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Appendix G: Numerical evaluation of wavefunction boundary integrals
Boundary methods are a central component of this thesis.
Closed integrals of a function
over the boundary coordinate are ubiquitous.
Generally
and
a square
matrix of integrals

(G.1) 
is required, where the indices label multiple functions.
For evaluation of the quantum band profile (Chapters 2 and
3),
the local density of states (Chapter 6),
the tension and areanorm matrices (Chapter 5), and
the Vergini matrix
and its derivative (Chapter 6),
and
are basis functions or eigenstates
which oscillate about zero on the length scale
, the quantum
(de Broglie) freespace wavelength.
The deformation function boundary integrals (Chapter 4)
do not involve any quantum scale, but are also evaluated using the method
below.
I will present only the case where boundary integrals over
become
line integrals over ; the generalization to higher is simple.
My tool for evaluation of an integral on a closed curve is the
discretization

(G.2) 
where is the range of , that is, the length of the line integral
(billiard perimeter).
The points are spread uniformly (equidistant in )
along the closed curve.
Because no point is special, no special quadrature [161]
weights arise near any endpoints: all weights are equal.
More sophisticated and accurate approximations
exist for closed line integral
evaluation [58], however this is sufficient for my needs
and is very simple to code.
Its errors will be discussed and tested below.
A single integral (G.2) requires function evaluations of .
Naively one might guess that filling a matrix using (G.1)
requires evaluations.
However, the correct way to compute (G.1) requires only
such evaluations:
First fill the rectangular matrices
and
, from which follows

(G.3) 
This matrix multiplication does require operations, but being
simple adds and multiplications (and using optimized library code
e.g. BLAS), it is very fast and does not affect the scaling.
If you like, the matrix multiply `performs' the integration over .
In the case where and are the same function, only evaluations
are required.
Note that if a general weighting function
is required in the
integrand (G.1), it can easily be incorporated into or ,
or equivalently be included as a diagonal matrix inserted between
and in (G.3).
Subsections
Next: Convergence with number of
Up: Dissipation in Deforming Chaotic
Previous: Appendix F: Cross correlations
Alex Barnett
20011003