Work at SourceForge, help us to make it a better place! We have an immediate need for a Support Technician in our San Francisco or Denver office.

Close

[a7fe2a]: inst / csape.m Maximize Restore History

Download this file

csape.m    276 lines (221 with data), 8.9 kB

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
## Copyright (C) 2000, 2001 Kai Habel
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{pp} = } csape (@var{x}, @var{y}, @var{cond}, @var{valc})
## cubic spline interpolation with various end conditions.
## creates the pp-form of the cubic spline.
##
## the following end conditions as given in @var{cond} are possible.
## @table @asis
## @item 'complete'
## match slopes at first and last point as given in @var{valc}
## @item 'not-a-knot'
## third derivatives are continuous at the second and second last point
## @item 'periodic'
## match first and second derivative of first and last point
## @item 'second'
## match second derivative at first and last point as given in @var{valc}
## @item 'variational'
## set second derivative at first and last point to zero (natural cubic spline)
## @end table
##
## @seealso{ppval, spline}
## @end deftypefn
## Author: Kai Habel <kai.habel@gmx.de>
## Date: 23. nov 2000
## Algorithms taken from G. Engeln-Muellges, F. Uhlig:
## "Numerical Algorithms with C", Springer, 1996
## Paul Kienzle, 19. feb 2001, csape supports now matrix y value
function pp = csape (x, y, cond, valc)
x = x(:);
n = length(x);
if (n < 3)
error("csape requires at least 3 points");
endif
## Check the size and shape of y
ndy = ndims (y);
szy = size (y);
if (ndy == 2 && (szy(1) == n || szy(2) == n))
if (szy(2) == n)
a = y.';
else
a = y;
szy = fliplr (szy);
endif
else
a = shiftdim (reshape (y, [prod(szy(1:end-1)), szy(end)]), 1);
endif
b = c = zeros (size (a));
h = diff (x);
idx = ones (columns(a),1);
if (nargin < 3 || strcmp(cond,"complete"))
# specified first derivative at end point
if (nargin < 4)
valc = [0, 0];
endif
if (n == 3)
dg = 1.5 * h(1) - 0.5 * h(2);
c(2:n - 1,:) = 1/dg(1);
else
dg = 2 * (h(1:n - 2) .+ h(2:n - 1));
dg(1) = dg(1) - 0.5 * h(1);
dg(n - 2) = dg(n-2) - 0.5 * h(n - 1);
e = h(2:n - 2);
g = 3 * diff (a(2:n,:)) ./ h(2:n - 1,idx)\
- 3 * diff (a(1:n - 1,:)) ./ h(1:n - 2,idx);
g(1,:) = 3 * (a(3,:) - a(2,:)) / h(2) \
- 3 / 2 * (3 * (a(2,:) - a(1,:)) / h(1) - valc(1));
g(n - 2,:) = 3 / 2 * (3 * (a(n,:) - a(n - 1,:)) / h(n - 1) - valc(2))\
- 3 * (a(n - 1,:) - a(n - 2,:)) / h(n - 2);
c(2:n - 1,:) = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-2,n-2) \ g;
end
c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * valc(1)
- c(2,:) * h(1)) / (2 * h(1));
c(n,:) = - (3 / h(n - 1) * (a(n,:) - a(n - 1,:)) - 3 * valc(2)
+ c(n - 1,:) * h(n - 1)) / (2 * h(n - 1));
b(1:n - 1,:) = diff (a) ./ h(1:n - 1, idx)\
- h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:));
d = diff (c) ./ (3 * h(1:n - 1, idx));
elseif (strcmp(cond,"variational") || strcmp(cond,"second"))
if ((nargin < 4) || strcmp(cond,"variational"))
## set second derivatives at end points to zero
valc = [0, 0];
endif
c(1,:) = valc(1) / 2;
c(n,:) = valc(2) / 2;
g = 3 * diff (a(2:n,:)) ./ h(2:n - 1, idx)\
- 3 * diff (a(1:n - 1,:)) ./ h(1:n - 2, idx);
g(1,:) = g(1,:) - h(1) * c(1,:);
g(n - 2,:) = g(n-2,:) - h(n - 1) * c(n,:);
if( n == 3)
dg = 2 * h(1);
c(2:n - 1,:) = g / dg;
else
dg = 2 * (h(1:n - 2) .+ h(2:n - 1));
e = h(2:n - 2);
c(2:n - 1,:) = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-2,n-2) \ g;
end
b(1:n - 1,:) = diff (a) ./ h(1:n - 1,idx)\
- h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:));
d = diff (c) ./ (3 * h(1:n - 1, idx));
elseif (strcmp(cond,"periodic"))
h = [h; h(1)];
## XXX FIXME XXX --- the following gives a smoother periodic transition:
## a(n,:) = a(1,:) = ( a(n,:) + a(1,:) ) / 2;
a(n,:) = a(1,:);
tmp = diff (shift ([a; a(2,:)], -1));
