Note:this routine usesoptional parametersto define choices in the problem specification. If you wish to use default settings for all of the optional parameters, then the option setting routine f12adf need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12adf for a detailed description of the specification of the optional parameters.
The routine may be called by the names f12acf or nagf_sparseig_real_proc.
3Description
The suite of routines is designed to calculate some of the eigenvalues, $\lambda $, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax=\lambda x$, or of a generalized eigenvalue problem $Ax=\lambda Bx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are sparse, real and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and nonsymmetric problems.
Following a call to f12abf, f12acf returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by real nonsymmetric matrices. There is negligible additional cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
f12acf is based on the routine dneupd from the ARPACK package, which uses the Implicitly Restarted Arnoldi iteration method. The method is described in Lehoucq and Sorensen (1996) and Lehoucq (2001) while its use within the ARPACK software is described in great detail in Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices is provided in Lehoucq and Scott (1996). This suite of routines offers the same functionality as the ARPACK software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
f12acf is a post-processing routine that must be called following a successful final exit from f12abf. f12acf uses data returned from f12abf and options, set either by default or explicitly by calling f12adf, to return the converged approximations to selected eigenvalues and (optionally):
–the corresponding approximate eigenvectors;
–an orthonormal basis for the associated approximate invariant subspace;
–both.
4References
Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications17 789–821
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: $\mathbf{nconv}$ – IntegerOutput
On exit: the number of converged eigenvalues as found by f12abf.
2: $\mathbf{dr}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array dr
must be at least
${\mathbf{nev}}+1$ (see f12aaf).
On exit: the first nconv locations of the array dr contain the real parts of the converged approximate eigenvalues.
3: $\mathbf{di}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array di
must be at least
${\mathbf{nev}}+1$ (see f12aaf).
On exit: the first nconv locations of the array di contain the imaginary parts of the converged approximate eigenvalues.
4: $\mathbf{z}({\mathbf{ldz}},*)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z
must be at least
${\mathbf{nev}}+1$ if the default option
${\mathbf{Vectors}}=\mathrm{RITZ}$
has been selected and at least $1$ if the option
${\mathbf{Vectors}}=\mathrm{NONE}$ or $\mathrm{SCHUR}$
has been selected (see f12aaf).
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see f12adf) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in dr and di. The complex eigenvector associated with the eigenvalue with positive imaginary part is stored in two consecutive columns. The first column holds the real part of the eigenvector and the second column holds the imaginary part. The eigenvector associated with the eigenvalue with negative imaginary part is simply the complex conjugate of the eigenvector associated with the positive imaginary part.
5: $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f12acf is called.
Constraints:
if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ has been selected, ${\mathbf{ldz}}\ge {\mathbf{n}}$;
if the option ${\mathbf{Vectors}}=\mathrm{NONE}$ or $\mathrm{SCHUR}$ has been selected, ${\mathbf{ldz}}\ge 1$.
6: $\mathbf{sigmar}$ – Real (Kind=nag_wp)Input
On entry: if one of the Shifted Inverse Real modes have been selected then sigmar contains the real part of the shift used; otherwise sigmar is not referenced.
7: $\mathbf{sigmai}$ – Real (Kind=nag_wp)Input
On entry: if one of the Shifted Inverse Real modes have been selected then sigmai contains the imaginary part of the shift used; otherwise sigmai is not referenced.
8: $\mathbf{resid}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array resid
must be at least
${\mathbf{n}}$ (see f12aaf).
On entry: must not be modified following a call to f12abf since it contains data required by f12acf.
9: $\mathbf{v}({\mathbf{ldv}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array v
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncv}})$ (see f12aaf).
On entry: the ncv columns of v contain the Arnoldi basis vectors for $\mathrm{op}$ as constructed by f12abf.
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ has been set, or the option ${\mathbf{Vectors}}=\mathrm{RITZ}$ has been set and a separate array z has been passed (i.e., z does not equal v), then the first nconv columns of v will contain approximate Schur vectors that span the desired invariant subspace.
10: $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f12acf is called.
Constraint:
${\mathbf{ldv}}\ge {\mathbf{n}}$.
11: $\mathbf{comm}\left(*\right)$ – Real (Kind=nag_wp) arrayCommunication Array
Note: the actual argument supplied must be the array comm supplied to the initialization routine f12aaf.
On initial entry: must remain unchanged from the prior call to f12abf.
On exit: contains data on the current state of the solution.
Note: the actual argument supplied must be the array icomm supplied to the initialization routine f12aaf.
On initial entry: must remain unchanged from the prior call to f12abf.
On exit: contains data on the current state of the solution.
13: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ 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 $\mathrm{-1}$, 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, ${\mathbf{ldz}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ in f12aaf, or ${\mathbf{ldz}}<1$ when no vectors required.
Constraint: ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$ (see n in f12aaf).
${\mathbf{ifail}}=2$
On entry, ${\mathbf{Vectors}}=\mathrm{SELECT}$, but this is not yet implemented.
${\mathbf{ifail}}=3$
The number of eigenvalues found to sufficient accuracy, as communicated through the argument icomm, is zero. You should experiment with different values of nev and ncv, or select a different computational mode or increase the maximum number of iterations prior to calling f12abf.
${\mathbf{ifail}}=4$
Got a different count of the number of converged Ritz values than the value passed to it through the argument icomm: number counted $=\u27e8\mathit{\text{value}}\u27e9$, number expected $=\u27e8\mathit{\text{value}}\u27e9$. This usually indicates that a communication array has been altered or has become corrupted between calls to f12abfandf12acf.
${\mathbf{ifail}}=5$
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.
${\mathbf{ifail}}=6$
The computed Schur form could not be reordered by an internal call. This routine returned with ${\mathbf{ifail}}=\u27e8\mathit{\text{value}}\u27e9$. Please contact NAG.
${\mathbf{ifail}}=7$
In calculating eigenvectors, an internal call returned with an error. Please contact NAG.
${\mathbf{ifail}}=8$
Either the solver routine has not been called prior to the call of this routine or a communication array has become corrupted.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
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, $\lambda $, is considered acceptable if its Ritz estimate $\le {\mathbf{Tolerance}}\times \left|\lambda \right|$. The default Tolerance used is the machine precision given by x02ajf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f12acf 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.
9Further Comments
None.
10Example
This example solves $Ax=\lambda Bx$ in regular-invert mode, where $A$ and $B$ are obtained from the standard central difference discretization of the one-dimensional convection-diffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\rho \frac{du}{dx}$
on $[0,1]$, with zero Dirichlet boundary conditions.