Mercurial > hg > octave-nkf
annotate scripts/polynomial/spline.m @ 9245:16f53d29049f
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Fri, 22 May 2009 10:46:00 -0400 |
| parents | f0c3d3fc4903 |
| children | 31900e17b5f5 |
| rev | line source |
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| 8920 | 1 ## Copyright (C) 2000, 2001, 2006, 2007, 2008, 2009 Kai Habel |
| 5824 | 2 ## Copyright (C) 2006 David Bateman |
| 3 ## | |
| 4 ## This file is part of Octave. | |
| 5 ## | |
| 6 ## Octave is free software; you can redistribute it and/or modify it | |
| 7 ## under the terms of the GNU General Public License as published by | |
| 7016 | 8 ## the Free Software Foundation; either version 3 of the License, or (at |
| 9 ## your option) any later version. | |
| 5824 | 10 ## |
| 11 ## Octave is distributed in the hope that it will be useful, but | |
| 12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 14 ## General Public License for more details. | |
| 15 ## | |
| 16 ## You should have received a copy of the GNU General Public License | |
| 7016 | 17 ## along with Octave; see the file COPYING. If not, see |
| 18 ## <http://www.gnu.org/licenses/>. | |
| 5824 | 19 |
| 20 ## -*- texinfo -*- | |
| 7650 | 21 ## @deftypefn {Function File} {@var{pp} =} spline (@var{x}, @var{y}) |
| 22 ## @deftypefnx {Function File} {@var{yi} =} spline (@var{x}, @var{y}, @var{xi}) | |
| 5824 | 23 ## |
| 8602 | 24 ## Return the cubic spline interpolant of @var{y} at points @var{x}. |
| 25 ## If called with two arguments, @code{spline} returns the piece-wise | |
| 26 ## polynomial @var{pp} that may later be used with @code{ppval} to | |
| 27 ## evaluate the polynomial at specific points. | |
| 28 ## If called with a third input argument, @code{spline} evaluates the | |
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29 ## spline at the points @var{xi}. There is an equivalence |
| 8602 | 30 ## between @code{ppval (spline (@var{x}, @var{y}), @var{xi})} and |
| 31 ## @code{spline (@var{x}, @var{y}, @var{xi})}. | |
| 5824 | 32 ## |
| 33 ## The variable @var{x} must be a vector of length @var{n}, and @var{y} | |
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34 ## can be either a vector or array. In the case where @var{y} is a |
| 5824 | 35 ## vector, it can have a length of either @var{n} or @code{@var{n} + 2}. |
| 36 ## If the length of @var{y} is @var{n}, then the 'not-a-knot' end | |
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37 ## condition is used. If the length of @var{y} is @code{@var{n} + 2}, |
| 8602 | 38 ## then the first and last values of the vector @var{y} are the values |
| 39 ## of the first derivative of the cubic spline at the end-points. | |
| 5824 | 40 ## |
| 41 ## If @var{y} is an array, then the size of @var{y} must have the form | |
| 42 ## @tex | |
| 43 ## $$[s_1, s_2, \cdots, s_k, n]$$ | |
| 44 ## @end tex | |
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45 ## @ifnottex |
| 5824 | 46 ## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n}]} |
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47 ## @end ifnottex |
| 5824 | 48 ## or |
| 49 ## @tex | |
| 8828 | 50 ## $$[s_1, s_2, \cdots, s_k, n + 2].$$ |
| 5824 | 51 ## @end tex |
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52 ## @ifnottex |
| 5824 | 53 ## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n} + 2]}. |
