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Cover -- Half-Title -- Title -- Copyright -- Contents -- Preface -- Chapter 1: Numerical Computations -- 1.1 Taylor's Theorem -- 1.2 Number Representation -- 1.3 Error Considerations -- 1.3.1 Absolute and Relative Errors -- 1.3.2 Inherent Errors -- 1.3.3 Round-off Errors -- 1.3.4 Truncation Errors -- 1.3.5 Machine Epsilon -- 1.3.6 Error Propagation -- 1.4 Error Estimation -- 1.5 General Error Formula -- 1.5.1 Function Approximation -- 1.5.2 Stability and Condition -- 1.5.3 Uncertainty in Data or Noise -- 1.6 Sequences -- 1.6.1 Linear Convergence -- 1.6.2 Quadratic Convergence -- 1.6.3 Aitken's Acceleration Formula -- 1.7 Summary -- Exercises -- Chapter 2: Linear System of Equations -- 2.1 Introduction -- 2.2 Methods of Solution -- 2.3 The Inverse of a Matrix -- 2.4 Matrix Inversion Method -- 2.4.1 Augmented Matrix -- 2.5 Gauss Elimination Method -- 2.5.1 MATLAB Program for the Gauss Elimination Method -- 2.6 Gauss-Jordan Method -- 2.6.1 MATLAB Program for the Gauss Jordan Method -- 2.7 Cholesky's Triangularization Method -- 2.8 Crout's Method -- 2.8.1 MATLAB Program for Crout's Method -- 2.9 Thomas Algorithm for Tridiagonal System -- 2.9.1 MATLAB Program for the Thomas Method for Tridiagonal Systems -- 2.10 Jacobi's Iteration Method -- 2.10.1 MATLAB Program for the Jacobi Iteration Method -- 2.11 Gauss-Seidel Iteration Method -- 2.11.1 MATLAB Program for the Gauss Seidel Method -- 2.12 Symmetric Matrix Eigenvalue Problems -- 2.12.1 The Jacobi Method -- 2.12.2 MATLAB Function for the Jacobi Method -- 2.12.3 Householder Reduction to Tridiagonal Form -- 2.12.4 Gerschgorin's Circle Theorem -- 2.12.5 Sturm Sequence -- 2.12.6 QR Method -- 2.12.7 Power Method -- 2.12.8 Inverse Power Method -- 2.13 Summary -- Exercises -- Chapter 3: Solution of Algebraic and Transcendental Equations -- 3.1 Introduction -- 3.2 Bisection Method -- 3.2.1 Error Bounds.
3.3 Method of False Position -- 3.3.1 MATLAB Program for the False Position Method -- 3.4 Newton-Raphson Method -- 3.4.1 Convergence of the Newton-Raphson Method -- 3.4.2 Rate of Convergence of the Newton-Raphson Method -- 3.4.3 MATLAB Program for the Newton Raphson Method -- 3.4.4 Modified Newton-Raphson Method -- 3.4.5 Rate of Convergence of Modified Newton-Raphson Method -- 3.5 Successive Approximation Method -- 3.5.1 Error Estimate in the Successive Approximation Method -- 3.6 Secant Method -- 3.6.1 Convergence of the Secant Method -- 3.6.2 MATLAB Program to Search for a Root of the Function f(x) in the Interval (a,b) -- 3.6.3 MATLAB Program for Secant Method -- 3.7 Muller's Method -- 3.7.1 MATLAB Program for Muller's Method -- 3.8 Chebyshev Method -- 3.9 Aitken's Δ2 Method -- 3.10 Brent's Method -- 3.10.1 MATLAB Program for Brent's Method -- 3.11 Newton Method for a System of Nonlinear Equations -- 3.12 Comparison of Iterative Methods -- 3.13 MATLAB Built-in Function: fzero -- 3.14 Summary -- Exercises -- Chapter 4: Numerical Differentiation -- 4.1 Introduction -- 4.2 Derivatives Based on Newton's Forward Integration Formula -- 4.2.1 MATLAB Program for Derivatives Based on Newton's Forward Integration Formula-Equally Spaced Points -- 4.3 Derivatives Based on Newton's Backward Interpolation Formula -- 4.4 Derivatives Based on Stirling's Interpolation Formula -- 4.5 Maxima and Minima of a Tabulated Function -- 4.6 Cubic Spline Method -- 4.7 Richardson Extrapolation -- 4.8 Differentiation of Unequally Spaced Data -- 4.9 MATLAB Built-in Functions: diff and gradient -- 4.10 Summary -- Exercises -- Chapter 5: Finite Differences and Interpolation -- 5.1 Introduction -- 5.2 Finite Difference Operators -- 5.2.1 Forward Differences -- 5.2.2 Backward Differences -- 5.2.3 Central Differences -- 5.2.4 Error Propagation in a Difference Table.
