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See the recommended documentation of this function
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 ton
.- 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 mostgtol
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
. ifepsfcn
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 tofactor
itself. in most casesfactor
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 mosttol
.- 2
number of calls to
fcn
reached- 3
tol
is too small. No further improvement in the approximate solutionx
is possible.- 4
iteration is not making good progress.
- 5
number of calls to
fcn
has reached or exceededmaxfev
- 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 calling sequence is
v=fct(x,m)
givenx
andm
.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)
wherem
,n
,iflag
are integers,x
a double precision vector of sizen
andv
a double precision vector of sizem
.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)
givenx
andm
.- 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)
wherem
,n
,iflag
are integers,x
a double precision vector of sizen
andjac
a double precision vector of sizem*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)
See Also
Used Functions
lmdif, lmder from minpack, Argonne National Laboratory.
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