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.TH "mdfl1d" 2 "March 12, 1998" "Fractales Group" "Scilab Function"
.so ../sci.an
.SH NAME
mdfl1d - Discrete Legendre spectrum estimation on 1d measure
.sp
Author: Christophe Canus
.sp
This routine estimates the discrete Legendre Spectrum on 1d measure.
.sp
.sp
.SH Usage
\f(CR[\fPalpha,f_alpha\f(CR]\fP=mdfl1d(mu_n,N,n)
.SH Input parameters
.RS
.TP
o
\fBmu_n\fP : strictly positive real vector \f(CR[\fP1,nu_n\f(CR]\fP
Contains the pre-multifractal measure.
.TP
o
\fBN\fP : strictly positive real (integer) scalar
Contains the number of Hoelder exponents.
.TP
o
\fBn\fP : strictly positive real (integer) scalar
Contains the final resolution.
.RE
.SH Output parameters
.RS
.TP
o
\fBalpha\fP : real vector \f(CR[\fP1,N\f(CR]\fP
Contains the Hoelder exponents.
.TP
o
\fBf_alpha\fP : real vector \f(CR[\fP1,N\f(CR]\fP
Contains the dimensions.
.RE
.SH Description
.SH Parameters
The discrete Legendre spectrum \fIf_alpha\fP is estimated on the finite
finer resolution of the pre-multifractal 1d measure \fImu_n\fP. The
three steps of the estimatation are:
.RS
.TP
o
estimation of the partition function;
.TP
o
estimation of the Reyni exponents;
.TP
o
estimation of the Legendre transform.
.RE
.SH Algorithm details
The discrete partition function is estimated by coarse-graining masses
mu_n into non-overlapping boxes of increasing diameter (box
method). If \fInu_n\fP is a power of 2, 2\(ha\fIn\fP
corresponds to the coarser scale.
The reyni exponents are estimated by least square
linear regression.
The Legendre transform of the mass exponent function is estimated with
the linear-time Legendre transform.
.SH See also
\fBmdzq1d,mdzq2d,reynitq,linearlt,mdfl2d\fP.
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