Newton Method for Nonlinear Systems
Newton's method applied to multivariate nonlinear functions.
newtonsys(Ffun, x0, Jfun = NULL, ...,
maxiter = 100, tol = .Machine$double.eps^(1/2))Ffun |
|
Jfun |
Function returning a square |
x0 |
Numeric vector of length |
maxiter |
Maximum number of iterations. |
tol |
Tolerance, relative accuracy. |
... |
Additional parameters to be passed to f. |
Solves the system of equations applying Newton's method with the univariate derivative replaced by the Jacobian.
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.
TODO: better error checking, e.g. when the Jacobian is not invertible.
Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg.
## 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
## Not run:
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''
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.