pracma (version 1.8.8)

newtonsys: Newton Method for Nonlinear Systems

Description

Newton's method applied to multivariate nonlinear functions.

Usage

newtonsys(Ffun, x0, Jfun = NULL, ...,
    	  maxiter = 100, tol = .Machine$double.eps^(1/2))

Arguments

Ffun

m functions of n variables.

Jfun

Function returning a square n-by-n matrix (of partial derivatives) or NULL, the default.

x0

Numeric vector of length n.

maxiter

Maximum number of iterations.

tol

Tolerance, relative accuracy.

...

Additional parameters to be passed to f.

Value

• List with components: zero the root found so far, fnorm the square root of sum of squares of the values of f, and iter the number of iterations needed.

Details

Solves the system of equations applying Newton's method with the univariate derivative replaced by the Jacobian.

References

Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg.

See Also

newtonRaphson, broyden

Examples

##  Example from Quarteroni & Saleri
F1 <- function(x) c(x[1]^2 + x[2]^2 - 1, sin(pi*x[1]/2) + x[2]^3)
newtonsys(F1, x0 = c(1, 1))  # zero: 0.4760958 -0.8793934

##  Find the roots of the complex function sin(z)^2 + sqrt(z) - log(z)
F2 <- function(x) {
    z  <- x[1] + x[2]*1i
    fz <- sin(z)^2 + sqrt(z) - log(z)
    c(Re(fz), Im(fz))
}
newtonsys(F2, c(1, 1))
# $zero   0.2555197 0.8948303 , i.e.  z0 = 0.2555 + 0.8948i
# $fnorm  2.220446e-16
# $niter  8

##  Two more problematic examples
F3 <- function(x)
        c(2*x[1] - x[2] - exp(-x[1]), -x[1] + 2*x[2] - exp(-x[2]))
newtonsys(F3, c(0, 0))
# $zero   0.5671433 0.5671433
# $fnorm  0
# $niter  4

F4 <- function(x)  # Dennis Schnabel
        c(x[1]^2 + x[2]^2 - 2, exp(x[1] - 1) + x[2]^3 - 2)
newtonsys(F4, c(2.0, 0.5))
# will result in an error ``missing value in  ... err<tol && niter<maxiter''