Mercurial > hg > octave-nkf
annotate doc/interpreter/poly.txi @ 10224:f6e0404421f4
point to polyint in @seealso, not polyinteg
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Fri, 29 Jan 2010 12:52:32 -0500 |
| parents | 923c7cb7f13f |
| children | 72585f1ca7a2 |
| rev | line source |
|---|---|
| 8920 | 1 @c Copyright (C) 1996, 1997, 1999, 2000, 2002, 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 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 | |
| 31 @ifinfo | |
| 32 @var{N} | |
| 33 | |
| 34 @example | |
|
9070
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Cleanup documentation for poly.texi, interp.texi, geometry.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
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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 |
| 37 @end ifinfo | |
| 38 | |
| 6850 | 39 @menu |
| 40 * Evaluating Polynomials:: | |
| 41 * Finding Roots:: | |
| 42 * Products of Polynomials:: | |
| 43 * Derivatives and Integrals:: | |
| 44 * Polynomial Interpolation:: | |
| 45 * Miscellaneous Functions:: | |
| 46 @end menu | |
| 47 | |
| 48 @node Evaluating Polynomials | |
| 49 @section Evaluating Polynomials | |
| 50 | |
| 51 The value of a polynomial represented by the vector @var{c} can be evaluated | |
| 8828 | 52 at the point @var{x} very easily, as the following example shows: |
| 6850 | 53 |
| 54 @example | |
|
9070
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Cleanup documentation for poly.texi, interp.texi, geometry.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
55 @group |
| 6850 | 56 N = length(c)-1; |
| 57 val = dot( x.^(N:-1:0), c ); | |
|
9070
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Cleanup documentation for poly.texi, interp.texi, geometry.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
58 @end group |
| 6850 | 59 @end example |
| 60 | |
| 61 @noindent | |
| 62 While the above example shows how easy it is to compute the value of a | |
| 63 polynomial, it isn't the most stable algorithm. With larger polynomials | |
| 64 you should use more elegant algorithms, such as Horner's Method, which | |
| 65 is exactly what the Octave function @code{polyval} does. | |
| 66 | |
| 67 In the case where @var{x} is a square matrix, the polynomial given by | |
| 68 @var{c} is still well-defined. As when @var{x} is a scalar the obvious | |
| 69 implementation is easily expressed in Octave, but also in this case | |
| 70 more elegant algorithms perform better. The @code{polyvalm} function | |
| 71 provides such an algorithm. | |
| 72 | |
| 73 @DOCSTRING(polyval) | |
| 74 | |
| 75 @DOCSTRING(polyvalm) | |
| 76 | |
| 77 @node Finding Roots | |
| 78 @section Finding Roots | |
| 79 | |
| 80 Octave can find the roots of a given polynomial. This is done by computing | |
| 81 the companion matrix of the polynomial (see the @code{compan} function | |
| 82 for a definition), and then finding its eigenvalues. | |
| 83 | |
| 84 @DOCSTRING(roots) | |
| 85 | |
| 3368 | 86 @DOCSTRING(compan) |
| 3294 | 87 |
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8286
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fix @seealso references to point to existing anchors
Thorsten Meyer <thorsten.meyier@gmx.de>
parents:
7640
diff
changeset
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88 @DOCSTRING(mpoles) |
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6f2d95255911
fix @seealso references to point to existing anchors
Thorsten Meyer <thorsten.meyier@gmx.de>
parents:
7640
diff
changeset
|
89 |
| 6850 | 90 @node Products of Polynomials |
| 91 @section Products of Polynomials | |
| 92 | |
| 3368 | 93 @DOCSTRING(conv) |
| 3294 | 94 |
| 7640 | 95 @DOCSTRING(convn) |
| 96 | |
| 3368 | 97 @DOCSTRING(deconv) |
| 3294 | 98 |
| 6549 | 99 @DOCSTRING(conv2) |
| 100 | |
| 6850 | 101 @DOCSTRING(polygcd) |
| 102 | |
| 103 @DOCSTRING(residue) | |