5.2.5 Properties of the Operator Δ -- 5.2.6 Difference Operators -- 5.2.7 Relation Among the Operators -- 5.2.8 Representation of a Polynomial using Factorial Notation -- 5.3 Interpolation with Equal Intervals -- 5.3.1 Missing Values -- 5.3.2 Newton's Binomial Expansion Formula -- 5.3.3 Newton's Forward Interpolation Formula -- 5.3.4 MATLAB M-file: Newtonint -- 5.3.5 Newton's Backward Interpolation Formula -- 5.3.6 Error in the Interpolation Formula -- 5.4 Interpolation with Unequal Intervals -- 5.4.1 Lagrange's Interpolating Polynomial for Equal Intervals -- 5.4.2 function yint = Lagrangeint (x,y,xx) -- 5.4.3 Lagrange's Formula for Unequal Intervals -- 5.4.4 Hermite's Interpolation Formula -- 5.4.5 Inverse Interpolation -- 5.4.6 Lagrange's Formula for Inverse Interpolation -- 5.5 Central Difference Interpolation Formulae -- 5.5.1 Gauss's Forward Interpolation Formula -- 5.5.2 Gauss Backward Interpolation Formula -- 5.5.3 Bessel's Formula -- 5.5.4 Stirling's Formula -- 5.5.5 Laplace-Everett's Formula -- 5.5.6 Selection of an Interpolation Formula -- 5.6 Divided Differences -- 5.6.1 Newton's Divided Difference Interpolation Formula -- 5.7 Cubic Spline Interpolation -- 5.8 Generalized Spline Method -- 5.8.1 Splines -- 5.8.2 Linear Splines -- 5.8.3 Quadratic Splines -- 5.8.4 Cubic Splines -- 5.8.5 End Conditions -- 5.8.6 MATLAB Built-in Function: spline -- 5.8.7 Multidimensional Interpolation -- 5.8.8 MATLAB Built-in Function: interpl -- 5.9 Summary -- Exercises -- Chapter 6: Curve Fitting, Regression, and Correlation -- Approximating Curves -- Linear Regression -- 6.1 Linear Equation -- 6.2 Curve Fitting With a Linear Equation -- 6.3 Criteria for a Best Fit -- 6.4 Linear Least-Squares Regression -- 6.5 Linear Regression Analysis -- 6.5.1 MATLAB built-in function: polyfit -- 6.5.2 MATLAB built-in function: polyval -- 6.6 Interpretation of a and b.
