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1 | #ifndef lint |
2 | static char *sccsid = "@(#)spline.c 4.3 (Berkeley) %G%"; | |
3 | #endif | |
4 | ||
365384c2 | 5 | #include <stdio.h> |
762ee6b6 | 6 | #include <math.h> |
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7 | |
8 | #define NP 1000 | |
762ee6b6 | 9 | #define INF HUGE |
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10 | |
11 | struct proj { int lbf,ubf; float a,b,lb,ub,quant,mult,val[NP]; } x,y; | |
12 | float *diag, *r; | |
13 | float dx = 1.; | |
14 | float ni = 100.; | |
15 | int n; | |
16 | int auta; | |
17 | int periodic; | |
18 | float konst = 0.0; | |
19 | float zero = 0.; | |
20 | ||
21 | /* Spline fit technique | |
22 | let x,y be vectors of abscissas and ordinates | |
ef461b9e | 23 | h be vector of differences hi=xi-xi-1 |
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24 | y" be vector of 2nd derivs of approx function |
25 | If the points are numbered 0,1,2,...,n+1 then y" satisfies | |
26 | (R W Hamming, Numerical Methods for Engineers and Scientists, | |
27 | 2nd Ed, p349ff) | |
ef461b9e | 28 | hiy"i-1+2(hi+hi+1)y"i+hi+1y"i+1 |
365384c2 | 29 | |
ef461b9e | 30 | = 6[(yi+1-yi)/hi+1-(yi-yi-1)/hi] i=1,2,...,n |
365384c2 | 31 | |
ef461b9e | 32 | where y"0 = y"n+1 = 0 |
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33 | This is a symmetric tridiagonal system of the form |
34 | ||
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35 | | a1 h2 | |y"1| |b1| |
36 | | h2 a2 h3 | |y"2| |b2| | |
37 | | h3 a3 h4 | |y"3| = |b3| | |
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38 | | . | | .| | .| |
39 | | . | | .| | .| | |
40 | It can be triangularized into | |
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41 | | d1 h2 | |y"1| |r1| |
42 | | d2 h3 | |y"2| |r2| | |
43 | | d3 h4 | |y"3| = |r3| | |
365384c2 BJ |
44 | | . | | .| | .| |
45 | | . | | .| | .| | |
46 | where | |
ef461b9e | 47 | d1 = a1 |
365384c2 | 48 | |
ef461b9e | 49 | r0 = 0 |
365384c2 | 50 | |
ef461b9e | 51 | di = ai - hi2/di-1 1<i<_n |
365384c2 | 52 | |
ef461b9e | 53 | ri = bi - hiri-1/di-1\ei 1<_i<_n |
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54 | |
55 | the back solution is | |
ef461b9e | 56 | y"n = rn/dn |
365384c2 | 57 | |
ef461b9e | 58 | y"i = (ri-hi+1y"i+1)/di 1<_i<n |
365384c2 | 59 | |
ef461b9e | 60 | superficially, di and ri don't have to be stored for they can be |
365384c2 BJ |
61 | recalculated backward by the formulas |
62 | ||
ef461b9e | 63 | di-1 = hi2/(ai-di) 1<i<_n |
365384c2 | 64 | |
ef461b9e | 65 | ri-1 = (bi-ri)di-1/hi 1<i<_n |
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66 | |
67 | unhappily it turns out that the recursion forward for d | |
68 | is quite strongly geometrically convergent--and is wildly | |
69 | unstable going backward. | |
70 | There's similar trouble with r, so the intermediate | |
71 | results must be kept. | |
72 | ||
73 | Note that n-1 in the program below plays the role of n+1 in the theory | |
74 | ||
ef461b9e | 75 | Other boundary conditions_________________________ |
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76 | |
77 | The boundary conditions are easily generalized to handle | |
78 | ||
ef461b9e | 79 | y0" = ky1", yn+1" = kyn" |
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80 | |
81 | for some constant k. The above analysis was for k = 0; | |
82 | k = 1 fits parabolas perfectly as well as stright lines; | |
83 | k = 1/2 has been recommended as somehow pleasant. | |
84 | ||
ef461b9e | 85 | All that is necessary is to add h1 to a1 and hn+1 to an. |
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86 | |
87 | ||
ef461b9e | 88 | Periodic case_____________ |
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89 | |
