1 /* fp2rat.c (convert floating-point number to rational number) */
2
3 /***********************************************************************
4 * This code is part of GLPK (GNU Linear Programming Kit).
5 * Copyright (C) 2000-2013 Free Software Foundation, Inc.
6 * Written by Andrew Makhorin <mao@gnu.org>.
7 *
8 * GLPK is free software: you can redistribute it and/or modify it
9 * under the terms of the GNU General Public License as published by
10 * the Free Software Foundation, either version 3 of the License, or
11 * (at your option) any later version.
12 *
13 * GLPK is distributed in the hope that it will be useful, but WITHOUT
14 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
16 * License for more details.
17 *
18 * You should have received a copy of the GNU General Public License
19 * along with GLPK. If not, see <http://www.gnu.org/licenses/>.
20 ***********************************************************************/
21
22 #include "env.h"
23 #include "misc.h"
24
25 /***********************************************************************
26 * NAME
27 *
28 * fp2rat - convert floating-point number to rational number
29 *
30 * SYNOPSIS
31 *
32 * #include "misc.h"
33 * int fp2rat(double x, double eps, double *p, double *q);
34 *
35 * DESCRIPTION
36 *
37 * Given a floating-point number 0 <= x < 1 the routine fp2rat finds
38 * its "best" rational approximation p / q, where p >= 0 and q > 0 are
39 * integer numbers, such that |x - p / q| <= eps.
40 *
41 * RETURNS
42 *
43 * The routine fp2rat returns the number of iterations used to achieve
44 * the specified precision eps.
45 *
46 * EXAMPLES
47 *
48 * For x = sqrt(2) - 1 = 0.414213562373095 and eps = 1e-6 the routine
49 * gives p = 408 and q = 985, where 408 / 985 = 0.414213197969543.
50 *
51 * BACKGROUND
52 *
53 * It is well known that every positive real number x can be expressed
54 * as the following continued fraction:
55 *
56 * x = b[0] + a[1]
57 * ------------------------
58 * b[1] + a[2]
59 * -----------------
60 * b[2] + a[3]
61 * ----------
62 * b[3] + ...
63 *
64 * where:
65 *
66 * a[k] = 1, k = 0, 1, 2, ...
67 *
68 * b[k] = floor(x[k]), k = 0, 1, 2, ...
69 *
70 * x[0] = x,
71 *
72 * x[k] = 1 / frac(x[k-1]), k = 1, 2, 3, ...
73 *
74 * To find the "best" rational approximation of x the routine computes
75 * partial fractions f[k] by dropping after k terms as follows:
76 *
77 * f[k] = A[k] / B[k],
78 *
79 * where:
80 *
81 * A[-1] = 1, A[0] = b[0], B[-1] = 0, B[0] = 1,
82 *
83 * A[k] = b[k] * A[k-1] + a[k] * A[k-2],
84 *
85 * B[k] = b[k] * B[k-1] + a[k] * B[k-2].
86 *
87 * Once the condition
88 *
89 * |x - f[k]| <= eps
90 *
91 * has been satisfied, the routine reports p = A[k] and q = B[k] as the
92 * final answer.
93 *
94 * In the table below here is some statistics obtained for one million
95 * random numbers uniformly distributed in the range [0, 1).
96 *
97 * eps max p mean p max q mean q max k mean k
98 * -------------------------------------------------------------
99 * 1e-1 8 1.6 9 3.2 3 1.4
100 * 1e-2 98 6.2 99 12.4 5 2.4
101 * 1e-3 997 20.7 998 41.5 8 3.4
102 * 1e-4 9959 66.6 9960 133.5 10 4.4
103 * 1e-5 97403 211.7 97404 424.2 13 5.3
104 * 1e-6 479669 669.9 479670 1342.9 15 6.3
105 * 1e-7 1579030 2127.3 3962146 4257.8 16 7.3
106 * 1e-8 26188823 6749.4 26188824 13503.4 19 8.2
107 *
108 * REFERENCES
109 *
110 * W. B. Jones and W. J. Thron, "Continued Fractions: Analytic Theory
111 * and Applications," Encyclopedia on Mathematics and Its Applications,
112 * Addison-Wesley, 1980. */
113
fp2rat(double x,double eps,double * p,double * q)114 int fp2rat(double x, double eps, double *p, double *q)
115 { int k;
116 double xk, Akm1, Ak, Bkm1, Bk, ak, bk, fk, temp;
117 xassert(0.0 <= x && x < 1.0);
118 for (k = 0; ; k++)
119 { xassert(k <= 100);
120 if (k == 0)
121 { /* x[0] = x */
122 xk = x;
123 /* A[-1] = 1 */
124 Akm1 = 1.0;
125 /* A[0] = b[0] = floor(x[0]) = 0 */
126 Ak = 0.0;
127 /* B[-1] = 0 */
128 Bkm1 = 0.0;
129 /* B[0] = 1 */
130 Bk = 1.0;
131 }
132 else
133 { /* x[k] = 1 / frac(x[k-1]) */
134 temp = xk - floor(xk);
135 xassert(temp != 0.0);
136 xk = 1.0 / temp;
137 /* a[k] = 1 */
138 ak = 1.0;
139 /* b[k] = floor(x[k]) */
140 bk = floor(xk);
141 /* A[k] = b[k] * A[k-1] + a[k] * A[k-2] */
142 temp = bk * Ak + ak * Akm1;
143 Akm1 = Ak, Ak = temp;
144 /* B[k] = b[k] * B[k-1] + a[k] * B[k-2] */
145 temp = bk * Bk + ak * Bkm1;
146 Bkm1 = Bk, Bk = temp;
147 }
148 /* f[k] = A[k] / B[k] */
149 fk = Ak / Bk;
150 #if 0
151 print("%.*g / %.*g = %.*g",
152 DBL_DIG, Ak, DBL_DIG, Bk, DBL_DIG, fk);
153 #endif
154 if (fabs(x - fk) <= eps)
155 break;
156 }
157 *p = Ak;
158 *q = Bk;
159 return k;
160 }
161
162 /* eof */
163