-function df = forward_euler(integer_order, h, f, x)
- ##
- ## Use the forward Euler method to compute the derivative of `f` at
- ## a point `x`.
- ##
- ## INPUTS:
- ##
- ## * ``integer_order`` - The order of the derivative.
- ##
- ## * ``h`` - The step size.
- ##
- ## * ``f`` - The function whose derivative we're computing.
- ##
- ## * ``x`` - The point at which to compute the derivative.
- ##
+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)
- df = x;
+ coefficients = x;
return;
end
-
- ## We need a few points around `x` to compute the derivative at `x`.
- ## The number of points depends on the order.
+
+ 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.
+ % 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
- backward_xs = [x-(offset_b*h) : h : x];
+ % 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);
- ## We'll always have at least one forward point, so start this vector
- ## from (x + h) and include `x` itself in the backward points.
- forward_xs = [x+h : h : x+(offset_f*h)];
+ targets = xs(first_nonzero_idx : last_nonzero_idx);
- xs = horzcat(backward_xs, forward_xs);
+ % The multiplier comes from the Taylor expansion.
+ multiplier = factorial(integer_order);
+ cs = divided_difference_coefficients(targets) * multiplier;
- ## Now that we have all of the points that we need in xs, we can use
- ## the divided_difference function to compute the derivative.
- df = divided_difference(f, xs);
+ coefficients = horzcat(leading_zeros, cs, trailing_zeros);
end