90 | To do this, add 1 more row and column thus | |
91 | ||
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92 | | a1 h2 h1 | |y1"| |b1| |
93 | | h2 a2 h3 | |y2"| |b2| | |
94 | | h3 a4 h4 | |y3"| |b3| | |
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95 | | | | .| = | .| |
96 | | . | | .| | .| | |
ef461b9e | 97 | | h1 h0 a0 | | .| | .| |
365384c2 | 98 | |
ef461b9e | 99 | where h0=_ hn+1 |
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100 | |
101 | The same diagonalization procedure works, except for | |
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102 | the effect of the 2 corner elements. Let si be the part |
103 | of the last element in the ith "diagonalized" row that | |
365384c2 BJ |
104 | arises from the extra top corner element. |
105 | ||
ef461b9e | 106 | s1 = h1 |
365384c2 | 107 | |
ef461b9e | 108 | si = -si-1hi/di-1 2<_i<_n+1 |
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109 | |
110 | After "diagonalizing", the lower corner element remains. | |
ef461b9e | 111 | Call ti the bottom element that appears in the ith colomn |
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112 | as the bottom element to its left is eliminated |
113 | ||
ef461b9e | 114 | t1 = h1 |
365384c2 | 115 | |
ef461b9e | 116 | ti = -ti-1hi/di-1 |
365384c2 | 117 | |
ef461b9e | 118 | Evidently ti = si. |
365384c2 BJ |
119 | Elimination along the bottom row |
120 | introduces further corrections to the bottom right element | |
121 | and to the last element of the right hand side. | |
122 | Call these corrections u and v. | |
123 | ||
ef461b9e | 124 | u1 = v1 = 0 |
365384c2 | 125 | |
ef461b9e | 126 | ui = ui-1-si-1*ti-1/di-1 |
365384c2 | 127 | |
ef461b9e | 128 | vi = vi-1-ri-1*ti-1/di-1 2<_i<_n+1 |
365384c2 BJ |
129 | |
130 | The back solution is now obtained as follows | |
131 | ||
ef461b9e | 132 | y"n+1 = (rn+1+vn+1)/(dn+1+sn+1+tn+1+un+1) |
365384c2 | 133 | |
ef461b9e | 134 | y"i = (ri-hi+1*yi+1-si*yn+1)/di 1<_i<_n |
365384c2 | 135 | |
ef461b9e | 136 | Interpolation in the interval xi<_x<_xi+1 is by the formula |
365384c2 | 137 | |
ef461b9e | 138 | y = yix+ + yi+1x- -(h2i+1/6)[y"i(x+-x+3)+y"i+1(x--x-3)] |
365384c2 | 139 | where |
ef461b9e | 140 | x+ = xi+1-x |
365384c2 | 141 | |
ef461b9e | 142 | x- = x-xi |
365384c2 BJ |
143 | */ |
144 | ||
145 | float | |
146 | rhs(i){ | |
147 | int i_; | |
148 | double zz; | |
149 | i_ = i==n-1?0:i; | |
150 | zz = (y.val[i]-y.val[i-1])/(x.val[i]-x.val[i-1]); | |
151 | return(6*((y.val[i_+1]-y.val[i_])/(x.val[i+1]-x.val[i]) - zz)); | |
152 | } | |
153 | ||
154 | spline(){ | |
155 | float d,s,u,v,hi,hi1; | |
156 | float h; | |
157 | float D2yi,D2yi1,D2yn1,x0,x1,yy,a; | |
158 | int end; | |
159 | float corr; | |
160 | int i,j,m; | |
161 | if(n<3) return(0); | |
162 | if(periodic) konst = 0; | |
163 | d = 1; | |
164 | r[0] = 0; | |
165 | s = periodic?-1:0; | |
166 | for(i=0;++i<n-!periodic;){ /* triangularize */ | |
167 | hi = x.val[i]-x.val[i-1]; | |
168 | hi1 = i==n-1?x.val[1]-x.val[0]: | |
169 | x.val[i+1]-x.val[i]; | |
170 | if(hi1*hi<=0) return(0); | |
171 | u = i==1?zero:u-s*s/d; | |
172 | v = i==1?zero:v-s*r[i-1]/d; | |
173 | r[i] = rhs(i)-hi*r[i-1]/d; | |
174 | s = -hi*s/d; | |
175 | a = 2*(hi+hi1); | |
176 | if(i==1) a += konst*hi; | |
177 | if(i==n-2) a += konst*hi1; | |
178 | diag[i] = d = i==1? a: | |
179 | a - hi*hi/d; | |
180 | } | |
181 | D2yi = D2yn1 = 0; | |
182 | for(i=n-!periodic;--i>=0;){ /* back substitute */ | |
183 | end = i==n-1; | |
184 | hi1 = end?x.val[1]-x.val[0]: | |
185 | x.val[i+1]-x.val[i]; | |
186 | D2yi1 = D2yi; | |
187 | if(i>0){ | |
188 | hi = x.val[i]-x.val[i-1]; | |
189 | corr = end?2*s+u:zero; | |
190 | D2yi = (end*v+r[i]-hi1*D2yi1-s*D2yn1)/ | |
191 | (diag[i]+corr); | |
192 | if(end) D2yn1 = D2yi; | |
193 | if(i>1){ | |
