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	1.1: 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.