解特殊指数方程

前几天在牛客上看到一个面试题:,给定x,求y

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
import java.util.Scanner;

/**
* y^7 + 0.5y = x,给定x,求y
*
* 分析:f(y)单调,而且还是奇函数
*
* 那么只考虑x > 0的情况:
* 由于f(y)单调,故考虑使用二分推出答案
* 并且由原式可知,0.5 * y = x - y ^ 7 < x
* ---> 0 < y < 2 * x
*
*/
public class 解指数方程 {

private static double accuracy = 1e-8;

public static void main(String[] args) {
Scanner sc = new Scanner(System.in);
while (sc.hasNext()) {
double x = sc.nextDouble();
double y = solve(x);
System.out.printf("x = %f, y = %f\n", x, y);
}
}

public static double solve(double x) {
if (x == 0) {
return 0;
}
boolean flag = x > 0;
x = Math.abs(x);
double left = 0, right = x;
double mid = 0;
while (left < right) {
mid = (right - left) / 2 + left;
double diff = calculate(mid) - x;
if (Math.abs(diff) <= accuracy) {
if (!flag) {
mid *= -1;
}
break;
} else if (diff > 0) {
right = mid;
} else {
left = mid;
}
}
return mid;
}

private static double calculate(double num) {
return Math.pow(num, 7) + 0.5 * num;
}
}


测试运行结果:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
-100
x = -100.000000, y = -1.928028
1
x = 1.000000, y = 0.916197
0
x = 0.000000, y = 0.000000
100
x = 100.000000, y = 1.928028
38291
x = 38291.000000, y = 4.515693
37261.77321
x = 37261.773210, y = 4.498150
32918429
x = 32918429.000000, y = 11.855500