- 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** - S2: Rounding error (cont). Forward and backward error,
examples. Condition number and properties of Condition
number. Computer arithmetics.
and . Total number of
numbers, OFL and UFL
**pp 13-18**. - S9: Rounding, Subnormals, Inf and NaN, Floating point
arithmetics,
, cancellations. Matlab introduction
and Matlab demonstration of infinite series sum.
**pp 19-27**. - S13 First Quiz.
**Linear Algebra**Linear Algebraic systems, Existence and Uniqueness, simple examples, matrices and vectors: 1,2 and 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** - S16 Properties of norms, 1,2 and norms for vectors. Matrix
norms. Condition number for a matrix.
Geometrial interpretation of a
condition number. Error Estimates, residual.
**pages 56-63** - 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** - S23 Gauss eliminations and LU decomposition, examples, partial
and complete pivoting, Cholesky factorisation.
**68-75,78-79,84-86** - S26
**Linear Least Squares**, set up of a problem, motivations and examples, overdetermined systems, Normal equations, geometric interpretation. - S29 Geometrical interpretation (cont), orthogonal projectors, pseudoinverse and condition number, error estimates, data fitting, augmented systems.
- Orthogonal transformations, QR factorization, orthogonal basis, Householder transformation,
- Givens rotations. GM-ortho-normalization.
**Midterm?****Eigenvalues and Eigenvectors**Eigenvalue and Eigenfunctions - definitions, examples, problem transofrmation. (direct, normalized and inverse) power iterations, deflation method.- O18
**Nonlinear system of equations**. Bisection method. - O21 Fixed point iterations. Newton method. Estimating the convergence rate.
- O25 Secant method. Linear and quadratic interpolation. Inverse interpolation. Linear Fractional interpolation.
- 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. - N1 Scaled monomials. Lagrange interpolation. Newton interpolation: triangular system, Incremental Newton interpolation.
- N4 Incremental Newton interpolation. Divided differences. Orthogonal Polynomials. Legendre and Chebyshev. Intrpolating continious funcitions.Piecewise polynomial, Hermite interpolation and Cubic splines.
- . N8 Cubic splines. Review of Splines
**Numerical Intergration**. Introduction to numerical integration. Quadrature rules that is based on Lagrange interpolation. “Baby” quadrature . Error estimate of this quadrature rule. - 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. - N15 Adaptive quadratures. Gauss quadratures. Review for midterm
- N18
**???** - N
**Midterm** - N29 Gauss quadrature. Derivation of Gauss quadratures using orthonormal polynomials. Improper and singular integrals.
- Lecture on
**Numerical differentiating** - D2
**Numerical Methods for Solving ODE's**Stable, unstable and Asymptotically stable ODE's. Numerical stability. Forward Euler method. - D6 Backward Euler method, Implicit methods, Trapezoid method, Taylor method, Runge-Kutta methods. Multi-step methods.
- D9 Concluding remarks