Approximating the Matrix Exponential

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Approximating the Matrix Exponential
By Nicholas Christian and Joshua Lee Fisher
http//www.ma.utexas.edu/users/nchristian/matrixex
p/matrixexp.html
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Goal

• Examine the accuracy and efficiency of methods
  for
• computing eA, where A is a nn matrix.
• Evaluation of two types of matrices positive
  definite
• and symmetric.
• We formally define eA by the series expansion,

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Methods

• Truncated Series Methods
• Truncated Taylor Series
• Padé Approximation
• Scaling and Squaring
• Ordinary Differential Equation (ODE) Methods
• General Purpose ODE Solver (Runge-Kutta-Fehlberg)
• Stiff Solver (R-K 3rd order solution 2nd order
  accurate)
• Multistep (Adams-Bashforth-Multon)

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Truncated Series Methods

• Truncated Taylor Series
• The accuracy decreases and the time efficiency
  increases as A increases.
• Padé Approximation
• The Padé Approximation is very accurate when
  A lt 1
• The loss of accuracy with these series is not the
  truncation, but the round off error from
  calculating the terms.

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Truncated Series Methods (continued)

• Scaling and Squaring
• Find the smallest power of 2 such that A lt 1
• Then compute e(A/n)
• Next form e A be repeatedly squaring e(A/n)
• By reducing the norm of the matrix, Scaling and
  Squaring improves the accuracy of any method.

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ODE Methods

• Motivation
• Linear first order constant coefficient ordinary
  differential equation,
• where A is a nn constant coefficient matrix
  and x(t) is a n1 function vector with respect to
  t.
• Initial conditions are the jth column of the
  identity matrix I.
• Using seperation of variables, the solution to
  the evolution equation is
• Therefore when t 1 we obtain the desired
  calculation,

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ODE Methods(continued)

• General Purpose ODE Solver
• Fifth order solution with fourth order accuracy
• Adaptable time step Runge-Kutta-Fehlberg
• Low order Stiff Solver
• Third order solution with second order accuracy
• Adaptable time step Runge-Kutta
• Multistep (Adams-Bashforth-Moulton)
• Adjustable order predictor-corrector method
• Fixed time step
• Open Adams-Bashforth explicit formula (Predictor)
• Closed Adams-Moulton implicit formula (Corrector)
• Adaptable time step improves the accuracy
• Stiff solver is the slowest

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Results Positive definite m50
x 10-3
x 10-9
Legend Taylor (blue), Padé (red), R-K-45(green),
R-K-23(black), A-B-M (cyan).
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Results Symmetric m50
Legend Taylor (blue), Padé (red), R-K-45(green),
R-K-23(black), A-B-M (cyan).
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Conclusion

• Series and ODE methods give more accurate results
  and are more time efficient for positive definite
  matrices.
• A clear example is Padé Approximation
• Runge-Kutta-Fehlberg method had average
  performance in accuracy and efficiency for both
  types of matrices.
• The stiff solver gave accurate results, however
  had the longest computation time. Taking up to
  30 sec. for the m50 symmetric case and over 5
  min. in the m100 symmetric case.
• Taylor series consistently gave good accuracy and
  increasing calculation time. However as A
  increases, Taylor gives less accurate results.

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References
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References(continued)