-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