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| Preface | |
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| Acknowledgments | |
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| Author | |
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| Background and Introduction | |
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| Background | |
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| Differential Equations | |
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| Linear First-Order ODEs | |
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| Second-Order ODEs with Constant Coefficients | |
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| Matrix Analysis | |
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| Matrix Operations | |
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| Special Matrices | |
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| Determinant of a Matrix | |
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| Inverse of a Matrix | |
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| Matrix Eigenvalue Problem | |
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| Solving the Eigenvalue Problem | |
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| Similarity Transformation | |
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| Introduction to Numerical Methods | |
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| Errors and Approximations | |
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| Computational Errors | |
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| Binary and Hexadecimal Numbers | |
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| Floating Point and Rounding Errors | |
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| Absolute and Relative Errors | |
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| Iterative Methods | |
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| A Fundamental Iterative Method | |
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| Rate of Convergence of an Iterative Method | |
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| Problem Set | |
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| Introduction to MATLAB� | |
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| MATLAB� Built-in Functions | |
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| Rounding Commands | |
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| Relational Operators | |
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| Format Options | |
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| Vectors and Matrices | |
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| Linspace | |
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| Matrices | |
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| Determinant, Transpose, and Inverse | |
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| Slash Operators | |
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| Element-by-Element Operations | |
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| User-Defined Functions and Script Files | |
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| Setting Default Values for Input Variables | |
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| Creating Script Files | |
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| Anonymous Functions | |
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| Inline | |
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| Program Flow Control | |
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| for Loop | |
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| if Command | |
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| while Loop | |
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| Displaying Formatted Data | |
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| Symbolic Toolbox | |
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| Differentiation | |
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| Integration | |
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| Differential Equations | |
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| Plotting | |
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| Subplot | |
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| Plotting Analytical Expressions | |
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| Multiple Plots | |
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| Problem Set | |
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| Solution of Equations of a Single Variable | |
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| Numerical Solution of Equations | |
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| Bisection Method | |
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| MATLAB� Built-in Function fzero | |
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| Regula Falsi Method (Method of False Position) | |
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| Modified Regula Falsi Method | |
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| Fixed-Point Method | |
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| Selection of a Suitable Iteration Function | |
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| A Note on Convergence | |
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| Rate of Convergence of the Fixed-Point Iteration | |
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| Newton's Method (Newton-Raphson Method) | |
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| Rate of Convergence of Newton's Method | |
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| A Few Notes on Newton's Method | |
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| Modified Newton's Method for Roots with Multiplicity 2 or Higher | |
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| Secant Method | |
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| Rate of Convergence of Secant Method | |
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| A Few Notes on Secant Method | |
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| Equations with Several Roots | |
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| Finding Zeros to the Right of a Specified Point | |
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| Finding Zeros on Two Sides of a Specified Point | |
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| Using f zero to Find Several Roots | |
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| Points of Discontinuity Mistaken for Roots | |
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| Problem Set | |
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| Solution of Systems of Equations | |
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| Linear Systems of Equations | |
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| Numerical Solution of Linear Systems | |
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| Gauss Elimination Method | |
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| Choosing the Pivot Row: Partial Pivoting with Row Scaling | |
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| Permutation Matrices | |
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| Counting the Number of Operations | |
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| Elimination | |
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| Back Substitution | |
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| Tridiagonal Systems | |
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| Thomas Method | |
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| MATLAB� Built-in Function "\" | |
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| LU Factorization Methods | |
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| Doolittle Factorization | |
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| Doolittle's Method to Solve a Linear System | |
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| Operations Count | |
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| Cholesky Factorization | |
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| Cholesky's Method to Solve a Linear System | |
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| Operations Count | |
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| MATLAB� Built-in Functions lu and chol | |
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| Iterative Solution of Linear Systems | |
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| Vector Norms | |
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| Matrix Norms | |
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| Compatibility of Vector and Matrix Norms | |
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| General Iterative Method | |
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| Convergence of the General Iterative Method | |
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| Jacobi Iteration Method | |
