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# lsqrsolve

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

### Calling Sequence

[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,`gtol`

measures 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
algorithm estimates that the relative error between

`x`

and the solution is at most`tol`

.- 2
number of calls to

`fcn`

reached- 3
`tol`

is too small. No further improvement in the approximate solution`x`

is possible.- 4
iteration is not making good progress.

- 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 solution`x`

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 calling sequence 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 calling sequence 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 calling sequence 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 calling sequence should be

`fjac(int *m, int *n, double x[],double v[],int *iflag)`

return -1 in iflag to stop the algoritm 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; deff('y=FF(x)','y=a*(x-b)+c*x.*x'); 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)

### Used Functions

lmdif, lmder from minpack, Argonne National Laboratory.

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