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.