An introduction to basic symbolic computation concepts using 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