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54 ## @end ifnottex |
| 8602 | 55 ## The array is then reshaped internally to a matrix where the leading |
| 5824 | 56 ## dimension is given by |
| 57 ## @tex | |
| 58 ## $$s_1 s_2 \cdots s_k$$ | |
| 59 ## @end tex | |
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60 ## @ifnottex |
| 5824 | 61 ## @code{@var{s1} * @var{s2} * @dots{} * @var{sk}} |
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62 ## @end ifnottex |
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63 ## and each row of this matrix is then treated separately. Note that this |
| 5824 | 64 ## is exactly the opposite treatment than @code{interp1} and is done |
| 7001 | 65 ## for compatibility. |
| 5824 | 66 ## @seealso{ppval, mkpp, unmkpp} |
| 67 ## @end deftypefn | |
| 68 | |
| 69 ## This code is based on csape.m from octave-forge, but has been | |
| 70 ## modified to use the sparse solver code in octave that itself allows | |
| 71 ## special casing of tri-diagonal matrices, modified for NDArrays and | |
| 72 ## for the treatment of vectors y 2 elements longer than x as complete | |
| 73 ## splines. | |
| 74 | |
| 75 function ret = spline (x, y, xi) | |
| 76 | |
| 77 x = x(:); | |
| 78 n = length (x); | |
| 79 if (n < 3) | |
| 80 error ("spline: requires at least 3 points"); | |
| 81 endif | |
| 82 | |
| 83 ## Check the size and shape of y | |
| 84 ndy = ndims (y); | |
| 85 szy = size (y); | |
| 86 if (ndy == 2 && (szy(1) == 1 || szy(2) == 1)) | |
| 87 if (szy(1) == 1) | |
| 6014 | 88 a = y.'; |
| 5824 | 89 else |
| 90 a = y; | |
| 91 szy = fliplr (szy); | |
| 92 endif | |
| 93 else | |
| 6014 | 94 a = reshape (y, [prod(szy(1:end-1)), szy(end)]).'; |
| 5824 | 95 endif |
| 96 complete = false; | |
| 97 if (size (a, 1) == n + 2) | |
| 98 complete = true; | |
| 99 dfs = a(1,:); | |
| 100 dfe = a(end,:); | |
| 101 a = a(2:end-1,:); | |
| 102 endif | |
| 103 | |
| 104 b = c = zeros (size (a)); | |
| 105 h = diff (x); | |
| 5838 | 106 idx = ones (columns (a), 1); |
| 5824 | 107 |
| 108 if (complete) | |
| 109 | |
| 110 if (n == 3) | |
| 111 dg = 1.5 * h(1) - 0.5 * h(2); | |
| 5838 | 112 c(2:n-1,:) = 1/dg(1); |
| 5824 | 113 else |
| 5838 | 114 dg = 2 * (h(1:n-2) .+ h(2:n-1)); |
| 5824 | 115 dg(1) = dg(1) - 0.5 * h(1); |
| 5838 | 116 dg(n-2) = dg(n-2) - 0.5 * h(n-1); |
| 5824 | 117 |
| 5838 | 118 e = h(2:n-2); |
| 5824 | 119 |
| 5838 | 120 g = 3 * diff (a(2:n,:)) ./ h(2:n-1,idx) ... |
| 121 - 3 * diff (a(1:n-1,:)) ./ h(1:n-2,idx); | |
| 122 g(1,:) = 3 * (a(3,:) - a(2,:)) / h(2) ... | |
| 5824 | 123 - 3 / 2 * (3 * (a(2,:) - a(1,:)) / h(1) - dfs); |
| 5838 | 124 g(n-2,:) = 3 / 2 * (3 * (a(n,:) - a(n-1,:)) / h(n-1) - dfe) ... |
| 125 - 3 * (a(n-1,:) - a(n-2,:)) / h(n-2); | |
| 5824 | 126 |
| 5838 | 127 c(2:n-1,:) = spdiags ([[e(:); 0], dg, [0; e(:)]], |
| 128 [-1, 0, 1], n-2, n-2) \ g; | |
| 5824 | 129 endif |
| 130 | |
| 131 c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * dfs | |
| 5838 | 132 - c(2,:) * h(1)) / (2 * h(1)); |
| 133 c(n,:) = - (3 / h(n-1) * (a(n,:) - a(n-1,:)) - 3 * dfe | |
| 134 + c(n-1,:) * h(n-1)) / (2 * h(n-1)); | |
| 135 b(1:n-1,:) = diff (a) ./ h(1:n-1, idx) ... | |
| 136 - h(1:n-1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); | |
| 137 d = diff (c) ./ (3 * h(1:n-1, idx)); | |
| 5824 | 138 |
| 139 else | |
| 140 | |
| 5838 | 141 g = zeros (n-2, columns (a)); |
| 142 g(1,:) = 3 / (h(1) + h(2)) ... | |
