leastsq - Solves non-linear least squares problems

Calling Sequence

[fopt,[xopt,[grdopt]]]=leastsq(fun, x0)

[fopt,[xopt,[grdopt]]]=leastsq(fun, dfun, x0)

[fopt,[xopt,[grdopt]]]=leastsq(fun, cstr, x0)

[fopt,[xopt,[grdopt]]]=leastsq(fun, dfun, cstr, x0)

[fopt,[xopt,[grdopt]]]=leastsq(fun, dfun, cstr, x0, algo)

[fopt,[xopt,[grdopt]]]=leastsq([imp], fun [,dfun] [,cstr],x0 [,algo],[df0,[mem]],[stop])

Parameters

• fopt : value of the function f(x)=||fun(x)||^2 at xopt
• xopt : best value of x found to minimize ||fun(x)||^2
• grdopt : gradient of f at xopt
• fun : a scilab function or a list defining a function from R^n to R^m (see more details in DESCRIPTION).
• x0 : real vector (initial guess of the variable to be minimized).
• dfun : a scilab function or a string defining the Jacobian matrix of fun (see more details in DESCRIPTION).
• cstr : 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 calling sequence). Those bounds are real vectors with same dimension than x0 (-%inf and +%inf may be used for dimension which are unrestricted).
• algo : 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 .
• imp : scalar argument used to set the trace mode. imp=0 nothing (except errors) is reported, imp=1 initial and final reports, imp=2 adds a report per iteration, imp>2 add reports on linear search. Warning, most of these reports are written on the Scilab standard output.
• df0 : real scalar. Guessed decreasing of ||fun||^2 at first iteration. (df0=1 is the default value).
• mem : integer, number of variables used to approximate the Hessean (second derivatives) of f when algo ='qn'. Default value is around 6.
• stop : 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]]]]
  • nap : maximum number of calls to fun allowed.
  • iter : maximum number of iterations allowed.
  • epsg : threshold on gradient norm.
  • epsf : threshold controlling decreasing of f
  • epsx : threshold controlling variation of x . This vector (possibly matrix) of same size as x0 can be used to scale x .

Description

fun being a function from R^n to R^m this routine tries to minimize w.r.t. x, the function:

                           _m_     
                      2    \      2
     f(x) = ||fun(x)||  =  /   fun (x)
                           ---    i
                           i=1     
          

How to provide fun and dfun

fun can be either a usual scilab function (case 1) or a fortran or a C routine linked to scilab (case 2). For most problems the definition of fun will need supplementary parameters and this can be done in both cases.

case 1: when fun is a Scilab function, its calling sequence must be: y=fun(x [,opt_par1,opt_par2,...]). When fun needs optional parameters it must appear as list(fun,opt_par1,opt_par2,...) in the calling sequence of leastsq .

case 2: when fun is defined by a Fortran or C routine it must appear as list(fun_name,m [,opt_par1,opt_par2,...]) in the calling sequence of leastsq , fun_name (a string) being the name of the routine which must be linked to Scilab (see link ). The generic calling sequences for this routine are:

In Fortran:    subroutine fun(m, n, x, params, y)
               integer m,n
               double precision x(n), params(*), y(m)

In C:          void fun(int *m, int *n, double *x, double *params, double *y)
         

where n is the dimension of vector x , m the dimension of vector y (which must store the evaluation of fun at x) and params is a vector which contains the optional parameters opt_par1, opt_par2, ... (each parameter may be a vector, for instance if opt_par1 has 3 components, the description of opt_par2 begin from params(4) (fortran case), and from params[3] (C case), etc... Note that even if fun doesn't need supplementary parameters you must anyway write the fortran code with a params argument (which is then unused in the subroutine core).

In many cases it is adviced to provide the Jacobian matrix dfun (dfun(i,j)=dfi/dxj) to the optimizer (which uses a finite difference approximation otherwise) and as for fun it may be given as a usual scilab function or as a fortran or a C routine linked to scilab.

case 1: when dfun is a scilab function, its calling sequence must be: y=dfun(x [, optional parameters]) (notes that even if dfun needs optional parameters it must appear simply as dfun in the calling sequence of leastsq ).

case 2: when dfun is defined by a Fortran or C routine it must appear as dfun_name (a string) in the calling sequence of leastsq (dfun_name being the name of the routine which must be linked to Scilab). The calling sequences for this routine are nearly the same than for fun :

In Fortran:    subroutine dfun(m, n, x, params, y)
               integer m,n
               double precision x(n), params(*), y(m,n)

In C:          void fun(int *m, int *n, double *x, double *params, double *y)
         

in the C case dfun(i,j)=dfi/dxj must be stored in y[m*(j-1)+i-1].

Remarks

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).

Examples

// We will show different calling possibilities of leastsq on one (trivial) example
// which is non linear but doesn't 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 unknow 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]';
wm = ones(m,1); // measure weights (here all equal to 1...)

// and we want to find the parameters x such that the model fits the given 
// datas 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, gropt] = leastsq(list(myfun,tm,ym,wm),x0)

// 2- we provide the Jacobian
[f,xopt, gropt] = 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);
xbasc()
plot2d(tm, ym, style=-2)
plot2d(tt, yy, style = 2)
legend(["measure points", "fitted curve"]);
xtitle("a simple fit with leastsq")

// 3- how to get some informations (we use imp=1)
[f,xopt, gropt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0)

// 4- using the conjugate gradient (instead of quasi Newton)
[f,xopt, gropt] = 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];
[f,xopt, gropt] = leastsq(list(myfun,tm,ym,wm),"b",xinf,xsup,x0) // without Jacobian
[f,xopt, gropt] = leastsq(list(myfun,tm,ym,wm),mydfun,"b",xinf,xsup,x0) // with Jacobian 

// 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, gropt] = leastsq(1,list(myfun,tm,ym,wm),mydfun,x0,"ar",40,8,0.01,0.1)


// 7 and 8: 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 independant (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).
 
// 7- 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"];
mputl(f_code,TMPDIR+'/myfun.f')

// 7-2/ compiles it. You need a fortran compiler !
names = ["myfun" "mydfun"]
flibname = ilib_for_link(names,"myfun.o",[],"f",TMPDIR+"/Makefile");

// 7-3/ link it to scilab (see link help page)
link(flibname,names,"f") 

// 7-4/ ready for the leastsq call: be carreful don't forget to
//      give the dimension m after the routine name !
[f,xopt, gropt] = leastsq(list("myfun",m,tm,ym,wm),x0)  // without Jacobian
[f,xopt, gropt] = leastsq(list("myfun",m,tm,ym,wm),"mydfun",x0) // with Jacobian


// 8- 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.o",[],"c",TMPDIR+"/Makefile");

// 8-3/ link it to scilab (see link help page)
link(clibname,names,"c") 

// 8-4/ ready for the leastsq call
[f,xopt, gropt] = leastsq(list("myfunc",m,tm,ym,wm),"mydfunc",x0)
   
  

See Also

lsqrsolve ,   optim ,   NDcost ,   datafit ,   external ,   quapro ,   linpro ,