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| Convergence of the Jacobi Iteration Method | |
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| Gauss-Seidel Iteration Method | |
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| Convergence of the Gauss-Seidel Iteration Method | |
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| Indirect Methods versus Direct Methods for Large Systems | |
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| Ill-Conditioning and Error Analysis | |
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| Condition Number | |
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| Ill-Conditioning | |
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| Indicators of Ill-Conditioning | |
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| Computational Error | |
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| Consequences of Ill-Conditioning | |
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| Effects of Parameter Changes on the Solution | |
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| Systems of NonLinear Equations | |
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| Newton's Method for a System of Nonlinear Equations | |
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| Method for Solving a System of Two Nonlinear Equations | |
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| Method for Solving a System of n Nonlinear Equations | |
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| Convergence of Newton's Method | |
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| Fixed-Point Iteration Method for a System of Nonlinear Equations | |
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| Convergence of the Fixed-Point Iteration Method | |
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| Problem Set | |
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| Curve Fitting (Approximation) and Interpolation | |
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| Least-Squares Regression | |
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| Linear Regression | |
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| Deciding a "Best" Fit Criterion | |
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| Linear Least-Squares Regression | |
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| Linearization of Nonlinear Data | |
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| Exponential Function | |
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| Power Function | |
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| Saturation Function | |
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| Polynomial Regression | |
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| Quadratic Least-Squares Regression | |
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| Cubic Least-Squares Regression | |
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| MATLAB� Built-in Functions polyfit and polyval | |
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| Polynomial Interpolation | |
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| Lagrange Interpolating Polynomials | |
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| Drawbacks of Lagrange Interpolation | |
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| Newton Divided-Difference Interpolating Polynomials | |
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| Special Case: Equally Spaced Data | |
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| Newton Forward-Difference Interpolating Polynomials | |
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| Spline Interpolation | |
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| Linear Splines | |
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| Quadratic Splines | |
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| Function Values at the Endpoints | |
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| Function Values at the Interior Knots | |
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| First Derivatives at the Interior Knots | |
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| Second Derivative at the Left Endpoint is Zero | |
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| Cubic Splines | |
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| Construction of Cubic Splines: Clamped Boundary Conditions | |
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| Construction of Cubic Splines: Free Boundary Conditions | |
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| MATLAB� Built-in Functions interp1 and spline | |
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| Boundary Conditions | |
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| Interactive Curve Fitting and Interpolation in MATLAB� | |
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| Fourier Approximation and Interpolation | |
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| Sinusoidal Curve Fitting | |
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| Linear Transformation of Data | |
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| Discrete Fourier Transform | |
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| Fast Fourier Transform | |
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| Sande-Tukey Algorithm (N = 2<sup>p</sup>, p = integer) | |
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| Cooley-Tukey Algorithm (N=2<sup>p</sup>, p = integer) | |
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| MATLAB� Built-in Function fft | |
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| Interpolation Using fft | |
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| Problem Set | |
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| Numerical Differentiation and Integration | |
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| Numerical Differentiation | |
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| Finite-Difference Formulas for Numerical Differentiation | |
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| Finite-Difference Formulas for the First Derivative | |
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| Two-Point Backward Difference Formula | |
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| Two-Point Forward Difference Formula | |
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| Two-Point Central Difference Formula | |
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| Three-Point Backward Difference Formula | |
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| Three-Point Forward Difference Formula | |
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| Finite-Difference Formulas for the Second Derivative | |
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| Three-Point Backward Difference Formula | |
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| Three-Point Forward Difference Formula | |
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| Three-Point Central Difference Formula | |
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| Summary of Finite-Difference Formulas for First to Fourth Derivatives | |
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| Estimate Improvement: Richardson's Extrapolation | |
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| Derivative Estimates for Nonevenly Spaced Data | |
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| MATLAB� Built-in Functions dif f and polyder | |
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| Numerical Integration: Newton-Cotes Formulas | |
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| Newton-Cotes Formulas | |
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| Rectangular Rule | |
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| Composite Rectangular Rule | |
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| Error Estimate for Composite Rectangular Rule | |
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| Trapezoidal Rule | |
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| Composite Trapezoidal Rule | |
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| Error Estimate for Composite Trapezoidal Rule | |
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| Simpson's Rules | |
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| Simpson's 1/3 Rule | |
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| Composite Simpson's 1/3 Rule | |
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| Error Estimate for Composite Simpson's 1/3 Rule | |
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| Simpson's 3/8 Rule | |
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| Composite Simpson's 3/8 Rule | |
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| Error Estimate for Composite Simpson's 3/8 Rule | |
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| MATLAB� Built-in Functions quad and trapz | |
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| Numerical Integration of Analytical Functions: Romberg Integration, Gaussian Quadrature | |
