Curriculum

In addition to the following list, the exercises that were given are also part of the curriculum.

There will be no questions concerning programming in the exam.

Chapter Section Keywords Pages
1: Mathematical Preliminaries and Floating-Point Representation 1.2: Mathematical Preliminaries Taylor's Theorem, Mean Value Theorem 20-23, 25-28
1.3: Floating-Point Representation Floating-point representation, single and double precision, machine epsilon, rounding, chopping 38-51
1.4: Loss of Significance Significant digits, range reduction 56-58, 60-63
2: Linear Systems 2.1: Naive Gaussian Elimination Linear system, naive Gaussian elimination, pivot, forward elimination, back substitution 69-78
2.2: Gaussian Elimination with Scaled Partial Pivoting Pivoting (partial, scaled partial, complete), index vector, long operation count, stability 82-87, 92-94
2.3: Tridiagonal and Banded Systems Banded matrix, diagonal matrix, tridiagonal matrix, (strict) diagonal dominance 103-106
3: Nonlinear Equations 3.1: Bisection Method Root/zero, bisection method, regula falsi 114-121
3.2: Newton's Method Newton's method, multiplicity, nonlinear system, Jacobian matrix, quadratic and linear convergence 125-134
3.3: Secant Method Secant method 142-144, 147
Note: Fixed Point Iterations Fixed point iteration, matrix norms, Banach's fixed point theorem, applications to the solution of equations, convergence of Newton's method Note
4: Interpolation and Numerical Differentiation 4.1: Polynomial Interpolation Interpolating polynomial, nodes, Lagrange form, cardinal polynomial, Newton form, divided differences 153-167
4.2: Errors in Polynomial Interpolation Runge function, interpolation error, Chebyshev nodes 178-185
4.3: Estimating Derivatives and Richardson Extrapolation Truncation error, forward difference, central difference, Richardson extrapolation, computational noise 187-197
5: Numerical Integration 5.1: Trapezoid Method Definite/indefinite integral, antiderivative, Fundamental Theorem of Calculus, trapezoid rule (basic, composite), recursive trapezoid formula 201-211
5.2: Romberg Algorithm Romberg algorithm, Euler-Maclaurin formula, general extrapolation (note that this is basically Richardson extrapolation in disguise) 217-224
5.3: Simpson's Rules and Newton-Cotes Rules Method of undetermined coefficients, Simpson's rule (basic, composite, adaptive), Newton-Cotes rules. 227-231, 235-236
5.4: Gaussian Quadrature Formulas Nodes, weights, linear transformation, Gaussian quadrature rules, Legendre polynomials 239-245
6: Spline Functions 6.1: First Degree and Second Degree Splines Spline (linear, quadratic), knots, interpolating spline, modulus of continuity 252-256
6.2: Natural Cubic Splines Spline (degree k), interpolation conditions, continuity conditions, natural cubic spline, smoothness of natural cubic splines 263-266, 274-276
7: Initial Value Problems 7.1 Taylor Series Methods Ordinary differential equation (ODE), initial value problem (IVP), solution, implicit/explicit formulas, vector field, Taylor series methods, Euler's method, order, local truncation error, accumulated global error, roundoff error 299-308
7.2: Runge-Kutta Methods Taylor series of f(x,y), Runge-Kutta methods of order 2 and 4 311-316
7.3: Adaptive Runge-Kutta and Multistep Methods Adams-Bashforth-Moulton Formulas 324-325
7.4: Methods for First and Higher Order Systems Coupled/uncoupled systems, systems of ODEs, vector notation, autonomous/nonautonomous ODE, higher order differential equation, transformation into autonomous and first order form 331-342
7.5: Adams-Bashforth-Moulton Methods A Predictor-Corrector Scheme, Stiff ODEs (implicit Euler method) 347-348, 353-354
8: More on Linear Systems 8.1: Matrix Factorizations LU factorization, elementary matrix, lower/upper triangular matrix, Doolittle factorization, LDLT factorization, Cholesky Factorization, symmetric positive definite (SPD) matrix 358-371, Note
8.2: Eigenvalues and Eigenvectors Eigenvalue, eigenvector, eigenspace, characteristic polynomial, multiplicity, direct method, Hermitian matrix, similar matrices 380-385
8.3: Power Method Mathematical derivation, inverse power method, shifted power method 396-399, 400-403
8.4: Iterative Solutions of Linear Systems Matrix/vector-norms (note that the definition of singular values is wrong; the correct one is in my notes on fixed point iterations), condition number, well/ill conditioned matrix, iterative method, Richardson iteration, Jacobi method, Gauss-Seidel method, SOR method 405-417
9: Least Squares Methods and Fourier Series 9.1: Method of Least Squares Minmínimization of error, linear least squares, normal equations, basis functions, linear independence 427-432
9.3: Examples of the Least-Squares Principle Inconsistent systems, modified Gram-Schmidt process 447-448

Note: If nothing else is stated, a page number that falls in the middle of a section has the following interpretation: You should start or stop at the subsection which begins on that page depending on whether the number is the initial or final one in a pair respectively. If no subsection starts on the page of a final page number, read the entire page. All page numbers refer to the 7th edition.

2014-04-08, Markus Grasmair