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Fix some segfaults
author Max Brister <max@2bass.com>
date Fri, 15 Jun 2012 13:15:48 -0500
parents 72c96de7a403
children e0525ecf156e
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72c96de7a403 maint: update copyright notices for 2012
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1 @c Copyright (C) 1996-2012 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 @menu
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25 * Solvers::
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26 * Minimizers::
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27 @end menu
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28
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29 @node Solvers
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30 @section Solvers
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31
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32 Octave can solve sets of nonlinear equations of the form
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33 @tex
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34 $$
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35 f (x) = 0
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36 $$
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37 @end tex
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38 @ifnottex
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39
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40 @example
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41 F (x) = 0
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42 @end example
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43
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44 @end ifnottex
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45
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46 @noindent
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47 using the function @code{fsolve}, which is based on the @sc{minpack}
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48 subroutine @code{hybrd}. This is an iterative technique so a starting
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49 point must be provided. This also has the consequence that
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50 convergence is not guaranteed even if a solution exists.
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51
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52 @DOCSTRING(fsolve)
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53
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54 The following is a complete example. To solve the set of equations
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55 @tex
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56 $$
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57 \eqalign{-2x^2 + 3xy + 4\sin(y) - 6 &= 0\cr
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58 3x^2 - 2xy^2 + 3\cos(x) + 4 &= 0}
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59 $$
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60 @end tex
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61 @ifnottex
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62
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63 @example
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64 @group
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65 -2x^2 + 3xy + 4 sin(y) = 6
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66 3x^2 - 2xy^2 + 3 cos(x) = -4
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67 @end group
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68 @end example
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69
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70 @end ifnottex
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71
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72 @noindent
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73 you first need to write a function to compute the value of the given
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74 function. For example:
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75
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76 @example
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77 @group
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78 function y = f (x)
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79 y = zeros (2, 1);
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80 y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6;
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81 y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
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82 endfunction
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83 @end group
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84 @end example
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85
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86 Then, call @code{fsolve} with a specified initial condition to find the
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87 roots of the system of equations. For example, given the function
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88 @code{f} defined above,
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89
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90 @example
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91 [x, fval, info] = fsolve (@@f, [1; 2])
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92 @end example
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93
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94 @noindent
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95 results in the solution
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96
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97 @example
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98 @group
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99 x =
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100
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101 0.57983
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102 2.54621
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103
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104 fval =
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105
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106 -5.7184e-10
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107 5.5460e-10
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108
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109 info = 1
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110 @end group
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111 @end example
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112
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113 @noindent
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114 A value of @code{info = 1} indicates that the solution has converged.
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115
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116 When no Jacobian is supplied (as in the example above) it is approximated
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117 numerically. This requires more function evaluations, and hence is
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118 less efficient. In the example above we could compute the Jacobian
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119 analytically as
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120
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121 @iftex
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122 @tex
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123 $$
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124 \left[\matrix{ {\partial f_1 \over \partial x_1} &
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125 {\partial f_1 \over \partial x_2} \cr
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126 {\partial f_2 \over \partial x_1} &
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127 {\partial f_2 \over \partial x_2} \cr}\right] =
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128 \left[\matrix{ 3 x_2 - 4 x_1 &
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129 4 \cos(x_2) + 3 x_1 \cr
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130 -2 x_2^2 - 3 \sin(x_1) + 6 x_1 &
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131 -4 x_1 x_2 \cr }\right]
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132 $$
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133 @end tex
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134 and compute it with the following Octave function
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135 @end iftex
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136
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137 @example
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138 @group
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139 function [y, jac] = f (x)
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140 y = zeros (2, 1);
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141 y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6;
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142 y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
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143 if (nargout == 2)
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144 jac = zeros (2, 2);
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145 jac(1,1) = 3*x(2) - 4*x(1);
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146 jac(1,2) = 4*cos(x(2)) + 3*x(1);
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147 jac(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1);
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148 jac(2,2) = -4*x(1)*x(2);
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149 endif
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150 endfunction
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151 @end group
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152 @end example
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153
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154 @noindent
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155 The Jacobian can then be used with the following call to @code{fsolve}:
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156
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157 @example
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158 [x, fval, info] = fsolve (@@f, [1; 2], optimset ("jacobian", "on"));
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159 @end example
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160
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161 @noindent
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162 which gives the same solution as before.
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163
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164 @DOCSTRING(fzero)
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165
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166 @node Minimizers
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167 @section Minimizers
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168 @cindex local minimum
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169 @cindex finding minimums
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170
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171 Often it is useful to find the minimum value of a function rather than just
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172 the zeroes where it crosses the x-axis. @code{fminbnd} is designed for the
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173 simpler, but very common, case of a univariate function where the interval
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174 to search is bounded. For unbounded minimization of a function with
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175 potentially many variables use @code{fminunc}. @xref{Optimization}, for
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176 minimization with the presence of constraint functions. Note that searches
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177 can be made for maxima by simply inverting the objective function
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178 @tex
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179 ($F_{max} = -F_{min}$).
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180 @end tex
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181 @ifnottex
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182 (@code{Fto_max = -Fto_min}).
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183 @end ifnottex
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184
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185 @DOCSTRING(fminbnd)
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186
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187 @DOCSTRING(fminunc)
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188