Mercurial > octave-nkf
annotate doc/interpreter/poly.txi @ 20595:c1a6c31ac29a
eliminate more simple uses of error_state
* ov-classdef.cc: Eliminate simple uses of error_state.
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Tue, 06 Oct 2015 00:20:02 -0400 |
| parents | 4197fc428c7d |
| children |
| rev | line source |
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1 @c Copyright (C) 1996-2015 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. | |
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9 @c |
| 7018 | 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. | |
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14 @c |
| 7018 | 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 |
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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 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 | |
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87 @DOCSTRING(polyeig) |
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88 |
| 3368 | 89 @DOCSTRING(compan) |
| 3294 | 90 |
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91 @DOCSTRING(mpoles) |
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92 |
| 6850 | 93 @node Products of Polynomials |
| 94 @section Products of Polynomials | |
| 95 | |
| 3368 | 96 @DOCSTRING(conv) |
| 3294 | 97 |
| 7640 | 98 @DOCSTRING(convn) |
| 99 | |
| 3368 | 100 @DOCSTRING(deconv) |
| 3294 | 101 |
| 6549 | 102 @DOCSTRING(conv2) |
| 103 | |
| 6850 | 104 @DOCSTRING(polygcd) |
| 105 | |
| 106 @DOCSTRING(residue) | |
| 107 | |
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108 @node Derivatives / Integrals / Transforms |
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109 @section Derivatives / Integrals / Transforms |
| 6850 | 110 |
| 111 Octave comes with functions for computing the derivative and the integral | |
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112 of a polynomial. The functions @code{polyder} and @code{polyint} |
| 6850 | 113 both return new polynomials describing the result. As an example we'll |
| 114 compute the definite integral of @math{p(x) = x^2 + 1} from 0 to 3. | |
| 115 | |
| 116 @example | |
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117 @group |
| 6850 | 118 c = [1, 0, 1]; |
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119 integral = polyint (c); |
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120 area = polyval (integral, 3) - polyval (integral, 0) |
| 6850 | 121 @result{} 12 |
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122 @end group |
| 6850 | 123 @end example |
| 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 | |
| 14509 | 137 a single polynomial, or a piecewise polynomial (spline) to some given |
| 138 data points. To avoid a highly fluctuating polynomial, one most often | |
| 139 wants to fit a low-order polynomial to data. This usually means that it | |
| 140 is necessary to fit the polynomial in a least-squares sense, which just | |
| 141 is what the @code{polyfit} function does. | |
| 3294 | 142 |
| 6850 | 143 @DOCSTRING(polyfit) |
| 3294 | 144 |
| 6850 | 145 In situations where a single polynomial isn't good enough, a solution |
| 14509 | 146 is to use several polynomials pieced together. The function |
| 147 @code{splinefit} fits a peicewise polynomial (spline) to a set of | |
| 148 data. | |
| 149 | |
| 150 @DOCSTRING(splinefit) | |
| 151 | |
| 152 The number of @var{breaks} (or knots) used to construct the piecewise | |
| 153 polynomial is a significant factor in suppressing the noise present in | |
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154 the input data, @var{x} and @var{y}. This is demonstrated by the example |
| 14509 | 155 below. |
| 156 | |
| 157 @example | |
| 158 @group | |
| 159 x = 2 * pi * rand (1, 200); | |
| 160 y = sin (x) + sin (2 * x) + 0.2 * randn (size (x)); | |
| 161 ## Uniform breaks | |
| 162 breaks = linspace (0, 2 * pi, 41); % 41 breaks, 40 pieces | |
| 163 pp1 = splinefit (x, y, breaks); | |
| 164 ## Breaks interpolated from data | |
| 165 pp2 = splinefit (x, y, 10); % 11 breaks, 10 pieces | |
| 166 ## Plot | |
| 167 xx = linspace (0, 2 * pi, 400); | |
| 168 y1 = ppval (pp1, xx); | |
| 169 y2 = ppval (pp2, xx); | |
| 170 plot (x, y, ".", xx, [y1; y2]) | |
| 171 axis tight | |
| 172 ylim auto | |
| 173 legend (@{"data", "41 breaks, 40 pieces", "11 breaks, 10 pieces"@}) | |
| 174 @end group | |
| 175 @end example | |
| 176 | |
| 177 @ifnotinfo | |
| 178 @noindent | |
| 179 The result of which can be seen in @ref{fig:splinefit1}. | |
| 180 | |
| 181 @float Figure,fig:splinefit1 | |
| 182 @center @image{splinefit1,4in} | |
| 183 @caption{Comparison of a fitting a piecewise polynomial with 41 breaks to one | |
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184 with 11 breaks. The fit with the large number of breaks exhibits a fast ripple |
