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 1.1: Introduction Significant digits, absolute and relative error, rounding, chopping, nested multiplication 3-9 (From "Significant Digits of Precision: Examples". Stop at "Horners algorithm…")
1.2: Mathematical Preliminaries Taylor's Theorem, Mean Value Theorem 20-23, 25-28
1.3: Floating-Point Representation Floating-point representation, single (has not been considered in detail) and double precision, machine epsilon, rounding, chopping 38-51
1.4: Loss of Significance Significant digits, loss of significance, range reduction 56-63
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 82-89, 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, linear convergence, regula falsi 114-121
3.2: Newton's Method Newton's method, multiplicity, nonlinear system, Jacobian matrix, quadratic convergence 125-134
3.3: Secant Method Secant method, superlinear convergence 142-144, 146 (From "the order of convergence …" to " … the convergence is superlinear.") 147
Note: Fixed Point Iterations Fixed point iteration, Banach's fixed point theorem, rate and order of convergence. Note
4: Interpolation and Numerical Differentiation 4.1: Polynomial Interpolation Interpolating polynomial, nodes, Lagrange form, cardinal polynomial, Newton form, divided differences, Neville's algorithm 153-167 170-172
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. 227-236
5.4: Gaussian Quadrature Formulas Nodes, weights, linear transformation, Gaussian quadrature rules, Legendre polynomials, integrals with singularities 239-246
6: Spline Functions 6.1: First Degree and Second Degree Splines Spline (linear), 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 263-276
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 4 311-315
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 Single-step and multi-step method, predictor-corrector scheme 347-348
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, permutation matrix 358-372 (all of 372)
8.2: Eigenvalues and Eigenvectors Eigenvalue, eigenvector, eigenspace, characteristic polynomial, multiplicity, direct method, Hermitian matrix, similar matrices 380-387
8.3: Power Method Mathematical derivation, Aitken acceleration inverse power method, shifted power method 396-403
8.4: Iterative Solutions of Linear Systems Matrix/vector-norms (note that the definition of singular values is wrong; The correct definition is given in the last sentence in the summary of section 8.2), condition number, well/ill conditioned matrix, iterative method, Richardson iteration, Jacobi method, Gauss-Seidel method, SOR method 405-417 (to "Another View of Overrelaxation")
9: Least Squares Methods and Fourier Series 9.1: Method of Least Squares Mínimization of error, linear least squares, normal equations, basis functions, linear independence 426-432
9.3: Examples of the Least-Squares Principle Inconsistent systems, modified Gram-Schmidt process 447-448 (to "Use of a Weight Function w(x)")

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 that 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.

2015-04-20, eirikhoe