Timothy Sauer earned his Ph.D. in mathematics at the University of California–Berkeley in 1982, and is currently a professor at George Mason University. He has published articles on a wide range of topics in applied mathematics, including dynamical systems, computational mathematics, and mathematical biology.

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

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 of root-finding

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

2.1 Gaussian Elimination

2.1.1 Naive Gaussian elimination

2.1.2 Operation counts

2.2 The LU Factorization

2.2.1 Matrix form of Gaussian elimination

2.2.2 Back substitution with the LU factorization

2.2.3 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

2.5 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

2.6 Methods for symmetric positive-definite matrices

2.6.1 Symmetric positive-definite matrices

2.6.2 Cholesky factorization

2.6.3 Conjugate Gradient Method

2.6.4 Preconditioning

2.7 Nonlinear Systems of Equations

2.7.1 Multivariate Newton’s Method

2.7.2 Broyden’s Method

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 Bézier Curves

4.1 Least Squares and the Normal Equations

4.1.1 Inconsistent systems of equations

4.1.2 Fitting models to data

4.1.3 Conditioning of least squares

4.2 A Survey of 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 Modified Gram—Schmidt orthogonalization

4.3.3 Householder reflectors

4.4 Generalized Minimum Residual (GMRES) Method

4.4.1 Krylov methods

4.4.2 Preconditioned GMRES

4.5 Nonlinear Least Squares

4.5.1 Gauss—Newton Method

4.5.2 Models with nonlinear parameters

4.5.3 The Levenberg—Marquardt Method

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 Trapezoid Rule

5.2.2 Simpson’s Rule

5.2.3 Composite Newton—Cotes formulas

5.2.4 Open Newton—Cotes Methods

5.3 Romberg Integration

5.4 Adaptive Quadrature

5.5 Gaussian Quadrature

6.1 Initial Value Problems

6.1.1 Euler’s Method

6.1.2 Existence, uniqueness, and continuity for solutions

6.1.3 First-order linear equations

6.2 Analysis of IVP Solvers

6.2.1 Local and global truncation error

6.2.2 The explicit Trapezoid Method

6.2.3 Taylor Methods

6.3 Systems of Ordinary Differential Equations

6.3.1 Higher order equations

6.3.2 Computer simulation: the pendulum

6.3.3 Computer simulation: orbital mechanics

6.4 Runge—Kutta Methods and Applications

6.4.1 The Runge—Kutta family

6.4.2 Computer simulation: the Hodgkin—Huxley neuron

6.4.3 Computer simulation: the Lorenz equations

6.5 Variable Step-Size Methods

6.5.1 Embedded Runge—Kutta pairs

6.5.2 Order 4/5 methods

6.6 Implicit Methods and Stiff Equations

6.7 Multistep Methods

6.7.1 Generating multistep methods

6.7.2 Explicit multistep methods

6.7.3 Implicit multistep methods

7.1 Shooting Method

7.1.1 Solutions of boundary value problems

7.1.2 Shooting Method implementation

7.2 Finite Difference Methods

7.2.1 Linear boundary value problems

7.2.2 Nonlinear boundary value problems

7.3 Collocation and the Finite Element Method

7.3.1 Collocation

7.3.2 Finite elements and the Galerkin Method

8.1 Parabolic Equations

8.1.1 Forward Difference Method

8.1.2 Stability analysis of Forward Difference Method

8.1.3 Backward Difference Method

8.1.4 Crank—Nicolson Method

8.2 Hyperbolic Equations

8.2.1 The wave equation

8.2.2 The CFL condition

8.3 Elliptic Equations

8.3.1 Finite Difference Method for elliptic equations

8.3.2 Finite Element Method for elliptic equations

8.4 Nonlinear partial differential equations

8.4.1 Implicit Newton solver

8.4.2 Nonlinear equations in two space dimensions

9.1 Random Numbers

9.1.1 Pseudo-random numbers

9.1.2 Exponential and normal random numbers

9.2 Monte Carlo Simulation

9.2.1 Power laws for Monte Carlo estimation

9.2.2 Quasi-random numbers

9.3 Discrete and Continuous Brownian Motion

9.3.1 Random walks

9.3.2 Continuous Brownian motion

9.4 Stochastic Differential Equations

9.4.1 Adding noise to differential equations

9.4.2 Numerical methods for SDEs

10.1 The Fourier Transform

10.1.1 Complex arithmetic

10.1.2 Discrete Fourier Transform

10.1.3 The Fast Fourier Transform

10.2 Trigonometric Interpolation

10.2.1 The DFT Interpolation Theorem

10.2.2 Efficient evaluation of trigonometric functions

10.3 The FFT and Signal Processing

10.3.1 Orthogonality and interpolation

10.3.2 Least squares fitting with trigonometric functions

10.3.3 Sound, noise, and filtering

11.1 The Discrete Cosine Transform

11.1.1 One-dimensional DCT

11.1.2 The DCT and least squares approximation

11.2 Two-Dimensional DCT and Image Compression

11.2.1 Two-dimensional DCT

11.2.2 Image compression

11.2.3 Quantization

11.3 Huffman Coding

11.3.1 Information theory and coding

11.3.2 Huffman coding for the JPEG format

11.4 Modified DCT and Audio Compression

11.4.1 Modified Discrete Cosine Transform

11.4.2 Bit quantization

12.1 Power Iteration Methods

12.1.1 Power Iteration

12.1.2 Convergence of Power Iteration

12.1.3 Inverse Power Iteration

12.1.4 Rayleigh Quotient Iteration

12.2 QR Algorithm

12.2.1 Simultaneous iteration

12.2.2 Real Schur form and the QR algorithm

12.2.3 Upper Hessenberg form

12.3 Singular Value Decomposition

12.3.1 Finding the SVD in general

12.3.2 Special case: symmetric matrices

12.4 Applications of the SVD

12.4.1 Properties of the SVD

12.4.2 Dimension reduction

12.4.3 Compression

12.4.4 Calculating the SVD

13.1 Unconstrained Optimization without Derivatives

13.1.1 Golden Section Search

13.1.2 Successive parabolic interpolation

13.1.3 Nelder—Mead search

13.2 Unconstrained Optimization with Derivatives

13.2.1 Newton’s Method

13.2.2 Steepest Descent

13.2.3 Conjugate Gradient Search

A.1 Matrix Fundamentals

A.2 Systems of linear equations

A.3 Block Multiplication

A.4 Eigenvalues and Eigenvectors

A.5 Symmetric Matrices

A.6 Vector Calculus

B.1 Starting MATLAB

B.2 Graphics

B.3 Programming in MATLAB

B.4 Flow Control

B.5 Functions

B.6 Matrix Operations

B.7 Animation and Movies

ANSWERS TO SELECTED EXERCISES

BIBLIOGRAPHY

INDEX

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