Note:this function 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 function f12adc need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12adc for a detailed description of the specification of the optional parameters.
f12agc is the main solver function in a suite of functions consisting of f12adc,f12afcandf12agc. It must be called following an initial call to f12afc and following any calls to f12adc.
f12agc returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by real banded nonsymmetric matrices. The banded matrix must be stored using the LAPACK
column ordered
storage format for real banded nonsymmetric
(see Section 3.4.4 in the F07 Chapter Introduction).
The function may be called by the names: f12agc, nag_sparseig_real_band_solve or nag_real_banded_sparse_eigensystem_sol.
3Description
The suite of functions 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 banded, real and nonsymmetric.
Following a call to the initialization function f12afc, f12agc 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 banded nonsymmetric matrices. There is negligible additional computational cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
The banded matrices $A$ and $B$ must be stored using the LAPACK column ordered storage format for banded nonsymmetric matrices; please refer to Section 3.4.2 in the F07 Chapter Introduction for details on this storage format.
f12agc is based on the banded driver functions dnbdr1 to dnbdr6 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 functions offers the same functionality as the ARPACK banded driver software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to combine the different drivers into a general purpose function.
f12agc, is a general purpose function that must be called following initialization by f12afc. f12agc uses options, set either by default or explicitly by calling f12adc, to return the converged approximations to selected eigenvalues and (optionally):
–the corresponding approximate eigenvectors;
–an orthonormal basis for the associated approximate invariant subspace;
–both.
Please note that for ${\mathbf{Generalized}}$ problems, the ${\mathbf{Shifted\; Inverse\; Imaginary}}$ and ${\mathbf{Shifted\; Inverse\; Real}}$ inverse modes are only appropriate if either $A$ or $B$ is symmetric semidefinite. Otherwise, if $A$ or $B$ is non-singular, the ${\mathbf{Standard}}$ problem can be solved using the matrix ${B}^{-1}A$ (say).
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: $\mathbf{kl}$ – IntegerInput
On entry: the number of subdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{kl}}\ge 0$.
2: $\mathbf{ku}$ – IntegerInput
On entry: the number of superdiagonals of the matrices $A$ and $B$.
Note: the dimension, dim, of the array ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}}\times (2\times {\mathbf{kl}}+{\mathbf{ku}}+1))$ (see f12afc).
On entry: must contain the matrix $A$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.4.4 in the F07 Chapter Introduction).
Note: the dimension, dim, of the array mb
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}}\times (2\times {\mathbf{kl}}+{\mathbf{ku}}+1))$ (see f12afc).
On entry: must contain the matrix $B$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.4.4 in the F07 Chapter Introduction).
5: $\mathbf{sigmar}$ – doubleInput
On entry: if one of the ${\mathbf{Shifted\; Inverse\; Real}}$ modes (see f12adc) have been selected then sigmar must contain the real part of the shift used; otherwise sigmar is not referenced. Section 4.2.3.4 in the F12 Chapter Introduction describes the use of shift and inverse transformations.
6: $\mathbf{sigmai}$ – doubleInput
On entry: if one of the ${\mathbf{Shifted\; Inverse\; Real}}$ modes (see f12adc) have been selected then sigmai must contain the imaginary part of the shift used; otherwise sigmai is not referenced. Section 4.2.3.4 in the F12 Chapter Introduction describes the use of shift and inverse transformations.
Note: the dimension, dim, of the array dr
must be at least
${\mathbf{nev}}+1$ (see f12afc).
On exit: the first nconv locations of the array dr contain the real parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by nev since complex values occur as conjugate pairs and the second in the pair can be returned in position ${\mathbf{nev}}+1$ of the array.
Note: the dimension, dim, of the array di
must be at least
${\mathbf{nev}}+1$ (see f12afc).
On exit: the first nconv locations of the array di contain the imaginary parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by nev since complex values occur as conjugate pairs and the second in the pair can be returned in position ${\mathbf{nev}}+1$ of the array.
