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1 @c Copyright (C) 1996, 1997 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Quadrature, Differential Equations, Nonlinear Equations, Top |
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6 @chapter Quadrature |
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7 |
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8 @menu |
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9 * Functions of One Variable:: |
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10 * Orthogonal Collocation:: |
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11 @end menu |
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12 |
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13 @node Functions of One Variable, Orthogonal Collocation, Quadrature, Quadrature |
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14 @section Functions of One Variable |
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15 |
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16 @DOCSTRING(quad) |
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17 |
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18 @DOCSTRING(quad_options) |
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19 |
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20 Here is an example of using @code{quad} to integrate the function |
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21 @iftex |
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22 @tex |
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23 $$ |
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24 f(x) = x \sin (1/x) \sqrt {|1 - x|} |
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25 $$ |
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26 from $x = 0$ to $x = 3$. |
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27 @end tex |
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28 @end iftex |
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29 @ifinfo |
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30 |
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31 @example |
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32 @var{f}(@var{x}) = @var{x} * sin (1/@var{x}) * sqrt (abs (1 - @var{x})) |
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33 @end example |
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34 |
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35 @noindent |
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36 from @var{x} = 0 to @var{x} = 3. |
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37 @end ifinfo |
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38 |
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39 This is a fairly difficult integration (plot the function over the range |
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40 of integration to see why). |
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41 |
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42 The first step is to define the function: |
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43 |
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44 @example |
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45 @group |
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46 function y = f (x) |
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47 y = x .* sin (1 ./ x) .* sqrt (abs (1 - x)); |
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48 endfunction |
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49 @end group |
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50 @end example |
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51 |
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52 Note the use of the `dot' forms of the operators. This is not necessary |
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53 for the call to @code{quad}, but it makes it much easier to generate a |
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54 set of points for plotting (because it makes it possible to call the |
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55 function with a vector argument to produce a vector result). |
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56 |
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57 Then we simply call quad: |
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58 |
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59 @example |
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60 @group |
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61 [v, ier, nfun, err] = quad ("f", 0, 3) |
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62 @result{} 1.9819 |
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63 @result{} 1 |
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64 @result{} 5061 |
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65 @result{} 1.1522e-07 |
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66 @end group |
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67 @end example |
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68 |
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69 Although @code{quad} returns a nonzero value for @var{ier}, the result |
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70 is reasonably accurate (to see why, examine what happens to the result |
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71 if you move the lower bound to 0.1, then 0.01, then 0.001, etc.). |
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72 |
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73 @node Orthogonal Collocation, , Functions of One Variable, Quadrature |
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74 @section Orthogonal Collocation |
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75 |
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76 @DOCSTRING(colloc) |
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77 |
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78 Here is an example of using @code{colloc} to generate weight matrices |
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79 for solving the second order differential equation |
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80 @iftex |
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81 @tex |
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82 $u^\prime - \alpha u^{\prime\prime} = 0$ with the boundary conditions |
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83 $u(0) = 0$ and $u(1) = 1$. |
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84 @end tex |
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85 @end iftex |
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86 @ifinfo |
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87 @var{u}' - @var{alpha} * @var{u}'' = 0 with the boundary conditions |
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88 @var{u}(0) = 0 and @var{u}(1) = 1. |
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89 @end ifinfo |
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90 |
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91 First, we can generate the weight matrices for @var{n} points (including |
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92 the endpoints of the interval), and incorporate the boundary conditions |
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93 in the right hand side (for a specific value of |
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94 @iftex |
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95 @tex |
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96 $\alpha$). |
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97 @end tex |
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98 @end iftex |
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99 @ifinfo |
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100 @var{alpha}). |
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101 @end ifinfo |
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102 |
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103 @example |
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104 @group |
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105 n = 7; |
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106 alpha = 0.1; |
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107 [r, a, b] = colloc (n-2, "left", "right"); |
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108 at = a(2:n-1,2:n-1); |
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109 bt = b(2:n-1,2:n-1); |
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110 rhs = alpha * b(2:n-1,n) - a(2:n-1,n); |
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111 @end group |
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112 @end example |
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113 |
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114 Then the solution at the roots @var{r} is |
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115 |
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116 @example |
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117 u = [ 0; (at - alpha * bt) \ rhs; 1] |
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118 @result{} [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ] |
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119 @end example |
