A Concise Introduction to Numerical Analysis

Fundamentals
Floating Point Arithmetic
Overflow and Underflow
Absolute, Relative Error, Machine Epsilon
Forward and Backward Error Analysis
Loss of Significance
Robustness
Error Testing and Order of Convergence
Computational Complexity
Condition
Revision Exercises

Linear Systems
Simultaneous Linear Equations
Gaussian Elimination and Pivoting
LU Factorization
Cholesky Factorization
QR Factorization
The Gram–Schmidt Algorithm
Givens Rotations
Householder Reflections
Linear Least Squares
Singular Value Decomposition
Iterative Schemes and Splitting
Jacobi and Gauss–Seidel Iterations
Relaxation
Steepest Descent Method
Conjugate Gradients
Krylov Subspaces and Pre-Conditioning
Eigenvalues and Eigenvectors
The Power Method
Inverse Iteration
Deflation
Revision Exercises

Interpolation and Approximation Theory
Lagrange Form of Polynomial Interpolation
Newton Form of Polynomial Interpolation
Polynomial Best Approximations
Orthogonal polynomials
Least-Squares Polynomial Fitting
The Peano Kernel Theorem
Splines
B-Spline
Revision Exercises

Non-Linear Systems
Bisection, Regula Falsi, and Secant Method
Newton’s Method
Broyden’s Method
Householder Methods
Müller’s Method
Inverse Quadratic Interpolation
Fixed Point Iteration Theory
Mixed Methods
Revision Exercises

Numerical Integration
Mid-Point and Trapezium Rule
The Peano Kernel Theorem
Simpson’s Rule
Newton–Cotes Rules
Gaussian Quadrature
Composite Rules
Multi-Dimensional Integration
Monte Carlo Methods
Revision Exercises

ODEs
One-Step Methods
Multistep Methods, Order, and Consistency
Order Conditions
Stiffness and A-Stability
Adams Methods
Backward Differentiation Formulae
The Milne and Zadunaisky Device
Rational Methods
Runge–Kutta Methods
Revision Exercises

Numerical Differentiation
Finite Differences
Differentiation of Incomplete or Inexact Data

PDEs
Classification of PDEs
Parabolic PDEs
Elliptic PDEs
Parabolic PDEs in Two Dimensions
Hyperbolic PDEs
Spectral Methods
Finite Element Method
Revision Exercises