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1 @c Copyright (C) 1996, 1997, 1999, 2000, 2002, 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 Polynomial Manipulations |
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20 @chapter Polynomial Manipulations |
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21 |
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22 In Octave, a polynomial is represented by its coefficients (arranged |
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23 in descending order). For example, a vector @var{c} of length |
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24 @math{N+1} corresponds to the following polynomial of order |
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25 @iftex |
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26 @tex |
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27 $N$ |
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28 $$ |
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29 p (x) = c_1 x^N + \ldots + c_N x + c_{N+1}. |
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30 $$ |
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31 @end tex |
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32 @end iftex |
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33 @ifinfo |
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34 @var{N} |
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35 |
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36 @example |
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37 p(x) = @var{c}(1) x^@var{N} + ... + @var{c}(@var{N}) x + @var{c}(@var{N}+1). |
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38 @end example |
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39 @end ifinfo |
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40 |
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41 @menu |
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42 * Evaluating Polynomials:: |
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43 * Finding Roots:: |
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44 * Products of Polynomials:: |
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45 * Derivatives and Integrals:: |
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46 * Polynomial Interpolation:: |
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47 * Miscellaneous Functions:: |
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48 @end menu |
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49 |
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50 @node Evaluating Polynomials |
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51 @section Evaluating Polynomials |
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52 |
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53 The value of a polynomial represented by the vector @var{c} can be evaluated |
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54 at the point @var{x} very easily, as the following example shows. |
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55 |
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56 @example |
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57 N = length(c)-1; |
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58 val = dot( x.^(N:-1:0), c ); |
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59 @end example |
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60 |
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61 @noindent |
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62 While the above example shows how easy it is to compute the value of a |
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63 polynomial, it isn't the most stable algorithm. With larger polynomials |
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64 you should use more elegant algorithms, such as Horner's Method, which |
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65 is exactly what the Octave function @code{polyval} does. |
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66 |
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67 In the case where @var{x} is a square matrix, the polynomial given by |
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68 @var{c} is still well-defined. As when @var{x} is a scalar the obvious |
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69 implementation is easily expressed in Octave, but also in this case |
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70 more elegant algorithms perform better. The @code{polyvalm} function |
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71 provides such an algorithm. |
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72 |
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73 @DOCSTRING(polyval) |
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74 |
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75 @DOCSTRING(polyvalm) |
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76 |
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77 @node Finding Roots |
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78 @section Finding Roots |
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79 |
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80 Octave can find the roots of a given polynomial. This is done by computing |
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81 the companion matrix of the polynomial (see the @code{compan} function |
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82 for a definition), and then finding its eigenvalues. |
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83 |
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84 @DOCSTRING(roots) |
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85 |
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86 @DOCSTRING(compan) |
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87 |
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88 @node Products of Polynomials |
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89 @section Products of Polynomials |
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90 |
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91 @DOCSTRING(conv) |
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92 |
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93 @DOCSTRING(convn) |
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94 |
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95 @DOCSTRING(deconv) |
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96 |
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97 @DOCSTRING(conv2) |
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98 |
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99 @DOCSTRING(polygcd) |
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100 |
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101 @DOCSTRING(residue) |
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102 |
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103 @node Derivatives and Integrals |
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104 @section Derivatives and Integrals |
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105 |
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106 Octave comes with functions for computing the derivative and the integral |
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107 of a polynomial. The functions @code{polyderiv} and @code{polyint} |
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108 both return new polynomials describing the result. As an example we'll |
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109 compute the definite integral of @math{p(x) = x^2 + 1} from 0 to 3. |
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110 |
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111 @example |
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112 c = [1, 0, 1]; |
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113 integral = polyint(c); |
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114 area = polyval(integral, 3) - polyval(integral, 0) |
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115 @result{} 12 |
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116 @end example |
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117 |
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118 @DOCSTRING(polyderiv) |
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119 |
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120 @DOCSTRING(polyder) |
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121 |
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122 @DOCSTRING(polyint) |
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123 |
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124 @node Polynomial Interpolation |
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125 @section Polynomial Interpolation |
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126 |
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127 Octave comes with good support for various kinds of interpolation, |
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128 most of which are described in @ref{Interpolation}. One simple alternative |
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129 to the functions described in the aforementioned chapter, is to fit |
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130 a single polynomial to some given data points. To avoid a highly |
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131 fluctuating polynomial, one most often wants to fit a low-order polynomial |
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132 to data. This usually means that it is necessary to fit the polynomial |
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133 in a least-squares sense, which is what the @code{polyfit} function does. |
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134 |
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135 @DOCSTRING(polyfit) |
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136 |
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137 In situations where a single polynomial isn't good enough, a solution |
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138 is to use several polynomials pieced together. The function @code{mkpp} |
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139 creates a piece-wise polynomial, @code{ppval} evaluates the function |
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140 created by @code{mkpp}, and @code{unmkpp} returns detailed information |
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141 about the function. |
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142 |
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143 The following example shows how to combine two linear functions and a |
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144 quadratic into one function. Each of these functions is expressed |
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145 on adjoined intervals. |
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146 |
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147 @example |
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148 x = [-2, -1, 1, 2]; |
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149 p = [ 0, 1, 0; |
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150 1, -2, 1; |
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151 0, -1, 1 ]; |
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152 pp = mkpp(x, p); |
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153 xi = linspace(-2, 2, 50); |
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154 yi = ppval(pp, xi); |
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155 plot(xi, yi); |
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156 @end example |
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157 |
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158 @DOCSTRING(ppval) |
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159 |
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160 @DOCSTRING(mkpp) |
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161 |
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162 @DOCSTRING(unmkpp) |
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163 |
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164 @node Miscellaneous Functions |
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165 @section Miscellaneous Functions |
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166 |
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167 @DOCSTRING(poly) |
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168 |
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169 @DOCSTRING(polyout) |
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170 |
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171 @DOCSTRING(polyreduce) |
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172 |
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173 |
