Mercurial > hg > octave-nkf
annotate doc/interpreter/nonlin.txi @ 8920:eb63fbe60fab
update copyright notices
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Sat, 07 Mar 2009 10:41:27 -0500 |
| parents | 8463d1a2e544 |
| children | 8207b833557f |
| rev | line source |
|---|---|
| 8920 | 1 @c Copyright (C) 1996, 1997, 2007, 2008, 2009 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 @iftex | |
| 26 @tex | |
| 27 $$ | |
| 28 f (x) = 0 | |
| 29 $$ | |
| 30 @end tex | |
| 31 @end iftex | |
| 6850 | 32 @ifnottex |
| 3294 | 33 |
| 34 @example | |
| 35 F (x) = 0 | |
| 36 @end example | |
| 6850 | 37 @end ifnottex |
| 3294 | 38 |
| 39 @noindent | |
| 40 using the function @code{fsolve}, which is based on the @sc{Minpack} | |
| 6850 | 41 subroutine @code{hybrd}. This is an iterative technique so a starting |
| 42 point will have to be provided. This also has the consequence that | |
|
8325
b93ac0586e4b
spelling corrections
Brian Gough<bjg@network-theory.co.uk>
parents:
7359
diff
changeset
|
43 convergence is not guaranteed even if a solution exists. |
| 3294 | 44 |
| 3368 | 45 @DOCSTRING(fsolve) |
| 3294 | 46 |
| 47 Here is a complete example. To solve the set of equations | |
| 48 @iftex | |
| 49 @tex | |
| 50 $$ | |
| 51 \eqalign{-2x^2 + 3xy + 4\sin(y) - 6 &= 0\cr | |
| 52 3x^2 - 2xy^2 + 3\cos(x) + 4 &= 0} | |
| 53 $$ | |
| 54 @end tex | |
| 55 @end iftex | |
| 56 @ifinfo | |
| 57 | |
| 58 @example | |
| 59 -2x^2 + 3xy + 4 sin(y) = 6 | |
| 60 3x^2 - 2xy^2 + 3 cos(x) = -4 | |
| 61 @end example | |
| 62 @end ifinfo | |
| 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 | |
| 69 function y = f (x) | |
| 70 y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6; | |
| 71 y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4; | |
| 72 endfunction | |
| 73 @end example | |
| 74 | |
| 75 Then, call @code{fsolve} with a specified initial condition to find the | |
| 76 roots of the system of equations. For example, given the function | |
| 77 @code{f} defined above, | |
| 78 | |
| 79 @example | |
| 7359 | 80 [x, fval, info] = fsolve (@@f, [1; 2]) |
| 3294 | 81 @end example |
| 82 | |
| 83 @noindent | |
| 84 results in the solution | |
| 85 | |
| 86 @example | |
| 87 x = | |
| 88 | |
| 89 0.57983 | |
| 90 2.54621 | |
| 91 | |
| 7359 | 92 fval = |
| 93 | |
| 94 -5.7184e-10 | |
| 95 5.5460e-10 | |
| 96 | |
| 3294 | 97 info = 1 |
| 98 @end example | |
| 99 | |
| 6850 | 100 @noindent |
| 3294 | 101 A value of @code{info = 1} indicates that the solution has converged. |
| 102 | |
| 103 The function @code{perror} may be used to print English messages | |
| 104 corresponding to the numeric error codes. For example, | |
| 105 | |
| 106 @example | |
| 107 @group | |
| 108 perror ("fsolve", 1) | |
| 109 @print{} solution converged to requested tolerance | |
| 110 @end group | |
| 111 @end example | |
| 112 | |
| 6850 | 113 When no Jacobian is supplied (as in the example above) it is approximated |
| 114 numerically. This requires more function evaluations, and hence is | |
| 115 less efficient. In the example above we could compute the Jacobian | |
| 116 analytically as | |
| 3294 | 117 |
| 6850 | 118 @iftex |
| 119 @tex | |
| 120 $$ | |
| 121 \left[\matrix{ {\partial f_1 \over \partial x_1} & | |
| 122 {\partial f_1 \over \partial x_2} \cr | |
| 123 {\partial f_2 \over \partial x_1} & | |
| 124 {\partial f_2 \over \partial x_2} \cr}\right] = | |
| 125 \left[\matrix{ 3 x_2 - 4 x_1 & | |
| 126 4 \cos(x_2) + 3 x_1 \cr | |
| 127 -2 x_2^2 - 3 \sin(x_1) + 6 x_1 & | |
| 128 -4 x_1 x_2 \cr }\right] | |
| 129 $$ | |
| 130 @end tex | |
| 8828 | 131 and compute it with the following Octave function |
| 6850 | 132 @end iftex |
| 133 | |
| 134 @example | |
| 135 function J = jacobian(x) | |
| 136 J(1,1) = 3*x(2) - 4*x(1); | |
| 137 J(1,2) = 4*cos(x(2)) + 3*x(1); | |
| 138 J(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1); | |
| 139 J(2,2) = -4*x(1)*x(2); | |
| 140 endfunction | |
| 141 @end example | |
| 142 | |
| 143 @noindent | |
| 8828 | 144 The Jacobian can then be used with the following call to @code{fsolve}: |
| 6850 | 145 |
| 146 @example | |
| 7359 | 147 [x, fval, info] = fsolve (@{@@f, @@jacobian@}, [1; 2]); |
| 6850 | 148 @end example |
| 149 | |
| 150 @noindent | |
| 151 which gives the same solution as before. | |
| 152 | |
|
8515
ec2715c76039
fzero.m, fsolve.m: additional doc fixes
John W. Eaton <jwe@octave.org>
parents:
8325
diff
changeset
|
153 @DOCSTRING(fzero) |