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All the
topics of this exam are covered in the undergraduate course MATE 4061,
Introduction to Numerical Analysis and the graduate course MATE 6025,
Numerical Linear Algebra. Elementary topics of Linear Algebra are covered
in the advanced undergraduate course MATE 5150, Linear Algebra.
Topics:
·
Floating
point numbers: representation of real numbers in finite precision, finite
precision arithmetic, error bounds, backward and forward error analysis.
·
Linear
algebra: real and complex vector spaces, families of linear independent
vectors, basis, dimension of a vector space. Emphasis will be given to
finite dimensional vector spaces.
·
Linear operators:
definition, matrix representation, definition of the kernel and rank of a
linear operator, Fundamental Theorem of Linear Algebra.
·
Eigenvalues and
eigenvectors: spectrum of a linear operator, spectral radius, invariant
spaces, diagonalizable matrices, Jordan decomposition Theorem
(a proof of the Jordan Theorem is not required however the student should
know how to apply it).
·
Normed spaces: vector and matrix norms, matrix norms
induced by vector norms, equivalent norms in finite dimensional vector
spaces. Convergence of sequences and series in normed
spaces, Cauchy sequences, complete normed
spaces.
·
Inner
products: norm induced by inner products, Cauchy-Schwarz inequality, orthogonality, adjoint
operator, orthogonal projection, norms induced by inner products,
Gram-Schmidt orthogonalization process, Givens
rotations, Househölder orthogonalization,
QR factorization, least squares problem.
·
Direct
methods for solving dense linear systems: Gaussian elimination and its
variants, LU and LUP factorization. Cholesky
factorization.
·
Iterative
methods for solving sparse linear systems: analysis of convergence of
Richardson, Jacobi, Gauss-Seidel, SOR, SSOR,
Steepest descent, Conjugate Gradient, GMRES, QMR, BICG, BICG-Stab
methods; Preconditioning techniques, computational aspects.
·
Singular
values: definition and decomposition of a matrix in singular values
(SVD).
·
Approximation
of eigenvalues and eigenvectors: power method,
Givens-Householder and QR, computational aspects.
Bibliography:
·
Matrix Analysis,
R. A. Horn and C.R. Johnson, Cambridge University Press, 1992.
·
Applied Numerical Linear Algebra, J.W. Demmel, SIAM, 1997
·
Introduction to matrix computations, G.W.
Stewart, Academic Press, 1973.
·
Matrix Computations, G.H. Golub
and C.F. Van Loan, third edition, John Hopkins University Press.
·
The symmetric eigenvalue problem, B.N. Parlett, Prentice Hall, Englewood Cliffs, 1980.
·
Iterative solution methods, O. Axelsson, Cambridge University Press.
·
Iterative methods for large sparse linear systems, Y. Saad, PWS Publishing Company, 1996.
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