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1 ## Copyright (C) 2000, 2001, 2006, 2007 Kai Habel |
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2 ## Copyright (C) 2006 David Bateman |
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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{pp} = } spline (@var{x}, @var{y}) |
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22 ## @deftypefnx {Function File} {@var{yi} = } spline (@var{x}, @var{y}, @var{xi}) |
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23 ## |
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24 ## Returns the cubic spline interpolation of @var{y} at the point |
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25 ## @var{x}. Called with two arguments the piece-wise polynomial @var{pp} |
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26 ## that may later be used with @code{ppval} to evaluate the polynomial |
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27 ## at specific points. |
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28 ## |
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29 ## The variable @var{x} must be a vector of length @var{n}, and @var{y} |
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30 ## can be either a vector or array. In the case where @var{y} is a |
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31 ## vector, it can have a length of either @var{n} or @code{@var{n} + 2}. |
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32 ## If the length of @var{y} is @var{n}, then the 'not-a-knot' end |
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33 ## condition is used. If the length of @var{y} is @code{@var{n} + 2}, |
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34 ## then the first and last values of the vector @var{y} are the first |
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35 ## derivative of the cubic spline at the end-points. |
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36 ## |
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37 ## If @var{y} is an array, then the size of @var{y} must have the form |
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38 ## @iftex |
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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 ## @end iftex |
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43 ## @ifinfo |
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44 ## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n}]} |
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45 ## @end ifinfo |
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46 ## or |
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47 ## @iftex |
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48 ## @tex |
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49 ## $$[s_1, s_2, \cdots, s_k, n + 2]$$. |
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50 ## @end tex |
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51 ## @end iftex |
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52 ## @ifinfo |
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53 ## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n} + 2]}. |
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54 ## @end ifinfo |
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55 ## The array is then reshaped internally to a matrix where to leading |
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56 ## dimension is given by |
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57 ## @iftex |
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58 ## @tex |
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59 ## $$s_1 s_2 \cdots s_k$$ |
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60 ## @end tex |
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61 ## @end iftex |
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62 ## @ifinfo |
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63 ## @code{@var{s1} * @var{s2} * @dots{} * @var{sk}} |
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64 ## @end ifinfo |
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65 ## and each row this matrix is then treated separately. Note that this |
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66 ## is exactly the opposite treatment than @code{interp1} and is done |
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67 ## for compatibility. |
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68 ## |
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69 ## Called with a third input argument, @code{spline} evaluates the |
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70 ## piece-wise spline at the points @var{xi}. There is an equivalence |
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71 ## between @code{ppval (spline (@var{x}, @var{y}), @var{xi})} and |
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72 ## @code{spline (@var{x}, @var{y}, @var{xi})}. |
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73 ## |
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74 ## @seealso{ppval, mkpp, unmkpp} |
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75 ## @end deftypefn |
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76 |
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77 ## This code is based on csape.m from octave-forge, but has been |
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78 ## modified to use the sparse solver code in octave that itself allows |
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79 ## special casing of tri-diagonal matrices, modified for NDArrays and |
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80 ## for the treatment of vectors y 2 elements longer than x as complete |
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81 ## splines. |
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82 |
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83 function ret = spline (x, y, xi) |
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84 |
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85 x = x(:); |
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86 n = length (x); |
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87 if (n < 3) |
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88 error ("spline: requires at least 3 points"); |
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89 endif |
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90 |
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91 ## Check the size and shape of y |
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92 ndy = ndims (y); |
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93 szy = size (y); |
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94 if (ndy == 2 && (szy(1) == 1 || szy(2) == 1)) |
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95 if (szy(1) == 1) |
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96 a = y.'; |
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97 else |
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98 a = y; |
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99 szy = fliplr (szy); |
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100 endif |
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101 else |
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102 a = reshape (y, [prod(szy(1:end-1)), szy(end)]).'; |
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103 endif |
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104 complete = false; |
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105 if (size (a, 1) == n + 2) |
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106 complete = true; |
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107 dfs = a(1,:); |
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108 dfe = a(end,:); |
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109 a = a(2:end-1,:); |
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110 endif |
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111 |
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112 b = c = zeros (size (a)); |
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113 h = diff (x); |
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114 idx = ones (columns (a), 1); |
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115 |
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116 if (complete) |
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117 |
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118 if (n == 3) |
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119 dg = 1.5 * h(1) - 0.5 * h(2); |
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120 c(2:n-1,:) = 1/dg(1); |
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121 else |
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122 dg = 2 * (h(1:n-2) .+ h(2:n-1)); |
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123 dg(1) = dg(1) - 0.5 * h(1); |
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124 dg(n-2) = dg(n-2) - 0.5 * h(n-1); |
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125 |
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126 e = h(2:n-2); |
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127 |
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128 size(a) |
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129 size(h) |
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130 n |
