Chapter 1 Introduction
1.2 Propagation of errors
1.2.1 Try to avoid subtracting two similar numbers.
1.2.2 Prevent numbers close to zero from being divided
1.2.3 Prevent large numbers from eating decimals
1.2.4 Simplify calculation steps and reduce the number of operations
1.3 Stability of numerical algorithms
Chapter 2 Solutions to Systems of Linear Equations
2.1 Gauss sequential elimination method
2.2 Gauss column pivot elimination method
2.3 Gauss-Jordan elimination method
2.4 LU decomposition method
2.6 Improved square root method
2.8 QR decomposition method
2.9 Behavior and error analysis of the system of equations
2.9.2 Iterative improvement
2.10 Jacobi iteration method
2.11 Gauss-Seidel iteration method
2.12 Relaxation iteration method
2.13 Convergence analysis of iterative method
Chapter 3 Interpolation of Functions
3.1 Lagrange interpolation
3.3 Hermite interpolation
3.4 Piecewise cubic Hermite interpolation
3.5 Cubic spline interpolation function
3.5.1 Squeezing spline interpolation function
3.5.2 End point curvature adjustment spline interpolation function
3.5.3 Non-nodal spline interpolation function
3.5.4 Periodic spline interpolation function
Chapter 4 Approximation of Functions
4.1 Best consistent approximation polynomial
4.2 Approximate best consistent approximation polynomial
4.3 Most square approximation polynomial
4.4 Use orthogonal polynomials for most square approximation
4.4.1 Use Legendre polynomials for most square approximation
4.4.2 Use Chebyshev polynomials for most square approximation
4.5 Least squares method for curve fitting
4.5.1 Linear least squares fitting
4.5.2 Least squares fitting using orthogonal polynomials
4.5.3 Example of nonlinear least squares fitting
4.6 Pade rational approximation
Chapter 5 Numerical Integration
5.1 Compound quadrature formula
5.1.1 Compound trapezoidal formula
5.1.2 Composite Simpson formula
5.1.3 Composite Cotes formula
5.2 Quadrature formula with variable step size
5.2.1 Trapezoidal formula with variable step size
5.2.2 Simpson’s formula with variable step size
5.2.3 Cotes formula with variable step size
5.3 Romberg integral method
5.4 Adaptive integration method
5.5 Gauss quadrature formula
5.5.1 Gauss-Legendre quadrature formula
5.5.2 Gauss-Chebyshev quadrature formula
5.5.3 Gauss-Laguerre quadrature formula
5.5.4 Gauss-Hermite quadrature formula
5.6 Gauss quadrature formula for pre-given nodes
5.6.1 Gauss-Radau quadrature formula
5.6.2 Gauss-Lobatto quadrature formula
5.7 Numerical calculation of double integrals
5.7.1 Composite Simpson formula
5.7.2 Simpson’s formula with variable step size
5.7.3 Composite Gauss formula
5.8 Numerical calculation of triple integrals
Chapter 6 Numerical Optimization
6.1 Golden section search method
6.2 Fibonacci search method
6.3 Quadratic approximation method
6.4 Cubic interpolation method
Chapter 7 Calculation of matrix eigenvalues and eigenvectors
7.1 Upper Hessenberg matrix and QR decomposition
7.1.1 Convert the matrix to the upper Hessenberg matrix
7.1.2 QR decomposition of matrix
7.2 Power method and inverse power method
7.2.2 Inverse exponentiation method
7.2.3 Shift inverse exponentiation method
7.5 QR method
7.5.1 Hessenberg’s QR method
7.5.2 QR method of origin shift
7.5.3 Dual-step QR method
Chapter 8 Finding Roots of Nonlinear Equations
8.2 Accelerated convergence of iterative methods
8.2.1 Aitken acceleration method
8.2.2 Steffensen acceleration method
8.4 Trial position method
8.5 Newton-Raphson method
8.7 Improved Newton method
Chapter 9 Numerical Solution of Nonlinear Equations
9.1 Fixed point iteration method
9.3 Modified Newton’s method
9.5 Numerical continuation method
9.6 Parametric Differentiation Method
Chapter 10 Numerical Solution to Initial Value Problem of Ordinary Differential Equations
10.1 Euler method
10.1.2 Improved Euler method
10.2 Runge-Kutta method
10.2.1 Second-order Runge-Kutta method
10.2.2 Third-order Runge-Kutta method
10.2.3 Fourth-order Runge-Kutta method
10.3 Higher-order Runge-Kutta method
10.3.1 Kutta-Nystrom fifth-order and sixth-order method
10.3.2 Huta six-level and eight-level method
10.4 Runge-Kutta-Fehlberg method
10.5 Linear multi-step method
10.6 Prediction-Correction Method
10.6.1 Fourth-order Adams prediction-correction method
10.6.2 Improved Adams fourth-order prediction-correction method
10.6.3 Hamming prediction-correction method
10.7 Multi-step method with variable step size
10.9 Numerical solutions to systems of ordinary differential equations and higher-order differential equations
10.9.1 Numerical solution of systems of ordinary differential equations
10.9.2 Numerical solutions of higher-order differential equations
Chapter 11 Numerical Solutions to Boundary Value Problems of Ordinary Differential Equations
11.1 Target practice
11.1.1 Target shooting method for linear boundary value problems
11.1.2 Targeting method for nonlinear boundary value problems
11.2 Finite difference method
11.2.1 Difference method for linear boundary value problems
11.2.2 Difference method for nonlinear boundary value problems
Chapter 12 Numerical Solution of Partial Differential Equations
12.2 Parabolic equation
12.2.1 Explicit forward Euler method
12.2.2 Implicit backward Euler methods
12.2.3 Crank-Nicholson method
12.2.4 Two-dimensional parabolic equation
12.3 Hyperbolic equations
12.3.1 One-dimensional wave equation
12.3.2 Two-dimensional wave equation
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