Numerical Analysis

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.

Preface

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

0.6 Software and Further Reading

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 Newtons 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

REALITY CHECK 1: Kinematics of the Stewart platform

1.6 Software and Further Reading

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

REALITY CHECK 2: The Euler-Bernoulli Beam

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

2.8 Software and Further Reading

3. Interpolation

3.1 Data and interpolating functions

   3.1.1 Lagrange interpolation

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