| 14509 | 185 that is not present in the underlying function.} |
| 186 @end float | |
| 187 @end ifnotinfo | |
| 188 | |
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189 The piecewise polynomial fit, provided by @code{splinefit}, has |
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190 continuous derivatives up to the @var{order}-1. For example, a cubic fit |
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191 has continuous first and second derivatives. This is demonstrated by |
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192 the code |
| 14509 | 193 |
| 194 @example | |
| 195 ## Data (200 points) | |
| 196 x = 2 * pi * rand (1, 200); | |
| 197 y = sin (x) + sin (2 * x) + 0.1 * randn (size (x)); | |
| 198 ## Piecewise constant | |
| 199 pp1 = splinefit (x, y, 8, "order", 0); | |
| 200 ## Piecewise linear | |
| 201 pp2 = splinefit (x, y, 8, "order", 1); | |
| 202 ## Piecewise quadratic | |
| 203 pp3 = splinefit (x, y, 8, "order", 2); | |
| 204 ## Piecewise cubic | |
| 205 pp4 = splinefit (x, y, 8, "order", 3); | |
| 206 ## Piecewise quartic | |
| 207 pp5 = splinefit (x, y, 8, "order", 4); | |
| 208 ## Plot | |
| 209 xx = linspace (0, 2 * pi, 400); | |
| 210 y1 = ppval (pp1, xx); | |
| 211 y2 = ppval (pp2, xx); | |
| 212 y3 = ppval (pp3, xx); | |
| 213 y4 = ppval (pp4, xx); | |
| 214 y5 = ppval (pp5, xx); | |
| 215 plot (x, y, ".", xx, [y1; y2; y3; y4; y5]) | |
| 216 axis tight | |
| 217 ylim auto | |
| 218 legend (@{"data", "order 0", "order 1", "order 2", "order 3", "order 4"@}) | |
| 219 @end example | |
| 220 | |
| 221 @ifnotinfo | |
| 222 @noindent | |
| 223 The result of which can be seen in @ref{fig:splinefit2}. | |
| 224 | |
| 225 @float Figure,fig:splinefit2 | |
| 226 @center @image{splinefit2,4in} | |
| 227 @caption{Comparison of a piecewise constant, linear, quadratic, cubic, and | |
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228 quartic polynomials with 8 breaks to noisy data. The higher order solutions |
| 14509 | 229 more accurately represent the underlying function, but come with the |
| 230 expense of computational complexity.} | |
| 231 @end float | |
| 232 @end ifnotinfo | |
| 233 | |
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234 When the underlying function to provide a fit to is periodic, @code{splinefit} |
| 14509 | 235 is able to apply the boundary conditions needed to manifest a periodic fit. |
| 236 This is demonstrated by the code below. | |
| 237 | |
| 238 @example | |
| 239 @group | |
| 240 ## Data (100 points) | |
| 241 x = 2 * pi * [0, (rand (1, 98)), 1]; | |
| 242 y = sin (x) - cos (2 * x) + 0.2 * randn (size (x)); | |
| 243 ## No constraints | |
| 244 pp1 = splinefit (x, y, 10, "order", 5); | |
| 245 ## Periodic boundaries | |
| 246 pp2 = splinefit (x, y, 10, "order", 5, "periodic", true); | |
| 247 ## Plot | |
| 248 xx = linspace (0, 2 * pi, 400); | |
| 249 y1 = ppval (pp1, xx); | |
| 250 y2 = ppval (pp2, xx); | |
| 251 plot (x, y, ".", xx, [y1; y2]) | |
| 252 axis tight | |
| 253 ylim auto | |
| 254 legend (@{"data", "no constraints", "periodic"@}) | |
| 255 @end group | |
| 256 @end example | |
| 257 | |
| 258 @ifnotinfo | |
| 259 @noindent | |
| 260 The result of which can be seen in @ref{fig:splinefit3}. | |
| 261 | |
| 262 @float Figure,fig:splinefit3 | |
| 263 @center @image{splinefit3,4in} | |
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264 @caption{Comparison of piecewise polynomial fits to a noisy periodic |
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265 function with, and without, periodic boundary conditions.} |
| 14509 | 266 @end float |
| 267 @end ifnotinfo | |
| 268 | |
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269 More complex constraints may be added as well. For example, the code below |
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270 illustrates a periodic fit with values that have been clamped at the endpoints, |
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271 and a second periodic fit which is hinged at the endpoints. |
| 14509 | 272 |
| 273 @example | |
| 274 ## Data (200 points) | |
| 275 x = 2 * pi * rand (1, 200); | |
| 276 y = sin (2 * x) + 0.1 * randn (size (x)); | |
| 277 ## Breaks | |
| 278 breaks = linspace (0, 2 * pi, 10); | |
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279 ## Clamped endpoints, y = y' = 0 |
| 14509 | 280 xc = [0, 0, 2*pi, 2*pi]; |
| 281 cc = [(eye (2)), (eye (2))]; | |
| 282 con = struct ("xc", xc, "cc", cc); | |
| 283 pp1 = splinefit (x, y, breaks, "constraints", con); | |
| 284 ## Hinged periodic endpoints, y = 0 | |
| 285 con = struct ("xc", 0); | |
| 286 pp2 = splinefit (x, y, breaks, "constraints", con, "periodic", true); | |
| 287 ## Plot | |
| 288 xx = linspace (0, 2 * pi, 400); | |
| 289 y1 = ppval (pp1, xx); | |
