Intro
Preface
Contents
1 Basics of Numerical Computation
1.1 How Computers Work
1.1.1 The Central Processing Unit
1.1.2 Code and Data
1.1.3 On Being Correct
1.1.4 On Being Efficient
1.1.5 Recursive Algorithms and Induction
1.1.6 Working in Groups: Parallel Computing
1.1.7 BLAS and LAPACK
Exercises
1.2 Programming Languages
1.2.1 MATLABTM
1.2.2 Julia
1.2.3 Python
1.2.4 C/C++ and Java
1.2.5 Fortran
Exercises
1.3 Floating Point Arithmetic
1.3.1 The IEEE Standards
1.3.2 Correctly Rounded Arithmetic

1.3.3 Future of Floating Point Arithmetic
Exercises
1.4 When Things Go Wrong
1.4.1 Underflow and Overflow
1.4.2 Subtracting Nearly Equal Quantities
1.4.3 Numerical Instability
1.4.4 Adding Many Numbers
Exercises
1.5 Measuring: Norms
1.5.1 What Is a Norm?
1.5.2 Norms of Functions
Exercises
1.6 Taylor Series and Taylor Polynomials
1.6.1 Taylor Series in One Variable
1.6.2 Taylor Series and Polynomials in More than One Variable
1.6.3 Vector-Valued Functions
Exercises
Project
2 Computing with Matrices and Vectors
2.1 Solving Linear Systems

2.1.1 Gaussian Elimination
2.1.2 LU Factorization
2.1.3 Errors in Solving Linear Systems
2.1.4 Pivoting and PA=LU
2.1.5 Variants of LU Factorization
Exercises
2.2 Least Squares Problems
2.2.1 The Normal Equations
2.2.2 QR Factorization
Exercises
2.3 Sparse Matrices
2.3.1 Tridiagonal Matrices
2.3.2 Data Structures for Sparse Matrices
2.3.3 Graph Models of Sparse Factorization
2.3.4 Unsymmetric Factorizations
Exercises
2.4 Iterations
2.4.1 Classical Iterations
2.4.2 Conjugate Gradients and Krylov Subspaces

2.4.3 Non-symmetric Krylov Subspace Methods
Exercises
2.5 Eigenvalues and Eigenvectors
2.5.1 The Power Method & Google
2.5.2 Schur Decomposition
2.5.3 The QR Algorithm
2.5.4 Singular Value Decomposition
2.5.5 The Lanczos and Arnoldi Methods
Exercises
3 Solving nonlinear equations
3.1 Bisection method
3.1.1 Convergence
3.1.2 Robustness and reliability
Exercises
3.2 Fixed-point iteration
3.2.1 Convergence
3.2.2 Robustness and reliability
3.2.3 Multivariate fixed-point iterations
Exercises
3.3 Newton's method

3.3.1 Convergence of Newton's method
3.3.2 Reliability of Newton's method
3.3.3 Variant: Guarded Newton method
3.3.4 Variant: Multivariate Newton method
Exercises
3.4 Secant and hybrid methods
3.4.1 Convenience: Secant method
3.4.2 Regula Falsi
3.4.3 Hybrid methods: Dekker's and Brent's methods
Exercises
3.5 Continuation methods
3.5.1 Following paths
3.5.2 Numerical methods to follow paths
Exercises
Project
4 Approximations and Interpolation
4.1 Interpolation-Polynomials
4.1.1 Polynomial Interpolation in One Variable