# 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