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1 @c Copyright (C) 1996, 1997, 2007 John W. Eaton |
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2 @c |
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3 @c This file is part of Octave. |
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4 @c |
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5 @c Octave is free software; you can redistribute it and/or modify it |
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6 @c under the terms of the GNU General Public License as published by the |
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7 @c Free Software Foundation; either version 3 of the License, or (at |
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8 @c your option) any later version. |
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9 @c |
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10 @c Octave is distributed in the hope that it will be useful, but WITHOUT |
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11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 @c for more details. |
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14 @c |
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15 @c You should have received a copy of the GNU General Public License |
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16 @c along with Octave; see the file COPYING. If not, see |
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17 @c <http://www.gnu.org/licenses/>. |
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18 |
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19 @node Nonlinear Equations |
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20 @chapter Nonlinear Equations |
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21 @cindex nonlinear equations |
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22 @cindex equations, nonlinear |
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23 |
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24 Octave can solve sets of nonlinear equations of the form |
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25 @iftex |
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26 @tex |
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27 $$ |
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28 f (x) = 0 |
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29 $$ |
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30 @end tex |
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31 @end iftex |
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32 @ifnottex |
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33 |
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34 @example |
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35 F (x) = 0 |
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36 @end example |
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37 @end ifnottex |
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38 |
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39 @noindent |
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40 using the function @code{fsolve}, which is based on the @sc{Minpack} |
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41 subroutine @code{hybrd}. This is an iterative technique so a starting |
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42 point will have to be provided. This also has the consequence that |
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43 convergence is not guarantied even if a solution exists. |
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44 |
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45 @DOCSTRING(fsolve) |
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46 |
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47 @DOCSTRING(fsolve_options) |
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48 |
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49 Here is a complete example. To solve the set of equations |
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50 @iftex |
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51 @tex |
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52 $$ |
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53 \eqalign{-2x^2 + 3xy + 4\sin(y) - 6 &= 0\cr |
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54 3x^2 - 2xy^2 + 3\cos(x) + 4 &= 0} |
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55 $$ |
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56 @end tex |
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57 @end iftex |
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58 @ifinfo |
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59 |
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60 @example |
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61 -2x^2 + 3xy + 4 sin(y) = 6 |
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62 3x^2 - 2xy^2 + 3 cos(x) = -4 |
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63 @end example |
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64 @end ifinfo |
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65 |
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66 @noindent |
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67 you first need to write a function to compute the value of the given |
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68 function. For example: |
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69 |
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70 @example |
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71 function y = f (x) |
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72 y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6; |
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73 y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4; |
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74 endfunction |
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75 @end example |
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76 |
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77 Then, call @code{fsolve} with a specified initial condition to find the |
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78 roots of the system of equations. For example, given the function |
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79 @code{f} defined above, |
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80 |
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81 @example |
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82 [x, info] = fsolve (@@f, [1; 2]) |
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83 @end example |
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84 |
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85 @noindent |
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86 results in the solution |
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87 |
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88 @example |
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89 x = |
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90 |
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91 0.57983 |
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92 2.54621 |
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93 |
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94 info = 1 |
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95 @end example |
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96 |
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97 @noindent |
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98 A value of @code{info = 1} indicates that the solution has converged. |
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99 |
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100 The function @code{perror} may be used to print English messages |
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101 corresponding to the numeric error codes. For example, |
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102 |
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103 @example |
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104 @group |
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105 perror ("fsolve", 1) |
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106 @print{} solution converged to requested tolerance |
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107 @end group |
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108 @end example |
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109 |
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110 When no Jacobian is supplied (as in the example above) it is approximated |
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111 numerically. This requires more function evaluations, and hence is |
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112 less efficient. In the example above we could compute the Jacobian |
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113 analytically as |
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114 |
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115 @iftex |
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116 @tex |
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117 $$ |
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118 \left[\matrix{ {\partial f_1 \over \partial x_1} & |
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119 {\partial f_1 \over \partial x_2} \cr |
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120 {\partial f_2 \over \partial x_1} & |
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121 {\partial f_2 \over \partial x_2} \cr}\right] = |
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122 \left[\matrix{ 3 x_2 - 4 x_1 & |
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123 4 \cos(x_2) + 3 x_1 \cr |
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124 -2 x_2^2 - 3 \sin(x_1) + 6 x_1 & |
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125 -4 x_1 x_2 \cr }\right] |
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126 $$ |
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127 @end tex |
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128 which is computed with the following Octave function |
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129 @end iftex |
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130 |
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131 @example |
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132 function J = jacobian(x) |
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133 J(1,1) = 3*x(2) - 4*x(1); |
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134 J(1,2) = 4*cos(x(2)) + 3*x(1); |
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135 J(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1); |
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136 J(2,2) = -4*x(1)*x(2); |
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137 endfunction |
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138 @end example |
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139 |
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140 @noindent |
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141 Using this Jacobian is done with the following code |
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142 |
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143 @example |
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144 [x, info] = fsolve (@{@@f, @@jacobian@}, [1; 2]); |
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145 @end example |
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146 |
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147 @noindent |
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148 which gives the same solution as before. |
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149 |
