# broyden: Broyden's Method In pracma: Practical Numerical Math Functions

## Description

Broyden's method for the numerical solution of nonlinear systems of `n` equations in `n` variables.

## Usage

 ```1 2``` ```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'.

## Value

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.

## Note

Applied to a system of `n` linear equations it will stop in `2n` steps

## References

Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg.

`newtonsys`, `fsolve`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```## 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) ```

### Example output

```\$zero
[1]  0.4760958 -0.8793934

\$fnorm
[1] 9.092626e-09

\$niter
[1] 13

\$zero
[1] 0.4407629 0.8660254 0.2360680

\$fnorm
[1] 1.34325e-08

\$niter
[1] 8

\$zero
[1] 0.2555197 0.8948303

\$fnorm
[1] 7.28437e-10

\$niter
[1] 13

\$zero
[1] 0.5671433 0.5671433

\$fnorm
[1] 4.677305e-12

\$niter
[1] 5

\$zero
[1] 1 1

\$fnorm
[1] 1.18821e-10

\$niter
[1] 43
```

pracma documentation built on Dec. 11, 2021, 9:57 a.m.