| 1 | #ifndef lint |
| 2 | static char *sccsid = "@(#)spline.c 4.5 (Berkeley) %G%"; |
| 3 | #endif |
| 4 | |
| 5 | #include <stdio.h> |
| 6 | #include <math.h> |
| 7 | |
| 8 | #define NP 1000 |
| 9 | #define INF HUGE |
| 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 |
| 23 | h be vector of differences hi=xi-xi-1 |
| 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) |
| 28 | hiy"i-1+2(hi+hi+1)y"i+hi+1y"i+1 |
| 29 | |
| 30 | = 6[(yi+1-yi)/hi+1-(yi-yi-1)/hi] i=1,2,...,n |
| 31 | |
| 32 | where y"0 = y"n+1 = 0 |
| 33 | This is a symmetric tridiagonal system of the form |
| 34 | |
| 35 | | a1 h2 | |y"1| |b1| |
| 36 | | h2 a2 h3 | |y"2| |b2| |
| 37 | | h3 a3 h4 | |y"3| = |b3| |
| 38 | | . | | .| | .| |
| 39 | | . | | .| | .| |
| 40 | It can be triangularized into |
| 41 | | d1 h2 | |y"1| |r1| |
| 42 | | d2 h3 | |y"2| |r2| |
| 43 | | d3 h4 | |y"3| = |r3| |
| 44 | | . | | .| | .| |
| 45 | | . | | .| | .| |
| 46 | where |
| 47 | d1 = a1 |
| 48 | |
| 49 | r0 = 0 |
| 50 | |
| 51 | di = ai - hi2/di-1 1<i<_n |
| 52 | |
| 53 | ri = bi - hiri-1/di-1i 1<_i<_n |
| 54 | |
| 55 | the back solution is |
| 56 | y"n = rn/dn |
| 57 | |
| 58 | y"i = (ri-hi+1y"i+1)/di 1<_i<n |
| 59 | |
| 60 | superficially, di and ri don't have to be stored for they can be |
| 61 | recalculated backward by the formulas |
| 62 | |
| 63 | di-1 = hi2/(ai-di) 1<i<_n |
| 64 | |
| 65 | ri-1 = (bi-ri)di-1/hi 1<i<_n |
| 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 | |
| 75 | Other boundary conditions_________________________ |
| 76 | |
| 77 | The boundary conditions are easily generalized to handle |
| 78 | |
| 79 | y0" = ky1", yn+1" = kyn" |
| 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 | |
| 85 | All that is necessary is to add h1 to a1 and hn+1 to an. |
| 86 | |
| 87 | |
| 88 | Periodic case_____________ |
| 89 | |
| 90 | To do this, add 1 more row and column thus |
| 91 | |
| 92 | | a1 h2 h1 | |y1"| |b1| |
| 93 | | h2 a2 h3 | |y2"| |b2| |
| 94 | | h3 a4 h4 | |y3"| |b3| |
| 95 | | | | .| = | .| |
| 96 | | . | | .| | .| |
| 97 | | h1 h0 a0 | | .| | .| |
| 98 | |
| 99 | where h0=_ hn+1 |
| 100 | |
| 101 | The same diagonalization procedure works, except for |
| 102 | the effect of the 2 corner elements. Let si be the part |
| 103 | of the last element in the ith "diagonalized" row that |
| 104 | arises from the extra top corner element. |
| 105 | |
| 106 | s1 = h1 |
| 107 | |
| 108 | si = -si-1hi/di-1 2<_i<_n+1 |
| 109 | |
| 110 | After "diagonalizing", the lower corner element remains. |
| 111 | Call ti the bottom element that appears in the ith colomn |
| 112 | as the bottom element to its left is eliminated |
| 113 | |
| 114 | t1 = h1 |
| 115 | |
| 116 | ti = -ti-1hi/di-1 |
| 117 | |
| 118 | Evidently ti = si. |
| 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 | |
| 124 | u1 = v1 = 0 |
| 125 | |
| 126 | ui = ui-1-si-1*ti-1/di-1 |
| 127 | |
| 128 | vi = vi-1-ri-1*ti-1/di-1 2<_i<_n+1 |
| 129 | |
| 130 | The back solution is now obtained as follows |
| 131 | |
| 132 | y"n+1 = (rn+1+vn+1)/(dn+1+sn+1+tn+1+un+1) |
| 133 | |
| 134 | y"i = (ri-hi+1*yi+1-si*yn+1)/di 1<_i<_n |
| 135 | |
| 136 | Interpolation in the interval xi<_x<_xi+1 is by the formula |
| 137 | |
| 138 | y = yix+ + yi+1x- -(h2i+1/6)[y"i(x+-x+3)+y"i+1(x--x-3)] |
| 139 | where |
| 140 | x+ = xi+1-x |
| 141 | |
| 142 | x- = x-xi |
| 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 | exit(0); |
| 325 | } |
| 326 | numb(np,argcp,argvp) |
| 327 | int *argcp; |
| 328 | float *np; |
| 329 | char ***argvp;{ |
| 330 | double atof(); |
| 331 | char c; |
| 332 | if(*argcp<=1) return(0); |
| 333 | c = (*argvp)[1][0]; |
| 334 | if(!('0'<=c&&c<='9' || c=='-' || c== '.' )) return(0); |
| 335 | *np = atof((*argvp)[1]); |
| 336 | (*argcp)--; |
| 337 | (*argvp)++; |
| 338 | return(1); } |