# Broyden's Method Broyden's method for the numerical solution of nonlinear systems of `n` equations in `n` variables. ```r broyden(Ffun, x0, J0 = NULL, ..., maxiter = 100, tol = .Machine$double.eps^(1/2)) ``` ## Arguments - `Ffun`: `n` functions of `n` variables. - `x0`: Numeric vector of length `n`. - `J0`: Jacobian of the function at `x0`. - `...`: additional parameters passed to the function. - `maxiter`: Maximum number of iterations. - `tol`: Tolerance, relative accuracy. ## Details 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'. ## Returns 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. ## References Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg. ## Note Applied to a system of `n` linear equations it will stop in `2n` steps ## See Also `newtonsys`, `fsolve` ## Examples ```r ## 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) ```

broyden function