# fndir

Directional derivative of function

df = fndir(f,y)

## Description

df = fndir(f,y) is the ppform of the directional derivative, of the function f in f, in the direction of the (column-)vector y. This means that df describes the function ${D}_{y}f\left(x\right):={\mathrm{lim}}_{t\to 0}\left(f\left(x+ty\right)-f\left(x\right)\right)/t$.

If y is a matrix, with n columns, and f is d-valued, then the function in df is prod(d)*n-valued. Its value at x, reshaped to be of size [d,n], has in its jth “column” the directional derivative of f at x in the direction of the jth column of y. If you prefer df to reflect explicitly the actual size of f, use instead

df = fnchg( fndir(f,y), 'dim',[fnbrk(f,'dim'),size(y,2)] );

Since fndir relies on the ppform of the function in f, it does not work for rational functions nor for functions in stform.

## Examples

For example, if f describes an m-variate d-vector-valued function and x is some point in its domain, then, e.g., with this particular ppform f that describes a scalar-valued bilinear polynomial,

f = ppmak({0:1,0:1},[1 0;0 1]); x = [0;0];
[d,m] = fnbrk(f,'dim','var');
jacobian = reshape(fnval(fndir(f,eye(m)),x),d,m)

is the Jacobian of that function at that point (which, for this particular scalar-valued function, is its gradient, and it is zero at the origin).

As a related example, the next statements plot the gradients of (a good approximation to) the Franke function at a regular mesh:

xx = linspace(-.1,1.1,13); yy = linspace(0,1,11);
[x,y] = ndgrid(xx,yy); z = franke(x,y);
pp2dir = fndir(csapi({xx,yy},z),eye(2));