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.
f12agf is the main solver routine in a suite of routines consisting of f12adf,f12affandf12agf. It must be called following an initial call to f12aff and following any calls to f12adf.
f12agf 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
storage format for real banded nonsymmetric matrices.
The routine may be called by the names f12agf or nagf_sparseig_real_band_solve.
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 banded, real and nonsymmetric.
Following a call to the initialization routine f12aff, f12agf 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 3.3.2 in the F07 Chapter Introduction for details on this storage format.
f12agf is based on the banded driver routines 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 routines 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 routine.
f12agf, is a general purpose routine that must be called following initialization by f12aff. f12agf uses 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.
Please note that for Generalized problems, the Shifted Inverse Imaginary and 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 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 $B$.
Constraint:
${\mathbf{ku}}\ge 0$.
3: $\mathbf{ab}({\mathbf{ldab}},*)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$ (see f12aff).
On entry: must contain the matrix $A$ in LAPACK banded storage format for nonsymmetric matrices (see Section 3.3.4 in the F07 Chapter Introduction).
4: $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f12agf is called.
On entry: if one of the Shifted Inverse Real modes (see f12adf) 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.
8: $\mathbf{sigmai}$ – Real (Kind=nag_wp)Input
On entry: if one of the Shifted Inverse Real modes (see f12adf) 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.
9: $\mathbf{nconv}$ – IntegerOutput
On exit: the number of converged eigenvalues.
10: $\mathbf{dr}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array dr
must be at least
${\mathbf{nev}}+1$ (see f12aff).
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.
11: $\mathbf{di}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array di
must be at least
${\mathbf{nev}}+1$ (see f12aff).
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.
12: $\mathbf{z}({\mathbf{ldz}},*)$ – 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 f12aff).
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 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.
13: $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f12agf 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, ${\mathbf{ldz}}\ge 1$.
14: $\mathbf{resid}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array resid
must be at least
${\mathbf{n}}$ (see f12aff).
On entry: need not be set unless the option Initial Residual has been set in a prior call to f12adf in which case resid must contain an initial residual vector.
On exit: contains the final residual vector.
15: $\mathbf{v}({\mathbf{ldv}},*)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array v
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncv}})$ (see f12aff).
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ (see f12adf) then the first ${\mathbf{nconv}}\times n$ elements of v will contain approximate Schur vectors that span the desired invariant subspace.
The $i$th Schur vector is stored in the $i$th column of v.
16: $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f12agf is called.
Constraint:
${\mathbf{ldv}}\ge {\mathbf{n}}$.
17: $\mathbf{comm}\left(*\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On entry: must remain unchanged from the prior call to f12adfandf12aff.
On entry: must remain unchanged from the prior call to f12adfandf12aff.
On exit: contains no useful information.
19: $\mathbf{ifail}$ – 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, ${\mathbf{kl}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{kl}}\ge 0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ku}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ku}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldab}}=\u27e8\mathit{\text{value}}\u27e9$, $2\times {\mathbf{kl}}+{\mathbf{ku}}+1=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldab}}\ge 2\times {\mathbf{kl}}+{\mathbf{ku}}+1$.
${\mathbf{ifail}}=4$
The option Shifted Inverse Imaginary has been selected and ${\mathbf{sigmai}}=\text{}$ zero on entry; sigmai must be nonzero for this mode of operation.
${\mathbf{ifail}}=5$
The maximum number of iterations $\text{}\le 0$, the option Iteration Limit has been set to $\u27e8\mathit{\text{value}}\u27e9$.
The option Initial Residual was selected but the starting vector held in resid is zero.
${\mathbf{ifail}}=8$
Either the initialization routine has not been called prior to the first call of this routine or a communication array has become corrupted.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{ldz}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ in f12aff.
Constraint: ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
${\mathbf{ifail}}=10$
On entry, ${\mathbf{Vectors}}=\text{Select}$, but this is not yet implemented.
${\mathbf{ifail}}=11$
The number of eigenvalues found to sufficient accuracy is zero.
${\mathbf{ifail}}=12$
Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=13$
Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact NAG.
${\mathbf{ifail}}=14$
During calculation of a real Schur form, there was a failure to compute a number of eigenvalues. Please contact NAG.
${\mathbf{ifail}}=15$
The computed Schur form could not be reordered by an internal call. Please contact NAG.
${\mathbf{ifail}}=16$
Error in internal call to compute eigenvectors. Please contact NAG.
${\mathbf{ifail}}=17$
Failure during internal factorization of real banded matrix. Please contact NAG.
${\mathbf{ifail}}=18$
Failure during internal solution of real banded matrix. Please contact NAG.
${\mathbf{ifail}}=19$
Failure during internal factorization of complex banded matrix. Please contact NAG.
${\mathbf{ifail}}=20$
Failure during internal solution of complex banded matrix. Please contact NAG.
${\mathbf{ifail}}=21$
The maximum number of iterations has been reached. The maximum number of $\text{iterations}=\u27e8\mathit{\text{value}}\u27e9$. The number of converged eigenvalues $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=22$
No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.
${\mathbf{ifail}}=23$
Overflow occurred during transformation of Ritz values to those of the original problem.
${\mathbf{ifail}}=-99$
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
f12agf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f12agf 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 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 $.