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 f12arc need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12arc for a detailed description of the specification of the optional parameters.
f12auc is the main solver function in a suite of functions consisting of f12arc,f12atcandf12auc. It must be called following an initial call to f12atc and following any calls to f12arc.
f12auc returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by complex banded non-Hermitian matrices. The banded matrix must be stored using the LAPACK
column ordered
storage format for complex banded non-Hermitian
(see Section 3.4.4 in the F07 Chapter Introduction).
The function may be called by the names: f12auc, nag_sparseig_complex_band_solve or nag_complex_banded_eigensystem_solve.
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, complex and non-Hermitian.
Following a call to the initialization function f12atc, f12auc returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or a unitary basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by complex banded non-Hermitian matrices. There is negligible additional computational cost to obtain eigenvectors; a unitary 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 non-Hermitian matrices; please refer to Section 3.4.4 in the F07 Chapter Introduction for details on this storage format.
f12auc is based on the banded driver functions znbdr1 to znbdr4 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 non-Hermitian matrices is provided in Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK banded driver software for complex non-Hermitian 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.
f12auc, is a general purpose function that must be called following initialization by f12atc. f12auc uses options, set either by default or explicitly by calling f12arc, to return the converged approximations to selected eigenvalues and (optionally):
–the corresponding approximate eigenvectors;
–a unitary 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
Note: in the following description n, nev and ncv appears. In every case they should be interpretted as the value associated with the identically named argument in a prior call to f12atc.
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
${\mathbf{n}}\times (2\times {\mathbf{kl}}+{\mathbf{ku}}+1)$.
On entry: must contain the matrix $A$ in LAPACK column-ordered banded storage format for non-Hermitian matrices; that is, element ${a}_{ij}$ is stored in ${\mathbf{ab}}\left[(j-1)\times (2\times {\mathbf{kl}}+{\mathbf{ku}}+1)+{\mathbf{kl}}+{\mathbf{ku}}+i-j\right]$, which may be written as ${\mathbf{ab}}\left[(2\times j-1)\times {\mathbf{kl}}+j\times {\mathbf{ku}}+i-1\right]$, for $\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,j-{\mathbf{ku}})\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(n,j+{\mathbf{kl}})$ and $j=1,2,\dots ,n$, (see Section 3.4.4 in the F07 Chapter Introduction).
Note: the dimension, dim, of the array mb
must be at least
${\mathbf{n}}\times (2\times {\mathbf{kl}}+{\mathbf{ku}}+1)$.
On entry: must contain the matrix $B$ in LAPACK column-ordered banded storage format for non-Hermitian matrices; that is, element ${a}_{ij}$ is stored in ${\mathbf{mb}}\left[(j-1)\times (2\times {\mathbf{kl}}+{\mathbf{ku}}+1)+{\mathbf{kl}}+{\mathbf{ku}}+i-j\right]$, which may be written as ${\mathbf{mb}}\left[(2\times j-1)\times {\mathbf{kl}}+j\times {\mathbf{ku}}+i-1\right]$, for $\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,j-{\mathbf{ku}})\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(n,j+{\mathbf{kl}})$ and $j=1,2,\dots ,n$, (see Section 3.4.4 in the F07 Chapter Introduction).
5: $\mathbf{sigma}$ – ComplexInput
On entry: if the ${\mathbf{Shifted\; Inverse}}$ mode (see f12arc) has been selected then sigma must contain the shift used; otherwise sigma is not referenced. Section 4.2.2 in the F12 Chapter Introduction describes the use of shift and invert transformations.
Note: the dimension, dim, of the array z
must be at least
${\mathbf{n}}\times {\mathbf{nev}}$ if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see f12arc) has been selected (see f12atc).
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see f12arc) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d, otherwise z is not referenced and may be NULL. The complex eigenvector associated with an eigenvalue ${\mathbf{d}}\left[j\right]$ is stored in the corresponding array section of z, namely
${\mathbf{z}}\left[{\mathbf{n}}\times \left(\mathit{j}-1\right)+\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nconv}}$.
Note: the dimension, dim, of the array resid
must be at least
${\mathbf{n}}$ (see f12atc).
On entry: need not be set unless the option ${\mathbf{Initial\; Residual}}$ has been set in a prior call to f12arc in which case resid must contain an initial residual vector.
On exit: contains the final residual vector. This can be used as the starting residual to improve convergence on the solution of a closely related eigenproblem. This has no relation to the error residual $Ax-\lambda x$ or $Ax-\lambda Bx$.
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ (see f12arc) has been set and a separate array z has been passed (i.e., z does not equal v), then the first nconv sections of v, of length $n$, will contain approximate Schur vectors that span the desired invariant subspace.
The $j$th Schur vector is stored in locations
${\mathbf{v}}\left[{\mathbf{n}}\times \left(\mathit{j}-1\right)+\mathit{i}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{nconv}}$ and $\mathit{i}=1,2,\dots ,n$.
On entry: must remain unchanged from the prior call to f12arcandf12atc.
13: $\mathbf{fail}$ – 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_EIGENVALUES
The number of eigenvalues found to sufficient accuracy is zero.
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: ${\mathbf{ku}}\ge 0$.
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.
The maximum number of iterations $\text{}\le 0$, the option ${\mathbf{Iteration\; Limit}}$ 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 $=\u27e8\mathit{\text{value}}\u27e9$.
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_OPT_INCOMPAT
The options ${\mathbf{Generalized}}$ and ${\mathbf{Regular}}$ are incompatible.
NE_OVERFLOW
Overflow occurred during transformation of Ritz values to those of the original problem.
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 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_TOO_MANY_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_ZERO_RESID
The option ${\mathbf{Initial\; Residual}}$ was selected but the starting vector held in resid is zero.
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
f12auc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f12auc 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 inverse mode using the complex shift $\sigma $.