| 104 | |
| 105 @node Derivatives and Integrals | |
| 106 @section Derivatives and Integrals | |
| 107 | |
| 108 Octave comes with functions for computing the derivative and the integral | |
| 6898 | 109 of a polynomial. The functions @code{polyderiv} and @code{polyint} |
| 6850 | 110 both return new polynomials describing the result. As an example we'll |
| 111 compute the definite integral of @math{p(x) = x^2 + 1} from 0 to 3. | |
| 112 | |
| 113 @example | |
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9070
e9dc2ed2ec0f
Cleanup documentation for poly.texi, interp.texi, geometry.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
114 @group |
| 6850 | 115 c = [1, 0, 1]; |
| 6898 | 116 integral = polyint(c); |
| 6850 | 117 area = polyval(integral, 3) - polyval(integral, 0) |
| 118 @result{} 12 | |
|
9070
e9dc2ed2ec0f
Cleanup documentation for poly.texi, interp.texi, geometry.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
119 @end group |
| 6850 | 120 @end example |
| 3294 | 121 |
| 3368 | 122 @DOCSTRING(polyderiv) |
| 3294 | 123 |
| 6502 | 124 @DOCSTRING(polyder) |
| 125 | |
| 6898 | 126 @DOCSTRING(polyint) |
| 3294 | 127 |
| 6850 | 128 @node Polynomial Interpolation |
| 129 @section Polynomial Interpolation | |
| 3294 | 130 |
| 6850 | 131 Octave comes with good support for various kinds of interpolation, |
| 132 most of which are described in @ref{Interpolation}. One simple alternative | |
| 133 to the functions described in the aforementioned chapter, is to fit | |
| 134 a single polynomial to some given data points. To avoid a highly | |
| 135 fluctuating polynomial, one most often wants to fit a low-order polynomial | |
| 136 to data. This usually means that it is necessary to fit the polynomial | |
| 137 in a least-squares sense, which is what the @code{polyfit} function does. | |
| 3294 | 138 |
| 6850 | 139 @DOCSTRING(polyfit) |
| 3294 | 140 |
| 6850 | 141 In situations where a single polynomial isn't good enough, a solution |
| 142 is to use several polynomials pieced together. The function @code{mkpp} | |
| 143 creates a piece-wise polynomial, @code{ppval} evaluates the function | |
| 144 created by @code{mkpp}, and @code{unmkpp} returns detailed information | |
| 145 about the function. | |
| 146 | |
| 147 The following example shows how to combine two linear functions and a | |
| 7001 | 148 quadratic into one function. Each of these functions is expressed |
| 6850 | 149 on adjoined intervals. |
| 3294 | 150 |
| 6850 | 151 @example |
|
9070
e9dc2ed2ec0f
Cleanup documentation for poly.texi, interp.texi, geometry.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
152 @group |
| 6850 | 153 x = [-2, -1, 1, 2]; |
| 154 p = [ 0, 1, 0; | |
| 155 1, -2, 1; | |
| 156 0, -1, 1 ]; | |
| 157 pp = mkpp(x, p); | |
| 158 xi = linspace(-2, 2, 50); | |
| 159 yi = ppval(pp, xi); | |
| 160 plot(xi, yi); | |
|
9070
e9dc2ed2ec0f
Cleanup documentation for poly.texi, interp.texi, geometry.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
161 @end group |
| 6850 | 162 @end example |
| 6502 | 163 |
| 164 @DOCSTRING(ppval) | |
| 165 | |
| 166 @DOCSTRING(mkpp) | |
| 167 | |
| 168 @DOCSTRING(unmkpp) | |
| 6850 | 169 |
| 170 @node Miscellaneous Functions | |
| 171 @section Miscellaneous Functions | |
| 172 | |
| 173 @DOCSTRING(poly) | |
| 174 | |
| 175 @DOCSTRING(polyout) | |
| 176 | |
| 177 @DOCSTRING(polyreduce) | |
| 178 | |
| 179 | |
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8286
6f2d95255911
fix @seealso references to point to existing anchors
Thorsten Meyer <thorsten.meyier@gmx.de>
parents:
7640
diff
changeset
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180 |
|
6f2d95255911
fix @seealso references to point to existing anchors
Thorsten Meyer <thorsten.meyier@gmx.de>
parents:
7640
diff
changeset
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181 |