# NAG C Library Function Document

## nag_eigen_real_symm_sparse_arnoldi (f02fkc)

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 nag_real_symm_sparse_eigensystem_option (f12fdc) from the user-supplied function option. Please refer to Section 11 for a detailed description of the specification of the optional parameters.

## 1Purpose

nag_eigen_real_symm_sparse_arnoldi (f02fkc) computes selected eigenvalues and eigenvectors of a real sparse symmetric matrix.

## 2Specification

 #include #include
void  nag_eigen_real_symm_sparse_arnoldi (Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], Integer nev, Integer ncv, double sigma,
 void (*monit)(Integer ncv, Integer niter, Integer nconv, const double w[], const double rzest[], Integer *istat, Nag_Comm *comm),
 void (*option)(Integer icom[], double com[], Integer *istat, Nag_Comm *comm),
Integer *nconv, double w[], double v[], Integer pdv, double resid[], Nag_Comm *comm, NagError *fail)

## 3Description

nag_eigen_real_symm_sparse_arnoldi (f02fkc) computes selected eigenvalues and the corresponding right eigenvectors of a real sparse symmetric matrix $A$:
 $Avi = λi vi .$
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 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 ($\sigma =0.0$). 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.
nag_eigen_real_symm_sparse_arnoldi (f02fkc) 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.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
HSL (2011) A collection of Fortran codes for large-scale scientific computation http://www.hsl.rl.ac.uk/
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

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}>0$.
2:    $\mathbf{nnz}$IntegerInput
On entry: the dimension of the array a.The number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
3:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const doubleInput
On entry: the array of nonzero elements of the lower triangular part of the $n$ by $n$ symmetric matrix $A$.
4:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
5:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const IntegerInput
On entry: the row and column indices of the elements supplied in array a.
If ${\mathbf{irow}}\left[k-1\right]=i$ and ${\mathbf{icol}}\left[k-1\right]=j$ then ${A}_{ij}$ is stored in ${\mathbf{a}}\left[k-1\right]$. irow does not need to be ordered: an internal sort will force the correct ordering.
Constraint:
irow and icol must satisfy these constraints:
$1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{irow}}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,{\mathbf{nnz}}-1$.
6:    $\mathbf{nev}$IntegerInput
On entry: the number of eigenvalues to be computed.
Constraint: $0<{\mathbf{nev}}<{\mathbf{n}}-1$.
7:    $\mathbf{ncv}$IntegerInput
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 ${\mathbf{ncv}}\ge 2×{\mathbf{nev}}+1$. 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 computation time is problem dependent and must be determined empirically.
Constraint: ${\mathbf{nev}}<{\mathbf{ncv}}\le {\mathbf{n}}$.
8:    $\mathbf{sigma}$doubleInput
On entry: if the ${\mathbf{Shifted Inverse}}$ 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 function option.
9:    $\mathbf{monit}$function, supplied by the userExternal Function
monit is used to monitor the progress of nag_eigen_real_symm_sparse_arnoldi (f02fkc). monit may be specified as NULLFN if no monitoring is actually required. monit is called after the solution of each eigenvalue sub-problem and also just prior to return from nag_eigen_real_symm_sparse_arnoldi (f02fkc).
The specification of monit is:
 void monit (Integer ncv, Integer niter, Integer nconv, const double w[], const double rzest[], Integer *istat, Nag_Comm *comm)
1:    $\mathbf{ncv}$IntegerInput
On entry: the dimension of the arrays w and rzest. The number of Arnoldi basis vectors used during the computation.
2:    $\mathbf{niter}$IntegerInput
On entry: the number of the current Arnoldi iteration.
3:    $\mathbf{nconv}$IntegerInput
On entry: the number of converged eigenvalues so far.
4:    $\mathbf{w}\left[{\mathbf{ncv}}\right]$const doubleInput
On entry: the first nconv elements of w contain the converged approximate eigenvalues.
5:    $\mathbf{rzest}\left[{\mathbf{ncv}}\right]$const doubleInput
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 nag_eigen_real_symm_sparse_arnoldi (f02fkc) 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.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_eigen_real_symm_sparse_arnoldi (f02fkc) you may allocate memory and initialize these pointers with various quantities for use by monit when called from nag_eigen_real_symm_sparse_arnoldi (f02fkc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
10:  $\mathbf{option}$function, supplied by the userExternal Function
You can supply non-default options to the Arnoldi eigensolver by repeated calls to nag_real_symm_sparse_eigensystem_option (f12fdc) from within option. (Please note that it is only necessary to call nag_real_symm_sparse_eigensystem_option (f12fdc); no call to nag_real_symm_sparse_eigensystem_init (f12fac) is required from within option.) For example, you can set the mode to ${\mathbf{Shifted Inverse}}$, 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 specified as NULLFN. See Section 10 for an example of using option to set some non-default options.
The specification of option is:
 void option (Integer icom[], double com[], Integer *istat, Nag_Comm *comm)
1:    $\mathbf{icom}\left[140\right]$IntegerCommunication Array
On entry: contains details of the default option set. This array must be passed as argument icomm in any call to nag_real_symm_sparse_eigensystem_option (f12fdc).
On exit: contains data on the current options set which may be altered from the default set via calls to nag_real_symm_sparse_eigensystem_option (f12fdc).
2:    $\mathbf{com}\left[60\right]$doubleCommunication Array
On entry: contains details of the default option set. This array must be passed as argument comm in any call to nag_real_symm_sparse_eigensystem_option (f12fdc).
On exit: contains data on the current options set which may be altered from the default set via calls to nag_real_symm_sparse_eigensystem_option (f12fdc).
3:    $\mathbf{istat}$Integer *Input/Output
On entry: set to zero.
On exit: if set to a nonzero value nag_eigen_real_symm_sparse_arnoldi (f02fkc) 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.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_eigen_real_symm_sparse_arnoldi (f02fkc) you may allocate memory and initialize these pointers with various quantities for use by option when called from nag_eigen_real_symm_sparse_arnoldi (f02fkc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
11:  $\mathbf{nconv}$Integer *Output
On exit: the number of converged approximations to the selected eigenvalues. On successful exit, this will normally be nev.
12:  $\mathbf{w}\left[{\mathbf{ncv}}\right]$doubleOutput
On exit: the first nconv elements contain the converged approximations to the selected eigenvalues.
13:  $\mathbf{v}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array v must be at least ${\mathbf{pdv}}×{\mathbf{ncv}}$.
On exit: contains the eigenvectors associated with the eigenvalue ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$ (stored in w). For eigenvalue, ${\lambda }_{j}$, the corresponding eigenvector is stored in ${\mathbf{v}}\left[\left(j-1\right)×{\mathbf{pdv}}+\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,n$.
14:  $\mathbf{pdv}$IntegerInput
On entry: the stride separating, in the array v, the elements of a real eigenvector from the corresponding elements of the next eigenvector.
Constraint: ${\mathbf{pdv}}\ge {\mathbf{n}}$.
15:  $\mathbf{resid}\left[{\mathbf{nev}}\right]$doubleOutput
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 ${\mathbf{resid}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$.
16:  $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
17:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_BOTH_ENDS_1
The option ${\mathbf{Both Ends}}$ has been set but only $1$ eigenvalue is requested.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{nev}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nev}}>0$.
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}>0$.
NE_INT_2
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: ${\mathbf{ncv}}>{\mathbf{nev}}$.
On entry, ${\mathbf{nev}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nev}}<\left({\mathbf{n}}-1\right)$.
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
On entry, ${\mathbf{pdv}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdv}}\ge {\mathbf{n}}$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
A serious error, code $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$, has occurred in an internal call. Check all function calls and array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_OPTION
The maximum number of iterations, through the optional parameter ${\mathbf{Iteration Limit}}$, has been set to a non-positive value.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
On entry, the matrix $\left(A-\sigma I\right)$ is numerically singular and could not be inverted. Try perturbing the value of $\sigma$.
NE_SPARSE_COL
On entry, for $i=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[i-1\right]=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{icol}}\left[i-1\right]\le {\mathbf{irow}}\left[i-1\right]$.
NE_SPARSE_ROW
On entry, for $i=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{irow}}\left[i-1\right]\le {\mathbf{n}}$.
NE_TOO_MANY_ITER
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 function document for further details.
NE_USER_STOP
User requested termination in monit, ${\mathbf{istat}}=〈\mathit{\text{value}}〉$.
User requested termination in option, ${\mathbf{istat}}=〈\mathit{\text{value}}〉$.

