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1.1 root 1: #include <stdio.h>
2:
3: #define NP 1000
4: #define INF 1.e37
5:
6: struct proj { int lbf,ubf; float a,b,lb,ub,quant,mult,val[NP]; } x,y;
7: float *diag, *r;
8: float dx = 1.;
9: float ni = 100.;
10: int n;
11: int auta;
12: int periodic;
13: float konst = 0.0;
14: float zero = 0.;
15:
16: /* Spline fit technique
17: let x,y be vectors of abscissas and ordinates
18: h be vector of differences h9i8=x9i8-x9i-1988
19: y" be vector of 2nd derivs of approx function
20: If the points are numbered 0,1,2,...,n+1 then y" satisfies
21: (R W Hamming, Numerical Methods for Engineers and Scientists,
22: 2nd Ed, p349ff)
23: h9i8y"9i-1988+2(h9i8+h9i+18)y"9i8+h9i+18y"9i+18
24:
25: = 6[(y9i+18-y9i8)/h9i+18-(y9i8-y9i-18)/h9i8] i=1,2,...,n
26:
27: where y"908 = y"9n+18 = 0
28: This is a symmetric tridiagonal system of the form
29:
30: | a918 h928 | |y"918| |b918|
31: | h928 a928 h938 | |y"928| |b928|
32: | h938 a938 h948 | |y"938| = |b938|
33: | . | | .| | .|
34: | . | | .| | .|
35: It can be triangularized into
36: | d918 h928 | |y"918| |r918|
37: | d928 h938 | |y"928| |r928|
38: | d938 h948 | |y"938| = |r938|
39: | . | | .| | .|
40: | . | | .| | .|
41: where
42: d918 = a918
43:
44: r908 = 0
45:
46: d9i8 = a9i8 - h9i8829/d9i-18 1<i<_n
47:
48: r9i8 = b9i8 - h9i8r9i-18/d9i-1i8 1<_i<_n
49:
50: the back solution is
51: y"9n8 = r9n8/d9n8
52:
53: y"9i8 = (r9i8-h9i+18y"9i+18)/d9i8 1<_i<n
54:
55: superficially, d9i8 and r9i8 don't have to be stored for they can be
56: recalculated backward by the formulas
57:
58: d9i-18 = h9i8829/(a9i8-d9i8) 1<i<_n
59:
60: r9i-18 = (b9i8-r9i8)d9i-18/h9i8 1<i<_n
61:
62: unhappily it turns out that the recursion forward for d
63: is quite strongly geometrically convergent--and is wildly
64: unstable going backward.
65: There's similar trouble with r, so the intermediate
66: results must be kept.
67:
68: Note that n-1 in the program below plays the role of n+1 in the theory
69:
70: Other boundary conditions_________________________
71:
72: The boundary conditions are easily generalized to handle
73:
74: y908" = ky918", y9n+18" = ky9n8"
75:
76: for some constant k. The above analysis was for k = 0;
77: k = 1 fits parabolas perfectly as well as stright lines;
78: k = 1/2 has been recommended as somehow pleasant.
79:
80: All that is necessary is to add h918 to a918 and h9n+18 to a9n8.
81:
82:
83: Periodic case_____________
84:
85: To do this, add 1 more row and column thus
86:
87: | a918 h928 h918 | |y918"| |b918|
88: | h928 a928 h938 | |y928"| |b928|
89: | h938 a948 h948 | |y938"| |b938|
90: | | | .| = | .|
91: | . | | .| | .|
92: | h918 h908 a908 | | .| | .|
93:
94: where h908=_ h9n+18
95:
96: The same diagonalization procedure works, except for
97: the effect of the 2 corner elements. Let s9i8 be the part
98: of the last element in the i8th9 "diagonalized" row that
99: arises from the extra top corner element.
100:
101: s918 = h918
102:
103: s9i8 = -s9i-18h9i8/d9i-18 2<_i<_n+1
104:
105: After "diagonalizing", the lower corner element remains.
106: Call t9i8 the bottom element that appears in the i8th9 colomn
107: as the bottom element to its left is eliminated
108:
109: t918 = h918
110:
111: t9i8 = -t9i-18h9i8/d9i-18
112:
113: Evidently t9i8 = s9i8.
114: Elimination along the bottom row
115: introduces further corrections to the bottom right element
116: and to the last element of the right hand side.
117: Call these corrections u and v.
118:
119: u918 = v918 = 0
120:
121: u9i8 = u9i-18-s9i-18*t9i-18/d9i-18
122:
123: v9i8 = v9i-18-r9i-18*t9i-18/d9i-18 2<_i<_n+1
124:
125: The back solution is now obtained as follows
126:
127: y"9n+18 = (r9n+18+v9n+18)/(d9n+18+s9n+18+t9n+18+u9n+18)
128:
129: y"9i8 = (r9i8-h9i+18*y9i+18-s9i8*y9n+18)/d9i8 1<_i<_n
130:
131: Interpolation in the interval x9i8<_x<_x9i+18 is by the formula
132:
133: y = y9i8x9+8 + y9i+18x9-8 -(h8299i+18/6)[y"9i8(x9+8-x9+8839)+y"9i+18(x9-8-x9-8839)]
134: where
135: x9+8 = x9i+18-x
136:
137: x9-8 = x-x9i8
138: */
139:
140: float
141: rhs(i){
142: int i_;
143: double zz;
144: i_ = i==n-1?0:i;
145: zz = (y.val[i]-y.val[i-1])/(x.val[i]-x.val[i-1]);
146: return(6*((y.val[i_+1]-y.val[i_])/(x.val[i+1]-x.val[i]) - zz));
147: }
148:
149: spline(){
150: float d,s,u,v,hi,hi1;
151: float h;
152: float D2yi,D2yi1,D2yn1,x0,x1,yy,a;
153: int end;
154: float corr;
155: int i,j,m;
156: if(n<3) return(0);
157: if(periodic) konst = 0;
158: d = 1;
159: r[0] = 0;
160: s = periodic?-1:0;
161: for(i=0;++i<n-!periodic;){ /* triangularize */
162: hi = x.val[i]-x.val[i-1];
163: hi1 = i==n-1?x.val[1]-x.val[0]:
164: x.val[i+1]-x.val[i];
165: if(hi1*hi<=0) return(0);
166: u = i==1?zero:u-s*s/d;
167: v = i==1?zero:v-s*r[i-1]/d;
168: r[i] = rhs(i)-hi*r[i-1]/d;
169: s = -hi*s/d;
170: a = 2*(hi+hi1);
171: if(i==1) a += konst*hi;
172: if(i==n-2) a += konst*hi1;
173: diag[i] = d = i==1? a:
174: a - hi*hi/d;
175: }
176: D2yi = D2yn1 = 0;
177: for(i=n-!periodic;--i>=0;){ /* back substitute */
178: end = i==n-1;
179: hi1 = end?x.val[1]-x.val[0]:
180: x.val[i+1]-x.val[i];
181: D2yi1 = D2yi;
182: if(i>0){
183: hi = x.val[i]-x.val[i-1];
184: corr = end?2*s+u:zero;
185: D2yi = (end*v+r[i]-hi1*D2yi1-s*D2yn1)/
186: (diag[i]+corr);
187: if(end) D2yn1 = D2yi;
188: if(i>1){
189: a = 2*(hi+hi1);
190: if(i==1) a += konst*hi;
191: if(i==n-2) a += konst*hi1;
192: d = diag[i-1];
193: s = -s*d/hi;
194: }}
195: else D2yi = D2yn1;
196: if(!periodic) {
197: if(i==0) D2yi = konst*D2yi1;
198: if(i==n-2) D2yi1 = konst*D2yi;
199: }
200: if(end) continue;
201: m = hi1>0?ni:-ni;
202: m = 1.001*m*hi1/(x.ub-x.lb);
203: if(m<=0) m = 1;
204: h = hi1/m;
205: for(j=m;j>0||i==0&&j==0;j--){ /* interpolate */
206: x0 = (m-j)*h/hi1;
207: x1 = j*h/hi1;
208: yy = D2yi*(x0-x0*x0*x0)+D2yi1*(x1-x1*x1*x1);
209: yy = y.val[i]*x0+y.val[i+1]*x1 -hi1*hi1*yy/6;
210: printf("%f ",x.val[i]+j*h);
211: printf("%f\n",yy);
212: }
213: }
214: return(1);
215: }
216: readin() {
217: for(n=0;n<NP;n++){
218: if(auta) x.val[n] = n*dx+x.lb;
219: else if(!getfloat(&x.val[n])) break;
220: if(!getfloat(&y.val[n])) break; } }
221:
222: getfloat(p)
223: float *p;{
224: char buf[30];
225: register c;
226: int i;
227: extern double atof();
228: for(;;){
229: c = getchar();
230: if (c==EOF) {
231: *buf = '\0';
232: return(0);
233: }
234: *buf = c;
235: switch(*buf){
236: case ' ':
237: case '\t':
238: case '\n':
239: continue;}
240: break;}
241: for(i=1;i<30;i++){
242: c = getchar();
243: if (c==EOF) {
244: buf[i] = '\0';
245: break;
246: }
247: buf[i] = c;
248: if('0'<=c && c<='9') continue;
249: switch(c) {
250: case '.':
251: case '+':
252: case '-':
253: case 'E':
254: case 'e':
255: continue;}
256: break; }
257: buf[i] = ' ';
258: *p = atof(buf);
259: return(1); }
260:
261: getlim(p)
262: struct proj *p; {
263: int i;
264: for(i=0;i<n;i++) {
265: if(!p->lbf && p->lb>(p->val[i])) p->lb = p->val[i];
266: if(!p->ubf && p->ub<(p->val[i])) p->ub = p->val[i]; }
267: }
268:
269:
270: main(argc,argv)
271: char *argv[];{
272: extern char *malloc();
273: int i;
274: x.lbf = x.ubf = y.lbf = y.ubf = 0;
275: x.lb = INF;
276: x.ub = -INF;
277: y.lb = INF;
278: y.ub = -INF;
279: while(--argc > 0) {
280: argv++;
281: again: switch(argv[0][0]) {
282: case '-':
283: argv[0]++;
284: goto again;
285: case 'a':
286: auta = 1;
287: numb(&dx,&argc,&argv);
288: break;
289: case 'k':
290: numb(&konst,&argc,&argv);
291: break;
292: case 'n':
293: numb(&ni,&argc,&argv);
294: break;
295: case 'p':
296: periodic = 1;
297: break;
298: case 'x':
299: if(!numb(&x.lb,&argc,&argv)) break;
300: x.lbf = 1;
301: if(!numb(&x.ub,&argc,&argv)) break;
302: x.ubf = 1;
303: break;
304: default:
305: fprintf(stderr, "Bad agrument\n");
306: exit(1);
307: }
308: }
309: if(auta&&!x.lbf) x.lb = 0;
310: readin();
311: getlim(&x);
312: getlim(&y);
313: i = (n+1)*sizeof(dx);
314: diag = (float *)malloc((unsigned)i);
315: r = (float *)malloc((unsigned)i);
316: if(r==NULL||!spline()) for(i=0;i<n;i++){
317: printf("%f ",x.val[i]);
318: printf("%f\n",y.val[i]); }
319: }
320: numb(np,argcp,argvp)
321: int *argcp;
322: float *np;
323: char ***argvp;{
324: double atof();
325: char c;
326: if(*argcp<=1) return(0);
327: c = (*argvp)[1][0];
328: if(!('0'<=c&&c<='9' || c=='-' || c== '.' )) return(0);
329: *np = atof((*argvp)[1]);
330: (*argcp)--;
331: (*argvp)++;
332: return(1); }
333:
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