Mercurial > octave-nkf
annotate doc/interpreter/poly.txi @ 11538:6eded7561d9d
Add undocumented polyaffine function to documentation.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Fri, 14 Jan 2011 16:58:44 -0800 |
| parents | 702dbd0c53f5 |
| children | 663594b481e5 |
| 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 Polynomial Manipulations |
| 3294 | 20 @chapter Polynomial Manipulations |
| 21 | |
| 22 In Octave, a polynomial is represented by its coefficients (arranged | |
| 6850 | 23 in descending order). For example, a vector @var{c} of length |
| 24 @math{N+1} corresponds to the following polynomial of order | |
| 3294 | 25 @tex |
| 26 $N$ | |
| 27 $$ | |
| 6850 | 28 p (x) = c_1 x^N + \ldots + c_N x + c_{N+1}. |
| 3294 | 29 $$ |
| 30 @end tex | |
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31 @ifnottex |
| 3294 | 32 @var{N} |
| 33 | |
| 34 @example | |
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35 p(x) = @var{c}(1) x^@var{N} + @dots{} + @var{c}(@var{N}) x + @var{c}(@var{N}+1). |
| 3294 | 36 @end example |
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37 |
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38 @end ifnottex |
| 3294 | 39 |
| 6850 | 40 @menu |
| 41 * Evaluating Polynomials:: | |
| 42 * Finding Roots:: | |
| 43 * Products of Polynomials:: | |
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44 * Derivatives / Integrals / Transforms:: |
| 6850 | 45 * Polynomial Interpolation:: |
| 46 * Miscellaneous Functions:: | |
| 47 @end menu | |
| 48 | |
| 49 @node Evaluating Polynomials | |
| 50 @section Evaluating Polynomials | |
| 51 | |
| 52 The value of a polynomial represented by the vector @var{c} can be evaluated | |
| 8828 | 53 at the point @var{x} very easily, as the following example shows: |
| 6850 | 54 |
| 55 @example | |
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56 @group |
| 6850 | 57 N = length(c)-1; |
| 58 val = dot( x.^(N:-1:0), c ); | |
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59 @end group |
| 6850 | 60 @end example |
| 61 | |
| 62 @noindent | |
| 63 While the above example shows how easy it is to compute the value of a | |
| 64 polynomial, it isn't the most stable algorithm. With larger polynomials | |
| 65 you should use more elegant algorithms, such as Horner's Method, which | |
| 66 is exactly what the Octave function @code{polyval} does. | |
| 67 | |
| 68 In the case where @var{x} is a square matrix, the polynomial given by | |
| 69 @var{c} is still well-defined. As when @var{x} is a scalar the obvious | |
| 70 implementation is easily expressed in Octave, but also in this case | |
| 71 more elegant algorithms perform better. The @code{polyvalm} function | |
| 72 provides such an algorithm. | |
| 73 | |
| 74 @DOCSTRING(polyval) | |
| 75 | |
| 76 @DOCSTRING(polyvalm) | |
| 77 | |
| 78 @node Finding Roots | |
| 79 @section Finding Roots | |
| 80 | |
| 81 Octave can find the roots of a given polynomial. This is done by computing | |
| 82 the companion matrix of the polynomial (see the @code{compan} function | |
| 83 for a definition), and then finding its eigenvalues. | |
| 84 | |
| 85 @DOCSTRING(roots) | |
| 86 | |
| 3368 | 87 @DOCSTRING(compan) |
| 3294 | 88 |
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89 @DOCSTRING(mpoles) |
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90 |
| 6850 | 91 @node Products of Polynomials |
| 92 @section Products of Polynomials | |
| 93 | |
| 3368 | 94 @DOCSTRING(conv) |
| 3294 | 95 |
| 7640 | 96 @DOCSTRING(convn) |
| 97 | |
| 3368 | 98 @DOCSTRING(deconv) |
| 3294 | 99 |
| 6549 | 100 @DOCSTRING(conv2) |
| 101 | |
| 6850 | 102 @DOCSTRING(polygcd) |
| 103 | |
| 104 @DOCSTRING(residue) | |
| 105 | |
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106 @node Derivatives / Integrals / Transforms |
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107 @section Derivatives / Integrals / Transforms |
| 6850 | 108 |
| 109 Octave comes with functions for computing the derivative and the integral | |
| 6898 | 110 of a polynomial. The functions @code{polyderiv} and @code{polyint} |
| 6850 | 111 both return new polynomials describing the result. As an example we'll |
| 112 compute the definite integral of @math{p(x) = x^2 + 1} from 0 to 3. | |
| 113 | |
| 114 @example | |
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115 @group |
| 6850 | 116 c = [1, 0, 1]; |
| 6898 | 117 integral = polyint(c); |
| 6850 | 118 area = polyval(integral, 3) - polyval(integral, 0) |
| 119 @result{} 12 | |
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120 @end group |
| 6850 | 121 @end example |
| 3294 | 122 |
| 3368 | 123 @DOCSTRING(polyderiv) |
| 3294 | 124 |
| 6502 | 125 @DOCSTRING(polyder) |
| 126 | |
| 6898 | 127 @DOCSTRING(polyint) |
| 3294 | 128 |
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129 @DOCSTRING(polyaffine) |
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130 |
| 6850 | 131 @node Polynomial Interpolation |
| 132 @section Polynomial Interpolation | |
| 3294 | 133 |
| 6850 | 134 Octave comes with good support for various kinds of interpolation, |
| 135 most of which are described in @ref{Interpolation}. One simple alternative | |
| 136 to the functions described in the aforementioned chapter, is to fit | |
| 137 a single polynomial to some given data points. To avoid a highly | |
| 138 fluctuating polynomial, one most often wants to fit a low-order polynomial | |
| 139 to data. This usually means that it is necessary to fit the polynomial | |
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140 in a least-squares sense, which just is what the @code{polyfit} function does. |
| 3294 | 141 |
| 6850 | 142 @DOCSTRING(polyfit) |
| 3294 | 143 |
| 6850 | 144 In situations where a single polynomial isn't good enough, a solution |
| 145 is to use several polynomials pieced together. The function @code{mkpp} | |
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146 creates a piecewise polynomial, @code{ppval} evaluates the function |
| 6850 | 147 created by @code{mkpp}, and @code{unmkpp} returns detailed information |
| 148 about the function. | |
| 149 | |
| 150 The following example shows how to combine two linear functions and a | |
| 7001 | 151 quadratic into one function. Each of these functions is expressed |
| 6850 | 152 on adjoined intervals. |
| 3294 | 153 |
| 6850 | 154 @example |
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155 @group |
| 6850 | 156 x = [-2, -1, 1, 2]; |
| 157 p = [ 0, 1, 0; | |
| 158 1, -2, 1; | |
| 159 0, -1, 1 ]; | |
| 160 pp = mkpp(x, p); | |
| 161 xi = linspace(-2, 2, 50); | |
| 162 yi = ppval(pp, xi); | |
| 163 plot(xi, yi); | |
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164 @end group |
| 6850 | 165 @end example |
| 6502 | 166 |
| 167 @DOCSTRING(mkpp) | |
| 168 | |
| 169 @DOCSTRING(unmkpp) | |
| 6850 | 170 |
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171 @DOCSTRING(ppval) |
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172 |
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173 @DOCSTRING(ppder) |
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174 |
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175 @DOCSTRING(ppint) |
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176 |
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177 @DOCSTRING(ppjumps) |
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178 |
| 6850 | 179 @node Miscellaneous Functions |
| 180 @section Miscellaneous Functions | |
| 181 | |
| 182 @DOCSTRING(poly) | |
| 183 | |
| 184 @DOCSTRING(polyout) | |
| 185 | |
| 186 @DOCSTRING(polyreduce) | |
| 187 | |
| 188 | |
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189 |
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190 |