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.TH leastsq G "April 1999" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
leastsq - Solves non-linear least squares problems
.SH CALLING SEQUENCE
.nf
[f,xopt]=leastsq([imp,] fun [,Dfun],x0)
[f,[xopt,[gradopt]]]=leastsq(fun [,Dfun],[contr],x0,['algo'],[df0,[mem]],
,[stop],['in'])
.fi
.SH PARAMETERS
.TP 10
imp
: scalar argument used to set the trace mode. \fVimp=0\fR nothing
(execpt errors) is reported, \fVimp=1\fR initial and final reports,
\fVimp=2\fR adds a report per iteration, \fVimp>2\fR add reports on
linear search. Warning, most of these reports are written on the
Scilab standard output.
.TP
fun
: external, i.e Scilab function or string (\fVfun\fR is the function
defining the least square probleme: see below.
.TP
x0
: real vector (initial value of variable to be minimized).
.TP
f
: value of optimal least square value.
.TP
xopt
: best value of \fVx\fR found.
.TP
contr
: \fV'b',binf,bsup\fR with \fVbinf\fR and \fVbsup\fR real vectors with same
dimension as \fVx0\fR. \fVbinf\fR and \fVbsup\fR are lower and upper bounds on \fVx\fR.
.TP
algo
: A string with possible values :\fV'qn'\fR or \fV'gc'\fR or \fV'nd'\fR . This string stands for quasi-Newton (default),
conjugate gradient or non-differentiable respectively.
Note that \fV'nd'\fR does not accept bounds on \fVx\fR ).
.TP
df0
: real scalar. Guessed decreasing of \fVf\fR at first iteration.
(\fVdf0=1\fR is the default value).
.TP
mem :
integer, number of variables used to approximate the
Hessian, (\fValgo='gc' or 'nd'\fR). Default value is around 6.
.TP
stop
: sequence of optional parameters controlling the
convergence of the algorithm.
\fV stop= 'ar',nap, [iter [,epsg [,epsf [,epsx]]]]\fR
.RS
.TP
"ar"
: reserved keyword for stopping rule selection defined as follows:
.TP
nap
: maximum number of calls to \fVfun\fR allowed.
.TP
iter
: maximum number of iterations allowed.
.TP
epsg
: threshold on gradient norm.
.TP
epsf
: threshold controlling decreasing of \fVf\fR
.TP
epsx
: threshold controlling variation of \fVx\fR.
This vector (possibly matrix) of same size as \fVx0\fR can be used
to scale \fVx\fR.
.RE
.TP
"in"
: reserved keyword for initialization of parameters
used when \fVfun\fR in given as a Fortran routine (see below).
.TP
gradopt
: gradient of \fVfun\fR at \fVxopt\fR
.SH DESCRIPTION
Non-linear optimization routine for programs without constraints or with bound constraints:
.nf
min sum(f(x).^2 w.r.t x.
.fi
.TP
\fVfun\fR
is an "external" i.e function, or list or Fortran routine (see "external").
This external must return (\fVf(x)\fR) for a given \fVx\fR.
.LP
If \fVfun\fR is a function, the calling sequence for \fVfun\fR must be:
.nf
[f]=fun(x, [optional parameters]).
.fi
Here, \fVfun\fR is a function from R^n to R^m which returns \fVf(x)\fR, a
real vector ( value of function at \fVx\fR)
.LP
If fun is defined by a Fortran or C routine first argument must be a
list:
\list(fun_name,m,...)
If \fVfun_name\fR is a character string, it refers to the name of the routine
which must be linked to Scilab.
.LP
Here, the generic calling sequence for the Fortran subroutine is:
\fVsubroutine fun(m,n,x,td,f)\fR
.LP
\fVn\fR is the dimension of \fVx\fR, \fVx\fR is an \fVn\fR vector,
\fVtd\fR are working arrays which may also be used to pass parameters
.LP
If \fVfun\fR is given as a Fortran routine, it is possible to initialize
parameters or to send Scilab variables to this routine using sequence
of arguments \fV'td' , valtd\fR. Then, the Fortran function
\fVfun(m,n , x, f, td)\fR is evaluated with \fVtd=valtd\fR.
Thus, the Scilab variables \fVvaltd\fR are sent to the Fortran
function \fVfun\fR.
.TP
\fVDfun\fR
is an "external".This external must return a matrix \fVg\fR such
as (\fVg(i,j)=dfi/dxj\fR) for a given \fVx\fR.
.LP
If \fVDfun\fR is a function, the calling sequence for \fVfun\fR must be:
.nf
[g]=Dfun(x, [optional parameters]).
.fi
.LP
If Dfun is defined by a Fortran or C routine first argument must be a
list:
\list(fun_name,m,...)
If \fVfun_name\fR is a character string, it refers to the name of the routine
which must be linked to Scilab.
.LP
Here, the generic calling sequence for the Fortran subroutine is:
\fVsubroutine dfun(m,n,x,td,g)\fR
.SH EXAMPLES
.nf
a=rand(3,2);b=[1;1;1];x0=[1;-1];
deff('f=fun(x,a,b)','f=a*x-b');
deff('g=dfun(x,a,b)','g=a');
[f,xopt]=leastsq(fun,x0) //Simplest call
xopt-a\b //compare with linear algebra solution
[f,xopt]=leastsq(fun,dfun,x0) //specify gradient
[f,xopt]=leastsq(list(fun,[1 2;3 4],[1;2]),x0)
deff('f=fun(x,a,b)','f=exp(a*x)-b');
deff('g=dfun(x,a,b)','g=a.*(exp(a*x)*ones(1,size(a,2)))');
[f,xopt]=leastsq(list(fun,[1 2;3 4],[1;2]),x0)
.fi
.SH SEE ALSO
external, quapro, linpro
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