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 f12fdf need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12fdf for a detailed description of the specification of the optional parameters.
f12fcf is a post-processing routine in a suite of routines which includes f12faf,f12fbf,f12fdfandf12fef. f12fcf must be called following a final exit from f12fbf.
The routine may be called by the names f12fcf or nagf_sparseig_real_symm_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 symmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and symmetric problems.
Following a call to f12fbf, f12fcf 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 symmetric 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.
f12fcf is based on the routine dseupd from the ARPACK package, which uses the Implicitly Restarted Lanczos 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 symmetric matrices is provided in Lehoucq and Scott (1996). This suite of routines offers the same functionality as the ARPACK software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
f12fcf, is a post-processing routine that must be called following a successful final exit from f12fbf. f12fcf uses data returned from f12fbf and options, set either by default or explicitly by calling f12fdf, 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 f12fbf.
2: $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array d
must be at least
${\mathbf{ncv}}$ (see f12faf).
On exit: the first nconv locations of the array d contain the converged approximate eigenvalues.
3: $\mathbf{z}({\mathbf{ldz}},*)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z
must be at least
${\mathbf{ncv}}$ 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 f12faf).
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see f12fdf) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d. The real eigenvector associated with an eigenvalue is stored in the corresponding column of z.
4: $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f12fcf 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, ${\mathbf{ldz}}\ge 1$.
5: $\mathbf{sigma}$ – Real (Kind=nag_wp)Input
On entry: if one of the Shifted Inverse (see f12fdf) modes has been selected then sigma contains the real shift used; otherwise sigma is not referenced.
6: $\mathbf{resid}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array resid
must be at least
${\mathbf{n}}$ (see f12faf).
On entry: must not be modified following a call to f12fbf since it contains data required by f12fcf.
7: $\mathbf{v}({\mathbf{ldv}},*)$ – 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 f12faf).
On entry: the ncv columns of v contain the Lanczos basis vectors for $\mathrm{op}$ as constructed by f12fbf.
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.
8: $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f12fcf is called.
Constraint:
${\mathbf{ldv}}\ge {\mathbf{n}}$.
9: $\mathbf{comm}\left(*\right)$ – Real (Kind=nag_wp) arrayCommunication Array
Note: the actual argument supplied must be the array comm supplied to the initialization routine f12faf.
On initial entry: must remain unchanged from the prior call to f12faf.
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 f12faf.
On initial entry: must remain unchanged from the prior call to f12faf.
On exit: contains data on the current state of the solution.
11: $\mathbf{ifail}$ – IntegerInput/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, ${\mathbf{ldz}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ in f12faf.
Constraint: ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$ (see n in f12faf).
${\mathbf{ifail}}=2$
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 f12fbf.
${\mathbf{ifail}}=4$
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 f12fbfandf12fcf.
${\mathbf{ifail}}=5$
During calculation of a tridiagonal form, there was a failure to compute $\u27e8\mathit{\text{value}}\u27e9$ eigenvalues in a total of $\u27e8\mathit{\text{value}}\u27e9$ iterations.
${\mathbf{ifail}}=8$
Either the routine was called out of sequence (following an initial call to the setup routine and following completion of calls to the reverse communication routine) or the communication arrays have 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
f12fcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f12fcf 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 mode, where $A$ and $B$ are obtained from the standard central difference discretization of the one-dimensional Laplacian operator $\frac{{d}^{2}u}{d{x}^{2}}$
on $[0,1]$, with zero Dirichlet boundary conditions.