| 143 * (a(3,:) - a(2,:) - h(2) / h(1) * (a(2,:) - a(1,:))); | |
| 144 g(n-2,:) = 3 / (h(n-1) + h(n-2)) ... | |
| 145 * (h(n-2) / h(n-1) * (a(n,:) - a(n-1,:)) - (a(n-1,:) - a(n-2,:))); | |
| 5824 | 146 |
| 147 if (n > 4) | |
| 148 | |
| 5838 | 149 g(2:n - 3,:) = 3 * diff (a(3:n-1,:)) ./ h(3:n-2,idx) ... |
| 150 - 3 * diff (a(2:n-2,:)) ./ h(2:n - 3,idx); | |
| 5824 | 151 |
| 5838 | 152 dg = 2 * (h(1:n-2) .+ h(2:n-1)); |
| 5824 | 153 dg(1) = dg(1) - h(1); |
| 5838 | 154 dg(n-2) = dg(n-2) - h(n-1); |
| 5824 | 155 |
| 5838 | 156 ldg = udg = h(2:n-2); |
| 5824 | 157 udg(1) = udg(1) - h(1); |
| 5838 | 158 ldg(n - 3) = ldg(n-3) - h(n-1); |
| 159 c(2:n-1,:) = spdiags ([[ldg(:); 0], dg, [0; udg(:)]], | |
| 160 [-1, 0, 1], n-2, n-2) \ g; | |
| 5824 | 161 |
| 162 elseif (n == 4) | |
| 163 | |
| 6248 | 164 dg = [h(1) + 2 * h(2); 2 * h(2) + h(3)]; |
| 5824 | 165 ldg = h(2) - h(3); |
| 166 udg = h(2) - h(1); | |
| 5838 | 167 c(2:n-1,:) = spdiags ([[ldg(:);0], dg, [0; udg(:)]], |
| 168 [-1, 0, 1], n-2, n-2) \ g; | |
| 5824 | 169 |
| 170 else # n == 3 | |
| 171 | |
| 5838 | 172 dg = h(1) + 2 * h(2); |
| 173 c(2:n-1,:) = g/dg(1); | |
| 5824 | 174 |
| 175 endif | |
| 176 | |
| 177 c(1,:) = c(2,:) + h(1) / h(2) * (c(2,:) - c(3,:)); | |
| 5838 | 178 c(n,:) = c(n-1,:) + h(n-1) / h(n-2) * (c(n-1,:) - c(n-2,:)); |
| 179 b = diff (a) ./ h(1:n-1, idx) ... | |
| 180 - h(1:n-1, idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); | |
| 181 d = diff (c) ./ (3 * h(1:n-1, idx)); | |
| 5824 | 182 |
| 183 endif | |
| 184 | |
| 5838 | 185 d = d(1:n-1,:); |
| 186 c = c(1:n-1,:); | |
| 187 b = b(1:n-1,:); | |
| 188 a = a(1:n-1,:); | |
| 5824 | 189 coeffs = [d(:), c(:), b(:), a(:)]; |
| 190 ret = mkpp (x, coeffs, szy(1:end-1)); | |
| 191 | |
| 192 if (nargin == 3) | |
| 193 ret = ppval (ret, xi); | |
| 194 endif | |
| 195 | |
| 196 endfunction | |
| 197 | |
| 198 %!demo | |
| 199 %! x = 0:10; y = sin(x); | |
| 200 %! xspline = 0:0.1:10; yspline = spline(x,y,xspline); | |
| 201 %! title("spline fit to points from sin(x)"); | |
| 6702 | 202 %! plot(xspline,sin(xspline),"r",xspline,yspline,"g-",x,y,"b+"); |
| 203 %! legend("original","interpolation","interpolation points"); | |
| 5824 | 204 %! %-------------------------------------------------------- |
| 205 %! % confirm that interpolated function matches the original | |
| 206 | |
| 6686 | 207 %!shared x,y,abserr |
| 208 %! x = [0:10]; y = sin(x); abserr = 1e-14; | |
| 209 %!assert (spline(x,y,x), y, abserr); | |
| 210 %!assert (spline(x,y,x'), y', abserr); | |
| 211 %!assert (spline(x',y',x'), y', abserr); | |
| 212 %!assert (spline(x',y',x), y, abserr); | |
| 5824 | 213 %!assert (isempty(spline(x',y',[]))); |
| 214 %!assert (isempty(spline(x,y,[]))); | |
| 6686 | 215 %!assert (spline(x,[y;y],x), [spline(x,y,x);spline(x,y,x)],abserr) |
| 6014 | 216 %! y = cos(x) + i*sin(x); |
| 6686 | 217 %!assert (spline(x,y,x), y, abserr) |
| 218 %!assert (real(spline(x,y,x)), real(y), abserr); | |
| 219 %!assert (real(spline(x,y,x.')), real(y).', abserr); | |
| 220 %!assert (real(spline(x.',y.',x.')), real(y).', abserr); | |
| 221 %!assert (real(spline(x.',y,x)), real(y), abserr); | |
| 222 %!assert (imag(spline(x,y,x)), imag(y), abserr); | |
| 223 %!assert (imag(spline(x,y,x.')), imag(y).', abserr); | |
| 224 %!assert (imag(spline(x.',y.',x.')), imag(y).', abserr); | |
| 225 %!assert (imag(spline(x.',y,x)), imag(y), abserr); |