function coefficients = forward_euler(integer_order, xs, x)
##
## Return the coefficients of u(x0), u(x1), ..., u(xn) as a vector.
## Take for example a first order approximation, with,
##
## xs = [x0,x1,x2,x3,x4]
##
## f'(x1) ~= [f(x2)-f(x1)]/(x2-x1)
##
## This would return [0, -1/(x2-x1), 2/(x2-x1), 0, 0]. This aids the
## solution of linear systems.
##
##
## INPUTS:
##
## * ``integer_order`` - The order of the derivative which we're
## approximating.
##
## * ``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`.
##
##
## OUTPUTS:
##
## * ``coefficients`` - The vector of coefficients, in order, of
## f(x0), f(x1), ..., f(xn).
##
if (integer_order < 0)
## You have made a grave mistake.
coefficients = NA;
return;
end
if (integer_order == 0)
coefficients = x;
return;
end
if (length(xs) < 2)
## You can't approximate a derivative of order greater than zero
## with zero or one points!
coefficients = NA
return;
end
if (even(integer_order))
offset_b = integer_order / 2;
offset_f = offset_b;
else
## When the order is odd, we need one more "forward" point than we
## do "backward" points.
offset_b = (integer_order - 1) / 2;
offset_f = offset_b + 1;
end
## Zero out the coefficients for terms that won't appear. We compute
## where `x` is, and we just computed how far back/forward we need to
## look from `x`, so we just need to make the rest zeros.
x_idx = find(xs == x);
first_nonzero_idx = x_idx - offset_b;
last_nonzero_idx = x_idx + offset_f;
leading_zero_count = first_nonzero_idx - 1;
leading_zeros = zeros(1, leading_zero_count);
trailing_zero_count = length(xs) - last_nonzero_idx;
trailing_zeros = zeros(1, trailing_zero_count);
targets = xs(first_nonzero_idx : last_nonzero_idx);
# The multiplier comes from the Taylor expansion.
multiplier = factorial(integer_order);
cs = divided_difference_coefficients(targets) * multiplier;
coefficients = horzcat(leading_zeros, cs, trailing_zeros);
end