Statistics

Problem Statement for "Mandelbrot"

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.

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

  1. 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. 2

    -1

    -1

    Returns: 2

    Z0 = -1 - 1i Z1 = -1 + 1i Z2 = -1 - 3i

  3. 30

    .25

    .25

    Returns: -1

    In this example, the magnitude of Zn is never more than 2.

  4. 30

    -1.9

    0

    Returns: -1

  5. 30

    .375

    .3

    Returns: 21

  6. 1

    2

    2

    Returns: 0

  7. 2

    -0.9757845334894477

    -0.5324823850973428

    Returns: -1

  8. 6

    -1.215710286583013

    -1.023275515748423

    Returns: 2

  9. 30

    -0.6032723480130606

    0.6801881709559305

    Returns: 12

  10. 9

    -0.1400492142091494

    0.9486376937662278

    Returns: -1

  11. 29

    -1.3881517891455584

    -0.02556210636440648

    Returns: 23

  12. 29

    0.07455255150701579

    0.6537762357563279

    Returns: -1

  13. 30

    -0.7486429223273539

    0.09778329030624722

    Returns: -1

  14. 30

    -0.1704792625621141

    1.0402943048285471

    Returns: 17

  15. 30

    0.807870298963598

    -0.6543660653618817

    Returns: 2

  16. 30

    0.48810360453387314

    -0.665488534645124

    Returns: 3

  17. 30

    -0.4385781787323486

    0.9158359631218591

    Returns: 4

  18. 30

    0.3876303092429134

    0.553235880211189

    Returns: 9

  19. 30

    -0.77194541564171

    -0.2104729508313854

    Returns: 13

  20. 30

    -0.6607090366845301

    0.43366731784830215

    Returns: 25

  21. 30

    0.36797794165192954

    0.5888317981912186

    Returns: 30

  22. 30

    0.9368843570133458

    0.2622863903243109

    Returns: 2

  23. 30

    0.020762888899839904

    -0.8665753621994896

    Returns: 10

  24. 30

    -0.5103992341377386

    -0.626336828815395

    Returns: 17

  25. 30

    -0.5433603111649807

    0.5278900520117933

    Returns: 23

  26. 30

    -0.18776392698249889

    0.8274060437584858

    Returns: 29

  27. 30

    0.27948801973818593

    -0.5899191256436842

    Returns: 30

  28. 30

    -0.6622628142150653

    -0.7829791656315668

    Returns: 3

  29. 30

    0.28406318697915256

    0.7072663999550006

    Returns: 5

  30. 30

    0.4229352491383338

    -0.12264605278203278

    Returns: 6

  31. 30

    0.42969095586857

    -0.15675007921442896

    Returns: 7

  32. 30

    0.3481849498123293

    -0.5625241775222809

    Returns: 21

  33. 30

    -0.5735481327981646

    0.6373126110310805

    Returns: 24

  34. 30

    -0.693206984256536

    0.46813084929404014

    Returns: 25

  35. 30

    -0.5512674608820742

    0.49656191346955136

    Returns: 28

  36. 30

    0.9734017363676726

    0.8845911530371948

    Returns: 1

  37. 30

    -0.5895060563118308

    0.47673774547673853

    Returns: 19

  38. 30

    -0.013764719206932785

    0.7469920439985955

    Returns: 28

  39. 30

    -0.8015334414990403

    -0.1834105013004188

    Returns: 29

  40. 30

    -1.9

    0.0

    Returns: -1

  41. 1

    2.0

    2.0

    Returns: 0

  42. 1

    1.5

    0.0

    Returns: 1

  43. 25

    0.35565

    -0.2545

    Returns: -1

  44. 28

    0.395

    0.28

    Returns: 20

  45. 28

    0.38

    0.33

    Returns: -1

  46. 30

    0.375

    0.3

    Returns: 21

  47. 21

    0.375

    0.3

    Returns: 21

  48. 1

    1.0

    1.0

    Returns: 1

  49. 10

    2.0

    2.0

    Returns: 0

  50. 10

    0.45

    0.0

    Returns: 5

  51. 2

    -1.0

    -1.0

    Returns: 2

  52. 1

    -1.0

    -1.0

    Returns: -1

  53. 29

    -1.3

    0.7

    Returns: 2

  54. 30

    1.1

    1.1

    Returns: 1

  55. 10

    0.5

    0.2

    Returns: 4

  56. 30

    -1.9

    0.0

    Returns: -1

  57. 1

    2.0

    2.0

    Returns: 0

  58. 1

    1.5

    0.0

    Returns: 1

  59. 25

    0.35565

    -0.2545

    Returns: -1

  60. 28

    0.395

    0.28

    Returns: 20

  61. 28

    0.38

    0.33

    Returns: -1

  62. 30

    0.375

    0.3

    Returns: 21

  63. 21

    0.375

    0.3

    Returns: 21

  64. 1

    1.0

    1.0

    Returns: 1

  65. 10

    2.0

    2.0

    Returns: 0

  66. 10

    0.45

    0.0

    Returns: 5

  67. 2

    -1.0

    -1.0

    Returns: 2

  68. 1

    -1.0

    -1.0

    Returns: -1

  69. 29

    -1.3

    0.7

    Returns: 2

  70. 30

    1.1

    1.1

    Returns: 1

  71. 10

    0.5

    0.2

    Returns: 4

  72. 30

    -1.9

    0.0

    Returns: -1

  73. 1

    2.0

    2.0

    Returns: 0

  74. 1

    1.5

    0.0

    Returns: 1

  75. 25

    0.35565

    -0.2545

    Returns: -1

  76. 28

    0.395

    0.28

    Returns: 20

  77. 28

    0.38

    0.33

    Returns: -1

  78. 30

    0.375

    0.3

    Returns: 21

  79. 21

    0.375

    0.3

    Returns: 21

  80. 1

    1.0

    1.0

    Returns: 1

  81. 10

    2.0

    2.0

    Returns: 0

  82. 10

    0.45

    0.0

    Returns: 5

  83. 2

    -1.0

    -1.0

    Returns: 2

  84. 1

    -1.0

    -1.0

    Returns: -1

  85. 29

    -1.3

    0.7

    Returns: 2

  86. 30

    1.1

    1.1

    Returns: 1

  87. 10

    0.5

    0.2

    Returns: 4

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