# NAG CL Interfacef12aqc (complex_​proc)

Note: this function uses optional parameters to 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.

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## 1Purpose

f12aqc is a post-processing function in a suite of functions consisting of f12anc, f12apc, f12aqc, f12arc and f12asc. It must be called following a final exit from f12apc.

## 2Specification

 #include
 void f12aqc (Integer *nconv, Complex d[], Complex z[], Complex sigma, const Complex resid[], Complex v[], Complex comm[], Integer icomm[], NagError *fail)
The function may be called by the names: f12aqc, nag_sparseig_complex_proc or nag_complex_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 sparse, complex and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, complex and nonsymmetric problems.
Following a call to f12apc, f12aqc 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 complex nonsymmetric 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.
f12aqc is based on the function zneupd 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 software for complex nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
f12aqc is a post-processing function that must be called following a successful final exit from f12apc. f12aqc uses data returned from f12apc and options, set either by default or explicitly by calling f12arc, 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 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, Philadelphia

## 5Arguments

1: $\mathbf{nconv}$Integer * Output
On exit: the number of converged eigenvalues as found by f12arc.
2: $\mathbf{d}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array d must be at least ${\mathbf{ncv}}$ (see f12anc).
On exit: the first nconv locations of the array d contain the converged approximate eigenvalues.
3: $\mathbf{z}\left[{\mathbf{n}}×{\mathbf{ncv}}\right]$Complex Output
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see f12adc) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d. The complex eigenvector associated with an eigenvalue is stored in the corresponding array section of z.
4: $\mathbf{sigma}$Complex Input
On entry: if one of the ${\mathbf{Shifted Inverse}}$ (see f12arc) modes has been selected then sigma contains the shift used; otherwise sigma is not referenced.
5: $\mathbf{resid}\left[\mathit{dim}\right]$const Complex Input
Note: the dimension, dim, of the array resid must be at least ${\mathbf{n}}$ (see f12anc).
On entry: must not be modified following a call to f12apc since it contains data required by f12aqc.
6: $\mathbf{v}\left[{\mathbf{n}}×{\mathbf{ncv}}\right]$Complex Input/Output
The $\mathit{i}$th element of the $\mathit{j}$th basis vector is stored in location ${\mathbf{v}}\left[{\mathbf{n}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{ncv}}$.
On entry: the ncv sections of v, of length $n$, contain the Arnoldi basis vectors for $\mathrm{op}$ as constructed by f12apc.
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ 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.
7: $\mathbf{comm}\left[\mathit{dim}\right]$Complex Communication Array
Note: the actual argument supplied must be the array comm supplied to the initialization routine f12anc.
On initial entry: must remain unchanged from the prior call to f12anc.
On exit: contains data on the current state of the solution.
8: $\mathbf{icomm}\left[\mathit{dim}\right]$Integer Communication Array
Note: the actual argument supplied must be the array icomm supplied to the initialization routine f12anc.
On initial entry: must remain unchanged from the prior call to f12anc.
On exit: contains data on the current state of the solution.
9: $\mathbf{fail}$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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INTERNAL_EIGVEC_FAIL
In calculating eigenvectors, an internal call returned with an error. The function returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=⟨\mathit{\text{value}}⟩$. 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_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_RITZ_COUNT
Got a different count of the number of converged Ritz values than the value passed to it through the argument icomm: number counted $=⟨\mathit{\text{value}}⟩$, number expected $=⟨\mathit{\text{value}}⟩$. This usually indicates that a communication array has been altered or has become corrupted between calls to f12apc and f12aqc.
NE_SCHUR_EIG_FAIL
During calculation of a Schur form, there was a failure to compute $⟨\mathit{\text{value}}⟩$ eigenvalues in a total of $⟨\mathit{\text{value}}⟩$ iterations.
NE_SCHUR_REORDER
The computed Schur form could not be reordered by an internal call. This function returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=⟨\mathit{\text{value}}⟩$. Please contact NAG.
NE_ZERO_EIGS_FOUND
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 f12apc.

## 7Accuracy

The relative accuracy of a Ritz value, $\lambda$, is considered acceptable if its Ritz estimate $\le {\mathbf{Tolerance}}×|\lambda |$. 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.
f12aqc 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.

None.

## 10Example

This example solves $Ax=\lambda Bx$ in regular-invert mode, where $A$ and $B$ are derived from the standard central difference discretization of the one-dimensional convection-diffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\rho \frac{du}{dx}$ on $\left[0,1\right]$, with zero Dirichlet boundary conditions.

### 10.1Program Text

Program Text (f12aqce.c)

### 10.2Program Data

Program Data (f12aqce.d)

### 10.3Program Results

Program Results (f12aqce.r)