Matrix computations by Monte Carlo optimization. Doctoral thesis

Technical Report · Wed Jan 01 04:00:00 EST 1986

OSTI ID:6628945

Murphy, J J

This research addresses the general problem of performing matrix computations by Monte Carlo optimization. Four problems are examined: (1) estimating the value of a matrix function having a matrix power series representation, (2) estimating the solution of a system of linear equations, (3) estimating eigenvalues and eigenvectors of a matrix, and (4) estimating the condition number, determinant, and 2-norms of a matrix. To estimate the value of F(A) of a matrix function F having a convergent matrix power series, the author derives and analyzes unbiased consistent estimators of a matrix polynomial approximating the power series. A notion of an absolutely-convergent matrix power series is proposed, and it is shown that if F has an absolutely-convergent matrix power series at A, then there exists an unbiased consistent estimator of F, thus eliminating the need for polynomial approximation. The methods may be applied to solve certain systems of differential equations. The Laplace method of approximating definite integrals is used to derive random variables for estimating the solution of a system of linear equations. They depend on a parameter which, if sufficiently large, yields an unbiased consistent estimator. Two methods are studied for sampling from the underlying population. The first uses Monte Carlo simulation of Markov processes. The second uses quasi-Monte Carlo methods adaptable to parallel computation.

• Research Organization: Air Force Inst. of Tech., Wright-Patterson AFB, OH (USA)
• OSTI ID: 6628945
• Report Number(s): AD-A-176555/1/XAB; AFIT/CI/NR-87-22D
• Country of Publication: United States
• Language: English

Related Subjects

657000* -- Theoretical & Mathematical Physics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS
990210 -- Supercomputers-- (1987-1989)
DIFFERENTIAL EQUATIONS
EIGENVALUES
EIGENVECTORS
EQUATIONS
FUNCTIONS
INTEGRAL TRANSFORMATIONS
INTEGRALS
LAPLACE TRANSFORMATION
MARKOV PROCESS
MATRICES
MONTE CARLO METHOD
OPTIMIZATION
PARALLEL PROCESSING
POLYNOMIALS
POWER SERIES
PROGRAMMING
SERIES EXPANSION
SIMULATION
STOCHASTIC PROCESSES
TRANSFORMATIONS