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

2. Overview Introduction Existing algorithms Busłowicz’s algorithm
Parallel algorithm
Results
Conclusions and future work

3. Definitions
A polynomial matrix is a matrix which has polynomials in all of its entries.
H(s) = Hnsn+Hn-1sn-1+Hn-2sn-2+…+Ho,
where Hi are constant r x r matrices,
i=0, …, n.

4. Definitions Example: s+2 s3+ 3s2+s s3 s2+1
n=3 – degree of the polynomial matrix
r=2 – the size of the matrix H
Ho= H1= …
2 0
0 1 1
0 0

5. 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)

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

7. Who Needs It??? Multivariable control systems
Analysis of power systems
Robust stability analysis
Design of linear decoupling controllers
… and many more areas.

8. Existing Algorithms Leverrier’s algorithm ( 1840) Exact algorithms
[sI-H] - resolvent matrix
Exact algorithms
Approximation methods

9. The Selection of the Algorithm
Large degree of polynomial operations
Lengthy calculations
Not very general
Before
Buslowicz’s algorithm (1980)
After
Some improvements at the cost of increased computational complexity

10. Buslowicz’s 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

11. Details of the Algorithm
Available upon request

12. Challenges Encountered (sequential)
Several inconsistencies in the original paper:

13. Challenges Encountered (parallel)
Dependent loops
for (i=2; i<r+1; i++) {
calculations requiring R[i-1][k] }
for(k=0; k<n*i+1; k++) { }
O(n2r4)

14. Challenges Encountered (parallel)
Loops of variable length
for(k=0; k<n*i+1; k++) {
for(ll=0; ll<min+1; ll++)
main calculations }
Varies with k

15. Shared and Distributed Memory
Main differences
Synchronization of the processes
Shared Memory (barrier)
Distributed memory (data exchange)
for (i=2; i<r+1; i++) {
calculations requiring R[i-1]
*Synchronization point }

16. Platforms Distributed memory platforms: SGI 02 NOW MIPS R5000 180MHz
P IV NOW GHz
P III Cluster 1GHz
P IV Cluster Zeon 2.2GHz

17. Platforms Shared memory platforms: SGI Power Challenge 10000
8 MPIS R10000
SGI Origin 2000
16 MPIS R MHz

18. Understanding the Results
n – degree of polynomial (<= 25)
r – size of a matrix (<=25)
Sequential algorithm – O(n2r5)
Average of multiple runs
Unloaded platforms

19. Sequential Run Times (n=25, r=25)
Platform
Times (sec)
SGI O2 NOW
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

20. Results – Distributed Memory
Speedup
SGI O2 NOW - slowdown
P IV NOW minimal speedup

21. Speedup (P III & P IV Clusters)

22. Results – Shared Memory
Excellent results!!!

23. Speedup (SGI Power Challenge)

24. Speedup (SGI Origin 2000)
Superlinear speedup!

25. Run times (SGI Power Challenge)
8 processors

26. Run times (SGI Origin 2000)
n =25

27. Run times (SGI Power Challenge)

28. Efficiency 2 4 6 8 16 24 P III Cluster 89.7% 76.5% 61.3% 58.5% 40.1%
25.0%
P IV
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
SGI
Origin 2000
99.9%
98.7%
99.0%
93.8%

29. Conclusions
We have performed an exhaustive search of all available algorithms;
We have implemented the sequential version of Busłowicz’s 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.

30. Future Work
Study the behavior of the algorithm for larger problem sizes (distributed memory).
Re-evaluate message passing in distributed memory implementation.
Extend Buslowicz’s algorithm to inverting multivariable polynomial matrices
H(s1, s2 … sk).

31. Questions