, niter, nconv Integer, Intent (Inout) :: istat, iuser(*) Real (Kind=nag_wp), Intent (In) :: rzest(ncv) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Complex (Kind=nag_wp), Intent (In) :: w(ncv)
 void monit (const Integer *ncv, const Integer *niter, const Integer *nconv, const Complex w[], const double rzest[], Integer *istat, Integer iuser[], double ruser[])
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: $\mathbf{w}\left({\mathbf{ncv}}\right)$Complex (Kind=nag_wp) array Input
On entry: the first nconv elements of w contain the converged approximate eigenvalues.
5: $\mathbf{rzest}\left({\mathbf{ncv}}\right)$Real (Kind=nag_wp) array 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 f02ekf returns immediately with ${\mathbf{ifail}}={\mathbf{9}}$.
7: $\mathbf{iuser}\left(*\right)$Integer array User Workspace
8: $\mathbf{ruser}\left(*\right)$Real (Kind=nag_wp) array User Workspace
monit is called with the arguments iuser and ruser as supplied to f02ekf. You should use the arrays iuser and ruser to supply information to monit.
monit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f02ekf is called. Arguments denoted as Input must not be changed by this procedure.
Note: monit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f02ekf. If your code inadvertently does return any NaNs or infinities, f02ekf is likely to produce unexpected results.
10: $\mathbf{option}$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. (Please note that it is only necessary to call f12adf; no call to f12aaf is required 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).
The specification of option is:
Fortran Interface
 Subroutine option ( comm,
 Integer, Intent (Inout) :: icomm(*), istat, iuser(*) Real (Kind=nag_wp), Intent (Inout) :: comm(*), ruser(*)
 void option (Integer icomm[], double comm[], Integer *istat, Integer iuser[], double ruser[])
1: $\mathbf{icomm}\left(*\right)$Integer array Communication Array
On entry: contains details of the default option set. This array must be passed as argument 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: $\mathbf{comm}\left(*\right)$Real (Kind=nag_wp) array Communication Array
On entry: contains details of the default option set. This array must be passed as argument comm 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: $\mathbf{istat}$Integer Input/Output
On entry: set to zero.
On exit: if set to a nonzero value f02ekf returns immediately with ${\mathbf{ifail}}={\mathbf{10}}$.
4: $\mathbf{iuser}\left(*\right)$Integer array User Workspace
5: $\mathbf{ruser}\left(*\right)$Real (Kind=nag_wp) array User Workspace
option is called with the arguments iuser and ruser as supplied to f02ekf. You should use the arrays iuser and ruser to supply information to option.
option must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f02ekf is called.
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 (Kind=nag_wp) array 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 f02ekf to return one more eigenvalue than requested.
13: $\mathbf{v}\left({\mathbf{ldv}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array v must be at least ${\mathbf{ncv}}$.
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 ${\mathbf{v}}\left(\mathit{i},j\right)$, 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 ${\mathbf{v}}\left(\mathit{i},j\right)$ and ${\mathbf{v}}\left(\mathit{i},j\right)$, 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: $\mathbf{ldv}$Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f02ekf is called.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{n}}$.
15: $\mathbf{resid}\left({\mathbf{nev}}+1\right)$Real (Kind=nag_wp) array Output
On exit: the residual ${‖A{w}_{\mathit{i}}-{\lambda }_{\mathit{i}}{w}_{\mathit{i}}‖}_{2}$ for the estimates to the eigenpair ${\lambda }_{\mathit{i}}$ and ${w}_{\mathit{i}}$ is returned in ${\mathbf{resid}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$.
16: $\mathbf{iuser}\left(*\right)$Integer array User 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.
17: $\mathbf{ruser}\left(*\right)$Real (Kind=nag_wp) array User 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.
18: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nnz}}>0$.
On entry, ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nnz}}\le {\mathbf{n}}×{\mathbf{n}}$.
${\mathbf{ifail}}=3$
On entry, in shifted inverse mode, the $j$th diagonal element of $A$ is not defined, for $j=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=4$
On entry, for $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{icolzp}}\left(i\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{icolzp}}\left(i+1\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{icolzp}}\left(i\right)<{\mathbf{icolzp}}\left(i+1\right)$.
On entry, ${\mathbf{icolzp}}\left(1\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{icolzp}}\left(1\right)=1$.
On entry, ${\mathbf{icolzp}}\left({\mathbf{n}}+1\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{icolzp}}\left({\mathbf{n}}+1\right)={\mathbf{nnz}}+1$.
${\mathbf{ifail}}=5$
On entry, in specification of column $⟨\mathit{\text{value}}⟩$, and for $j=⟨\mathit{\text{value}}⟩$, ${\mathbf{irowix}}\left(j\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{irowix}}\left(j+1\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irowix}}\left(j\right)<{\mathbf{irowix}}\left(j+1\right)$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{nev}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nev}}>0$.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{ncv}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ncv}}\le {\mathbf{n}}$.
On entry, ${\mathbf{ncv}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nev}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ncv}}>{\mathbf{nev}}+1$.
${\mathbf{ifail}}=8$
On entry, the matrix $A-\sigma ×I$ is nearly numerically singular and could not be inverted. Try perturbing the value of $\sigma$. Norm of matrix $\text{}=⟨\mathit{\text{value}}⟩$, Reciprocal condition number $\text{}=⟨\mathit{\text{value}}⟩$.
On entry, the matrix $A-\sigma ×I$ is numerically singular and could not be inverted. Try perturbing the value of $\sigma$.
${\mathbf{ifail}}=9$
User requested termination in monit, ${\mathbf{istat}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=10$
User requested termination in option, ${\mathbf{istat}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=14$
On entry, ${\mathbf{ldv}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=21$
The maximum number of iterations $\text{}\le 0$, the optional parameter Iteration Limit has been set to $⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=22$
An unexpected internal error occurred when solving an eigenproblem.
${\mathbf{ifail}}=23$
An unexpected internal error occurred when solving an eigenproblem.
${\mathbf{ifail}}=24$
The maximum number of iterations has been reached.
The maximum number of iterations $\text{}=⟨\mathit{\text{value}}⟩$.
The number of converged eigenvalues $\text{}=⟨\mathit{\text{value}}⟩$.
See the routine document for further details.
${\mathbf{ifail}}=25$
No shifts could be applied during a cycle of the implicitly restarted Arnoldi iteration.
${\mathbf{ifail}}=26$
Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $\text{}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=27$
Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix.
${\mathbf{ifail}}=32$
An unexpected internal error occurred when postprocessing an eigenproblem.
${\mathbf{ifail}}=33$
The number of eigenvalues found to sufficient accuracy is zero.
${\mathbf{ifail}}=34$
Internal inconsistency in the number of converged Ritz values. Number counted $\text{}=⟨\mathit{\text{value}}⟩$, number expected $\text{}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=35$
During calculation of a real Schur form, there was a failure to compute $⟨\mathit{\text{value}}⟩$ eigenvalues in a total of $⟨\mathit{\text{value}}⟩$ iterations.
${\mathbf{ifail}}=36$
The computed Schur form could not be reordered by an internal call.
This routine returned with ${\mathbf{ifail}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=37$
In calculating eigenvectors, an internal call returned with an error.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

The relative accuracy of a Ritz value (eigenvalue approximation), $\lambda$, is considered acceptable if its Ritz estimate $\le {\mathbf{Tolerance}}×\lambda$. 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.

## 8Parallelism and Performance

f02ekf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f02ekf 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.

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 $A-\sigma I$ is used in linear system solves by the Arnoldi process. This is first factorized internally using the direct $LU$ factorization routine f11mef. The condition number of $A-\sigma I$ 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., $\sigma$ is an approximate eigenvalue of $A$, and the routine exits with an error. In this situation it is normally sufficient to perturb $\sigma$ 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.
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 monitoring file by setting the option to a file identification (unit) number associated with Monitoring (see x04acf).

## 10Example

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
 $aij = { 2+i, j=i 3, j = i-1 -1 + ρ / (2n+2), j = i+1 ​ with ​ ρ = 10.0.$

### 10.1Program Text

Program Text (f02ekfe.f90)

### 10.2Program Data

Program Data (f02ekfe.d)

### 10.3Program Results

Program Results (f02ekfe.r)

## 11Optional 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 arguments 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 11.1.

### 11.1Description 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 x02ajf).
Keywords and character values are case and white space insensitive.
 Advisory $i$ Default $\text{}=0$
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 $\text{}=300$
The limit on the number of Arnoldi iterations that can be performed before f02ekf exits with ${\mathbf{ifail}}\ne {\mathbf{0}}$.
 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 $\mathrm{op}$ the linear operator determined by the computational mode and problem type.
 Nolist Default
 List
Optional parameter List enables printing of each optional parameter specification as it is supplied. Nolist suppresses this printing.
 Monitoring $i$ Default $\text{}=-1$
If $i>0$, monitoring information is output to unit number $i$ during the solution of each problem; this may be the same as the Advisory unit 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 unit number.
 Print Level $i$ Default $\text{}=0$
This controls the amount of printing produced by f02ekf as follows.
$\mathbit{i}$ Output
$=0$ No output except error messages.
$>0$ The set of selected options.
$=2$ Problem and timing statistics when all calls to f12abf have been completed.
$\ge 5$ A single line of summary output at each Arnoldi iteration.
$\ge 10$ If ${\mathbf{Monitoring}}>0$, 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.
$\ge 20$ Problem and timing statistics on final exit from f12abf. If ${\mathbf{Monitoring}}>0$, 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.
$\ge 30$ If ${\mathbf{Monitoring}}>0$, 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.
$\ge 40$ If ${\mathbf{Monitoring}}>0$, 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 $H$.
$\ge 50$ If ${\mathbf{Monitoring}}>0$, 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 $\mathrm{op}\left(x\right)$ to be performed.
Given a standard eigenvalue problem in the form $Ax=\lambda x$ then the following modes are available with the appropriate operator $\mathrm{op}\left(x\right)$.
 Regular $\mathrm{op}=A$ 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 Tolerance relative to the magnitude of the eigenvalue.