Mandelbrot | Topcoder Archive

Problem Statement

A complex number is one of the form a+bi, where a and b are real numbers, and i represents the square root of -1. We call a the real part of the complex number, and b the imaginary part. When multiplying two complex numbers a+bi and c+di, the result is (a*c-b*d)+(a*d+b*c)i. When adding two complex numbers, a+bi and c+di, the result is (a+c)+(b+d)i. Finally, the magnitude of a complex number, a+bi, is sqrt(a*a+b*b).

The Mandelbrot set is a set of complex numbers. To determine whether or not a complex number, C = a+bi, is in the Mandelbrot set, we use the following recurrence:
Z₀ = C
Z_n+1 = Z_n*Z_n+C

If, for all n greater than or equal to 0, the magnitude of Z_n is less than or equal to 2, then the complex number represented by C = a+bi is in the Mandelbrot set.

You will be given an int, max, representing the maximum value of n to check. For example, if max were 10, you would be required to check Z_n for all 0 <= n <= 10. You will also be given two doubles, a and b, representing the real and imaginary parts of Z₀, respectively. If the magnitude of Z_n is always less than or equal to 2, for all 0 <= n <= max, you should return -1. Otherwise, you should return the smallest n such that the magnitude of Z_n is greater than 2.

Definition

Class:: Mandelbrot
Method:: iterations
Parameters:: int, double, double
Returns:: int
Method signature:: int iterations(int max, double a, double b)
(be sure your method is public)

Constraints

max will be between 1 and 30, inclusive.
a will be between -2 and 2, inclusive.
b will be between -2 and 2, inclusive.
To prevent rounding errors, the magnitude of Zn will not be within 1e-3 of 2, for any n between 0 and n', inclusive, where n' is the smallest number (or max) such that the magnitude of Zn' is greater than 2.

Examples

```
5
```
```
1
```
```
1
```
```
Returns: 1
```
Here, Z0 = 1 + 1i Thus, Z1 = 1 - 1 + 1i + 1i + 1 + 1i = 1 + 3i. The magnitude of 1+3i is sqrt(1*1 + 3*3) = sqrt(10) > 2. Therefore, the return is 1.
```
2
```
```
-1
```
```
-1
```
```
Returns: 2
```
Z0 = -1 - 1i Z1 = -1 + 1i Z2 = -1 - 3i
```
30
```
```
.25
```
```
.25
```
```
Returns: -1
```
In this example, the magnitude of Zn is never more than 2.
```
30
```
```
-1.9
```
```
0
```
```
Returns: -1
```
```
30
```
```
.375
```
```
.3
```
```
Returns: 21
```
```
1
```
```
2
```
```
2
```
```
Returns: 0
```

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