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| Richardson's Extrapolation, Romberg Integration | |
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| Richardson's Extrapolation | |
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| Romberg Integration | |
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| Gaussian Quadrature | |
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| Improper Integrals | |
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| Problem Set | |
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| Numerical Solution of Initial-Value Problems | |
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| One-Step Methods | |
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| Euler's Method | |
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| Error Analysis for Euler's Method | |
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| Calculation of Local and Global Truncation Errors | |
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| Higher-Order Taylor Methods | |
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| Runge-Kutta Methods | |
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| Second-Order Runge-Kutta Methods | |
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| Improved Euler's Method | |
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| Heun's Method | |
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| Ralston's Method | |
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| Graphical Representation of Heun's Method | |
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| Third-Order Runge-Kutta Methods | |
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| Classical RK3 Method | |
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| Heun's RK3 Method | |
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| Fourth-Order Runge-Kutta Methods | |
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| Classical RK4 Method | |
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| Higher-Order Runge-Kutta Methods | |
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| Runge-Kutta-Fehlberg Method | |
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| Multistep Methods | |
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| Adams-Bashforth Method | |
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| Second-Order Adams-Bashforth Formula | |
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| Third-Order Adams-Bashforth Formula | |
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| Fourth-Order Adams-Bashforth Formula | |
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| Adams-Moulton Method | |
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| Second-Order Adams-Moulton Formula | |
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| Third-Order Adams-Moulton Formula | |
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| Fourth-Order Adams-Moulton Formula | |
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| Predictor-Corrector Methods | |
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| Heun's Predictor-Corrector Method | |
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| Adams-Bashforth-Moulton Predictor-Corrector Method | |
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| Systems of Ordinary Differential Equations | |
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| Transformation into a System of First-Order ODEs | |
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| State Variables | |
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| Notation | |
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| State-Variable Equations | |
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| Numerical Solution of a System of First-Order ODEs | |
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| Euler's Method for Systems | |
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| Heun's Method for Systems | |
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| Classical RK4 Method for Systems | |
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| Stability | |
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| Euler's Method | |
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| Euler's Implicit Method | |
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| Stiff Differential Equations | |
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| MATLAB� Built-in Functions for Initial-Value Problems | |
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| Nonstiff Equations | |
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| Single First-Order IVP | |
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| Setting ODE Solver Options | |
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| System of First-Order IVPs | |
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| Stiff Equations | |
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| Problem Set | |
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| Numerical Solution of Boundary-Value Problems | |
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| Shooting Method | |
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| Finite-Difference Method | |
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| BVPs with Mixed Boundary Conditions | |
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| MATLAB� Built-in Function bvp4c for BVPs | |
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| Second-Order BVP | |
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| Problem Set | |
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| Matrix Eigenvalue Problem | |
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| Power Method: Estimation of the Dominant Eigenvalue | |
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| Inverse Power Method: Estimation of the Smallest Eigenvalue | |
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| Shifted Inverse Power Method: Estimation of the Eigenvalue Nearest a Specified Value | |
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| Shifted Power Method | |
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| MATLAB� Built-in Function eig | |
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| Deflation Methods | |
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| Wielandt's Deflation Method | |
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| Deflation Process | |
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| Householder Tridiagonalization and QR Factorization Methods | |
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| Householder's Tridiagonalization Method (Symmetric Matrices) | |
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| Determination of Symmetric Orthogonal P<sub>k</sub> (k = 1,2,…, n-2) | |
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| QR Factorization Method | |
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| Determination of Q<sub>k</sub> and R<sub>k</sub> Matrices | |
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| Structure of L<sub>k</sub> (k = 2,3,…, n) | |
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| MATLAB� Built-in Function qr | |
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| Transformation to Hessenberg Form (Nonsymmetric Matrices) | |
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| Problem Set | |
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| Numerical Solution of Partial Differential Equations | |
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| Introduction | |
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| Elliptic PDEs | |
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| Dirichlet Problem | |
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| Alternating Direction-Implicit Methods | |
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| Peaceman-Rachford Alternating Direction-Implicit Method | |
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| Neumann Problem | |
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| Existence of Solution for Neumann Problem | |
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| Mixed Problem | |
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| More Complex Regions | |
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| Parabolic PDEs | |
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| Finite-Difference (FD) Method | |
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| Stability and Convergence of the FD Method | |
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| Crank-Nicolson (CN) Method | |
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| CN Method versus FD Method | |
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| Hyperbolic PDEs | |
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| Starting the Procedure | |
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| Problem Set | |
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| Bibliography | |
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| Index | |