Symbolic LU Decomposition With Large Expression Management

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Symbolic LU Decomposition With Large Expression
Management
PHD Talk 2005

• Speaker Wenqin Zhou
• Supervisors David Jeffrey and Greg Reid
• March 3, 2005

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Outline

• Introduction on LU decomposition

• Symbolic LU decomposition and its main challenge

• Existing tools in Maple for handling expression
  swell

• LargeExpressions Package (LEM)

• LU LEM decomposition advantages

• Bench mark files and applications

• Future work and References

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Introduction On LU Decomposition

• Solve system of linear equations
• 
  Axb,
• ( A square matrix, singular or
  non-singular, b vector .)

PALU, ( L lower matrix with 1s as
diagonal entries, U upper matrix, P
permutation matrix. )

LUxPb,
Forward Substitution Ly Pb,
Backward Substitution Ux y.
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Symbolic LU Decomposition

• The advantages for symbolic computation
• Always exact
• Good for the later use, like doing some
  differentiations, or integrations.
• Easy to manipulate or find the relations between
  the different variables or parameters.

• The main challenge for symbolic computation
• How to handle the intermediate expression
  swell?
• Originally this problem is from DynaFlex.
• (Please see Maple worksheet.)

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Tools for Handling Expression Swell

• There are four main tools in Maple for handling
  the large expressions
• Veil and Unveil in LargeExpressions package.
• Freeze/thaw commands
• Compseq command
• Optimize in Codegen package

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LargeExpressions Package (LEM)

• Veil/Unveil command hide/show the complicated
  expressions.
• More controllable and can be adjusted based on
  users requirement. It works during the
  expression generation.
• User can have options to isolate the global
  variables and Unveil them independently to verify
  the correctness at anytime.
• User can collect the expressions as a functional
  argument, replacing the complicated coefficients
  by simple labels.

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LU LEM Decomposition Algorithm

• Main features
• ---- Symbolic LU decomposition with all kinds of
  different strategies
• Large Expression Management strategies
• ( LEMStrategy_n max number of indets,
• LEMStrategy_L max length. )

• Symbolic partial pivoting strategies
• ( PivotingStrategy_Llength max length,
• PivotingStrategy_Lindets max number of
  indets. )

• Zero recognition strategies
• ( ZeroStrategy_Normalizer normalize the
  expression,
• ZeroStrategy_Simplify simplify the
  expression.)

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The Advantages

• Improve the intermediate computation speed and
  the whole computation speed
• The representation ways are much neater than
  expanding them all
• Solve problems which are too large to be solved
  by Maple LUDecomposition
• Please see the Maple worksheet.

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Bench Mark files (1)

• Matrix A assumed to be full.
• Results using Maple 9.5 on Pentium 3, 1GHz.

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Bench Mark files (2)

• Matrix A assumed to be sparse matrix with
  density 0.3.
• Results using Maple 9.5 on Pentium 3, 1GHz.

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Applications

• Originally, the question is from DynaFlex, a
  Maple package for making symbolic models. LU
  LEM algorithm can speed up the model construction
  and obtain more models.
• Solving the linear equations with different
  vector b or different numeric entries of A
  without doing decomposition again.
• Easy for finding the relations between the matrix
  A, vector b and solution x.
• etc

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Future work

• Symbolic LU decomposition with numeric
  evaluation.
• Parallel computation for symbolic matrix LU
  decomposition.
• The complexity analysis for the LU decomposition
  and LU LEM decomposition.

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References

• 1 R. M. Corless, D. J. Jeffrey, M. B. Monagan,
  Pratibha. Two Perturbation Calculation in Fluid
  Mechanics Using Large-Expression Management. J.
  Symbolic Computation, 11, 1--17, (1996).
• 2 R. M. Corless. Essential Maple 7.
  Springer-Verlag.
• 3 J. J. Dongarra, ,I. S. Duff, D.C. Sorensen,
  H. A.Van der
• Vorst. Solving Linear Systems on Vector and
  Shared
• Memory Computers. SIAM Publications (1991).
• 4 G. T. Fernando, Monica Denham, Armando De
  Giusti. Parallel Matrix Multiplication and LU
  Factorization on Ethernet-Based Clusters. ISHPC
  2003, Springer-Verlag Berlin Heidelberg, 2003, Pp
  431-439.

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ACA Special Session

• Title Computation Tools for Large Expressions
• Abstract Large Expressions appear in many
  symbolic contexts. In this session, we invite
  papers expressing different viewpoints concerning
  large expressions. Papers describing applications
  in which large expressions arise are also
  welcome.
• More details, please visit the conference web
• http//www.jssac.org/Conference/ACA/

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Thank you for your attention!