# Gauss-Newton Function Minimization Gauss-Newton method of minimizing a term $f_1(x)^2 + \ldots + f_m(x)^2$ or $F' F$ where $F = (f_1, \ldots, f_m)$ is a multivariate function of $n$ variables, not necessarily $n = m$. ```r gaussNewton(x0, Ffun, Jfun = NULL, maxiter =100, tol = .Machine$double.eps^(1/2), ...) ``` ## Arguments - `Ffun`: `m` functions of `n` variables. - `Jfun`: function returning the Jacobian matrix of `Ffun`; if `NULL`, the default, the Jacobian will be computed numerically. The gradient of `f` will be computed internally from the Jacobian (i.e., cannot be supplied). - `x0`: Numeric vector of length `n`. - `maxiter`: Maximum number of iterations. - `tol`: Tolerance, relative accuracy. - `...`: Additional parameters to be passed to f. ## Details Solves the system of equations applying the Gauss-Newton's method. It is especially designed for minimizing a sum-of-squares of functions and can be used to find a common zero of several function. This algorithm is described in detail in the textbook by Antoniou and Lu, incl. different ways to modify and remedy the Hessian if not being positive definite. Here, the approach by Goldfeld, Quandt and Trotter is used, and the hessian modified by the Matthews and Davies algorithm if still not invertible. To accelerate the iteration, an inexact linesearch is applied. ## Returns List with components: `xs` the minimum or root found so far, `fs` the square root of sum of squares of the values of f, `iter` the number of iterations needed, and `relerr` the absoulte distance between the last two solutions. ## References Antoniou, A., and W.-S. Lu (2007). Practical Optimization: Algorithms and Engineering Applications. Springer Business+Science, New York. ## Note If `n=m` then directly applying the `newtonsys` function might be a better alternative. ## See Also `newtonsys`, `softline` ## Examples ```r f1 <- function(x) c(x[1]^2 + x[2]^2 - 1, x[1] + x[2] - 1) gaussNewton(c(4, 4), f1) f2 <- function(x) c( x[1] + 10*x[2], sqrt(5)*(x[] - x[4]), (x[2] - 2*x[3])^2, 10*(x[1] - x[4])^2) gaussNewton(c(-2, -1, 1, 2), f2) f3 <- function(x) c(2*x[1] - x[2] - exp(-x[1]), -x[1] + 2*x[2] - exp(-x[2])) gaussNewton(c(0, 0), f3) # $xs 0.5671433 0.5671433 f4 <- function(x) # Dennis Schnabel c(x[1]^2 + x[2]^2 - 2, exp(x[1] - 1) + x[2]^3 - 2) gaussNewton(c(2.0, 0.5), f4) # $xs 1 1 ## Examples (from Matlab) F1 <- function(x) c(2*x[1]-x[2]-exp(-x[1]), -x[1]+2*x[2]-exp(-x[2])) gaussNewton(c(-5, -5), F1) # Find a matrix X such that X %*% X %*% X = [1 2; 3 4] F2 <- function(x) { X <- matrix(x, 2, 2) D <- X %*% X %*% X - matrix(c(1,3,2,4), 2, 2) return(c(D)) } sol <- gaussNewton(ones(2,2), F2) (X <- matrix(sol$xs, 2, 2)) # [,1] [,2] # [1,] -0.1291489 0.8602157 # [2,] 1.2903236 1.1611747 X %*% X %*% X ```

gaussNewton function