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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 @deftypefn {Loadable Function} {[@var{v}, @var{ier}, @var{nfun}, @var{err}] =} quad (@var{f}, @var{a}, @var{b}, @var{tol}, @var{sing}) |
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17 Integrate a nonlinear function of one variable using Quadpack. |
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18 The first argument is the name of the function to call to compute the |
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19 value of the integrand. It must have the form |
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20 |
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21 @example |
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22 y = f (x) |
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23 @end example |
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24 |
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25 @noindent |
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26 where @var{y} and @var{x} are scalars. |
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27 |
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28 The second and third arguments are limits of integration. Either or |
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29 both may be infinite. |
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30 |
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31 The optional argument @var{tol} is a vector that specifies the desired |
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32 accuracy of the result. The first element of the vector is the desired |
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33 absolute tolerance, and the second element is the desired relative |
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34 tolerance. To choose a relative test only, set the absolute |
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35 tolerance to zero. To choose an absolute test only, set the relative |
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36 tolerance to zero. |
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37 |
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38 The optional argument @var{sing} is a vector of values at which the |
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39 integrand is known to be singular. |
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40 |
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41 The result of the integration is returned in @var{v} and @var{ier} |
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42 contains an integer error code (0 indicates a successful integration). |
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43 The value of @var{nfun} indicates how many function evaluations were |
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44 required, and @var{err} contains an estimate of the error in the |
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45 solution. |
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46 @end deftypefn |
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47 |
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48 @deftypefn {Loadable Function} {} quad_options (@var{opt}, @var{val}) |
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49 When called with two arguments, this function allows you set options |
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50 parameters for the function @code{quad}. Given one argument, |
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51 @code{quad_options} returns the value of the corresponding option. If |
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52 no arguments are supplied, the names of all the available options and |
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53 their current values are displayed. |
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54 @end deftypefn |
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55 |
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56 Here is an example of using @code{quad} to integrate the function |
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57 @iftex |
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58 @tex |
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59 $$ |
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60 f(x) = x \sin (1/x) \sqrt {|1 - x|} |
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61 $$ |
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62 from $x = 0$ to $x = 3$. |
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63 @end tex |
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64 @end iftex |
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65 @ifinfo |
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66 |
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67 @example |
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68 @var{f}(@var{x}) = @var{x} * sin (1/@var{x}) * sqrt (abs (1 - @var{x})) |
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69 @end example |
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70 |
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71 @noindent |
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72 from @var{x} = 0 to @var{x} = 3. |
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73 @end ifinfo |
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74 |
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75 This is a fairly difficult integration (plot the function over the range |
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76 of integration to see why). |
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77 |
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78 The first step is to define the function: |
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79 |
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80 @example |
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81 @group |
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82 function y = f (x) |
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83 y = x .* sin (1 ./ x) .* sqrt (abs (1 - x)); |
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84 endfunction |
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85 @end group |
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86 @end example |
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87 |
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88 Note the use of the `dot' forms of the operators. This is not necessary |
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89 for the call to @code{quad}, but it makes it much easier to generate a |
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90 set of points for plotting (because it makes it possible to call the |
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91 function with a vector argument to produce a vector result). |
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92 |
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93 Then we simply call quad: |
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94 |
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95 @example |
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96 @group |
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97 [v, ier, nfun, err] = quad ("f", 0, 3) |
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98 @result{} 1.9819 |
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99 @result{} 1 |
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100 @result{} 5061 |
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101 @result{} 1.1522e-07 |
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102 @end group |
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103 @end example |
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104 |
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105 Although @code{quad} returns a nonzero value for @var{ier}, the result |
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106 is reasonably accurate (to see why, examine what happens to the result |
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107 if you move the lower bound to 0.1, then 0.01, then 0.001, etc.). |
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108 |
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109 @node Orthogonal Collocation, , Functions of One Variable, Quadrature |
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110 @section Orthogonal Collocation |
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111 |
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112 @deftypefn {Loadable Function} {[@var{r}, @var{A}, @var{B}, @var{q}] =} colloc (@var{n}, "left", "right") |
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113 Compute derivative and integral weight matrices for orthogonal |
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114 collocation using the subroutines given in J. Villadsen and |
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115 M. L. Michelsen, @cite{Solution of Differential Equation Models by |
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116 Polynomial Approximation}. |
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117 @end deftypefn |
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118 |
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119 Here is an example of using @code{colloc} to generate weight matrices |
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120 for solving the second order differential equation |
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121 @iftex |
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122 @tex |
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123 $u^\prime - \alpha u^{\prime\prime} = 0$ with the boundary conditions |
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124 $u(0) = 0$ and $u(1) = 1$. |
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125 @end tex |
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126 @end iftex |
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127 @ifinfo |
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128 @var{u}' - @var{alpha} * @var{u}'' = 0 with the boundary conditions |
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129 @var{u}(0) = 0 and @var{u}(1) = 1. |
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130 @end ifinfo |
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131 |
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132 First, we can generate the weight matrices for @var{n} points (including |
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133 the endpoints of the interval), and incorporate the boundary conditions |
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134 in the right hand side (for a specific value of |
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135 @iftex |
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136 @tex |
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137 $\alpha$). |
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138 @end tex |
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139 @end iftex |
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140 @ifinfo |
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141 @var{alpha}). |
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142 @end ifinfo |
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143 |
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144 @example |
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145 @group |
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146 n = 7; |
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147 alpha = 0.1; |
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148 [r, a, b] = colloc (n-2, "left", "right"); |
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149 at = a(2:n-1,2:n-1); |
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150 bt = b(2:n-1,2:n-1); |
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151 rhs = alpha * b(2:n-1,n) - a(2:n-1,n); |
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152 @end group |
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153 @end example |
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154 |
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155 Then the solution at the roots @var{r} is |
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156 |
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157 @example |
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158 u = [ 0; (at - alpha * bt) \ rhs; 1] |
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159 @result{} [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ] |
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160 @end example |
