Solves non-linear least squares problems
fopt=leastsq(fun, x0) fopt=leastsq(fun, x0) fopt=leastsq(fun, dfun, x0) fopt=leastsq(fun, cstr, x0) fopt=leastsq(fun, dfun, cstr, x0) fopt=leastsq(fun, dfun, cstr, x0, algo) fopt=leastsq([iprint], fun [,dfun] [,cstr],x0 [,algo],[df0,[mem]],[stop]) [fopt,xopt] = leastsq(...) [fopt,xopt,gopt] = = leastsq(...)
value of the function f(x)=||fun(x)||^2
at xopt
best value of x
found to minimize
||fun(x)||^2
gradient of f
at
xopt
a scilab function or a list defining a function from
R^n
to R^m
(see more
details in DESCRIPTION).
real vector (initial guess of the variable to be minimized).
a scilab function or a string defining the Jacobian matrix of
fun
(see more details in DESCRIPTION).
bound constraints on x
. They must be
introduced by the string keyword 'b'
followed by
the lower bound binf
then by the upper bound
bsup
(so cstr
appears as
'b',binf,bsup
in the syntax). Those
bounds are real vectors with same dimension than
x0
(-%inf and +%inf may be used for dimension
which are unrestricted).
a string with possible values: 'qn'
or
'gc'
or 'nd'
. These strings
stand for quasi-Newton (default), conjugate gradient or
non-differentiable respectively. Note that 'nd'
does not accept bounds on x
.
scalar argument used to set the trace mode.
iprint=0
nothing (except errors) is reported,
iprint=1
initial and final reports,
iprint=2
adds a report per iteration,
iprint>2
add reports on linear search. Warning,
most of these reports are written on the Scilab standard
output.
real scalar. Guessed decreasing of
||fun||^2
at first iteration.
(df0=1
is the default value).
integer, number of variables used to approximate the Hessian
(second derivatives) of f
when
algo
='qn'
. Default value is
10.
sequence of optional parameters controlling the convergence of
the algorithm. They are introduced by the keyword
'ar'
, the sequence being of the form
'ar',nap, [iter [,epsg [,epsf [,epsx]]]]
maximum number of calls to fun
allowed.
maximum number of iterations allowed.
threshold on gradient norm.
threshold controlling decreasing of
f
threshold controlling variation of x
.
This vector (possibly matrix) of same size as
x0
can be used to scale
x
.
The leastsq
function
solves the problem
where f
is a function from
R^n
to R^m
.
Bound constraints cab be imposed on x
.
fun
can be a scilab function (case
1) or a fortran or a C routine linked to scilab (case 2).
When fun
is a Scilab function, its calling
sequence must be:
y=fun(x)
y=f(x,a1,a2,...)
fun
as a list,
which contains list(f,a1,a2,...)
.When fun
is a Fortran or C
routine, it must be list(fun_name,m[,a1,a2,...])
in the syntax of
leastsq
, where fun_name
is
a 1-by-1 matrix of strings, the name of the routine which must be linked to Scilab (see
link). The header must be, in Fortran:
subroutine fun(m, n, x, params, y) integer m,n double precision x(n), params(*), y(m)
void fun(int *m, int *n, double *x, double *params, double *y)
n
is the dimension of vector
x
, m
the dimension of vector
y
, with y=fun(x)
, and
params
is a vector which contains the optional
parameters a1, a2, ...
. Each
parameter may be a vector, for instance if
a1
has 3 components, the description of
a2
begin from
params(4)
(in fortran), and from
params[3]
(in C).
Note that even if fun
does not need supplementary parameters you
must anyway write the fortran code with a params
argument (which is then unused in the subroutine core).By default, the algorithm uses a finite difference approximation
of the Jacobian matrix.
The Jacobian matrix can be provided by defining the function
dfun
, where to the
optimizer it may be given as a usual scilab function or
as a fortran or a C routine linked to scilab.
when dfun
is a scilab function, its calling
sequence must be:
y=dfun(x)
y(i,j)=dfi/dxj
.
If extra parameters are required by fun
, i.e. if arguments
a1,a2,...
are required, they are passed also to
dfun
, which must have header
y=dfun(x,a1,a2,...)
dfun
needs extra parameters, it must appear simply as
dfun
in the syntax of
leastsq
.When dfun
is defined by a Fortran or C
routine it must be a string, the name of the function linked to
Scilab.
The calling sequences must be, in Fortran:
subroutine dfun(m, n, x, params, y) integer m,n double precision x(n), params(*), y(m,n)
void fun(int *m, int *n, double *x, double *params, double *y)
y(i,j)=dfi/dxj
must be
stored in y[m*(j-1)+i-1]
.Like datafit,
leastsq
is a front end onto the optim function. If you want to try the
Levenberg-Marquard method instead, use lsqrsolve.
A least squares problem may be solved directly with the optim function ; in this case the function NDcost may be useful to compute the derivatives (see the NDcost help page which provides a simple example for parameters identification of a differential equation).
