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annotate scripts/ode/private/ode_rk_interpolate.m @ 20598:ea6a1c00763a

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