In addition to the following list, the exercises given are also part of the curriculum.
| 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, error vector, residual vector | 69-79 |
| 2.2: Gaussian Elimination with Scaled Partial Pivoting | Pivoting (partial, scaled partial, complete), index vector, long operation, condition number | 82-97 | |
| 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, false position method | 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 iteraton, contraction | Note | |
| 4: Interpolation and Numerical Differentiation | 4.1: Polynomial Interpolation | Interpolating polynomial, nodes, Lagrange form, cardinal polynomial, Newton form, divided differences, Neville's algorithm | 153-173 |
| 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, multidimensional integration | 201-212 |
| 5.2: Romberg Algorithm | Romberg algorithm, Euler-Maclaurin formula, general extrapolation | 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, weighted Guassian quadrature, weight function | 239-246 | |
| 6: Spline Functions | 6.1: First Degree and Second Degree Splines | Spline (linear, quadratic), knots, interpolating spline, modulus of continuity | 252-258 |
| 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 | Runge-Kutta method, two variable Taylor series | 311-316 | |
| 7.3: Adaptive Runge-Kutta and Multistep Methods | Adaptive Runge-Kutta-Fehlberg Method, automatic step size adjustment, stability, convergent/divergent solution curves | 320-324, 325-327 | |
| 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 | |
| 8: More on Linear Systems | 8.1: Matrix Factorizations | LU factorization, elementary matrix, lower/upper triangular matrix, Doolittle factorization, LDLT factorization, Crout factorization, Cholesky Factorization, symmetric positive definite (SPD) matrix, permutation matrix | 358-373 |
| 8.2: Eigenvalues and Eigenvectors | Eigenvalue, eigenvector, eigenspace, characteristic polynomial, multiplicity, direct method, Hermitian matrix, similar matrices | 380-385 | |
| 8.4: Iterative Solutions of Linear Systems | Matrix/vector-norms (l_1, l_2, l_inf), spectral radius, condiiton number, well/ill conditioned matrix, iterative method, Richardson iteration, Jacobi method, Gauss-Seidel method, SOR method, overrelaxation, sparse system | 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 |
| 9.2: Orthogonal Systems and Chebyshev Polynomials | Orthogonality, orthonormality, basis, Chebyshev polynomials, polynomial fitting, inner product, Gram-Schmidt process | 435-439, 441-443 (From equation (7) to equation (10)), 444-445 | |
| 9.3: Examples of the Least-Squares Principle | Inconsistent systems, approximation of functions on intervals with weight function | 447-449 (Skip "Modified Gram-Schmidt Process") |
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. E.g. 25-28 means start reading from the subsection "Taylor's Theorem in Terms of (x-c)" on page 25 and up to the subsection "Alternating Series" on page 28. If no subsection starts on the page of a final page number, read the entire page. All page numbers refer to the 7th edition.