g = 3 * tmp(1:n - 1,:) ./ h(2:n,idx)\
- 3 * diff (a) ./ h(1:n - 1,idx);
if (n > 3)
dg = 2 * (h(1:n - 1) .+ h(2:n));
e = h(2:n - 1);
## Use Sherman-Morrison formula to extend the solution
## to the cyclic system. See Numerical Recipes in C, pp 73-75
gamma = - dg(1);
dg(1) -= gamma;
dg(end) -= h(1) * h(1) / gamma;
z = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-1,n-1) \ ...
[[gamma; zeros(n-3,1); h(1)],g];
fact = (z(1,2:end) + h(1) * z(end,2:end) / gamma) / ...
(1.0 + z(1,1) + h(1) * z(end,1) / gamma);
c(2:n,idx) = z(:,2:end) - z(:,1) * fact;
endif
c(1,:) = c(n,:);
b = diff (a) ./ h(1:n - 1,idx)\
- h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:));
b(n,:) = b(1,:);
d = diff (c) ./ (3 * h(1:n - 1, idx));
d(n,:) = d(1,:);
elseif (strcmp(cond,"not-a-knot"))
g = zeros(n - 2,columns(a));
g(1,:) = 3 / (h(1) + h(2)) * (a(3,:) - a(2,:)\
- h(2) / h(1) * (a(2,:) - a(1,:)));
g(n - 2,:) = 3 / (h(n - 1) + h(n - 2)) *\
(h(n - 2) / h(n - 1) * (a(n,:) - a(n - 1,:)) -\
(a(n - 1,:) - a(n - 2,:)));
if (n > 4)
g(2:n - 3,:) = 3 * diff (a(3:n - 1,:)) ./ h(3:n - 2,idx)\
- 3 * diff (a(2:n - 2,:)) ./ h(2:n - 3,idx);
dg = 2 * (h(1:n - 2) .+ h(2:n - 1));
dg(1) = dg(1) - h(1);
dg(n - 2) = dg(n-2) - h(n - 1);
ldg = udg = h(2:n - 2);
udg(1) = udg(1) - h(1);
ldg(n - 3) = ldg(n-3) - h(n - 1);
c(2:n - 1,:) = spdiags([[ldg(:);0],dg,[0;udg(:)]],[-1,0,1],n-2,n-2) \ g;
elseif (n == 4)
dg = [h(1) + 2 * h(2), 2 * h(2) + h(3)];
ldg = h(2) - h(3);
udg = h(2) - h(1);
c(2:n - 1,:) = spdiags([[ldg(:);0],dg,[0;udg(:)]],[-1,0,1],n-2,n-2) \ g;
else # n == 3
dg= [h(1) + 2 * h(2)];
c(2:n - 1,:) = g/dg(1);
endif
c(1,:) = c(2,:) + h(1) / h(2) * (c(2,:) - c(3,:));
c(n,:) = c(n - 1,:) + h(n - 1) / h(n - 2) * (c(n - 1,:) - c(n - 2,:));
b = diff (a) ./ h(1:n - 1, idx)\
- h(1:n - 1, idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:));
d = diff (c) ./ (3 * h(1:n - 1, idx));
else
msg = sprintf("unknown end condition: %s",cond);
error (msg);
endif
d = d(1:n-1,:); c=c(1:n-1,:); b=b(1:n-1,:); a=a(1:n-1,:);
pp = mkpp (x, cat (2, d'(:), c'(:), b'(:), a'(:)), szy(1:end-1));
endfunction
%!shared x,y,cond
%! x = linspace(0,2*pi,15); y = sin(x);
%!assert (ppval(csape(x,y),x), y, 10*eps);
%!assert (ppval(csape(x,y),x'), y', 10*eps);
%!assert (ppval(csape(x',y'),x'), y', 10*eps);
%!assert (ppval(csape(x',y'),x), y, 10*eps);
%!assert (ppval(csape(x,[y;y]),x), \
%! [ppval(csape(x,y),x);ppval(csape(x,y),x)], 10*eps)
%!test cond='complete';
%!assert (ppval(csape(x,y,cond),x), y, 10*eps);
%!assert (ppval(csape(x,y,cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x), y, 10*eps);
%!assert (ppval(csape(x,[y;y],cond),x), \
%! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps)
%!test cond='variational';
%!assert (ppval(csape(x,y,cond),x), y, 10*eps);
%!assert (ppval(csape(x,y,cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x), y, 10*eps);
%!assert (ppval(csape(x,[y;y],cond),x), \
%! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps)
%!test cond='second';
%!assert (ppval(csape(x,y,cond),x), y, 10*eps);
%!assert (ppval(csape(x,y,cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x), y, 10*eps);
%!assert (ppval(csape(x,[y;y],cond),x), \
%! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps)
%!test cond='periodic';
%!assert (ppval(csape(x,y,cond),x), y, 10*eps);
%!assert (ppval(csape(x,y,cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x), y, 10*eps);
%!assert (ppval(csape(x,[y;y],cond),x), \
%! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps)
%!test cond='not-a-knot';
%!assert (ppval(csape(x,y,cond),x), y, 10*eps);
%!assert (ppval(csape(x,y,cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x'), y', 10*eps);
%!assert (ppval(csape(x',y',cond),x), y, 10*eps);
%!assert (ppval(csape(x,[y;y],cond),x), \
%! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps)