This document contains C++ code that implements an adaptive Runge-Kutta Fehlberg method (RKFM) for numerically solving ordinary differential equations. It defines classes and functions for initializing the RKFM algorithm, calculating successive approximations, and checking error tolerances to adapt the step size accordingly. The main function initializes an RKFM object and calls its runIt() method to iteratively apply the algorithm up to 200 times or until the integration range is complete.

1. 1 //humanity is a very illusive quality
2
3 /*
4 Name:
5 OJ:
6 Link:
7 Algorithm:
8 Type:
9 Difficulty:
10 Interest:
11 Additional:
12 */
13 #include<cstdio>
14 #include<cstring>
15 #include<cstdlib>
16 #include<cctype>
17
18 #include<cmath>
19 #include<iostream>
20 #include<fstream>
21
22 #include<string>
23 #include<vector>
24 #include<queue>
25 #include<map>
26 #include<algorithm>
27 #include<set>
28 #include<sstream>
29 #include<stack>
30 #include <ctime>
31 using namespace std;
32
33 class TimeLogger {
34 clock_t st, nd; double elapsed;
35 public:
36 TimeLogger() { st = clock(); }
37 ~TimeLogger() {
38 nd = clock(); elapsed = (nd - st)/(double)CLOCKS_PER_SEC;
39 printf("nYour program took %.3lf secondsn", elapsed);
40 }
41 };
42
43 const int inf = (1<<28);
44 const double pi = (2.0*acos(0.0));
45 const double eps = 1e-9;
46 const double eps2 = 1e-12; // printf rounder
47 const double sensitiveEPS = 1e-14;// depends on digits after . ex 7
digit
48
49 typedef long long lli;
50 //typedef __int64 lli;
51 //typedef unsigned long long llu;
52 //typedef unsigned __int64 llu;
53 //typedef pair < int , int > pii;
54 //typedef vector < int > vi;
55 //typedef vector < string > vs;
56
57 #define isp2( a ) (!(a & (a-1)))
58 #define CLR( a ) memset(a , 0 , sizeof (a))
59 #define SET( a , b) memset(a , b , sizeof (a))
60 #define SZ( a ) ((int)a.size())
61 #define all( a ) a.begin(),a.end()
62
63
64 //#define _rep( i, a, b, x ) for( __typeof(b) i = ( a ); i <= ( b );
i += x )

2. 65 #define _rep( i, a, b, x ) for( i = ( a ) ; i <= ( b ) ; i += x )
66 #define rep( i, n ) _rep( i, 0, n - 1, 1 )
67 #define _rrep( i, a, b, x ) for( i = (a) ; i >= (b) ; i -= x )
68 #define rrep( i, a, b) _rrep( i, a, b, 1)
69 #define xrep( i, a, b) _rep( i, a, b, 1)
70
71 #define SD( a ) scanf("%d",&a)
72 #define SL( a ) scanf("%lld",&a)
73 #define SI( a ) scanf("%I64d",&a)
74 #define SS( a ) scanf("%s",a)
75 #define SF( a ) scanf("%lf",&a)
76
77 #define pb push_back
78 #define ff first
79 #define ss second
80 ///Comparision macros
81 #define _aEb(a,b) (fabs((a)-(b))<eps)
82 #define _aGb(a,b) ((a)>(b)+eps)
83 #define _aLb(a,b) ((a)+eps<(b))
84 #define _aLEb(a,b) (_aLb(a,b) || _aEb(a,b))
85 #define _aGEb(a,b) (_aGb(a,b) || _aEb(a,b))
86 #define _minf(a,b) ((a)+eps<(b)?(a):(b))
87 #define _maxf(a,b) ((a)+eps<(b)?(b):(a))
88 #define _sq(x) ((x)*(x))
89
90 const int MX = 1000;
91 struct RKFMY
92 {
93 double t[ MX + 10 ];
94 double y[ MX + 10 ];
95 double w[ MX + 10 ];
96 double ww[ MX + 10 ];
97 double h[ MX + 10 ];
98 double R[ MX + 10 ];
99 double a , b , alpha , hmax, hmin;
100 double TOL;
101 bool flag;
102
103 double k[7];
104
105 double f( double T , double Y )
106 {
107 return ( Y - T*T + 1 );
108 }
109 void init()
110 {
111 hmax = .25;
112 hmin = .01;
113 a = 0.;
114 b = 2.;
115 alpha = .5;
116 t[ 0 ] = a;
117 w[ 0 ] = alpha;
118 h[ 0 ] = hmax;
119 flag = true;
120 TOL = 1.e-5;
121 }
122 double runIt()
123 {
124 int i = 0,j;
125 while( flag && i<200 )
126 {
127 k[1] = h[i] * f( t[i] , w[i] );
128 k[2] = h[i] * f( t[i] + (1./4.) * h[i] , w[i] + (1./4.) *
k[1] );
129 k[3] = h[i] * f( t[i] + (3./8.) * h[i] , w[i] + (3./32.)*

3. k[1] + (9/32.)*k[2] );
130 k[4] = h[i] * f( t[i] + (12./13.)*h[i] , w[i] + (1932./
2197.)*k[1]
131 - (7200./2197)*k[2] + (7296./
2197)*k[3] );
132 k[5] = h[i] * f( t[i] + h[i] , w[i] + (439./
216.)*k[1] - 8.*k[2]
133 + (3680./513)*k[3] - (845./
4104)*k[4] ) ;
134 k[6] = h[i] * f( t[i] + .5*h[i] , w[i] - (8./27.)*
k[1] + 2.*k[2]
135 - (3544./2565.)*k[3] + (1859./
4104.)*k[4] - (11./40.)*k[5] );
136 i++;
137
138 R[i] = (1./h[i-1]) * fabs( (1./360.)*k[1] - (128./4275.)*
k[3] - (2197./75240.)*k[4]
139 + (1./50.)*k[5]
+ (2./55.)*k[6] );
140
141 if( _aLEb(R[i],TOL) )
142 {
143 t[i] = t[i-1] + h[i-1];
144 w[i] = w[i-1] + (25./216.)*k[1] + (1408./2565.)*k[3]
+ (2197./4104.)*k[4] - (1./5.)*k[5];
145 printf("%.12lf %.12lf %.12lf %.12lfn",t[i],w[i],h[
i-1],R[i]);
146 }
147
148 double del = .84 * pow( double(TOL / R[i]) , double(1./4.
) );
149
150 if( _aLb( del , .1 ) ) h[i] = .1 *h[i-1];
151 else if ( _aGb( del , 4. ) ) h[i] = 4 *h[i-1];
152 else h[i] = del*h[i-1];
153
154 if( _aGb( h[i] , hmax ) ) h[i] = hmax ;
155 if( _aGb( t[i] , b ) ) flag = 0;
156 else if( _aGb( t[i] + h[i] , b ) ) h[i] = b-t[i];
157 else if( _aLb( h[i] , hmin ) )
158 {
159 flag = 0;
160 printf("min h exceededn");
161 }
162 }
163 }
164
165 };
166 RKFMY rk;
167 int main(void)
168 {
169 //freopen("in.txt","r",stdin);
170 //freopen("out.txt","w",stdout);
171 //TimeLogger tm;
172 int i,j,k,kase=0;
173
174 rk.init();
175 rk.runIt();
176
177 return 0;
178 }