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annotate scripts/polynomial/pchip.m @ 20828:a3b9ee5c040a

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Replace bsxfun with broadcasting for performance with complex inputs (bug #38628). cumtrapz.m, quadgk.m, trapz.m, center.m, zscore.m: Replace bsxfun with broadcasting for performance where inputs might be complex.
author Rik <rik@octave.org>
date Mon, 12 Oct 2015 21:28:32 -0700
parents 83792dd9bcc1
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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4197fc428c7d maint: Update copyright notices for 2015.
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1 ## Copyright (C) 2001-2015 Kai Habel
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {@var{pp} =} pchip (@var{x}, @var{y})
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21 ## @deftypefnx {Function File} {@var{yi} =} pchip (@var{x}, @var{y}, @var{xi})
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22 ## Return the Piecewise Cubic Hermite Interpolating Polynomial (pchip) of
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23 ## points @var{x} and @var{y}.
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24 ##
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25 ## If called with two arguments, return the piecewise polynomial @var{pp}
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26 ## that may be used with @code{ppval} to evaluate the polynomial at specific
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27 ## points.
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28 ##
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29 ## When called with a third input argument, @code{pchip} evaluates the pchip
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30 ## polynomial at the points @var{xi}. The third calling form is equivalent to
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31 ## @code{ppval (pchip (@var{x}, @var{y}), @var{xi})}.
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32 ##
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33 ## The variable @var{x} must be a strictly monotonic vector (either increasing
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34 ## or decreasing) of length @var{n}.
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35 ##
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36 ## @var{y} can be either a vector or array. If @var{y} is a vector then it
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37 ## must be the same length @var{n} as @var{x}. If @var{y} is an array then
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38 ## the size of @var{y} must have the form
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39 ## @tex
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40 ## $$[s_1, s_2, \cdots, s_k, n]$$
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41 ## @end tex
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42 ## @ifnottex
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43 ## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n}]}
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44 ## @end ifnottex
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45 ## The array is reshaped internally to a matrix where the leading dimension is
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46 ## given by
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47 ## @tex
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48 ## $$s_1 s_2 \cdots s_k$$
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49 ## @end tex
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50 ## @ifnottex
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51 ## @code{@var{s1} * @var{s2} * @dots{} * @var{sk}}
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52 ## @end ifnottex
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53 ## and each row of this matrix is then treated separately. Note that this is
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54 ## exactly opposite to @code{interp1} but is done for @sc{matlab} compatibility.
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55 ##
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56 ## @seealso{spline, ppval, mkpp, unmkpp}
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57 ## @end deftypefn
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58
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59 ## Author: Kai Habel <kai.habel@gmx.de>
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60 ## Date: 9. mar 2001
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61 ##
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62 ## S_k = a_k + b_k*x + c_k*x^2 + d_k*x^3; (spline polynom)
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63 ##
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64 ## 4 conditions:
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65 ## S_k(x_k) = y_k;
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66 ## S_k(x_k+1) = y_k+1;
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67 ## S_k'(x_k) = y_k';
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68 ## S_k'(x_k+1) = y_k+1';
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69
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70 function ret = pchip (x, y, xi)
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71
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72 if (nargin < 2 || nargin > 3)
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73 print_usage ();
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74 endif
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75
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76 ## make row vector
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77 x = x(:).';
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78 n = length (x);
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79
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80 ## Check the size and shape of y
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81 if (isvector (y))
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82 y = y(:).'; # force row vector
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83 szy = size (y);
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84 if (! size_equal (x, y))
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85 error ("pchip: length of X and Y must match");
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86 endif
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87 else
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88 szy = size (y);
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89 if (n != szy(end))
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90 error ("pchip: length of X and last dimension of Y must match");
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91 endif
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92 y = reshape (y, [prod(szy(1:end-1)), szy(end)]);
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93 endif
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94
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95 h = diff (x);
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96 if (all (h < 0))
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97 x = fliplr (x);
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98 h = diff (x);
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99 y = fliplr (y);
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100 elseif (any (h <= 0))
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101 error ("pchip: X must be strictly monotonic");
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102 endif
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103
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104 f1 = y(:, 1:n-1);
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105
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106 ## Compute derivatives.
