Mercurial > octave-nkf
view doc/interpreter/nonlin.txi @ 8828:8463d1a2e544
Doc fixes.
* 2]$$. => 2].$$
* @var{extrapval} => @var{extrapval}.
* call helloworld.oct => called @file{helloworld.oct}
* @itemize => @table @code
* shows. => shows:
* save => @code{save}
* @ref{Breakpoints} => @pxref{Breakpoints}
* add @noindent following example
* which is computed => and compute it
* clarify wording
* remove comma
* good => well
* set => number
* by writing => with the command
* has the option of directly calling => can call
* [-like-] {+of the right size,+}
* solvers => routines
* handle => test for
* add introductory section
* add following
* {+the+} [0..bitmax] => [0,bitmax]
* of the => with
* number => value
* add usual
* Besides when doing comparisons, logical => Logical {+also+}
* array comparison => array, comparisons
* param => parameter
* works very similar => is similar
* strings, => strings
* most simple => simplest
* easier => more easily
* like => as
* called => called,
* clarify wording
* you should simply type => use
* clarify wording
* means => way
* equally => also
* [-way much-] {+way+}
* add with mean value parameter given by the first argument, @var{l}
* add Functions described as @dfn{mapping functions} apply the given
operation to each element when given a matrix argument.
* in this brief introduction => here
* It is worth noticing => Note
* add following
* means => ways
author | Brian Gough <bjg@network-theory.co.uk> |
---|---|
date | Fri, 20 Feb 2009 11:17:01 -0500 |
parents | e8cb7f97131b |
children | eb63fbe60fab |
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@c Copyright (C) 1996, 1997, 2007 John W. Eaton @c @c This file is part of Octave. @c @c Octave is free software; you can redistribute it and/or modify it @c under the terms of the GNU General Public License as published by the @c Free Software Foundation; either version 3 of the License, or (at @c your option) any later version. @c @c Octave is distributed in the hope that it will be useful, but WITHOUT @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License @c for more details. @c @c You should have received a copy of the GNU General Public License @c along with Octave; see the file COPYING. If not, see @c <http://www.gnu.org/licenses/>. @node Nonlinear Equations @chapter Nonlinear Equations @cindex nonlinear equations @cindex equations, nonlinear Octave can solve sets of nonlinear equations of the form @iftex @tex $$ f (x) = 0 $$ @end tex @end iftex @ifnottex @example F (x) = 0 @end example @end ifnottex @noindent using the function @code{fsolve}, which is based on the @sc{Minpack} subroutine @code{hybrd}. This is an iterative technique so a starting point will have to be provided. This also has the consequence that convergence is not guaranteed even if a solution exists. @DOCSTRING(fsolve) Here is a complete example. To solve the set of equations @iftex @tex $$ \eqalign{-2x^2 + 3xy + 4\sin(y) - 6 &= 0\cr 3x^2 - 2xy^2 + 3\cos(x) + 4 &= 0} $$ @end tex @end iftex @ifinfo @example -2x^2 + 3xy + 4 sin(y) = 6 3x^2 - 2xy^2 + 3 cos(x) = -4 @end example @end ifinfo @noindent you first need to write a function to compute the value of the given function. For example: @example function y = f (x) y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6; y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4; endfunction @end example Then, call @code{fsolve} with a specified initial condition to find the roots of the system of equations. For example, given the function @code{f} defined above, @example [x, fval, info] = fsolve (@@f, [1; 2]) @end example @noindent results in the solution @example x = 0.57983 2.54621 fval = -5.7184e-10 5.5460e-10 info = 1 @end example @noindent A value of @code{info = 1} indicates that the solution has converged. The function @code{perror} may be used to print English messages corresponding to the numeric error codes. For example, @example @group perror ("fsolve", 1) @print{} solution converged to requested tolerance @end group @end example When no Jacobian is supplied (as in the example above) it is approximated numerically. This requires more function evaluations, and hence is less efficient. In the example above we could compute the Jacobian analytically as @iftex @tex $$ \left[\matrix{ {\partial f_1 \over \partial x_1} & {\partial f_1 \over \partial x_2} \cr {\partial f_2 \over \partial x_1} & {\partial f_2 \over \partial x_2} \cr}\right] = \left[\matrix{ 3 x_2 - 4 x_1 & 4 \cos(x_2) + 3 x_1 \cr -2 x_2^2 - 3 \sin(x_1) + 6 x_1 & -4 x_1 x_2 \cr }\right] $$ @end tex and compute it with the following Octave function @end iftex @example function J = jacobian(x) J(1,1) = 3*x(2) - 4*x(1); J(1,2) = 4*cos(x(2)) + 3*x(1); J(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1); J(2,2) = -4*x(1)*x(2); endfunction @end example @noindent The Jacobian can then be used with the following call to @code{fsolve}: @example [x, fval, info] = fsolve (@{@@f, @@jacobian@}, [1; 2]); @end example @noindent which gives the same solution as before. @DOCSTRING(fzero)