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131 |
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132 g = 3 * diff (a(2:n,:)) ./ h(2:n-1,idx) ... |
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133 - 3 * diff (a(1:n-1,:)) ./ h(1:n-2,idx); |
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134 g(1,:) = 3 * (a(3,:) - a(2,:)) / h(2) ... |
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135 - 3 / 2 * (3 * (a(2,:) - a(1,:)) / h(1) - dfs); |
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136 g(n-2,:) = 3 / 2 * (3 * (a(n,:) - a(n-1,:)) / h(n-1) - dfe) ... |
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137 - 3 * (a(n-1,:) - a(n-2,:)) / h(n-2); |
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138 |
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139 c(2:n-1,:) = spdiags ([[e(:); 0], dg, [0; e(:)]], |
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140 [-1, 0, 1], n-2, n-2) \ g; |
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141 endif |
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142 |
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143 c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * dfs |
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144 - c(2,:) * h(1)) / (2 * h(1)); |
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145 c(n,:) = - (3 / h(n-1) * (a(n,:) - a(n-1,:)) - 3 * dfe |
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146 + c(n-1,:) * h(n-1)) / (2 * h(n-1)); |
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147 b(1:n-1,:) = diff (a) ./ h(1:n-1, idx) ... |
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148 - h(1:n-1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); |
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149 d = diff (c) ./ (3 * h(1:n-1, idx)); |
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150 |
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151 else |
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152 |
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153 g = zeros (n-2, columns (a)); |
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154 g(1,:) = 3 / (h(1) + h(2)) ... |
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155 * (a(3,:) - a(2,:) - h(2) / h(1) * (a(2,:) - a(1,:))); |
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156 g(n-2,:) = 3 / (h(n-1) + h(n-2)) ... |
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157 * (h(n-2) / h(n-1) * (a(n,:) - a(n-1,:)) - (a(n-1,:) - a(n-2,:))); |
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158 |
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159 if (n > 4) |
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160 |
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161 g(2:n - 3,:) = 3 * diff (a(3:n-1,:)) ./ h(3:n-2,idx) ... |
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162 - 3 * diff (a(2:n-2,:)) ./ h(2:n - 3,idx); |
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163 |
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164 dg = 2 * (h(1:n-2) .+ h(2:n-1)); |
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165 dg(1) = dg(1) - h(1); |
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166 dg(n-2) = dg(n-2) - h(n-1); |
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167 |
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168 ldg = udg = h(2:n-2); |
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169 udg(1) = udg(1) - h(1); |
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170 ldg(n - 3) = ldg(n-3) - h(n-1); |
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171 c(2:n-1,:) = spdiags ([[ldg(:); 0], dg, [0; udg(:)]], |
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172 [-1, 0, 1], n-2, n-2) \ g; |
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173 |
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174 elseif (n == 4) |
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175 |
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176 dg = [h(1) + 2 * h(2); 2 * h(2) + h(3)]; |
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177 ldg = h(2) - h(3); |
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178 udg = h(2) - h(1); |
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179 c(2:n-1,:) = spdiags ([[ldg(:);0], dg, [0; udg(:)]], |
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180 [-1, 0, 1], n-2, n-2) \ g; |
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181 |
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182 else # n == 3 |
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183 |
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184 dg = h(1) + 2 * h(2); |
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185 c(2:n-1,:) = g/dg(1); |
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186 |
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187 endif |
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188 |
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189 c(1,:) = c(2,:) + h(1) / h(2) * (c(2,:) - c(3,:)); |
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190 c(n,:) = c(n-1,:) + h(n-1) / h(n-2) * (c(n-1,:) - c(n-2,:)); |
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191 b = diff (a) ./ h(1:n-1, idx) ... |
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192 - h(1:n-1, idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); |
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193 d = diff (c) ./ (3 * h(1:n-1, idx)); |
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194 |
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195 endif |
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196 |
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197 d = d(1:n-1,:); |
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198 c = c(1:n-1,:); |
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199 b = b(1:n-1,:); |
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200 a = a(1:n-1,:); |
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201 coeffs = [d(:), c(:), b(:), a(:)]; |
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202 ret = mkpp (x, coeffs, szy(1:end-1)); |
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203 |
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204 if (nargin == 3) |
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205 ret = ppval (ret, xi); |
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206 endif |
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207 |
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208 endfunction |
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209 |
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210 %!demo |
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211 %! x = 0:10; y = sin(x); |
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212 %! xspline = 0:0.1:10; yspline = spline(x,y,xspline); |
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213 %! title("spline fit to points from sin(x)"); |
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214 %! plot(xspline,sin(xspline),"r",xspline,yspline,"g-",x,y,"b+"); |
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215 %! legend("original","interpolation","interpolation points"); |
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216 %! %-------------------------------------------------------- |
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217 %! % confirm that interpolated function matches the original |
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218 |
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219 %!shared x,y,abserr |
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220 %! x = [0:10]; y = sin(x); abserr = 1e-14; |
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221 %!assert (spline(x,y,x), y, abserr); |
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222 %!assert (spline(x,y,x'), y', abserr); |
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223 %!assert (spline(x',y',x'), y', abserr); |
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224 %!assert (spline(x',y',x), y, abserr); |
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225 %!assert (isempty(spline(x',y',[]))); |
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226 %!assert (isempty(spline(x,y,[]))); |
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227 %!assert (spline(x,[y;y],x), [spline(x,y,x);spline(x,y,x)],abserr) |
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228 %! y = cos(x) + i*sin(x); |
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229 %!assert (spline(x,y,x), y, abserr) |
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230 %!assert (real(spline(x,y,x)), real(y), abserr); |
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231 %!assert (real(spline(x,y,x.')), real(y).', abserr); |
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232 %!assert (real(spline(x.',y.',x.')), real(y).', abserr); |
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233 %!assert (real(spline(x.',y,x)), real(y), abserr); |
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234 %!assert (imag(spline(x,y,x)), imag(y), abserr); |
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235 %!assert (imag(spline(x,y,x.')), imag(y).', abserr); |
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236 %!assert (imag(spline(x.',y.',x.')), imag(y).', abserr); |
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237 %!assert (imag(spline(x.',y,x)), imag(y), abserr); |
