PIGALE 1.3.9 Documentation

void CalcNorm	(	double	p,
		double	q,
		double &	norm,
		double &	c,
		double &	s
	)			`[static]`

Computes ${\rm norm}=\sqrt{{\rm p}^2+{\rm q}^2}, {\rm c}=\frac{\rm p}{\rm norm},{\rm s}=\frac{\rm q}{\rm norm}$ .

int ComputeBilinearForm	(	double **	dis,
		int	ni
	)			`[static]`

Computes the bilinear form to be put in diagonal form from the distance matrix dis.

Computes the bilinear form:

${\rm dis}\leftarrow -\frac{1}{2}\left(I-\frac{J}{\rm ni}\right){\rm dis}\left(I-\frac{J}{\rm ni}\right)$

from the developped formula:

${\rm dis}_{i,j}\leftarrow -\frac{1}{2}\left( {\rm dis}_{i,j}-\frac{1}{\rm ni}\sum_{\alpha=1}^{\rm ni} {\rm dis}_{i,\alpha} -\frac{1}{\rm ni}\sum_{\alpha=1}^{\rm ni} {\rm dis}_{\alpha,j} +\frac{1}{{\rm ni}^2}\sum_{\alpha=1}^{\rm ni}\sum_{\beta=1}^{\rm ni}{\rm dis}_{\alpha,\beta}\right)$

int diag	(	double **	dis,
		int	NumberOfPoints,
		double **	Distances,
		svector< double > &	EigenValues,
		bool	project
	)

Isometrically embed a set of points with given distances among them in the Euclidean space $\mathbb{R}^n$ .

Parameters:

	dis	coordinates of the points (returned value)
	NumberOfPoints	number of points > 2
	Distances	distances among the points
	EigenValues	eigenvalues of `dis` (returned value)
	project	indicates if one should project the matrix of distances before putting it in diagonal form (always true except when using the Laplacian)

Actions:

copies the Distances matrix into the dis matrix,
call ComputeBilinearForm() to compute the projection matrix dis to be put in diagonal form
compute the trace of dis
call symqr() to compute the spectrum of dis
normalise the coordinates
compute the multiplicities of the eigenvalues and print some useful information.

See also:: ComputeBilinearForm() and symqr()

int symqr	(	double **	dis,
		int	ni,
		int	nf,
		double &	trace,
		svector< double > &	EigenValues
	)			`[static]`

Computes the eigenvalues of the ni $\times$ ni matrix dis.

const double epsilon = 1.E-12

rounding precision

Diagonalise.cpp File Reference

Detailed Description

Functions

Variables

Function Documentation

Variable Documentation