| 290 y2 = ppval (pp2, xx); | |
| 291 plot (x, y, ".", xx, [y1; y2]) | |
| 292 axis tight | |
| 293 ylim auto | |
| 294 legend (@{"data", "clamped", "hinged periodic"@}) | |
| 295 @end example | |
| 296 | |
| 297 @ifnotinfo | |
| 298 @noindent | |
| 299 The result of which can be seen in @ref{fig:splinefit4}. | |
| 300 | |
| 301 @float Figure,fig:splinefit4 | |
| 302 @center @image{splinefit4,4in} | |
| 303 @caption{Comparison of two periodic piecewise cubic fits to a noisy periodic | |
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304 signal. One fit has its endpoints clamped and the second has its endpoints |
| 14509 | 305 hinged.} |
| 306 @end float | |
| 307 @end ifnotinfo | |
| 308 | |
| 309 The @code{splinefit} function also provides the convenience of a @var{robust} | |
| 310 fitting, where the effect of outlying data is reduced. In the example below, | |
| 311 three different fits are provided. Two with differing levels of outlier | |
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312 suppression and a third illustrating the non-robust solution. |
| 14509 | 313 |
| 314 @example | |
| 315 ## Data | |
| 316 x = linspace (0, 2*pi, 200); | |
| 317 y = sin (x) + sin (2 * x) + 0.05 * randn (size (x)); | |
| 318 ## Add outliers | |
| 319 x = [x, linspace(0,2*pi,60)]; | |
| 320 y = [y, -ones(1,60)]; | |
| 321 ## Fit splines with hinged conditions | |
| 322 con = struct ("xc", [0, 2*pi]); | |
| 323 ## Robust fitting, beta = 0.25 | |
| 324 pp1 = splinefit (x, y, 8, "constraints", con, "beta", 0.25); | |
| 325 ## Robust fitting, beta = 0.75 | |
| 326 pp2 = splinefit (x, y, 8, "constraints", con, "beta", 0.75); | |
| 327 ## No robust fitting | |
| 328 pp3 = splinefit (x, y, 8, "constraints", con); | |
| 329 ## Plot | |
| 330 xx = linspace (0, 2*pi, 400); | |
| 331 y1 = ppval (pp1, xx); | |
| 332 y2 = ppval (pp2, xx); | |
| 333 y3 = ppval (pp3, xx); | |
| 334 plot (x, y, ".", xx, [y1; y2; y3]) | |
| 335 legend (@{"data with outliers","robust, beta = 0.25", ... | |
| 336 "robust, beta = 0.75", "no robust fitting"@}) | |
| 337 axis tight | |
| 338 ylim auto | |
| 339 @end example | |
| 340 | |
| 341 @ifnotinfo | |
| 342 @noindent | |
| 343 The result of which can be seen in @ref{fig:splinefit6}. | |
| 344 | |
| 345 @float Figure,fig:splinefit6 | |
| 346 @center @image{splinefit6,4in} | |
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347 @caption{Comparison of two different levels of robust fitting (@var{beta} = 0.25 and 0.75) to noisy data combined with outlying data. A conventional fit, without |
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348 robust fitting (@var{beta} = 0) is also included.} |
| 14509 | 349 @end float |
| 350 @end ifnotinfo | |
| 351 | |
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352 The function, @code{ppval}, evaluates the piecewise polynomials, created |
| 14509 | 353 by @code{mkpp} or other means, and @code{unmkpp} returns detailed |
| 354 information about the piecewise polynomial. | |
| 6850 | 355 |
| 356 The following example shows how to combine two linear functions and a | |
| 7001 | 357 quadratic into one function. Each of these functions is expressed |
| 6850 | 358 on adjoined intervals. |
| 3294 | 359 |
| 6850 | 360 @example |
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361 @group |
| 6850 | 362 x = [-2, -1, 1, 2]; |
| 363 p = [ 0, 1, 0; | |
| 364 1, -2, 1; | |
| 365 0, -1, 1 ]; | |
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366 pp = mkpp (x, p); |
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367 xi = linspace (-2, 2, 50); |
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368 yi = ppval (pp, xi); |
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369 plot (xi, yi); |
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370 @end group |
| 6850 | 371 @end example |
| 6502 | 372 |
| 373 @DOCSTRING(mkpp) | |
| 374 | |
| 375 @DOCSTRING(unmkpp) | |
| 6850 | 376 |
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377 @DOCSTRING(ppval) |
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378 |
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379 @DOCSTRING(ppder) |
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380 |
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381 @DOCSTRING(ppint) |
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382 |
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383 @DOCSTRING(ppjumps) |
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384 |
| 6850 | 385 @node Miscellaneous Functions |
| 386 @section Miscellaneous Functions | |
| 387 | |
| 388 @DOCSTRING(poly) | |
| 389 | |
| 390 @DOCSTRING(polyout) | |
| 391 | |
| 392 @DOCSTRING(polyreduce) | |
| 393 | |
| 394 | |
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395 |
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396 |