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title: An Introduction to Numerical Computing
subtitle: Tips, Tricks, and Guidelines
author: Jeffrey B. Armstrong
company: Approximatrix, LLC
footer: Columbus Code Camp
subfooter: 13 October 2012
h1. Introduction
* Different Goals
* Different Techniques
* Different Concerns
h1. About the Speaker
* Experience with small-scale numerical computing
* Worked in numerical computing for ~12 years
* Still works with C, Fortran, Python, and Ruby regularly
h1. Outline
* Language Choice
* Central Processing Units
* Arrays
* Mathematics
h1. Choosing a Language
1. Speed
2. Ease
3. Maintainability
4. Popularity
h1. Choosing a Language
* Fortran
** Inherently vectorized
** Multidimensional arrays
** Not so popular
h1. Choosing a Language
* Python
** Interpretted
** Multidimensional arrays with addons
** Popular
h1. Choosing a Language
* Ruby
** Interpretted
** Multidimensional arrays (Matrix)
** Not popular for numerics
h1. Choosing a Language
* Bad Choices
** Fast!
*** Is it actually faster? Maintainable?
** Popular
*** Is it popular in numerics? Does it make life easier?
** Obscure
*** Does anyone use APL? What about Ada?
** Proprietary Math Packages
*** Fast? Correct? Upgrade path? Cross-platform? Availability?
h1. Know Your Processor
_Your Commodore 64 is really neato / What kind of chip you got in there, a Dorito?_
p{font-size: 0.7em; text-align: right;}. - "Wierd Al" Yankovic, _It's All About the Pentiums_
h1. Know Your Processor
When writing code, the developer should be aware of his or her CPU.
* How fast is integer and floating point math?
* Do I have multiple cores?
* Do I have vector instructions?
h1. Know Your Processor
Integer Math: blazing fast on x86
!{border:0px; height:12em;}dhrystone.svg!
h1. Know Your Processor
Multiple "Cores"
* Use your whole CPU!
* Understand what a real "CPU core" is:
** -Hyperthreading-
** -Cell SPEs-
** -Your GPU-
* Other processing elements are extremely useful, but they aren't necessarily general purpose
h1. Know Your Processor
Multiple "Cores"
* Multithreading
** Easy to do in most languages
** Look for easy-to-parallelize tasks
*** Monte-Carlo Simulations
*** Explicit time-stepping finite difference/volume/etc.
** Avoid the need for complex synchronization
h1. Know Your Processor
Vector Instructions
* Need double precision
** Scientific computing usually requires this
* Program with vectors in mind
* Compilers are often smarter than you
h1. Big Ol' Arrays
_Nobody will ever have an image bigger than 512 x 512, so a fixed-size array is OK. For best precision, make it an array of doubles._
p{font-size: 0.7em; text-align: right;}. - Static Arrays, _How To Write Unmaintainable Code_, "http://tx0.org/4g5":http://tx0.org/4g5
h1. Big Ol' Arrays
Example: Computational Fluid Dynamics
!{border:0px; float:right; height:11em}cfd.svg!
* Finite Volume techniques
* Object-oriented!
{{{
class Cell():
def __init__(self):
self.u0 = 0
self.u1 = 0
self.v0 = 0
self.v1 = 0
self.p = 14.697
def update(self):
pass
}}}
h1. Big Ol' Arrays
{{{
class Cell():
def __init__(self):
self.u0 = 0
self.u1 = 0
self.v0 = 0
self.v1 = 0
self.p = 14.697
def update(self):
pass
}}}
* Terrible Idea!
** 1 million cells, requiring 1 million separate memory accesses
*** No guarantee about memory layout
** Problem gets worse with getters/setters
h1. Big Ol' Arrays
With arrays, we can:
* Abuse vector instructions, usually automatically
* Use pointer arithmetic for fast, in-place operations
** Vector/matrix interpolation
h1. Conditionals
* Why use conditionals?
** _Seriously?_
h1. Conditionals
* Why avoid conditionals?
