Avoid divide-by-zero in step_length_positive_definite.m.

[octave.git] / forward_euler.m

diff --git a/forward_euler.m b/forward_euler.m
index ec678a15c741ee29bb7aec50929e6330959a7adc..6984cfc8dc9c14963bc1f8f1eb83b67f36869419 100644 (file)
--- a/forward_euler.m
+++ b/forward_euler.m
@@ -1,17 +1,49 @@
 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'(x=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.
-  ##
+  %
+  % 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
+
+  if (length(xs) < 2)
+    % You can't approximate a derivative of order greater than zero
+    % with zero or one points!
+    coefficients = NA
     return;
   end
 
@@ -19,15 +51,15 @@ function coefficients = forward_euler(integer_order, xs, x)
     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
 
-  ## 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.
+  % 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;
@@ -37,7 +69,10 @@ function coefficients = forward_euler(integer_order, xs, x)
   trailing_zeros = zeros(1, trailing_zero_count);
 
   targets = xs(first_nonzero_idx : last_nonzero_idx);
-  cs = divided_difference_coefficients(targets);
 
-  coefficients = horzcat(leading_zeros, cs, trailing_zeros);  
+  % 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