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Solutions of Nonlinear Equations
Computer Arithmetics
Review of Calculus
The Bisection Method
Fixed Point Iteration
Newton’s, Secant, and False Position Methods
Accelerating Convergence
Horner’s Method and the Synthetic Division
Muller’s Method
Interpolation and Extrapolation
Lagrange Interpolating Polynomial
Newton’s Divided Difference Interpolating Polynomial
Gregory–Newton Forward-Difference Polynomial
Gregory–Newton Backward-Difference Polynomial
Hermite Interpolating Polynomial
Cubic Spline Interpolation
Numerical Differentiation and Integration
Numerical Differentiation
The Effect of Roundoff and Truncation Errors
Richardson’s Extrapolation
Basic Numerical Integration Rules
The Composite Midpoint Rule
The Composite Trapezoidal Rule
The Composite Simpson’s Rule
Romberg Integration for the Trapezoidal Rule
Adaptive Quadrature Methods
Matrix Computations
LU Solution of Ax = b
Cholesky Decomposition
Matrix Norms
Iterative Methods
Overdetermined Systems
Matrix Eigenvalues and Eigenvectors
The QR Decomposition
The QR algorithm
The Singular Value Decomposition
Numerical Solution of Differential Equations
Initial Value Problems
Euler’s and Improved Euler’s Method
Low-Order Explicit Runge–Kutta Methods
Convergence of Numerical Methods
Absolutely Stable Numerical Methods
Stability of Runge–Kutta methods
Embedded Pairs of Runge–Kutta methods
Multistep Predictor-Corrector Methods
Stiff Systems of Differential Equations
The Matlab ODE Suite
The Methods in the Matlab ODE Suite
The ode set Options
Nonstiff Problems of the Matlab ode demo
Stiff Problems of the Matlab ode demo
Concluding Remarks
Orthogonal polynomials
Fourier–Legendre Series
Derivation of Gaussian Quadratures
Numerical Solution of Integral Equations of the Second Kind
Formulae and Tables
Legendre Polynomials Pn(x) on [−1, 1]
Laguerre Polynomials on 0 ≤ x ∞
Fourier–Legendre Series Expansion
Exercises for Numerical Methods
Solutions to Exercises for Numerical Methods