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Overview . Floating Point ComputationExact Real ArithmeticSolid Modelling
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1. Doing it without Floating Real and Solid Computing
2. Overview Floating Point Computation
Exact Real Arithmetic
Solid Modelling & Computational Geometry
A New Integration
The Moral of Our Story
3. Decimal System Foundation of our computer revolution.
Imagine computing in the Roman system CCXXXII times XLVIII, i.e. 232 ? 48.
Zero was invented by Indian mathematicians, who were inspired by the Babylonian and the Chinese number systems, particularly as used in abacuses.
4. The Discovery of Decimal Fractions
5. The Long Journey
6. Khwarizmi (780 850) Settled in the House of Wisdom (Baghdad).
Wrote three books:
Hindu Arithmetic
Al-jabr va Al-Moghabela
Astronomical Tables
The established words: Algorithm from Al-Khwarizmi and Algebra from Al-jabr testify to his fundamental contribution to human thought.
7. The Long Journey
8. Adelard of Bath (1080 1160) First English Scientist.
Translated from Arabic to Latin Khwarizmis astronomical tables with their use of zero.
9. The Long Journey
10. Kashani (1380 1429) Developed arithmetic algorithms for fractions, that we use today.
11. Kashani invented the first mechanical special purpose computers:
to find when the planets are closest,
to calculate longitudes of planets,
to predict lunar eclipses. Kashani (1380 1429)
13. Mechanical Computers in Europe
14. Modern Computers: Floating Point Numbers Any other number like ? is rounded or approximated to a close floating point number.
15. Floating Point Arithmetic is not sound
16. Floating Point Arithmetic is not sound
17. Failure of Floating Point Computation
18. Depending on the floating point format, the sequence tends to 1 or 2 or 3 or 4.
In reality, it oscillates about 1.51 and 2.37. Failure of Floating Point Computation
19. Failure of Floating Point Computation In any floating point format, the sequence converges to 100.
In reality, it converges to 6.
20. Failure of Floating Point Computation In any floating point format, the sequence converges to 100.
In reality, it converges to 6.
21. Failure of Floating Point Computation In any floating point format, the sequence converges to 100.
In reality, it converges to 6.
22. Bankers Example
23. Bankers Example
24. Bankers Example
25. Bankers Example
26. Bankers Example
27. He finds out that he would have an overdraft of
2,000,000,000.00 !! Bankers Example
28. Suspicious about this astonishing result, he buys a better computer. Bankers Example
29. The clients balance is: Bankers Example
30. Pilots dilemma
31. Evaluate numerical expressions correctly up to any given number of decimal places. Exact Real Arithmetic
32. Exact Real Arithmetic Evaluate numerical expressions correctly up to any given number of decimal places.
33. Exact Real Arithmetic Evaluate numerical expressions correctly up to any given number of decimal places.
34. A computation is possible only if any output digit can be calculated from a finite number of the input digits. Exact Real Arithmetic
35. The Signed Decimal System
36. The Signed Decimal System Gives a redundant representation.
37. Numbers as Sequences of Operations Signed binary system:
38. Numbers as Sequences of Operations Signed binary system:
39. Numbers as Sequences of Operations Signed binary system:
40. Numbers as Sequences of Operations
41. Numbers as Sequences of Operations
42. Numbers as Sequences of Operations
43. Numbers as Sequences of Operations
44. Numbers as Sequences of Operations
45. Numbers as Sequences of Operations
46. Numbers as Sequences of Operations
47. Numbers as Sequences of Operations
48. Numbers as Sequences of Operations
49. Numbers as Sequences of Operations
50. Numbers as Sequences of Operations
51. Numbers as Sequences of Operations
52. Numbers as Sequences of Operations
53. Numbers as Sequences of Operations
54. Numbers as Sequences of Operations
55. Numbers as Sequences of Operations
56. Numbers as Sequences of Operations
57. Numbers as Sequences of Operations
58. Basic Arithmetic Operations
59. Addition
60. Addition
61. Addition
62. Addition
63. Addition
64. Elementary Functions
65. Elementary Functions
66. Elementary Functions
67. Elementary Functions
68. Elementary Functions
69. Elementary Functions
70. Domain of Intervals
71. Correct geometric algorithms become unreliable when implemented in floating point. Solid Modelling / Computational Geometry
72. With floating point arithmetic, find the point P of the intersection of L1 and L2. Then: minimum_distance(P, L1) > 0 minimum_distance(P, L2) > 0 Solid Modelling / Computational Geometry
73. The Convex Hull Algorithm
74. The Convex Hull Algorithm
75. The Convex Hull Algorithm
76. The Convex Hull Algorithm
77. The Convex Hull Algorithm
78. A Fundamental Problem
79. A Fundamental Problem
80. A Fundamental Problem
81. Intersection of Two Cubes
82. Intersection of Two Cubes
83. This is Really Ironical !
84. Foundation of a Computable Geometry Reconsider the membership predicate:
85. A Three-Valued Logic
86. Computing a Solid Object In this model, a solid object is represented by its interior and exterior,
87. Computing a Solid Object Kashanis computation of ?
88. Computable Predicates & Operations This gives a model for geometry and topology in which all the basic building blocks (membership, intersection, union) are continuous and computable.
89. The Convex Hull Algorithm
90. The Convex Hull Algorithm
91. The Convex Hull Algorithm
92. The Convex Hull Algorithm
93. Calculating the Number of Holes For a computable solid with computable volume, one can calculate the number of holes with volume greater than any desired value.
94. The Riemann Integral
95. The Riemann Integral
96. The Riemann Integral
97. The Riemann Integral
98. The Riemann Integral
99. The Generalized Riemann Integral The generalized Riemann integral has been applied to compute physical quantities in chaotic systems:
100. The Real and Solid People
101. The Long Journey
102. The Moral of Our Story The ever increasing power of computer technology enables us to perform exact computation efficiently, in the spirit of Kashani.
People from many nations have contributed to the present achievements of science and technology.
History has imposed a reversal of fortune: Nations who developed the foundation of our present computer revolution in the very dark ages of Europe, later experienced a much stifled development.
The Internet can be a global equaliser if, and only if, we make it available to the youth of the developing countries.
103. Empowering the Youth in the Developing World
104. THE END
