| Usage: |
z = nmnewton(func, fder, x0{, epsf, epsx, maxiter, checkcond})
|
| Input: |
| func | name of function(s) (string or vector of strings);
strings describe functions func_i, their
common root is to be found; format of func should be:
1. func is one string describing a function whose
output is n x 1 vector (in this case, derivative
of func, fder, has to be given)
2. func is a vector of strings, each of them describes
one of functions in the system (i.e. its output is
a real number)
The function should have just one
parameter x, which is:
a. vector n x 1, if the fder is given (see below)
b. matrix n x k (for any k >= 1) containing
x = (x1,x2,...,xk), if the jacobian
is to be computed automatically;
xi (n x 1 vector; i=1..k) represent points
in which the function should be evaluated.
|
| fder | derivatives of func; there are several possible
formats:
1. name of function(s) (string or vector of strings);
strings describe gradients of func_i;
fder is either one string describing the function computing
jacobian of func and its output is n x n vector, or
fder is a vector of strings, each of them describing
the gradient of one of functions in the system
(i.e. its output is n x 1 vector)
2. empty string; in this case the jacobian will be
computed automatically using the quantlet nmjacobian
with a default step h
3. a number or n x 1 vector h; the jacobian will be computed
automatically using the quantlet nmjacobian with
the given step h
|
| x0 | n x 1 vector, the initial estimate of the solution
|
| epsf | optional number; iterative process ends if sum of
absolute values of func will be less or equal to epsf;
default value is 1e-7
|
| epsx | optional number; iterative process ends if sum of
absolute values of corrections to x will be less
or equal to epsx;
default value is 1e-7
|
| maxiter | optional number; maximal number of iterations;
default value is 100
|
| checkcond | optional number; if given, the jacobian will be
tested in each step; a warning will be written if
its condition number is greater then checkcond
|
| Output: |
| z.x | n x 1 vector, estimated solution |
| z.fval | n x 1 vector, values of func in the point x |
| z.fderval | n x n matrix, values of gradients of func in the point x |
- Example:
library("nummath")
;
; definition of vector function f(x1,x2) =(x1+x2,x1-x2)
; and its jacobian
;
proc(func)=f(x)
func=#(x[1,]+x[2,],x[1,]-x[2,])
endp
proc(fder)=d(x)
fder=#(1,1)~#(1,-1)
endp
;
nmnewton("f","d",#(1,1))
- Result:
Contents of _tmp.x
[1,] 0
[2,] 0
Contents of _tmp.fval
[1,] 0
[2,] 0
Contents of _tmp.fderval
[1,] 1 1
[2,] 1 -1
- Example:
library("nummath")
;
; definition of function f(x1,x2) =(sin(x1+x2),cos(x1-x2))
; and of gradients of its components
;
proc(func)=f(x)
func=#(sin(x[1,]+x[2,]),cos(x[1,]-x[2,]))
endp
proc(fder1)=d1(x)
fder1=#(cos(x[1,]+x[2,]),cos(x[1,]+x[2,]))
endp
proc(fder2)=d2(x)
fder2=#(-sin(x[1,]-x[2,]),sin(x[1,]-x[2,]))
endp
;
nmnewton("f",#("d1","d2"),#(1,-1))
; the nearest solution is(pi/4,-pi/4)
- Result:
Contents of _tmp.x
[1,] 0.7854
[2,] -0.7854
Contents of _tmp.fval
[1,] 0
[2,] 6.1257e-17
Contents of _tmp.fderval
[1,] 1 1
[2,] -1 1
- Example:
library("nummath")
;
; definition of functions f1(x1,x2) = x1^2 - 2*x2 - 2
; f2(x1,x2) = x1 - x2^3 - 1
; and of their gradients
;
proc(func1)=f1(x)
func1=x[1,]^2-2*x[2,]-2
endp
proc(func2)=f2(x)
func2=x[1,]-x[2,]^3-1
endp
proc(fder1)=d1(x)
fder1=#(2*x[1,],-2)
endp
proc(fder2)=d2(x)
fder2=#(1,-3*x[2,]^2)
endp
sol = nmnewton(#("f1","f2"),#("d1","d2"),#(5,5),1e-15,1e-15,200)
sol
- Result:
Contents of sol.x
[1,] 2
[2,] 1
Contents of sol.fval
[1,] 0
[2,] 0
Contents of sol.fderval
[1,] 4 -2
[2,] 1 -3
- Example:
library("nummath")
;
; definition of functions f1(x1,x2) = x1^2 - 2*x2 - 2
; f2(x1,x2) = x1 - x2^2 - 1
; jacobian will be computed automatically
;
proc(func1)=f1(x)
func1=x[1,]^2-2*x[2,]-2
endp
proc(func2)=f2(x)
func2=x[1,]-x[2,]^3-1
endp
sol = nmnewton(#("f1","f2"),"",#(0,0))
sol
- Result:
Contents of sol.x
[1,] -1.7807e-15
[2,] -1
Contents of sol.fval
[1,] 1.5987e-14
[2,] 2.2204e-14
Contents of sol.fderval
[1,] 0 -2
[2,] 1 -3