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lsqrsolve

minimize the sum of the squares of nonlinear functions, levenberg-marquardt algorithm

Syntax

[x [,v [,info]]]=lsqrsolve(x0,fct,m [,stop [,diag]])
[x [,v [,info]]]=lsqrsolve(x0,fct,m ,fjac [,stop [,diag]])

Arguments

x0

real vector of size n(initial estimate of the solution vector).

fct

external (i.e function or list or string).

m

integer, the number of functions. m must be greater than or equal to n.

fjac

external (i.e function or list or string).

stop

optional vector [ftol,xtol,gtol,maxfev,epsfcn,factor] the default value is [1.d-8,1.d-8,1.d-5,1000,0,100]

ftol

A positive real number,termination occurs when both the actual and predicted relative reductions in the sum of squares are at most ftol. therefore, ftol measures the relative error desired in the sum of squares.

xtol

A positive real number, termination occurs when the relative error between two consecutive iterates is at most xtol. therefore, xtol measures the relative error desired in the approximate solution.

gtol

A nonnegative input variable. termination occurs when the cosine of the angle between fct(x) and any column of the jacobian is at most gtol in absolute value. therefore, gtolmeasures the orthogonality desired between the function vector and the columns of the jacobian.

maxfev

A positive integer, termination occurs when the number of calls to fct is at least maxfev by the end of an iteration.

epsfcn

A positive real number, used in determining a suitable step length for the forward-difference approximation. this approximation assumes that the relative errors in the functions are of the order of epsfcn. if epsfcn is less than the machine precision, it is assumed that the relative errors in the functions are of the order of the machine precision.

factor

A positive real number, used in determining the initial step bound. this bound is set to the product of factor and the euclidean norm of diag*x if nonzero, or else to factor itself. in most cases factor should lie in the interval (0.1,100). 100 is a generally recommended value.

diag

is an array of length n. diag must contain positive entries that serve as multiplicative scale factors for the variables.

x :

real vector (final estimate of the solution vector).

v :

real vector (value of fct(x)).

info

termination indicator

0

improper input parameters.

1

both actual and predicted relative reductions in the sum of squares are at most ftol.

2

relative error between two consecutive iterates is at most xtol.

3

conditions for info = 1 and info = 2 both hold.

4

the cosine of the angle between fvec and any column of the jacobian is at most gtol in c absolute value.

5

number of calls to fcn has reached or exceeded maxfev

6

ftol is too small. no further reduction in the sum of squares is possible.

7

xtol is too small. no further improvement in the approximate solutionx is possible.

8

gtol is too small. fvec is orthogonal to the columns of the jacobian to machine precision.

Description

minimize the sum of the squares of m nonlinear functions in n variables by a modification of the levenberg-marquardt algorithm. the user must provide a subroutine which calculates the functions. the jacobian is then calculated by a forward-difference approximation.

minimize sum(fct(x,m).^2) where fct is function from R^n to R^m

fct should be :

• a Scilab function whose syntax is v=fct(x,m) given x and m.

• a character string which refers to a C or Fortran routine which must be linked to Scilab.

Fortran calling sequence should be fct(m,n,x,v,iflag) where m, n, iflag are integers, x a double precision vector of size n and v a double precision vector of size m.

C syntax should be fct(int *m, int *n, double x[],double v[],int *iflag)

fjac is an external which returns v=d(fct)/dx (x). it should be :

a Scilab function

whose syntax is J=fjac(x,m) given x and m.

a character string

it refers to a C or Fortran routine which must be linked to Scilab.

Fortran calling sequence should be fjac(m,n,x,jac,iflag) where m, n, iflag are integers, x a double precision vector of size n and jac a double precision vector of size m*n.

C syntax should be fjac(int *m, int *n, double x[],double v[],int *iflag)

return -1 in iflag to stop the algorithm if the function or jacobian could not be evaluated.

Examples

// A simple example with lsqrsolve
a=[1,7;
2,8
4 3];
b=[10;11;-1];

function y=f1(x, m)
y=a*x+b;
endfunction

[xsol,v]=lsqrsolve([100;100],f1,3)
xsol+a\b

function y=fj1(x, m)
y=a;
endfunction

[xsol,v]=lsqrsolve([100;100],f1,3,fj1)
xsol+a\b

// Data fitting problem
// 1 build the data
a=34;
b=12;
c=14;

function y=FF(x)
y=a*(x-b)+c*x.*x
endfunction
X=(0:.1:3)';
Y=FF(X)+100*(rand()-.5);

//solve
function e=f1(abc, m)
a=abc(1);
b=abc(2);
c=abc(3);
e=Y-(a*(X-b)+c*X.*X);
endfunction

[abc,v]=lsqrsolve([10;10;10],f1,size(X,1));
abc
norm(v)

• external — Scilab Object, external function or routine
• qpsolve — linear quadratic programming solver
• optim — non-linear optimization routine
• fsolve — find a zero of a system of n nonlinear functions

Used Functions

lmdif, lmder from minpack, Argonne National Laboratory.