f12jgc is a setup function in a suite of functions consisting of f12jac, f12jbc, f12jgc, f12jkc, f12jsc, f12jtc, f12juc and f12jvc. It is used to find some of the eigenvalues, and the corresponding eigenvectors, of a standard, generalized or polynomial eigenvalue problem. The initialization function f12jac must have been called prior to calling f12jgc. In addition calls to f12jbc can be made to supply individual optional parameters to f12jgc.

The suite of functions is suitable for the solution of large sparse eigenproblems where only a few eigenvalues from a selected range of the spectrum are required.

2 Specification

#include <nag.h>

void	f12jgc (void handle, Integer ccn, const Integer nedge[], const Integer tedge[], const Complex zedge[], NagError fail)

The function may be called by the names: f12jgc or nag_sparseig_feast_custom_contour.

3 Description

The suite of functions is designed to calculate some of the eigenvalues,

λ

, and the corresponding eigenvectors,

x

, of a standard eigenvalue problem

A x = λ x

, a generalized eigenvalue problem

A x = λ B x

, or a polynomial eigenvalue problem

\sum_{i} λ^{i} A_{i} x = 0

, where the coefficient matrices are large and sparse. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense problems.

f12jgc is used to specify a closed contour in the complex plane within which eigenvalues will be sought. The contour can be made up of a combination of line segments and half ellipses. f12jgc uses this information to create a polygonal representation of the contour and to then define the integration nodes and weights to be used by the solvers f12jkc, f12jsc, f12jtc, f12juc or f12jvc.

The arrays zedge, tedge and nedge are used to define the geometry of your contour. Each array is of size ccn, where ccn is the number of pieces that make up the contour. The entries in zedge specify the endpoints in the complex plane of each piece of the contour. The entries in tedge specify whether each piece of the contour is a line segment or a half ellipse. Finally, entries in nedge specify the number of integration points to use for each piece of the contour. See the individual argument descriptions in Section 5 for further details.

Prior to calling f12jgc, the option setting function f12jbc can be called to specify various optional parameters for the solution of the eigenproblem. For details of the options available and how to set them see Section 11.1 in f12jbc.

4 References

Polizzi E (2009) Density-Matrix-Based Algorithms for Solving Eigenvalue Problems Phys. Rev. B. 79 115112

5 Arguments

1: $handle$ – void * Input

On entry: the handle to the internal data structure used by the NAG FEAST suite. It needs to be initialized by f12jac. It must not be changed between calls to the NAG FEAST suite.

2: $ccn$ – Integer Input

On entry: the number of pieces that make up the contour.

Constraint:

ccn > 1

3: $nedge [ccn]$ – const Integer Input

On entry:

nedge [i - 1]

specifies how many integration nodes and weights f12jgc should use for the

i

th piece of the contour.

Constraint:

nedge [i - 1] > 0

, for

i = 1, 2, \dots, ccn

4: $tedge [ccn]$ – const Integer Input

On entry:

tedge [i - 1]

specifies what shape the

i

th piece of the contour should be.

$tedge [i - 1] = 0$: The $i$ th piece of the contour is straight.
$tedge [i - 1] > 0$: The $i$ th piece of the contour is a (convex) half-ellipse, with $tedge [i - 1] / 100 = a / b$ , where $a$ is the primary radius from the endpoints of the piece, and $b$ is the radius perpendicular to this. Thus, if $tedge [i - 1] = 100$ , then the $i$ th piece of the contour is a semicircle.

Constraint:

tedge [i - 1] \geq 0

, for

i = 1, 2, \dots, ccn

5: $zedge [ccn]$ – const Complex Input

On entry: zedge specifies the endpoints of the contour piece.

The

i

th piece has endpoints at

zedge [i - 1]

and

zedge [i]

, for

i = 1, \dots, ccn - 1

The last piece has endpoints at

zedge [ccn - 1]

and

zedge [0]

Note: the contour should be described in a clockwise direction..

6: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 $⟨ value ⟩$ had an illegal value.
NE_HANDLE: The supplied handle does not define a valid handle to the data structure used by the NAG FEAST suite. It has not been properly initialized or it has been corrupted.
NE_INT: On entry, $ccn = ⟨ value ⟩$ .
Constraint: $ccn > 1$ .
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_ARRAY: On entry, one or more elements of nedge were less than or equal to zero.

On entry, one or more elements of tedge were negative.
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.

7 Accuracy

Not applicable.

8 Parallelism and Performance

f12jgc is not threaded in any implementation.

9 Further Comments

The contour you specify must be convex and must not self-intersect. f12jgc does not explicitly test for either condition, but if the contour contains concave pieces or self-intersects then spurious eigenvalues may arise and stochastic estimation of the number of eigenvalues within the contour will not be accurate.