Parallel Pi Implementation

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Parallel Pi Implementation

• Ken Edwards

2
Overview

• Algorithms chosen
• Implementation
• Performance results

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Monte Carlo

• Randomly select values for x and y
• 0 lt x,y lt1
• Pi/4 inside points / total points

Source http//www.modula2.org/projects/pi_by_mont
ecarlo/est_pi.gif
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Monte Carlo Continued

• Pros
• Straightforward implementation
• Monte Carlo algorithms parallelize well
• Cons
• Very slow convergence

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Machins Formula

• p 4 arctan(1/5) arctan(1/239)
• arctan(x) x x3/3 x5/5 - x7/7 x9/9 -
• Infinite number of similar formulas
• p 4 arctan(1)
• p 4 arctan(1/2) 4 arctan(1/3)
• p 4 arctan(1/2) 4 arctan(1/5) arctan(1/8)
• p 16 arctan(1/5) 4 arctan(1/70) 4
  arctan(1/99)
• p 24 arctan(1/8) 8 arctan(1/57) 4
  arctan(1/239)
• p 48 arctan(1/18) 4 arctan(1/57) - 20
  arctan(1/239)

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Machins Formula

• Pros
• Much faster convergence
• Variety of ways to parallelize
• Cons
• More complex than Monte Carlo
• Assigning equal loads across processors
  challenging

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What I Did

• Monte Carlo
• Serial version
• Pthread version
• OpenMP version
• Machins
• Serial version
• Several Pthread versions
• OpenMP version

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Machins Formula - Pthreads

• Each arctan calculation is its own thread
• Two threads
• p 4 arctan(1/5) arctan(1/239)
• Three threads
• p 4 arctan(1/2) arctan(1/5) arctan(1/8)
• p 48 arctan(1/18) 32 arctan(1/57) 20
  arctan(1/239)
• Four threads
• p 7 arctan(1/13) 8 arctan(1/32) -
• 2 arctan(1/132) 5 arctan(1/378)
• p 14 arctan(1/4) 10 arctan(1/32)
• 6 arctan(1/132) - 8 arctan(1/378)

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Initial Pthread Performance
Tests run on 4-way machine with 2.20 GHz Intel
Xeons
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Why Such Variable Results?
Source http//turner.faculty.swau.edu/pi/piforms.
html
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Monte Carlo Performance

• Initially, concurrent version performed worse!
• rand() is not thread-safe
• used rand_r() instead
• Lesson learned make sure functions you call are
  thread-safe before using them!

Tests run on 4-way machine with 2.20 GHz Intel
Xeons
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Whats Left

• Try more formulas to achieve faster converging 3
  and 4-threaded versions
• Profile the code to find additional performance
  bottlenecks

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Thank You