## 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 ${\mathbf{Tolerance}}$ is the machine precision given by nag_machine_precision (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.

## 8Parallelism and Performance

nag_eigen_real_symm_sparse_arnoldi (f02fkc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_eigen_real_symm_sparse_arnoldi (f02fkc) 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 implementation-specific information.

nag_eigen_real_symm_sparse_arnoldi (f02fkc) calls functions based on the ARPACK suite in Chapter f12. These functions 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 ${\mathbf{Regular}}$ mode, only matrix-vector multiplications are performed using the sparse matrix $A$ during the Lanczos process; nag_sparse_sym_matvec (f11xec) can be used to perform this task. Each iteration is therefore cheap computationally, relative to the alternative, ${\mathbf{Shifted Inverse}}$, 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 ${\mathbf{Smallest Magnitude}}$ 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 ${\mathbf{Shifted Inverse}}$ mode should be used.
If the ${\mathbf{Shifted Inverse}}$ mode is used (via a call to nag_real_symm_sparse_eigensystem_option (f12fdc) in option) then the matrix $A-\sigma I$ is used in linear system solves by the Lanczos process. This is first factorized internally using a direct sparse $LD{L}^{\mathrm{T}}$ factorization under the assumption that the matrix is indefinite. If the factorization determines that the matrix is numerically singular then the function exits with an error. In this situation it is normally sufficient to perturb $\sigma$ by a small amount and call nag_eigen_real_symm_sparse_arnoldi (f02fkc) again. After successful factorization, subsequent solves are performed by backsubstitution using the sparse factorization.
Finally, nag_eigen_real_symm_sparse_arnoldi (f02fkc) transforms the eigenvectors. Each eigenvector $w$ is normalized so that ${‖w‖}_{2}=1$.
The monitoring function 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 ${\mathbf{Print Level}}$. 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 ${\mathbf{Monitoring}}$ option.

