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annotate scripts/polynomial/polyval.m @ 11916:377d908f7e40 release-3-0-x

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use QR decomposition and normalization for polyfit; normalization for polyval
author Ben Abbott <bpabbott@mac.com>
date Mon, 12 Jan 2009 12:09:30 +0100
parents a1dbe9d80eee
children deb777a926ee
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1 ## Copyright (C) 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2002, 2004,
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2 ## 2005, 2006, 2007 John W. Eaton
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3 ##
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4 ## This file is part of Octave.
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5 ##
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6 ## Octave is free software; you can redistribute it and/or modify it
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7 ## under the terms of the GNU General Public License as published by
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8 ## the Free Software Foundation; either version 3 of the License, or (at
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9 ## your option) any later version.
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10 ##
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11 ## Octave is distributed in the hope that it will be useful, but
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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14 ## General Public License for more details.
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15 ##
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16 ## You should have received a copy of the GNU General Public License
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17 ## along with Octave; see the file COPYING. If not, see
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18 ## <http://www.gnu.org/licenses/>.
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19
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20 ## -*- texinfo -*-
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21 ## @deftypefn {Function File} {@var{y}=} polyval (@var{p}, @var{x})
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22 ## @deftypefnx {Function File} {@var{y}=} polyval (@var{p}, @var{x}, [], @var{mu})
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23 ## Evaluate the polynomial at of the specified values for @var{x}. When @var{mu}
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24 ## is present evaluate the polynomial for (@var{x}-@var{mu}(1))/@var{mu}(2).
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25 ## If @var{x} is a vector or matrix, the polynomial is evaluated for each of
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26 ## the elements of @var{x}.
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27 ## @deftypefnx {Function File} {[@var{y}, @var{dy}] =} polyval (@var{p}, @var{x}, @var{S})
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28 ## @deftypefnx {Function File} {[@var{y}, @var{dy}] =} polyval (@var{p}, @var{x}, @var{S}, @var{mu})
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29 ## In addition to evaluating the polynomial, the second output
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30 ## represents the prediction interval, @var{y} +/- @var{dy}, which
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31 ## contains at least 50% of the future predictions. To calculate the
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32 ## prediction interval, the structured variable @var{s}, originating
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33 ## form `polyfit', must be present.
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34 ## @seealso{polyfit, polyvalm, poly, roots, conv, deconv, residue, filter,
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35 ## polyderiv, polyinteg}
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36 ## @end deftypefn
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37
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38 ## Author: Tony Richardson <arichard@stark.cc.oh.us>
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39 ## Created: June 1994
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40 ## Adapted-By: jwe
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41
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42 function [y, dy] = polyval (p, x, s, mu)
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43
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44 if (nargin < 2 || nargin > 4 || (nargout == 2 && nargin < 3))
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45 print_usage ();
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46 endif
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47
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48 if (nargin < 3)
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49 s = [];
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50 endif
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51
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52 if (! (isvector (p) || isempty (p)))
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53 error ("polyval: first argument must be a vector");
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54 endif
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55
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56 if (nargin < 4)
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57 mu = [0, 1];
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58 endif
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59
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60 if (isempty (x))
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61 y = [];
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62 return;
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63 endif
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64
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65 if (length (p) == 0)
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66 y = p;
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67 return;
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68 endif
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69
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70 n = length (p) - 1;
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71 k = numel (x);
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72 x = (x - mu(1)) / mu(2);
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73 A = (x(:) * ones (1, n+1)) .^ (ones (k, 1) * (n:-1:0));
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74 y(:) = A * p(:);
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75 y = reshape (y, size (x));
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76
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77 if (nargout == 2)
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78 ## The line below is *not* the result of a conceptual grasp of statistics.
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79 ## Instead, after reading the links below and comparing to the output of Matlab's polyval.m,
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80 ## http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/finv.html
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81 ## http://www.mathworks.com/access/helpdesk/help/toolbox/curvefit/index.html?/access/helpdesk/help/toolbox/curvefit/bq_5ka6-1_1.html
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82 ## Note: the F-Distribution is generally considered to be single-sided.
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83 ## http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm
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84 ## t = finv (1-alpha, s.df, s.df);
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85 ## dy = t * sqrt (1 + sumsq (A/s.R, 2)) * s.normr / sqrt (s.df)
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86 ## If my inference is correct, then t must equal 1 for polyval.
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87 ## This is because finv (0.5, n, n) = 1.0 for any n.
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88 dy = sqrt (1 + sumsq (A/s.R, 2)) * s.normr / sqrt (s.df);
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89 dy = reshape (dy, size (x));
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90 endif
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91
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92 endfunction
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93
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94 %!test
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95 %! fail("polyval([1,0;0,1],0:10)");
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96
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97 %!test
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98 %! r = 0:10:50;
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99 %! p = poly (r);
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100 %! p = p / max(abs(p));
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101 %! x = linspace(0,50,11);
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102 %! y = polyval(p,x) + 0.25*sin(100*x);
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103 %! [pf, s] = polyfit (x, y, numel(r));
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104 %! [y1, delta] = polyval (pf, x, s);
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105 %! expected = [0.37235, 0.35854, 0.32231, 0.32448, 0.31328, ...
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106 %! 0.32036, 0.31328, 0.32448, 0.32231, 0.35854, 0.37235];
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107 %! assert (delta, expected, 0.00001)
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108
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109 %!test
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110 %! x = 10 + (-2:2);
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111 %! y = [0, 0, 1, 0, 2];
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112 %! p = polyfit (x, y, numel (x) - 1);
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113 %! [pn, s, mu] = polyfit (x, y, numel (x) - 1);
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114 %! y1 = polyval (p, x);
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115 %! yn = polyval (pn, x, [], mu);
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116 %! assert (y1, y, sqrt(eps))
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117 %! assert (yn, y, sqrt(eps))
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118
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119 %!test
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120 %! p = [0, 1, 0];
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121 %! x = 1:10;
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122 %! assert (x, polyval(p,x), eps)
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123 %! x = x(:);
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124 %! assert (x, polyval(p,x), eps)
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125 %! x = reshape(x, [2, 5]);
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126 %! assert (x, polyval(p,x), eps)
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127 %! x = reshape(x, [5, 2]);
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128 %! assert (x, polyval(p,x), eps)
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129 %! x = reshape(x, [1, 1, 5, 2]);
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130 %! assert (x, polyval(p,x), eps)