We will show different calling possibilities of leastsq on one (trivial) example which is non linear but does not really need to be solved with leastsq (applying log linearizes the model and the problem may be solved with linear algebra). In this example we look for the 2 parameters x(1) and x(2) of a simple exponential decay model (x(1) being the unknown initial value and x(2) the decay constant):
function y=yth(t, x) y = x(1)*exp(-x(2)*t) endfunction // we have the m measures (ti, yi): m = 10; tm = [0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5]'; ym = [0.79, 0.59, 0.47, 0.36, 0.29, 0.23, 0.17, 0.15, 0.12, 0.08]'; // measure weights (here all equal to 1...) wm = ones(m,1); // and we want to find the parameters x such that the model fits the given // data in the least square sense: // // minimize f(x) = sum_i wm(i)^2 ( yth(tm(i),x) - ym(i) )^2 // initial parameters guess x0 = [1.5 ; 0.8]; // in the first examples, we define the function fun and dfun // in scilab language function e=myfun(x, tm, ym, wm) e = wm.*( yth(tm, x) - ym ) endfunction function g=mydfun(x, tm, ym, wm) v = wm.*exp(-x(2)*tm) g = [v , -x(1)*tm.*v] endfunction // now we could call leastsq: // 1- the simplest call [f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),x0) // 2- we provide the Jacobian [f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),mydfun,x0) // a small graphic (before showing other calling features) tt = linspace(0,1.1*max(tm),100)'; yy = yth(tt, xopt); scf(); plot(tm, ym, "kx") plot(tt, yy, "b-") legend(["measure points", "fitted curve"]); xtitle("a simple fit with leastsq") // 3- how to get some information (we use iprint=1) [f,xopt, gopt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0) // 4- using the conjugate gradient (instead of quasi Newton) [f,xopt, gopt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0,"gc") // 5- how to provide bound constraints (not useful here !) xinf = [-%inf,-%inf]; xsup = [%inf, %inf]; // without Jacobian: [f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),"b",xinf,xsup,x0) // with Jacobian : [f,xopt, gopt] = leastsq(list(myfun,tm,ym,wm),mydfun,"b",xinf,xsup,x0) // 6- playing with some stopping parameters of the algorithm // (allows only 40 function calls, 8 iterations and set epsg=0.01, epsf=0.1) [f,xopt, gopt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0,"ar",40,8,0.01,0.1) | ![]() | ![]() |
Now we want to define fun and dfun in Fortran, then in C. Note that the "compile and link to scilab" method used here is believed to be OS independent (but there are some requirements, in particular you need a C and a fortran compiler, and they must be compatible with the ones used to build your scilab binary).
Let us begin by an example with fun and dfun in fortran
// 7-1/ Let 's Scilab write the fortran code (in the TMPDIR directory): f_code = [" subroutine myfun(m,n,x,param,f)" "* param(i) = tm(i), param(m+i) = ym(i), param(2m+i) = wm(i)" " implicit none" " integer n,m" " double precision x(n), param(*), f(m)" " integer i" " do i = 1,m" " f(i) = param(2*m+i)*( x(1)*exp(-x(2)*param(i)) - param(m+i) )" " enddo" " end ! subroutine fun" "" " subroutine mydfun(m,n,x,param,df)" "* param(i) = tm(i), param(m+i) = ym(i), param(2m+i) = wm(i)" " implicit none" " integer n,m" " double precision x(n), param(*), df(m,n)" " integer i" " do i = 1,m" " df(i,1) = param(2*m+i)*exp(-x(2)*param(i))" " df(i,2) = -x(1)*param(i)*df(i,1)" " enddo" " end ! subroutine dfun"]; cd TMPDIR; mputl(f_code,TMPDIR+'/myfun.f') // 7-2/ compiles it. You need a fortran compiler ! names = ["myfun" "mydfun"] flibname = ilib_for_link(names,"myfun.f",[],"f"); // 7-3/ link it to scilab (see link help page) link(flibname,names,"f") // 7-4/ ready for the leastsq call: be carreful do not forget to // give the dimension m after the routine name ! [f,xopt, gopt] = leastsq(list("myfun",m,tm,ym,wm),x0) // without Jacobian [f,xopt, gopt] = leastsq(list("myfun",m,tm,ym,wm),"mydfun",x0) // with Jacobian | ![]() | ![]() |
Last example: fun and dfun in C.
// 8-1/ Let 's Scilab write the C code (in the TMPDIR directory): c_code = ["#include <math.h>" "void myfunc(int *m,int *n, double *x, double *param, double *f)" "{" " /* param[i] = tm[i], param[m+i] = ym[i], param[2m+i] = wm[i] */" " int i;" " for ( i = 0 ; i < *m ; i++ )" " f[i] = param[2*(*m)+i]*( x[0]*exp(-x[1]*param[i]) - param[(*m)+i] );" " return;" "}" "" "void mydfunc(int *m,int *n, double *x, double *param, double *df)" "{" " /* param[i] = tm[i], param[m+i] = ym[i], param[2m+i] = wm[i] */" " int i;" " for ( i = 0 ; i < *m ; i++ )" " {" " df[i] = param[2*(*m)+i]*exp(-x[1]*param[i]);" " df[i+(*m)] = -x[0]*param[i]*df[i];" " }" " return;" "}"]; mputl(c_code,TMPDIR+'/myfunc.c') // 8-2/ compiles it. You need a C compiler ! names = ["myfunc" "mydfunc"] clibname = ilib_for_link(names,"myfunc.c",[],"c"); // 8-3/ link it to scilab (see link help page) link(clibname,names,"c") // 8-4/ ready for the leastsq call [f,xopt, gopt] = leastsq(list("myfunc",m,tm,ym,wm),"mydfunc",x0) | ![]() | ![]() |