Broyden's Method
Broyden's method for the numerical solution of nonlinear systems of
n equations in n variables.
broyden(Ffun, x0, J0 = NULL, ...,
maxiter = 100, tol = .Machine$double.eps^(1/2))Ffun |
|
x0 |
Numeric vector of length |
J0 |
Jacobian of the function at |
... |
additional parameters passed to the function. |
maxiter |
Maximum number of iterations. |
tol |
Tolerance, relative accuracy. |
F as a function must return a vector of length n, and accept an
n-dim. vector or column vector as input. F must not be univariate,
that is n must be greater than 1.
Broyden's method computes the Jacobian and its inverse only at the first iteration, and does a rank-one update thereafter, applying the so-called Sherman-Morrison formula that computes the inverse of the sum of an invertible matrix A and the dyadic product, uv', of a column vector u and a row vector v'.
List with components: zero the best root found so far, fnorm
the square root of sum of squares of the values of f, and niter the
number of iterations needed.
Applied to a system of n linear equations it will stop in
2n steps
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)
broyden(F1, x0 = c(1, 1))
# zero: 0.4760958 -0.8793934; fnorm: 9.092626e-09; niter: 13
F <- function(x) {
x1 <- x[1]; x2 <- x[2]; x3 <- x[3]
as.matrix(c(x1^2 + x2^2 + x3^2 - 1,
x1^2 + x3^2 - 0.25,
x1^2 + x2^2 - 4*x3), ncol = 1)
}
x0 <- as.matrix(c(1, 1, 1))
broyden(F, x0)
# zero: 0.4407629 0.8660254 0.2360680; fnorm: 1.34325e-08; niter: 8
## 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))
}
broyden(F2, c(1, 1))
# zero 0.2555197 0.8948303 , i.e. z0 = 0.2555 + 0.8948i
# fnorm 7.284374e-10
# niter 13
## Two more problematic examples
F3 <- function(x)
c(2*x[1] - x[2] - exp(-x[1]), -x[1] + 2*x[2] - exp(-x[2]))
broyden(F3, c(0, 0))
# $zero 0.5671433 0.5671433 # x = exp(-x)
F4 <- function(x) # Dennis Schnabel
c(x[1]^2 + x[2]^2 - 2, exp(x[1] - 1) + x[2]^3 - 2)
broyden(F4, c(2.0, 0.5), maxiter = 100)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.