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comparison scripts/ode/private/ode_rk_interpolate.m @ 20596:87b557ee8e5d

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clean up and vectorize code for dense output in ode45 * scripts/ode/private/ode_rk_interpolate.m: new file * scripts/ode/private/ode_rk_interpolate.m(hermite_quartic_interpolation): move to internal function, use vectorization and broadcasting. * scripts/ode/private/hermite_quartic_interpolation.m: remove file * scripts/ode/module.mk: list added and removed files * scripts/ode/private/integrate_adaptive.m: use new interpolation code.
author Carlo de Falco <carlo.defalco@polimi.it>
date Tue, 06 Oct 2015 19:28:59 +0200
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children ea6a1c00763a
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20595:c1a6c31ac29a 20596:87b557ee8e5d
1 ## Copyright (C) 2015 Carlo de Falco
2 ## Copyright (C) 2015 Jacopo Corno <jacopo.corno@gmail.com>
3 ##
4 ## This file is part of Octave.
5 ##
6 ## Octave is free software; you can redistribute it and/or modify it
7 ## under the terms of the GNU General Public License as published by
8 ## the Free Software Foundation; either version 3 of the License, or (at
9 ## your option) any later version.
10 ##
11 ## Octave is distributed in the hope that it will be useful, but
12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 ## General Public License for more details.
15 ##
16 ## You should have received a copy of the GNU General Public License
17 ## along with Octave; see the file COPYING. If not, see
18 ## <http://www.gnu.org/licenses/>.
19
20 function u_interp = ode_rk_interpolate (order, z, u, t, k_vals, dt, args)
21
22 switch order
23
24 #{
25 case 1
26 u_interp = linear_interpolation (z, u, t);
27 case 2
28 if (! isempty (k_vals))
29 der = k_vals(:,1);
30 else
31 der = feval (func, z(1) , u(:,1), args);
32 endif
33 u_interp = quadratic_interpolation (z, u, der, t);
34 case 3
35 u_interp = ...
36 hermite_cubic_interpolation (z, u, k_vals, t);
37 case 4
38 ## if ode45 is used without local extrapolation this function
39 ## doesn't require a new function evaluation.
40 u_interp = dorpri_interpolation ([z(i-1) z(i)],
41 [u(:,i-1) u(:,i)],
42 k_vals, tspan(counter));
43
44 #}
45
46 case 5
47 ## ode45 with Dormand-Prince scheme:
48 u_interp = ...
49 hermite_quartic_interpolation (z, u, k_vals, t);
50
51 ## it is also possible to do a new function evaluation and use
52 ## the quintic hermite interpolator
53 ## f_half = feval (func, t+1/2*dt, u_half,
54 ## options.vfunarguments{:});
55 ## u_interp =
56 ## hermite_quintic_interpolation ([z(i-1) z(i)],
57 ## [u(:,i-1) u_half u(:,i)],
58 ## [k_vals(:,1) f_half ...
59 ## k_vals(:,end)],
60 ## tspan(counter));
61 otherwise
62 warning ("High order interpolation not yet implemented: ",
63 "using cubic interpolation instead");
64 der(:,1) = feval (func, z(1) , u(:,1), args);
65 der(:,2) = feval (func, z(2) , u(:,2), args);
66 u_interp = hermite_cubic_interpolation (z, u, der, t);
67 endswitch
68
69 endfunction
70
71
72
73 ## The function below can be used in an ODE solver to
74 ## interpolate the solution at the time t_out using 4th order
75 ## hermite interpolation.
76 function x_out = hermite_quartic_interpolation (t, x, der, t_out)
77
78 persistent coefs_u_half = ...
79 [(6025192743/30085553152), 0, (51252292925/65400821598), ...
80 (-2691868925/45128329728), (187940372067/1594534317056), ...
81 (-1776094331/19743644256), (11237099/235043384)].';
82
83 ## 4th order approximation of y in t+dt/2 as proposed by
84 ## Shampine in Lawrence, Shampine, "Some Practical
85 ## Runge-Kutta Formulas", 1986.
86 dt = t(2) - t(1);
87 u_half = x(:,1) + (1/2) * dt * (der(:,1:7) * coefs_u_half);
88
89 ## Rescale time on [0,1]
90 s = (t_out - t) / dt;
91
92 ## Hermite basis functions
93 ## H0 = 1 - 11*s.^2 + 18*s.^3 - 8*s.^4;
94 ## H1 = s - 4*s.^2 + 5*s.^3 - 2*s.^4;
95 ## H2 = 16*s.^2 - 32*s.^3 + 16*s.^4;
96 ## H3 = - 5*s.^2 + 14*s.^3 - 8*s.^4;
97 ## H4 = s.^2 - 3*s.^3 + 2*s.^4;
98
99 x_out = zeros (rows (x), length (t_out));
100 x_out = (1 - 11*s.^2 + 18*s.^3 - 8*s.^4) .* x(:,1) + ...
101 ( s - 4*s.^2 + 5*s.^3 - 2*s.^4) .* (dt * der(:,1)) + ...
102 ( 16*s.^2 - 32*s.^3 + 16*s.^4) .* u_half + ...
103 ( - 5*s.^2 + 14*s.^3 - 8*s.^4) .* x(:,2) + ...
104 ( s.^2 - 3*s.^3 + 2*s.^4) .* (dt * der(:,end));
105
106 endfunction