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Numerical Linear Algebra

Numerical Linear Algebra. IKI40600. Course outline. Review linear algebra Square linear systems Least Square Problems Eigen Problems Text: Applied Numerical Linear Algebra, C.W. Hager Matrix Computations, G.Golub & R. van Loan. Review Linear Algebra. Matrix arithmetic operations

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Numerical Linear Algebra

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  1. Numerical Linear Algebra IKI40600

  2. Course outline • Review linear algebra • Square linear systems • Least Square Problems • Eigen Problems Text: Applied Numerical Linear Algebra, C.W. Hager Matrix Computations, G.Golub & R. van Loan

  3. Review Linear Algebra • Matrix arithmetic operations • BLAS level-1,2,3 • Matrix representation of a linear system • linear dependencies, rank, determinant • elementary row operations (RE, RRE form) • existence and uniqueness of solution • Matrix norms • Lp norm, Frobenius Norm

  4. Computational issues in NLA • Numerical stability • Accuracy • Efficiency • run-time • storage & memory • Special systems • sparse and large

  5. Square linear systems • Methods of solving • direct methods vs iterative methods • Types of problems • dense vs. sparse • general vs. special

  6. Direct Methods • Mostly suitable for small to medium scale • Produce ‘accurate’ solution • Cost: O(n3) • Gaussian • LU factorization • Pivoting • Orthogonal factorization • Givens, Householder, SVD

  7. Iterative Methods • Suitable for large systems • Accuracy adaptive to user needs • Cost: O(kn2), k=#iterations • Stationary methods • Splitting: Gauss-Seidel, Jacobi, SOR • Evolutionary methods • GMRES, CG, • Preconditioning

  8. Sparse systems • Level of sparsity 1% or less • Dynamic data structures • Direct method • minimum fill-in; ordering • Iterative methods • pack storage, preconditioning

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