** An "if" statement isn't vectorized
** Causes a virtual traffic jam of vectorized operations
* Explains why highly specialized numerical software isn't flexible
** Staticly allocated or fixed-size arrays
** Hard-coded boundary conditions
h1. Math and Other Lies
* Integer math is your friend
* Example: Calculating a percent complete in a loop
Wrong: `percent = (i / n) * 100.0`
Right: `percent = (i * 100) / n`
h1. Math and Other Lies
* Floating point mathematics is far from perfect
* IEEE Floating Point (IEEE-754):
p=. _sign * 2 ^exponent^ * mantissa_
<table width="75%">
<tr><td>Precision</td><td>Sign</td><td>Exponent</td><td>Mantissa</td><td>Bias</td></tr>
<tr><td>Single</td><td>1 bit</td><td>8 bits</td><td>23 bits</td><td>127</td></tr>
<tr><td>Double</td><td>1 bit</td><td>11 bits</td><td>52 bits</td><td>1023</td></tr>
</table>
h1. Math and Other Lies
* IEEE Floating Point (IEEE-754):
p{font-size: 0.9em;}. 0.125 = `2^{(124 - 127)}` = | 0 | 01111100 | 00000000000000000000000 |
p{font-size: 0.9em;}. 0.3 = `2^{(125 - 127)}*1.2` = | 0 | 01111101 | 00110011001100110011010 | != 0.3
h1. Distrusting Others
* Dangerous to blindly accept the output of "black box" math packages
** Especially dangerous for complex functions/methods
* Example: SciPy discrete algebraic Riccati equation solver:
p=. `X = A^T X A - ( A^T X B) (R + B^T X B)^-1 (B^T X A) + Q`
h1. Distrusting Others
{{{
failed DARE...
}}}
* Lies!
h1. Distrusting Others
* Simpler example: exponential function
p=. `f(x) = e^x
h1. Catastrophic Subtractive Cancelation
_If I were to get a tatoo, it would say "Catastrophic Subtractive Cancelation."_
p{font-size: 0.7em; text-align: right;}. - Me
h1. Catastrophic Subtractive Cancelation
* Side effect of floating point mathematics
* Single precision example:
p=. `5 * 10^8 - 1 * 10^-17 = 4.999999999999999999999999999 * 10^8`
h1. Catastrophic Subtractive Cancelation
* Exponential function
** Often estimated by Taylor series
`e^x = sum_(n=0)^oo( \frac {x^n} {n!} ) = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \frac {x^4} {4!} + \frac {x^5} {5!} + ...`
`e^10 = 1 + 10 + \frac 100 2 + \frac 1000 6 + \frac 10000 24 + \frac 100000 120 + ...`
`e^-10 = 1 - 10 + \frac 100 2 - \frac 1000 6 + \frac 10000 24 - \frac 100000 120 + ...`
h1. Catastrophic Subtractive Cancelation
Taylor series results (128 steps):
<table width="90%">
<tr><td>Method</td><td>exp(-10)</td><td>exp(-100)</td></tr>
<tr><td> `\frac 1 {e^{|x|}}` </td><td>4.5399929762484861e-05</td><td>3.7349958663065218e-44</td></tr>
<tr><td> `e^x` </td><td>4.5399929434053169e-05</td><td>-1.4586490453281426e+40</td></tr>
</table>
h1. Catastrophic Subtractive Cancelation
* Where can this occur?
** Math libraries
** Numerical integration
** Linear algebra
* How is it avoided?
** Testing
*** Unit tests
*** Sensitivity Analysis
*** Corroboration
** Avoidance
** Knowledge
h1. Summary
* "Trendy" languages might be wrong, or right, it depends...
* It's all about the vectors!
* Floating point math is easy to get wrong
h1. More Information
* "http://jeff.rainbow-100.com":http://jeff.rainbow-100.com
* @fortranjeff on Twitter
* ArmstrongJ on GitHub
|
