Mercurial > hg > octave-nkf
annotate doc/interpreter/nonlin.txi @ 12385:c468c5b902b3 release-3-4-x ss-3-3-92
version is now 3.3.92
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Sun, 06 Feb 2011 06:27:19 -0500 |
| parents | fd0a3ac60b0e |
| children | 02e48856e486 |
| rev | line source |
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| 11523 | 1 @c Copyright (C) 1996-2011 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 18 |
| 4167 | 19 @node Nonlinear Equations |
| 3294 | 20 @chapter Nonlinear Equations |
| 21 @cindex nonlinear equations | |
| 22 @cindex equations, nonlinear | |
| 23 | |
| 24 Octave can solve sets of nonlinear equations of the form | |
| 25 @tex | |
| 26 $$ | |
| 27 f (x) = 0 | |
| 28 $$ | |
| 29 @end tex | |
| 6850 | 30 @ifnottex |
| 3294 | 31 |
| 32 @example | |
| 33 F (x) = 0 | |
| 34 @end example | |
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35 |
| 6850 | 36 @end ifnottex |
| 3294 | 37 |
| 38 @noindent | |
| 39 using the function @code{fsolve}, which is based on the @sc{Minpack} | |
| 6850 | 40 subroutine @code{hybrd}. This is an iterative technique so a starting |
| 41 point will have to be provided. This also has the consequence that | |
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42 convergence is not guaranteed even if a solution exists. |
| 3294 | 43 |
| 3368 | 44 @DOCSTRING(fsolve) |
| 3294 | 45 |
| 46 Here is a complete example. To solve the set of equations | |
| 47 @tex | |
| 48 $$ | |
| 49 \eqalign{-2x^2 + 3xy + 4\sin(y) - 6 &= 0\cr | |
| 50 3x^2 - 2xy^2 + 3\cos(x) + 4 &= 0} | |
| 51 $$ | |
| 52 @end tex | |
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53 @ifnottex |
| 3294 | 54 |
| 55 @example | |
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56 @group |
| 3294 | 57 -2x^2 + 3xy + 4 sin(y) = 6 |
| 58 3x^2 - 2xy^2 + 3 cos(x) = -4 | |
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59 @end group |
| 3294 | 60 @end example |
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61 |
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62 @end ifnottex |
| 3294 | 63 |
| 64 @noindent | |
| 65 you first need to write a function to compute the value of the given | |
| 66 function. For example: | |
| 67 | |
| 68 @example | |
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69 @group |
| 3294 | 70 function y = f (x) |
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71 y = zeros (2, 1); |
| 3294 | 72 y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6; |
| 73 y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4; | |
| 74 endfunction | |
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75 @end group |
| 3294 | 76 @end example |
| 77 | |
| 78 Then, call @code{fsolve} with a specified initial condition to find the | |
| 79 roots of the system of equations. For example, given the function | |
| 80 @code{f} defined above, | |
| 81 | |
| 82 @example | |
| 7359 | 83 [x, fval, info] = fsolve (@@f, [1; 2]) |
| 3294 | 84 @end example |
| 85 | |
| 86 @noindent | |
| 87 results in the solution | |
| 88 | |
| 89 @example | |
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90 @group |
| 3294 | 91 x = |
| 92 | |
| 93 0.57983 | |
| 94 2.54621 | |
| 95 | |
| 7359 | 96 fval = |
| 97 | |
| 98 -5.7184e-10 | |
| 99 5.5460e-10 | |
| 100 | |
| 3294 | 101 info = 1 |
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102 @end group |
| 3294 | 103 @end example |
| 104 | |
| 6850 | 105 @noindent |
| 3294 | 106 A value of @code{info = 1} indicates that the solution has converged. |
| 107 | |
| 108 The function @code{perror} may be used to print English messages | |
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109 corresponding to the numeric error codes. For example: |
| 3294 | 110 |
| 111 @example | |
| 112 @group | |
| 113 perror ("fsolve", 1) | |
| 114 @print{} solution converged to requested tolerance | |
| 115 @end group | |
| 116 @end example | |
| 117 | |
| 6850 | 118 When no Jacobian is supplied (as in the example above) it is approximated |
| 119 numerically. This requires more function evaluations, and hence is | |
| 120 less efficient. In the example above we could compute the Jacobian | |
| 121 analytically as | |
| 3294 | 122 |
| 6850 | 123 @iftex |
| 124 @tex | |
| 125 $$ | |
| 126 \left[\matrix{ {\partial f_1 \over \partial x_1} & | |
| 127 {\partial f_1 \over \partial x_2} \cr | |
| 128 {\partial f_2 \over \partial x_1} & | |
| 129 {\partial f_2 \over \partial x_2} \cr}\right] = | |
| 130 \left[\matrix{ 3 x_2 - 4 x_1 & | |
| 131 4 \cos(x_2) + 3 x_1 \cr | |
| 132 -2 x_2^2 - 3 \sin(x_1) + 6 x_1 & | |
| 133 -4 x_1 x_2 \cr }\right] | |
| 134 $$ | |
| 135 @end tex | |
| 8828 | 136 and compute it with the following Octave function |
| 6850 | 137 @end iftex |
| 138 | |
| 139 @example | |
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140 @group |
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141 function [y, jac] = f (x) |
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142 y = zeros (2, 1); |
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143 y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6; |
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144 y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4; |
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145 if (nargout == 2) |
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146 jac = zeros (2, 2); |
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147 jac(1,1) = 3*x(2) - 4*x(1); |
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148 jac(1,2) = 4*cos(x(2)) + 3*x(1); |
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149 jac(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1); |
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150 jac(2,2) = -4*x(1)*x(2); |
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151 endif |
| 6850 | 152 endfunction |
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153 @end group |
| 6850 | 154 @end example |
| 155 | |
| 156 @noindent | |
| 8828 | 157 The Jacobian can then be used with the following call to @code{fsolve}: |
| 6850 | 158 |
| 159 @example | |
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160 [x, fval, info] = fsolve (@@f, [1; 2], optimset ("jacobian", "on")); |
| 6850 | 161 @end example |
| 162 | |
| 163 @noindent | |
| 164 which gives the same solution as before. | |
| 165 | |
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166 @DOCSTRING(fzero) |