Finding Eigen Values

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Finding Eigen ValuesEigen VectorsUsing
Parallel Programs

• Saketha Nath J

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Physical Meaning

• Rotate a coin
• Spread butter on bread
• Stretch a band
• H(x) k x

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Importance of the problem

• Solving a set of ODEs
• Decoupling equations
• Evaluating functions of a matrix
• Finding Vibration frequencies modes
• Finding roots of any polynomial
• Finding principal stress, strain directions
• Concept can be extended to functions
• Second Largest Eigenvalue effects the Convergence
  Rate of Genetic Algorithms!
• Solutions exist for the time independent
  Schrodinger equation only for certain values of
  energy, and these values are called "eigenvalues"
  of energy

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What is the need for parallelization?

• Sequential algorithms run in O(n3)
• Frequent need of solving problem
• Huge problems encountered in areas like
  Acoustics, Vibrations

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Sequential Algorithms

• Jacobi Symmetric matrix only
• QR Any square matrix
• QR factorization using householder matrices
• QR factorization using gram schmidt method
• Power Method Useful in finding largest or
  smallest few
• Lanczos Method With any symmetric matrix gives
  tridiagonal matrix

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Project Goal

• To implement Jacobi method
• MPI
• OpenMP
• Compare the results on a Shared memory
  architecture
• Compare with existing algorithms

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Inspirations

• Prof. Micheal T Heath explains that though QR
  performs better than Jacobi, Jacobi is more
  parallelizable.
• Gutheil, says though jacobi does not suite
  generic problem, it works well for some specific
  cases.
• Zhou Brent also support fact that Jacobi is
  more parallelizable and infact more accurate

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References

• An efficient method for computing eigenvalues of
  a real normal matrix Zhou Brent
• www.ittiam.com/pages/competency/eigen_mmcsp_2001.p
  df
• Explains cyclic jacobi
• mrccs.man.ac.uk/mpp-workshop6/proc/pdf/gutheil.pdf
  
• Compares various existing libraries in detail
• ipdps.eece.unm.edu/1996/PAPERS/S15/ZHOU/ZHOU.PDF
• Suggests a jacobi-like algorithm