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
Zn+1 = Zn*Zn+C
If, for all n greater than or equal to 0, the magnitude of Zn 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 Zn 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 Zn 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 Zn is greater than 2. |
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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) |
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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. |
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Examples |
| 0) | |
| | | 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. |
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| 1) | |
| | | Returns: 2 | Z0 = -1 - 1i
Z1 = -1 + 1i
Z2 = -1 - 3i
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| 2) | |
| | | Returns: -1 | | In this example, the magnitude of Zn is never more than 2. |
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