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 –L p 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(n 3 ) Gaussian –LU factorization –Pivoting Orthogonal factorization –Givens, Householder, SVD

7. Iterative Methods Suitable for large systems Accuracy adaptive to user needs Cost: O(kn 2 ), 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