This document discusses using symbolic computation in Perl to represent and manipulate mathematical objects like rational numbers, complex numbers, polynomials, and linear/quadratic equations. It presents examples of implementing rational numbers as ordered pairs of integers, complex numbers as ordered pairs of reals, and polynomials as ordered tuples. The key steps are to figure out the rules/properties of the mathematical objects, eliminate extraneous details, and find a representation to encode the objects as data structures that support defined arithmetic operations and algorithms. This allows performing symbolic rather than numeric computations natively in Perl.

1. Symbolic Computation
in Perl
Mike Friedman
(friedo)
YAPC::NA 2015
Salt Lake City, UT
@friedo

2. Me:
Mike Friedman
(friedo)
@friedo

3. http://quire.com/
@quire

4. The Problem:

5. The Problem:
We have computers.

6. The Problem:
We have computers.
But they don't really compute.

7. The Problem:
We have computers.
But they don't really compute.

8. The Problem:
We have computers.
But they don't really compute.
==

9. The Problem:
Native number types in most programming languages
(including Perl)
are ints and floats.

10. The Problem:
Native number types in most programming languages
(including Perl)
are ints and floats.
Because that's what the microprocessor can do.

11. The Problem:
What about rational numbers?

12. The Problem:
What about rational numbers?

13. The Problem:
What about rational numbers?
These can only be approximated
by floating point.

14. The Problem:
What about irrational numbers?

15. The Problem:
What about irrational numbers?
Same problem.

16. The Problem:
What about imaginary and complex numbers?

17. The Problem:
What about imaginary and complex numbers?
Floats can't even approximate these.

18. The Solution:
We do not want to calculate with numbers.

19. The Solution:
We do not want to calculate with numbers.
We want to compute with symbols.

20. The Solution:
We do not want to calculate with numbers.
We want to compute with symbols.
We do that in software.

21. The Solution:
>

22. Example 1
Rational numbers

23. Example 1
Rational numbers
We start with integers.

24. Example 1
Rational numbers
We start with integers.

25. Example 1
For division, we must extend the integers.

26. Example 1
For division, we must extend the integers.

27. Example 1
We can think of them as ordered pairs.

28. Example 1
Rational object as an ordered pair:
Divide by greatest common factor $f to put
in lowest terms.

29. Example 1
Convenience subs for a, b, c, d terms:

30. Example 1
Arithmetic operators

31. Example 1
Stringification operator

32. Example 1
Let's try it!

33. Example 2
Complex numbers

34. Example 2
Rational numbers
We start with the real numbers.

35. Example 2
Rational numbers
We start with the real numbers.

36. Example 2
Rational numbers
We start with the real numbers.

37. Example 2
To solve, we must extend the real numbers.

38. Example 2
We can think of them as ordered pairs.

39. Example 2
Complex object as an ordered pair:

40. Example 2
Convenience subs for a, b, c, d terms:
Remember these?

41. Example 2
Arithmetic operators

42. Example 2
Stringification operator

43. Example 2
Let's try it!

44. Example 3
Linear equations

45. Example 3
Linear equations
standard form:

46. Example 3
Linear equations
standard form:
solution:

47. Example 3
We can think of them as ordered pairs.

48. Example 3
Linear object as an ordered pair:

49. Example 3
Solving algorithm
That's it!

50. Example 3
Let's try it!

51. Example 3
What about Rationals?
Changed integer 0 to Rational 0/1.
Overloading handles the rest.

52. Example 3
Let's try it!

53. Example 4
Quadratic equations

54. Example 4
Quadratic equations
standard form:

55. Example 4
Quadratic equations
standard form:
solution:

56. Example 4
We can think of them as ordered triplets.

57. Example 4
Quadratic object as an ordered triplet:

58. Example 4
Solving algorithm

59. Example 4
Let's try it!
Now make it work with Complex objects!

60. Example 4
Let's try it!
Now make it work with Complex objects!
(I'm too lazy.)

61. Example 5
Polynomials

62. Example 5
Polynomials
standard form:

63. Example 5
Polynomials

64. Example 5
We can think of them as ordered tuples
of size n+1, where n is the degree of the
polynomial.

65. Example 5
Polynomial object as an ordered tuple:

66. Example 5
Addition operator:

67. Example 5
Let's try it!

68. Example 5
How about derivatives?

69. Example 5
Derivative operator:

70. Example 5
Let's try it!

71. Conclusions
1. Figure out the rules for your thing.

72. Conclusions
Complex numbers extend the reals.
They have self-consistent rules for
arithmetic.

73. Conclusions
2. Eliminate extraneous information.

74. Conclusions
Rationals are just pairs of integers.
Complex are just pairs of reals.
Fraction bars and i are just notation.

75. Conclusions
3. Find a representation for your thing.

76. Conclusions
Ordered pairs, triplets, and tuples.
For more interesting things, you'll use
trees.

77. Thanks!
Symbolic Computation
in Perl
Mike Friedman
(@friedo)
github.com/friedo/perl-symbolic