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107 d = __pchip_deriv__ (x, y, 2);
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108 d1 = d(:, 1:n-1);
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109 d2 = d(:, 2:n);
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110
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111 ## This is taken from SLATEC.
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112 h = diag (h);
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113
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114 delta = diff (y, 1, 2) / h;
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115 del1 = (d1 - delta) / h;
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116 del2 = (d2 - delta) / h;
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117 c3 = del1 + del2;
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118 c2 = -c3 - del1;
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119 c3 /= h;
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120 coeffs = cat (3, c3, c2, d1, f1);
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121
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122 ret = mkpp (x, coeffs, szy(1:end-1));
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123
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124 if (nargin == 3)
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125 ret = ppval (ret, xi);
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126 endif
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127
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128 endfunction
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129
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130
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131 %!demo
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132 %! x = 0:8;
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133 %! y = [1, 1, 1, 1, 0.5, 0, 0, 0, 0];
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134 %! xi = 0:0.01:8;
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135 %! yspline = spline (x,y,xi);
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136 %! ypchip = pchip (x,y,xi);
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137 %! title ("pchip and spline fit to discontinuous function");
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138 %! plot (xi,yspline, xi,ypchip,"-", x,y,"+");
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139 %! legend ("spline", "pchip", "data");
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140 %! %-------------------------------------------------------------------
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141 %! % confirm that pchip agreed better to discontinuous data than spline
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142
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143 %!shared x, y, y2, pp, yi1, yi2, yi3
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144 %! x = 0:8;
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145 %! y = [1, 1, 1, 1, 0.5, 0, 0, 0, 0];
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146 %!assert (pchip (x,y,x), y)
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147 %!assert (pchip (x,y,x'), y')
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148 %!assert (pchip (x',y',x'), y')
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149 %!assert (pchip (x',y',x), y)
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150 %!assert (isempty (pchip (x',y',[])))
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151 %!assert (isempty (pchip (x,y,[])))
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152 %!assert (pchip (x,[y;y],x), [pchip(x,y,x);pchip(x,y,x)])
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153 %!assert (pchip (x,[y;y],x'), [pchip(x,y,x);pchip(x,y,x)])
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154 %!assert (pchip (x',[y;y],x), [pchip(x,y,x);pchip(x,y,x)])
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155 %!assert (pchip (x',[y;y],x'), [pchip(x,y,x);pchip(x,y,x)])
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156 %!test
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157 %! x = (0:8)*pi/4; y = [sin(x); cos(x)];
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158 %! y2(:,:,1) = y; y2(:,:,2) = y+1; y2(:,:,3) = y-1;
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159 %! pp = pchip (x, shiftdim (y2,2));
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160 %! yi1 = ppval (pp, (1:4)*pi/4);
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161 %! yi2 = ppval (pp, repmat ((1:4)*pi/4, [5,1]));
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162 %! yi3 = ppval (pp, [pi/2,pi]);
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163 %!assert (size (pp.coefs), [48,4])
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164 %!assert (pp.pieces, 8)
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165 %!assert (pp.order, 4)
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166 %!assert (pp.dim, [3,2])
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167 %!assert (ppval (pp,pi), [0,-1;1,0;-1,-2], 1e-14)
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168 %!assert (yi3(:,:,2), ppval (pp,pi), 1e-14)
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169 %!assert (yi3(:,:,1), [1,0;2,1;0,-1], 1e-14)
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170 %!assert (squeeze (yi1(1,2,:)), [1/sqrt(2); 0; -1/sqrt(2);-1], 1e-14)
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171 %!assert (size (yi2), [3,2,5,4])
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172 %!assert (squeeze (yi2(1,2,3,:)), [1/sqrt(2); 0; -1/sqrt(2);-1], 1e-14)
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173
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174 %!error (pchip (1,2));
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175 %!error (pchip (1,2,3));
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176