Lecture-by-Lecture Outline of the course Fall2005

1. A30: General introduction, motivation for numerical computing, strategies and phylosophy of in NC, absolute and relative error, computational and propagated data error, truncation and rounding error. pp 1-12
2. S2: Rounding error (cont). Forward and backward error, examples. Condition number and properties of Condition number. Computer arithmetics. $\beta, p, L$ and $U$. Total number of numbers, OFL and UFL pp 13-18.
3. S9: Rounding, Subnormals, Inf and NaN, Floating point arithmetics, $+, -, *, .$, cancellations. Matlab introduction and Matlab demonstration of infinite series sum. pp 19-27.
4. S13 First Quiz. Linear Algebra Linear Algebraic systems, Existence and Uniqueness, simple examples, matrices and vectors: 1,2 and $\infty$ norms. Cramer rule for solving linear systems. Matlab demonstration: matrices, their inverses, transposed, condition number, norms, determinant, operations on matrices, code for implementing Cramer rule. pages 49-56
5. S16 Properties of norms, 1,2 and $\infty$ norms for vectors. Matrix norms. Condition number for a matrix. Geometrial interpretation of a condition number. Error Estimates, residual.pages 56-63
6. S20 Theory: premultiplying and postmultiplying systems of linear equations, diagonal scaling, triangular matrices, upper- and lower-diagonal matrices, forward and backward substitutions, elementary elimination matrices and their properties. pages 64-68
7. S23 Gauss eliminations and LU decomposition, examples, partial and complete pivoting, Cholesky factorisation. 68-75,78-79,84-86

8. S26 Linear Least Squares, set up of a problem, motivations and examples, overdetermined systems, Normal equations, geometric interpretation.
9. S29 Geometrical interpretation (cont), orthogonal projectors, pseudoinverse and condition number, error estimates, data fitting, augmented systems.
10. Orthogonal transformations, QR factorization, orthogonal basis, Householder transformation,
11. Givens rotations. GM-ortho-normalization.
12. Midterm?
13. Eigenvalues and Eigenvectors Eigenvalue and Eigenfunctions - definitions, examples, problem transofrmation. (direct, normalized and inverse) power iterations, deflation method.
14. O18 Nonlinear system of equations. Bisection method.
15. O21 Fixed point iterations. Newton method. Estimating the convergence rate.
16. O25 Secant method. Linear and quadratic interpolation. Inverse interpolation. Linear Fractional interpolation.
17. O28 Generalization to N dimensions. Fixed point iterations. Newton method, simplified formulation. General formulation, Jacobians, etc. Stopping criteria. Interpolation. Interpolation, general formulation. Monominal interpolation. Scaled monomials.
18. N1 Scaled monomials. Lagrange interpolation. Newton interpolation: triangular system, Incremental Newton interpolation.
19. N4 Incremental Newton interpolation. Divided differences. Orthogonal Polynomials. Legendre and Chebyshev. Intrpolating continious funcitions.Piecewise polynomial, Hermite interpolation and Cubic splines.
20. . N8 Cubic splines. Review of Splines Numerical Intergration. Introduction to numerical integration. Quadrature rules that is based on Lagrange interpolation. “Baby” quadrature $\int_a^b f(x) d x\simeq (b-a) f(a)$. Error estimate of this quadrature rule.
21. N11 Midpoint method. Error estimate for midpoint. Trapezoid method. Error Estimate for trapezoid method.
    item N15. Simpson method as a weigted average of Trapezoid and Midpoint. Method of undetermined coefficients. Composite and progressive quadratures.
22. N15 Adaptive quadratures. Gauss quadratures. Review for midterm
23. N18 ???
24. N Midterm
25. N29 Gauss quadrature. Derivation of Gauss quadratures using orthonormal polynomials. Improper and singular integrals.
26. Lecture on Numerical differentiating
27. D2 Numerical Methods for Solving ODE's Stable, unstable and Asymptotically stable ODE's. Numerical stability. Forward Euler method.
28. D6 Backward Euler method, Implicit methods, Trapezoid method, Taylor method, Runge-Kutta methods. Multi-step methods.
29. D9 Concluding remarks