CHAPTER 0. Fundamentals

0.1 Evaluating a polynomial

0.2 Binary numbers

0.2.1 Decimal to binary

0.2.2 Binary to decimal

0.3 Floating point representation of real numbers

0.3.1 Floating point formats

0.3.2 Machine representation

0.3.3 Addition of floating point numbers

0.4 Loss of significance

0.5 Review of calculus (independent reading)

0.6 Software and Further Reading (independent reading)

CHAPTER 1. Solving Equations

1.1 The Bisection Method

1.1.1 Bracketing a root

1.1.2 How accurate and how fast?

1.2 Fixed point iteration

1.2.1 Fixed points of a function

1.2.2 Geometry of Fixed Point Iteration

1.2.3 Linear Convergence of Fixed Point Iteration

1.2.4 Stopping criteria

1.3 Limits of accuracy

1.3.1 Forward and backward error

1.3.2 The Wilkinson polynomial

1.3.3 Sensitivity and error magnification

1.4 Newton's Method

1.4.1 Quadratic convergence of Newton's method

1.4.2 Linear convergence of Newton's method

1.5 Root-finding without derivatives

1.5.1 Secant method and variants

1.5.2 Brent's Method (optional, independent reading)

REALITY CHECK 1: Kinematics of the Stewart platform (independent reading)

1.6 Software and Further Reading (independent reading)

CHAPTER 2. Systems of Equations

2.1 Gaussian elimination

2.1.1 Naive Gaussian elimination

2.1.2 Operation counts

2.2 The LU factorization

2.2.1 Backsolving with the LU factorization

2.2.2 Complexity of the LU factorization

2.3 Sources of error

2.3.1 Error magnification and condition number

2.3.2 Swamping

2.4 The PA=LU factorization

2.4.1 Partial pivoting

2.4.2 Permutation matrices

2.4.3 PA = LU factorization

2.4.4 Matlab commands for linear systems

2.5 Iterative methods

2.5.1 Jacobi Method

2.5.2 Gauss-Seidel Method and SOR

2.5.3 Convergence of iterative methods

2.5.4 Sparse matrix computations

REALITY CHECK 2: The Euler-Bernoulli Beam (optional, independent reading)

2.6 Conjugate Gradient Method (optional, independent reading)

2.6.1 Positive-definite matrices (optional, independent reading)

2.6.2 Conjugate Gradient Method (optional, independent reading)

2.7 Nonlinear systems of equations (optional, independent reading)

2.7.1 Multivariate Newton's method (optional, independent reading)

2.7.2 Broyden's method (optional, independent reading)

2.8 Software and Further Reading (independent reading)

CHAPTER 3. Interpolation

3.1 Data and interpolating functions

3.1.1 Lagrange interpolation

3.1.2 Newton's divided differences

3.1.3 How many degree d polynomials pass through n points?

3.1.4 Code for interpolation

3.1.5 Representing functions by approximating polynomials

3.2 Interpolation error

3.2.1 Interpolation error formula

3.2.2 Proof of Newton form and error formula

3.2.3 Runge phenomenon

3.3 Chebyshev interpolation

3.3.1 Chebyshev's Theorem

3.3.2 Chebyshev polynomials

3.3.3 Change of interval

3.4 Cubic splines

3.4.1 Properties of splines

3.4.2 Endpoint conditions

3.5 B´ezier curves (optional, independent reading)

REALITY CHECK 3: Constructing fonts from B´ezier splines (optional, independent reading)

3.6 Software and Further Reading (independent reading)

CHAPTER 4. Least Squares

4.1 Inconsistent systems of equations

4.2 Linear and nonlinear models

4.2.1 Periodic data

4.2.2 Data linearization

4.3 QR factorization

4.3.1 Gram-Schmidt orthogonalization and least squares

4.3.2 Householder reflectors

4.4 Nonlinear least squares (optional, independent reading)

4.4.1 Gauss-Newton method (optional, independent reading)

4.4.2 Models with nonlinear coefficients (optional, independent reading)

REALITY CHECK 4: GPS, conditioning and nonlinear least squares (optional, independent reading)

4.5 Software and Further Reading (independent reading)

CHAPTER 5. Numerical Differentiation and Integration

5.1 Numerical differentiation

5.1.1 Finite difference formulas

5.1.2 Rounding error

5.1.3 Extrapolation

5.1.4 Symbolic differentiation and integration

5.2 Newton-Cotes formulas for numerical integration

5.2.1 Three simple integrals for Newton-Cotes Formulas

5.2.2 Trapezoid rule

5.2.3 Simpson's Rule

5.2.4 Composite Newton-Cotes Formulas

5.2.5 Open Newton-Cotes methods

5.3 Romberg integration

5.4 Adaptive quadrature

5.5 Gaussian quadrature

REALITY CHECK 5: Motion control in computer-aided modelling (optional, independent reading)

5.6 Software and Further Reading (independent reading)

APPENDIX B. Introduction to MATLAB (independent reading)