Parallel Inversion of Polynomial Matrices

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Parallel Inversion of Polynomial Matrices
Alina Solovyova-Vincent Frederick C. Harris,
Jr. M. Sami Fadali
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Overview

• Introduction
• Existing algorithms
• Buslowiczs algorithm
• Parallel algorithm
• Results
• Conclusions and future work

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Definitions

• A polynomial matrix is a matrix which has
  polynomials in all of its entries.
• H(s) HnsnHn-1sn-1Hn-2sn-2Ho,
• where Hi are constant r x r matrices,
• i0, , n.

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Definitions

• Example
• s2 s3 3s2s
• s3 s21
• n3 degree of the polynomial matrix
• r2 the size of the matrix H
• Ho H1

• 2 0
• 0 1

• 1
• 0 0

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Definitions

• H-1(s) inverse of the matrix H(s)
• One of the ways to calculate it
• H-1(s) adj H(s) /det H(s)

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Definitions

• A rational matrix can be expressed as a ration of
  a numerator polynomial matrix and a denominator
  scalar polynomial.

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Who Needs It???

• Multivariable control systems
• Analysis of power systems
• Robust stability analysis
• Design of linear decoupling controllers
• and many more areas.

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Existing Algorithms

• Leverriers algorithm ( 1840)
• sI-H - resolvent matrix
• Exact algorithms
• Approximation methods

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The Selection of the Algorithm

• Large degree of polynomial operations
• Lengthy calculations
• Not very general

• Before
• Buslowiczs algorithm (1980)
• After

• Some improvements at the cost of increased
  computational complexity

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Buslowiczs Algorithm

• Benefits
• More general than methods proposed earlier
• Only requires operations on constant matrices
• Suitable for computer programming
• Drawback
• the irreducible form cannot be ensured in general

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Details of the Algorithm

• Available upon request

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Challenges Encountered (sequential)

• Several inconsistencies in the original paper

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Challenges Encountered (parallel)

• Dependent loops

for (i2 iltr1 i) calculations
requiring Ri-1k

• for(k0 kltni1 k)

O(n2r4)
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Challenges Encountered (parallel)

• Loops of variable length
• for(k0 kltni1 k)
• for(ll0 llltmin1 ll)
• main calculations

Varies with k
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Shared and Distributed Memory

• Main differences
• Synchronization of the processes
• Shared Memory (barrier)
• Distributed memory (data exchange)

for (i2 iltr1 i) calculations requiring Ri-1 Synchronization point
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Platforms

• Distributed memory platforms
• SGI 02 NOW MIPS R5000 180MHz
• P IV NOW 1.8 GHz
• P III Cluster 1GHz
• P IV Cluster Zeon 2.2GHz

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Platforms

• Shared memory platforms
• SGI Power Challenge 10000
• 8 MPIS R10000
• SGI Origin 2000
• 16 MPIS R12000 300MHz

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Understanding the Results

• n degree of polynomial (lt 25)
• r size of a matrix (lt25)
• Sequential algorithm O(n2r5)
• Average of multiple runs
• Unloaded platforms

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Sequential Run Times (n25, r25)
Platform Times (sec)
SGI O2 NOW 2645.30
P IV NOW 22.94
P III Cluster 26.10
P IV Cluster 18.75
SGI Power Challenge 913.99
SGI Origin 2000 552.95
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Results Distributed Memory

• Speedup
• SGI O2 NOW - slowdown
• P IV NOW - minimal speedup

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Speedup (P III P IV Clusters)
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Results Shared Memory

• Excellent results!!!

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Speedup (SGI Power Challenge)
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Speedup (SGI Origin 2000)
Superlinear speedup!
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Run times (SGI Power Challenge)
8 processors
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Run times (SGI Origin 2000)
n 25
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Run times (SGI Power Challenge)
r 20
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Efficiency
2 4 6 8 16 24
P III Cluster 89.7 76.5 61.3 58.5 40.1 25.0
P IV Cluster 88.3 68.2 49.9 46.9 26.1 15.5
SGI Power Challenge 99.7 98.2 97.9 95.8 n/a n/a
SGI Origin 2000 99.9 98.7 99.0 98.2 93.8 n/a
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Conclusions

• We have performed an exhaustive search of all
  available algorithms
• We have implemented the sequential version of
  Buslowiczs algorithm
• We have implemented two versions of the parallel
  algorithm
• We have tested parallel algorithm on 6 different
  platforms
• We have obtained excellent speedup and efficiency
  in a shared memory environment.

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Future Work

• Study the behavior of the algorithm for larger
  problem sizes (distributed memory).
• Re-evaluate message passing in distributed memory
  implementation.
• Extend Buslowiczs algorithm to inverting
  multivariable polynomial matrices
• H(s1, s2 sk).

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Questions