Preface -- Contents -- 1 Introduction -- 1.1 Why study numerical methods? -- 1.2 Terminology -- 1.3 Convergence terminology -- 1.4 Exercises -- 2 Computer representation of numbers and roundoff error -- 2.1 Examples of the effects of roundoff error -- 2.2 Binary numbers -- 2.3 64 bit floating point numbers -- 2.3.1 Avoid adding large and small numbers -- 2.3.2 Subtracting two nearly equal numbers is bad -- 2.4 Exercises -- 3 Solving linear systems of equations -- 3.1 Linear systems of equations and solvability -- 3.2 Solving triangular systems -- 3.3 Gaussian elimination -- 3.4 The backslash operator -- 3.5 LU decomposition -- 3.6 Exercises -- 4 Finite difference methods -- 4.1 Approximating the first derivative -- 4.1.1 Forward and backward differences -- 4.1.2 Centered difference -- 4.1.3 Three point difference formulas -- 4.1.4 Further notes -- 4.2 Approximating the second derivative -- 4.3 Application: Initial value ODE’s using the forward Euler method -- 4.4 Application: Boundary value ODE's -- 4.5 Exercises -- 5 Solving nonlinear equations -- 5.1 The bisection method -- 5.2 Newton's method -- 5.3 Secant method -- 5.4 Comparing bisection, Newton, secant method -- 5.5 Combining secant and bisection and the fzero command -- 5.6 Equation solving in higher dimensions -- 5.7 Exercises -- 6 Accuracy in solving linear systems -- 6.1 Gauss-Jordan elimination and finding matrix inverses -- 6.2 Matrix and vector norms and condition number -- 6.3 Sensitivity in linear system solving -- 6.4 Exercises -- 7 Eigenvalues and eigenvectors -- 7.1 Mathematical definition -- 7.2 Power method -- 7.3 Application: Population dynamics -- 7.4 Exercises -- 8 Fitting curves to data -- 8.1 Interpolation -- 8.1.1 Interpolation by a single polynomial -- 8.1.2 Piecewise polynomial interpolation -- 8.2 Curve fitting -- 8.2.1 Line of best fit.

8.2.2 Curve of best fit -- 8.3 Exercises -- 9 Numerical integration -- 9.1 Newton-Cotes methods -- 9.2 Composite rules -- 9.3 MATLAB's integral function -- 9.4 Gauss quadrature -- 9.5 Exercises -- 10 Initial value ODEs -- 10.1 Reduction of higher order ODEs to first order -- 10.2 Common methods and derivation from integration rules -- 10.2.1 Backward Euler -- 10.2.2 Crank-Nicolson -- 10.2.3 Runge-Kutta 4 -- 10.3 Comparison of speed of implicit versus explicit solvers -- 10.4 Stability of ODE solvers -- 10.4.1 Stability of forward Euler -- 10.4.2 Stability of backward Euler -- 10.4.3 Stability of Crank-Nicolson -- 10.4.4 Stability of Runge-Kutta 4 -- 10.5 Accuracy of ODE solvers -- 10.5.1 Forward Euler -- 10.5.2 Backward Euler -- 10.5.3 Crank-Nicolson -- 10.5.4 Runge-Kutta 4 -- 10.6 Summary, general strategy, and MATLAB ODE solvers -- 10.7 Exercises -- A Getting started with Octave and MATLAB -- A.1 Basic operations -- A.2 Arrays -- A.3 Operating on arrays -- A.4 Script files -- A.5 Function files -- A.5.1 Inline functions -- A.5.2 Passing functions to other functions -- A.6 Outputting information -- A.7 Programming in MATLAB -- A.8 Plotting -- A.9 Exercises.