194 | a = 2*(hi+hi1); | |
195 | if(i==1) a += konst*hi; | |
196 | if(i==n-2) a += konst*hi1; | |
197 | d = diag[i-1]; | |
198 | s = -s*d/hi; | |
199 | }} | |
200 | else D2yi = D2yn1; | |
201 | if(!periodic) { | |
202 | if(i==0) D2yi = konst*D2yi1; | |
203 | if(i==n-2) D2yi1 = konst*D2yi; | |
204 | } | |
205 | if(end) continue; | |
206 | m = hi1>0?ni:-ni; | |
207 | m = 1.001*m*hi1/(x.ub-x.lb); | |
208 | if(m<=0) m = 1; | |
209 | h = hi1/m; | |
210 | for(j=m;j>0||i==0&&j==0;j--){ /* interpolate */ | |
211 | x0 = (m-j)*h/hi1; | |
212 | x1 = j*h/hi1; | |
213 | yy = D2yi*(x0-x0*x0*x0)+D2yi1*(x1-x1*x1*x1); | |
214 | yy = y.val[i]*x0+y.val[i+1]*x1 -hi1*hi1*yy/6; | |
215 | printf("%f ",x.val[i]+j*h); | |
216 | printf("%f\n",yy); | |
217 | } | |
218 | } | |
219 | return(1); | |
220 | } | |
221 | readin() { | |
222 | for(n=0;n<NP;n++){ | |
223 | if(auta) x.val[n] = n*dx+x.lb; | |
224 | else if(!getfloat(&x.val[n])) break; | |
225 | if(!getfloat(&y.val[n])) break; } } | |
226 | ||
227 | getfloat(p) | |
228 | float *p;{ | |
229 | char buf[30]; | |
230 | register c; | |
231 | int i; | |
232 | extern double atof(); | |
233 | for(;;){ | |
234 | c = getchar(); | |
235 | if (c==EOF) { | |
236 | *buf = '\0'; | |
237 | return(0); | |
238 | } | |
239 | *buf = c; | |
240 | switch(*buf){ | |
241 | case ' ': | |
242 | case '\t': | |
243 | case '\n': | |
244 | continue;} | |
245 | break;} | |
246 | for(i=1;i<30;i++){ | |
247 | c = getchar(); | |
248 | if (c==EOF) { | |
249 | buf[i] = '\0'; | |
250 | break; | |
251 | } | |
252 | buf[i] = c; | |
253 | if('0'<=c && c<='9') continue; | |
254 | switch(c) { | |
255 | case '.': | |
256 | case '+': | |
257 | case '-': | |
258 | case 'E': | |
259 | case 'e': | |
260 | continue;} | |
261 | break; } | |
262 | buf[i] = ' '; | |
263 | *p = atof(buf); | |
264 | return(1); } | |
265 | ||
266 | getlim(p) | |
267 | struct proj *p; { | |
268 | int i; | |
269 | for(i=0;i<n;i++) { | |
270 | if(!p->lbf && p->lb>(p->val[i])) p->lb = p->val[i]; | |
271 | if(!p->ubf && p->ub<(p->val[i])) p->ub = p->val[i]; } | |
272 | } | |
273 | ||
274 | ||
275 | main(argc,argv) | |
276 | char *argv[];{ | |
277 | extern char *malloc(); | |
278 | int i; | |
279 | x.lbf = x.ubf = y.lbf = y.ubf = 0; | |
280 | x.lb = INF; | |
281 | x.ub = -INF; | |
282 | y.lb = INF; | |
283 | y.ub = -INF; | |
284 | while(--argc > 0) { | |
285 | argv++; | |
286 | again: switch(argv[0][0]) { | |
287 | case '-': | |
288 | argv[0]++; | |
289 | goto again; | |
290 | case 'a': | |
291 | auta = 1; | |
292 | numb(&dx,&argc,&argv); | |
293 | break; | |
294 | case 'k': | |
295 | numb(&konst,&argc,&argv); | |
296 | break; | |
297 | case 'n': | |
298 | numb(&ni,&argc,&argv); | |
299 | break; | |
300 | case 'p': | |
301 | periodic = 1; | |
302 | break; | |
303 | case 'x': | |
304 | if(!numb(&x.lb,&argc,&argv)) break; | |
305 | x.lbf = 1; | |
306 | if(!numb(&x.ub,&argc,&argv)) break; | |
307 | x.ubf = 1; | |
308 | break; | |
309 | default: | |
310 | fprintf(stderr, "Bad agrument\n"); | |
311 | exit(1); | |
312 | } | |
313 | } | |
314 | if(auta&&!x.lbf) x.lb = 0; | |
315 | readin(); | |
316 | getlim(&x); | |
317 | getlim(&y); | |
318 | i = (n+1)*sizeof(dx); | |
319 | diag = (float *)malloc((unsigned)i); | |
320 | r = (float *)malloc((unsigned)i); | |
321 | if(r==NULL||!spline()) for(i=0;i<n;i++){ | |
322 | printf("%f ",x.val[i]); | |
323 | printf("%f\n",y.val[i]); } | |
324 | } | |
325 | numb(np,argcp,argvp) | |
326 | int *argcp; | |
327 | float *np; | |
328 | char ***argvp;{ | |
329 | double atof(); | |
330 | char c; | |
331 | if(*argcp<=1) return(0); | |
332 | c = (*argvp)[1][0]; | |
333 | if(!('0'<=c&&c<='9' || c=='-' || c== '.' )) return(0); | |
334 | *np = atof((*argvp)[1]); | |
335 | (*argcp)--; | |
336 | (*argvp)++; | |
337 | return(1); } | |
338 |