The direct sparse factorization is performed by an implementation of HSL_MA97 (see HSL (2011)).

## 10Example

This example solves $Ax=\lambda x$ in ${\mathbf{Shifted Inverse}}$ mode, where $A$ is obtained from the standard central difference discretization of the one-dimensional Laplacian operator $\frac{{\partial }^{2}u}{\partial {x}^{2}}$ on $\left[0,1\right]$, with zero Dirichlet boundary conditions.

### 10.1Program Text

Program Text (f02fkce.c)

### 10.2Program Data

Program Data (f02fkce.d)

### 10.3Program Results

Program Results (f02fkce.r)

## 11Optional Parameters

Internally nag_eigen_real_symm_sparse_arnoldi (f02fkc) calls functions from the suite nag_real_symm_sparse_eigensystem_init (f12fac), nag_real_symm_sparse_eigensystem_iter (f12fbc), nag_real_symm_sparse_eigensystem_sol (f12fcc), nag_real_symm_sparse_eigensystem_option (f12fdc) and nag_real_symm_sparse_eigensystem_monit (f12fec). 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 nag_eigen_real_symm_sparse_arnoldi (f02fkc) 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 function option in the call to nag_eigen_real_symm_sparse_arnoldi (f02fkc). option must be coded such that one call to nag_real_symm_sparse_eigensystem_option (f12fdc) 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 nag_machine_precision (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 nag_open_file (x04acc). See Section 10 for an example of the use of this facility.
 Advisory Default $\text{}=0$
(See Section 3.3.1.1 in How to Use the NAG Library and its Documentation for further information on NAG data types.)
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 ${\mathbf{Advisory}}$ messages (see nag_open_file (x04acc)).
 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 Lanczos iterations that can be performed before nag_real_symm_sparse_eigensystem_iter (f12fbc) exits. If not all requested eigenvalues have converged to within ${\mathbf{Tolerance}}$ and the number of Lanczos iterations has reached this limit then nag_real_symm_sparse_eigensystem_iter (f12fbc) exits with an error; nag_real_symm_sparse_eigensystem_sol (f12fcc) 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 nag_real_symm_sparse_eigensystem_iter (f12fbc) to compute the eigenvalues of largest magnitude using ${\mathbf{Largest Magnitude}}$. Alternatively, eigenvalues may be chosen which have ${\mathbf{Largest Algebraic}}$ part, ${\mathbf{Smallest Magnitude}}$, or ${\mathbf{Smallest Algebraic}}$ part; or eigenvalues which are from ${\mathbf{Both Ends}}$ 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 ${\mathbf{List}}$ and ${\mathbf{Nolist}}$ options.
 Monitoring Default $\text{}=-1$
(See Section 3.3.1.1 in How to Use the NAG Library and its Documentation for further information on NAG data types.)
Unless ${\mathbf{Monitoring}}$ is set to $-1$ (the default), monitoring information is output to Nag_FileID ${\mathbf{Monitoring}}$ 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 nag_open_file (x04acc) to associate a file with a given Nag_FileID.
 Print Level $i$ Default $\text{}=0$
This controls the amount of printing produced by nag_eigen_real_symm_sparse_arnoldi (f02fkc) as follows.
 $=0$ No output except error messages. If you want to suppress all output, set ${\mathbf{Print Level}}=0$. $>0$ The set of selected options. $=2$ Problem and timing statistics on final exit from nag_real_symm_sparse_eigensystem_iter (f12fbc). $\ge 5$ A single line of summary output at each Lanczos iteration. $\ge 10$ If ${\mathbf{Monitoring}}$ 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. $\ge 20$ Problem and timing statistics on final exit from nag_real_symm_sparse_eigensystem_iter (f12fbc). If ${\mathbf{Monitoring}}$ 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. $\ge 30$ If ${\mathbf{Monitoring}}$ 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. $\ge 40$ If ${\mathbf{Monitoring}}$ 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 $H$. $\ge 50$ If ${\mathbf{Monitoring}}$ 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 ${\mathbf{Print Level}}\ge 30$ can result in very lengthy ${\mathbf{Monitoring}}$ output.
 Regular Default
 Regular Inverse
 Shifted Inverse
These options define the computational mode which in turn defines the form of operation $\mathrm{OP}\left(x\right)$ to be performed.
 ${\mathbf{Regular}}$ $\mathrm{OP}=A$ ${\mathbf{Shifted Inverse}}$ $\mathrm{OP}={\left(A-\sigma I\right)}^{-1}$ where $\sigma$ is real ${\mathbf{Regular Inverse}}$ $\mathrm{OP}={A}^{-1}$
 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.