Jacobian Matrix

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

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.

Details

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

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 \(x_1, \ldots, x_n\).

References

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

Note

Obviously, this function is not vectorized.

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)))
#> [1] TRUE
# TRUE