Assumptions in the Regression Model -- 6.7 Standard Deviation of Random Errors -- 6.8 Coefficient of Determination -- 6.9 Linear Correlation -- Properties of the Linear Correlation Coefficient r -- Explained and Unexplained Variation -- 6.10 Linearization of Nonlinear Relationships -- 6.11 Polynomial Regression -- 6.11.1 Polynomial Fit -- 6.11.2 MATLAB Built-in Functions for Polynomial Fit -- 6.12 Quantification of Error of Linear Regression -- 6.13 Multiple Linear Regression -- 6.14 Weighted Least-Squares Method -- 6.15 Orthogonal Polynomials and Least-Squares Approximation -- 6.16 Least-Squares Method for Continuous Data -- 6.17 Approximation Using Orthogonal Polynomials -- 6.18 Gram-Schmidt Orthogonalization Process -- 6.19 Fitting a Function Having a Specified Power -- 6.20 Fitting a Cubic Spring Model -- 6.21 Additional Example Problems and Solutions -- 6.22 Summary -- Exercises -- Chapter 7: Numerical Integration -- 7.1 Introduction -- 7.1.1 Relative Error -- 7.2 Newton-Cotes Closed Quadrature Formula -- 7.3 Trapezoidal Rule -- 7.3.1 Error Estimate in Trapezoidal Rule -- 7.3.2 MATLAB Functions: trapz and cumtrapz -- 7.4 Simpson's 1/3 Rule -- 7.4.1 Error Estimate in Simpson's 1/3 Rule -- 7.4.2 MATLAB Program for Simpson's Integration: simpsonint -- 7.4.3 MATLAB Built-in Functions: quad and quad1 -- 7.5 Simpson's 3/8 Rule -- 7.6 Boole's and Weddle's Rules -- 7.6.1 Boole's Rule -- 7.6.2 Weddle's Rule -- 7.7 Romberg's Integration -- 7.7.1 Richardson's Extrapolation -- 7.7.2 Romberg Integration Formula -- 7.7.3 MATLAB Program for Romberg Integration: Romberg -- 7.8 Gaussian Quadrature -- 7.8.1 Gaussian Integration Formulas -- 7.8.2 Orthogonal Polynomials -- 7.8.3 Gauss-Lagendre Quadrature -- 7.8.4 Gauss-Chebyshev Quadrature Method -- 7.8.5 Gauss-Laguerre Quadrature -- 7.8.6 Gauss-Hermite Quadrature.
7.8.7 MATLAB Programs for Gaussian Quadrature: gaussnodes and gaussquad -- 7.9 Double Integration -- 7.9.1 Trapezoidal Method -- 7.9.2 Simpson's 1/3 Rule -- 7.9.3 MATLAB Built-in Function for Double Integration: dblquad -- 7.10 Summary -- Exercises -- Chapter 8: Numerical Solution of Ordinary Differential Equations -- 8.1 Introduction -- 8.2 One-Step Methods or Single-Step Methods -- 8.2.1 Picard's Method of Successive Approximation -- 8.2.2 The Taylor's Series Method -- 8.3 Step-by-Step Methods or Marching Methods -- 8.3.1 Euler's Method -- 8.3.2 MATLAB Program for Euler's Method: euler -- 8.3.3 Modified Euler's Method -- 8.3.4 MATLAB Program for the Modified Euler's Method: modeuler -- 8.3.5 Runge-Kutta Methods -- 8.3.6 Predictor-Corrector Methods -- 8.4 MATLAB Functions for Ordinary Differential Equations: ode45, ode23, ode113, ode15s, ode23s, ode23t, ode23tb -- 8.5 System of First-order Ordinary Differential Equations -- 8.6 Initial Value Problems -- 8.6.1 The Taylor Series Method -- 8.6.2 Picard's Method -- 8.6.3 Second-Order Runge-Kutta Method -- 8.6.4 Fourth-Order Runge-Kutta Method -- 8.6.5 Euler's Formula -- 8.6.6 Modified Euler's Formula -- 8.6.7 Burlirsch-Stoer Method (Mid-Point Method) -- 8.6.8 The Runge-Kutta-Fehlberg Method -- 8.6.9 The Runge-Kutta-Butcher Method -- 8.7 Two-Point Boundary Value Problems -- 8.7.1 Finite Difference Method -- 8.7.2 Second-Order Differential Equations -- 8.7.3 The Shooting Method -- 8.8 Second-Order Initial Value Problem (IVP) -- 8.9 Second-Order Boundary Value Problem (BVP) -- 8.10 MATLAB Built-in Functions -- 8.11 Summary -- Exercises -- Chapter 9: Direct Numerical Integration Methods -- 9.1 Introduction -- 9.2 Single Degree of Freedom System -- 9.2.1 Finite Difference Method -- 9.2.2 Central Difference Method -- 9.2.3 The Runge-Kutta Method -- 9.3 Multi-degree of Freedom Systems -- 9.4 Explicit Schemes.
9.4.1 Central Difference Method.
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Applied Numerical Methods Using MATLAB
