pracma (version 1.9.9)

jacobian: Jacobian Matrix

Description

Jacobian matrix of a function R^n --> R^m .

Usage

jacobian(f, x0, heps = .Machine$double.eps^(1/3), ...)

Arguments

f

m functions of n variables.

x0

Numeric vector of length n.

heps

This is h in the derivative formula.

...

parameters to be passed to f.

Value

Numeric m-by-n matrix J where the entry J[j, i] is $\frac{\partial f_j}{\partial x_i}$, i.e. the derivatives of function $f_j$ line up in row $i$ for $x_1, \ldots, x_n$.

Details

Computes the derivative of each funktion $f_j$ by variable $x_i$ separately, taking the discrete step $h$.

References

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

See Also

gradient

Examples

##  Example function from Quarteroni & Saleri
f <- function(x) c(x[1]^2 + x[2]^2 - 1, sin(pi*x[1]/2) + x[2]^3)
jf <- function(x) 
          matrix( c(2*x[1], pi/2 * cos(pi*x[1]/2), 2*x[2], 3*x[2]^2), 2, 2)
all.equal(jf(c(1,1)), jacobian(f, c(1,1)))
# TRUE