f12 Chapter Contents
f12 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_complex_banded_eigensystem_init (f12atc)

## 1  Purpose

nag_complex_banded_eigensystem_init (f12atc) is a setup function for nag_complex_banded_eigensystem_solve (f12auc) which may be used for finding some 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 matrices (see Section 3.3.4 in the f07 Chapter Introduction).

## 2  Specification

 #include #include
 void nag_complex_banded_eigensystem_init (Integer n, Integer nev, Integer ncv, Integer icomm[], Integer licomm, Complex comm[], Integer lcomm, NagError *fail)

## 3  Description

The pair of functions nag_complex_banded_eigensystem_init (f12atc) and nag_complex_banded_eigensystem_solve (f12auc) together with the option setting function nag_complex_sparse_eigensystem_option (f12arc) are 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.
nag_complex_banded_eigensystem_init (f12atc) is a setup function which must be called before the option setting function nag_complex_sparse_eigensystem_option (f12arc) and the solver function nag_complex_banded_eigensystem_solve (f12auc). Internally, nag_complex_banded_eigensystem_solve (f12auc) makes calls to nag_complex_sparse_eigensystem_iter (f12apc) and nag_complex_sparse_eigensystem_sol (f12aqc); the function documents for nag_complex_sparse_eigensystem_iter (f12apc) and nag_complex_sparse_eigensystem_sol (f12aqc) should be consulted for details of the algorithm used.
This setup function initializes the communication arrays, sets (to their default values) all options that can be set by you via the option setting function nag_complex_sparse_eigensystem_option (f12arc), and checks that the lengths of the communication arrays as passed by you are of sufficient length. For details of the options available and how to set them, see Section 11.1 in nag_complex_sparse_eigensystem_option (f12arc).

## 4  References

Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562
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 Applications 17 789–821
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, Philidelphia

## 5  Arguments

1:     nIntegerInput
On entry: the order of the matrix $A$ (and the order of the matrix $B$ for the generalized problem) that defines the eigenvalue problem.
Constraint: ${\mathbf{n}}>0$.
2:     nevIntegerInput
On entry: the number of eigenvalues to be computed.
Constraint: $0<{\mathbf{nev}}<{\mathbf{n}}-1$.
3:     ncvIntegerInput
On entry: the number of Lanczos basis vectors to use during the computation.
At present there is no a priori analysis to guide the selection of ncv relative to nev. However, it is recommended that ${\mathbf{ncv}}\ge 2×{\mathbf{nev}}+1$. If many problems of the same type are to be solved, you should experiment with increasing ncv while keeping nev fixed for a given test problem. This will usually decrease the required number of matrix-vector operations but it also increases the work and storage required to maintain the orthogonal basis vectors. The optimal ‘cross-over’ with respect to CPU time is problem dependent and must be determined empirically.
Constraint: ${\mathbf{nev}}+1<{\mathbf{ncv}}\le {\mathbf{n}}$.
4:     icomm[$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$]IntegerCommunication Array
On exit: contains data to be communicated to nag_complex_banded_eigensystem_solve (f12auc).
5:     licommIntegerInput
On entry: the dimension of the array icomm.
If ${\mathbf{licomm}}=-1$, a workspace query is assumed and the function only calculates the required dimensions of icomm and comm, which it returns in ${\mathbf{icomm}}\left[0\right]$ and ${\mathbf{comm}}\left[0\right]$ respectively.
Constraint: ${\mathbf{licomm}}\ge 140\text{​ or ​}{\mathbf{licomm}}=-1$.
6:     comm[$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$]ComplexCommunication Array
On exit: contains data to be communicated to nag_complex_banded_eigensystem_solve (f12auc).
7:     lcommIntegerInput
On entry: the dimension of the array comm.
If ${\mathbf{lcomm}}=-1$, a workspace query is assumed and the function only calculates the dimensions of icomm and comm required by nag_complex_banded_eigensystem_solve (f12auc), which it returns in ${\mathbf{icomm}}\left[0\right]$ and ${\mathbf{comm}}\left[0\right]$ respectively.
Constraint: ${\mathbf{lcomm}}\ge 60\text{​ or ​}{\mathbf{lcomm}}=-1$.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{nev}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nev}}>0$.
The length of the complex array comm is too small ${\mathbf{lcomm}}=⟨\mathit{\text{value}}⟩$, but must be at least $⟨\mathit{\text{value}}⟩$.
The length of the integer array icomm is too small ${\mathbf{licomm}}=⟨\mathit{\text{value}}⟩$, but must be at least $⟨\mathit{\text{value}}⟩$.
NE_INT_3
On entry, ${\mathbf{ncv}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{nev}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ncv}}>{\mathbf{nev}}+1$ and ${\mathbf{ncv}}\le {\mathbf{n}}$.
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.

Not applicable.

Not applicable.

None.

## 10  Example

The use of nag_complex_banded_eigensystem_init (f12atc) is illustrated in Section 10 in nag_complex_banded_eigensystem_solve (f12auc).