Add the advection_matrix() function and a test for it.

[octave.git] / forward_euler.m

diff --git a/forward_euler.m b/forward_euler.m
index c6ef21722dbbd3fa1c405f2a594ad793d0c48f79..ff9c5eb2a498bc2be65d366fa9ba497d568354b7 100644 (file)
--- a/forward_euler.m
+++ b/forward_euler.m
@@ -1,26 +1,52 @@
-function df = forward_euler(integer_order, h, f, x)
+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.
   ##
-  ## Use the forward Euler method to compute the derivative of `f` at
-  ## a point `x`.
   ##
   ## INPUTS:
   ##
-  ##   * ``integer_order`` - The order of the derivative.
+  ##   * ``integer_order`` - The order of the derivative which we're
+  ##     approximating.
   ##
-  ##   * ``h`` - The step size.
+  ##   * ``xs`` - The vector of x-coordinates.
   ##
-  ##   * ``f`` - The function whose derivative we're computing.
+  ##   * ``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`.
   ##
-  ##   * ``x`` - The point at which to compute the derivative.
+  ##
+  ## 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
+
+  if (length(xs) < 2)
+    ## You can't approximate a derivative of order greater than zero
+    ## with zero or one points!
+    coefficients = NA
     return;
   end
-    
-  ## We need a few points around `x` to compute the derivative at `x`.
-  ## The number of points depends on the order.
+
   if (even(integer_order))
     offset_b = integer_order / 2;
     offset_f = offset_b;
@@ -31,15 +57,22 @@ function df = forward_euler(integer_order, h, f, x)
     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