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# 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])

### Arguments

- 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:

which is the sum of the squares of the components of
*fun*. Bound constraints may be imposed on
`x`

.

### 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`

does not 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 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 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); clf() 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 information (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"]; 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, 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.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, gropt] = leastsq(list("myfunc",m,tm,ym,wm),"mydfunc",x0)

### See Also

- lsqrsolve — minimize the sum of the squares of nonlinear functions, levenberg-marquardt algorithm
- optim — non-linear optimization routine
- NDcost — generic external for optim computing gradient using finite differences
- datafit — Parameter identification based on measured data
- external — Scilabオブジェクト, 外部関数またはルーチン
- qpsolve — 線形二次計画ソルバー

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