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prescrib(2) Scilab Function prescrib(2)
NAME
prescrib - Generation of signals with prescribed Holder function
Author: Khalid Daoudi
Using the GIFS method, this routine generates a continous function with
prescribed Holder function, while interpolating a set of point.
Usage
[x,y]=prescrib(Interp_pts, Holder_funct, nbr_iter)
Input parameters
o Interp_pts : Real matrix [n,2] Contains the interpolation points in
the format : abscissa-ordinate.
o Holder_funct : Character string Specifies the Holder function you
want to prescribe. It must have the form of compositions of matlab
functions of variable t ('2*sqrt(1-t)' for instance). The use of the
variable t is crucial. For shake of simplicity, this variable t is
supposed to vary in [0,1].
o nbr_iter : integer Number of iteration wanted in the generation
process of the GIFS attractor.
Output parameters
o x : Real vector Contains the abscissa of the attractor graph.
o y : Real vector Contains the ordinates of the attractor graph.
Description
Parameters
o Interp_pts is a real matrix [n,2] containing the cordinates of the
interpolation points.
o Holder_funct is a character string specifying the Holder function
you want to prescribe. This means that GIFS attrcator will have, at
a point t, a Holder exponent equal to the value of this function at
pint t.
o nbr_iter is the number of iteration wanted in the generation pro-
cess of the GIFS attractor.
o x and y contain the cordinates of the GIFS attractor.
Algorithm details
Generalized Iterated Functions Systems (GIFS) are a generalization of the
usual IFS. This generalization consists in allowing the contarations to
change at each step (scale) of the attractor generation process. We also
allow their number and their support to change. Here, we use the GIFS to
construct continuous function with prescribed local regularity. More pre-
cisely, if H(t) is the prescribed Holder function, then for each
j=1,...,nbr_iter-1, and for each k=0,...,pow(m,j)-1, the GIFS coefficient
c_kj is definied as : c_kj = pow(m,H(k*pow(m,-j))), where m+1 is the number
of interpolation points. The resulting attractor is the graph of a con-
tinuous function F such that the Holder exponent of F, at each point t, is
H(t). Moreover, if {(t_i, y_i), i=1,...,m+1} is the set of interpolation
points, then F(t_i)=y_i for each i=1,...,m+1.
See also:
gifs and alphagifs
Example:
I = [0 0 1 0]; [x,y] = prescrib(I,'abs(sin(3*pi*t))',10); plot(x,y)
[x,y] is the graph of a continuous function F which interpolates {(0,0);
(0.5 1); (1,0)} and such that the Holder exponent of F, at each point t, is
abs(sin(3*pi*t)).
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