NAG CL Interface f02ekc (real_gen_sparse_arnoldi)
Note:this function usesoptional parametersto 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 user-supplied function option. Please refer to Section 11 for a detailed description of the specification of the optional parameters.
The function may be called by the names: f02ekc or nag_eigen_real_gen_sparse_arnoldi.
3Description
f02ekc computes selected eigenvalues and the corresponding right eigenvectors of a real sparse general matrix $A$:
$$A{w}_{i}={\lambda}_{i}{w}_{i}\text{.}$$
A specified number, ${n}_{ev}$, of eigenvalues ${\lambda}_{i}$, or the shifted inverses ${\mu}_{i}=1/({\lambda}_{i}-\sigma )$, 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.
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: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
2: $\mathbf{nnz}$ – IntegerInput
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.
On entry: ${\mathbf{icolzp}}\left[i-1\right]$ 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 ${\mathbf{icolzp}}\left[i\right]-{\mathbf{icolzp}}\left[i-1\right]$; when shifts are applied this includes diagonal elements irrespective of value. See Section 2.1.3 in the F11 Chapter Introduction.
On entry: ${\mathbf{irowix}}\left[i-1\right]$ contains the row index for each entry in a. See Section 2.1.3 in the F11 Chapter Introduction.
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 ${\mathbf{ncv}}\ge 2\times {\mathbf{nev}}+1$. 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 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.
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 user-supplied function option.
9: $\mathbf{monit}$ – function, supplied by the userExternal 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 sub-problem and also just prior to return from f02ekc.
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 floating-point 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 userExternal Function
You can supply non-default 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.
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.
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.
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
${\mathbf{v}}\left[\left(j-1\right)\times {\mathbf{pdv}}+\mathit{i}-1\right]$, 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
${\mathbf{v}}\left[\left(j-1\right)\times {\mathbf{pdv}}+\mathit{i}-1\right]$ 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: $\mathbf{pdv}$ – IntegerInput
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.
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).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_DIAG_ELEMENTS
On entry, in shifted inverse mode, the $j$th diagonal element of $A$ is not defined, for $j=\u27e8\mathit{\text{value}}\u27e9$.
NE_EIGENVALUES
The number of eigenvalues found to sufficient accuracy is zero.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{nev}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nev}}>0$.
On entry, ${\mathbf{nnz}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{nnz}}>0$.
NE_INT_2
On entry, ${\mathbf{ncv}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ncv}}\le {\mathbf{n}}$.
On entry, ${\mathbf{ncv}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nev}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ncv}}>{\mathbf{nev}}+1$.
On entry, ${\mathbf{nnz}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nnz}}\le {\mathbf{n}}\times {\mathbf{n}}$.
On entry, ${\mathbf{pdv}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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 $\text{}=\u27e8\mathit{\text{value}}\u27e9$, number expected $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
NE_INVALID_OPTION
The maximum number of iterations $\text{}\le 0$, the optional parameter ${\mathbf{Iteration\; Limit}}$ has been set to $\u27e8\mathit{\text{value}}\u27e9$.
NE_NO_ARNOLDI_FAC
Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
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 $\u27e8\mathit{\text{value}}\u27e9$ eigenvalues in a total of $\u27e8\mathit{\text{value}}\u27e9$ iterations.
NE_SCHUR_REORDER
The computed Schur form could not be reordered by an internal call. This routine returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=\u27e8\mathit{\text{value}}\u27e9$. 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 $\text{}=\u27e8\mathit{\text{value}}\u27e9$, Reciprocal condition number $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
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 $i=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{icolzp}}\left[i-1\right]=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{icolzp}}\left[i\right]=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{icolzp}}\left[i-1\right]<{\mathbf{icolzp}}\left[i\right]$.
On entry, ${\mathbf{icolzp}}\left[0\right]=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{icolzp}}\left[0\right]=1$.
On entry, ${\mathbf{icolzp}}\left[{\mathbf{n}}\right]=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nnz}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{icolzp}}\left[{\mathbf{n}}\right]={\mathbf{nnz}}+1$.
NE_SPARSE_ROW
On entry, in specification of column $\u27e8\mathit{\text{value}}\u27e9$, and for $j=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{irowix}}\left[j-1\right]=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{irowix}}\left[j\right]=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{irowix}}\left[j-1\right]<{\mathbf{irowix}}\left[j\right]$.
NE_TOO_MANY_ITER
The maximum number of iterations has been reached. The maximum number of iterations $\text{}=\u27e8\mathit{\text{value}}\u27e9$. The number of converged eigenvalues $\text{}=\u27e8\mathit{\text{value}}\u27e9$. See the function document for further details.
NE_USER_STOP
User requested termination in monit, ${\mathbf{istat}}=\u27e8\mathit{\text{value}}\u27e9$.
User requested termination in option, ${\mathbf{istat}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_ZERO_RESID
An unexpected internal error occurred when solving an eigenproblem. This error should not occur. Please contact NAG.
7Accuracy
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.
8Parallelism 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 implementation-specific information.
9Further Comments
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 matrix-vector 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).
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
Internally f02ekc calls functions from the suite f12aac,f12abc,f12acc,f12adcandf12aec. 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 user-supplied 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.
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).
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 f02ekc exits with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_TOO_MANY_ITER.
LargestMagnitude
Default
LargestImaginary
LargestReal
SmallestImaginary
SmallestMagnitude
SmallestReal
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.
Nolist
Default
List
Optional parameter ${\mathbf{List}}$ enables printing of each optional parameter specification as it is supplied. ${\mathbf{Nolist}}$ suppresses this printing.
Monitoring
$i$
Default $\text{}=\mathrm{-1}$
Unless ${\mathbf{Monitoring}}$ is set to $\mathrm{-1}$ (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).
PrintLevel
$i$
Default $\text{}=0$
This controls the amount of printing produced by f02ekc as follows.
$\mathit{i}$
Output
$=0$
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 re-orthogonalization, 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 re-orthogonalization.
$\ge 30$
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 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 ${\mathbf{Monitoring}}$ 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.
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 $\mathrm{op}\left(x\right)$.
${\mathbf{Regular}}$
$\mathrm{op}=A$
${\mathbf{Shifted\; Inverse\; Real}}$
$\mathrm{op}={(A-\sigma I)}^{\mathrm{-1}}$ 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.