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ 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 array segments. The first segment holds the real part of the eigenvector and the second 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.
For example, if ${\mathbf{di}}\left[0\right]$ is nonzero, the first eigenvector has real parts stored in locations
${\mathbf{z}}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,{\mathbf{n}}-1$ and imaginary parts stored in
${\mathbf{z}}\left[\mathit{i}\right]$, for $\mathit{i}={\mathbf{n}},\dots ,2{\mathbf{n}}-1$.
Note: the dimension, dim, of the array resid
must be at least
${\mathbf{n}}$ (see f12afc).
On entry: need not be set unless the option ${\mathbf{Initial\; Residual}}$ has been set in a prior call to f12adc in which case resid must contain an initial residual vector.
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ (see f12adc) then the first nconv sections of v, of length $n$, will contain approximate Schur vectors that span the desired invariant subspace.
The $i$th Schur vector is stored in locations
${\mathbf{v}}\left[{\mathbf{n}}\times \left(\mathit{i}-1\right)+\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$ and $\mathit{j}=1,2,\dots ,n$.
Note: the actual argument supplied must be the array icomm supplied to the initialization routine f12afc.
On entry: must remain unchanged from the prior call to f12adcandf12afc.
On exit: contains no useful information.
15: $\mathbf{fail}$ – 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_COMP_BAND_FAC
Failure during internal factorization of complex banded matrix. Please contact NAG.
NE_COMP_BAND_SOL
Failure during internal solution of complex banded matrix. Please contact NAG.
NE_INITIALIZATION
Either the initialization function has not been called prior to the first call of this function or a communication array has become corrupted.
NE_INT
On entry, ${\mathbf{kl}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ku}}\ge 0$.
The maximum number of iterations $\text{}\le 0$, the option ${\mathbf{Iteration\; Limit}}$ has been set to $\u27e8\mathit{\text{value}}\u27e9$.
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
Error in internal call to compute eigenvectors. 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.
NE_INVALID_OPTION
On entry, ${\mathbf{Vectors}}=\text{Select}$, but this is not yet implemented.
NE_MAX_ITER
The maximum number of iterations has been reached. The maximum number of $\text{iterations}=\u27e8\mathit{\text{value}}\u27e9$. The number of converged eigenvalues $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
NE_NO_ARNOLDI_FAC
Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $=\u27e8\mathit{\text{value}}\u27e9$.
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 Lanczos iteration.
NE_OPT_INCOMPAT
The options ${\mathbf{Generalized}}$ and ${\mathbf{Regular}}$ are incompatible.
NE_REAL_BAND_FAC
Failure during internal factorization of real banded matrix. Please contact NAG.
NE_REAL_BAND_SOL
Failure during internal solution of real banded matrix. Please contact NAG.
NE_SCHUR_EIG_FAIL
During calculation of a real Schur form, there was a failure to compute a number of eigenvalues. Please contact NAG.
NE_SCHUR_REORDER
The computed Schur form could not be reordered by an internal call. Please contact NAG.
NE_TRANSFORM_OVFL
Overflow occurred during transformation of Ritz values to those of the original problem.
NE_ZERO_EIGS_FOUND
The number of eigenvalues found to sufficient accuracy is zero.
NE_ZERO_INIT_RESID
The option ${\mathbf{Initial\; Residual}}$ was selected but the starting vector held in resid is zero.
NE_ZERO_SHIFT
The option ${\mathbf{Shifted\; Inverse\; Imaginary}}$ has been selected and ${\mathbf{sigmai}}=\text{}$ zero on entry; sigmai must be nonzero for this mode of operation.
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 ${\mathbf{Tolerance}}$ used is the machine precision given by X02AJC.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f12agc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f12agc 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
None.
10Example
This example constructs the matrices $A$ and $B$ using LAPACK band storage format and solves $Ax=\lambda Bx$ in shifted imaginary mode using the complex shift $\sigma $.