function coefficients = central_difference(xs, x)
##
## The first order central difference at x1 is,
##
## f'(x1) = (f(x2) - f(x0))/2
##
## where the index x1 is of course arbitrary but x2, x0 are adjacent
## to x1. The coefficients we seek are the coefficients of f(xj) for
## j = 1,...,N-2, where N is the length of ``xs``. We omit the first
## and last coefficient because at x0 and xN, the previous/next
## value is not available.
##
## This should probably take an 'order' parameter as well; see
## forward_euler().
##
## INPUT:
##
## * ``xs`` - The vector of x-coordinates.
##
## * ``x`` - The point `x` at which you'd like to evaluate the
## derivative of the specified `integer_order`. This should be an
## element of `xs`.
##
## OUTPUT:
##
## * ``coefficients`` - The vector of coefficients, in order, of
## f(x0), f(x1), ..., f(xn).
##
if (length(xs) < 3)
## We need at least one point other than the first and last.
coefficients = NA;
return;
end
x_idx = find(xs == x);
if (x_idx == 1 || x_idx == length(xs))
## You asked for the difference at the first or last element, which
## we can't do.
coefficients = NA;
return;
end
## Start with a vector of zeros.
coefficients = zeros(1, length(xs));
## And fill in the two values that we know.
coefficients(x_idx - 1) = -1/2;
coefficients(x_idx + 1) = 1/2;
end