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

functions of n variables.

Jfun

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

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.

Examples

Run this code

##  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

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