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| Preface | |
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| Foundations | |
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| Introductory Examples | |
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| Nonlinear Equations | |
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| Linear Systems | |
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| Numerical Integration | |
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| Useful Background | |
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| Results from Calculus | |
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| Results from Linear Algebra | |
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| A Little Information about Computers | |
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| Some Basic Issues | |
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| Error | |
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| Convergence | |
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| Getting Better Results | |
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| Using MATLAB | |
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| Command Window Computations | |
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| M-Files | |
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| Programming in MATLAB | |
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| Matrix Multiplication | |
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| Chapter Wrap-Up | |
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| Functions of One Variable | |
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| Bisection Method | |
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| Secant-Type Methods | |
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| Regula Falsi | |
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| Secant Method | |
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| Analysis | |
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| Newton's Method | |
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| Muller's Method | |
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| Minimization | |
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| Golden-Section Search | |
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| Brent's Method | |
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| Beyond the Basics | |
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| Using MATLAB's Functions | |
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| Laguerre's Method | |
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| Zeros of a Nonlinear Function | |
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| Chapter Wrap-Up | |
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| Solving Linear Systems: Direct Methods | |
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| Gaussian Elimination | |
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| Basic Method | |
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| Row Pivoting | |
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| Gauss-Jordan | |
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| Inverse of a Matrix | |
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| Tridiagonal Systems | |
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| Further Topics | |
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| MATLAB's Methods | |
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| Condition of a Matrix | |
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| Iterative Refinement | |
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| Chapter Wrap-Up | |
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| LU and QR Factorization | |
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| LU Factorization | |
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| Using Gaussian Elimination | |
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| Direct LU Factorization | |
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| Applications | |
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| Matrix Transformations | |
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| Householder Transformation | |
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| Givens Rotations | |
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| QR Factorization | |
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| Using Householder Transformations | |
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| Using Givens Rotations | |
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| Beyond the Basics | |
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| LU Factorization with Implicit Row Pivoting | |
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| Efficient Conversion to Hessenberg Form | |
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| Using MATLAB's Functions | |
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| Chapter Wrap-Up | |
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| Eigenvalues and Eigenvectors | |
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| Power Method | |
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| Basic Power Method | |
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| Rayleigh Quotient | |
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| Shifted Power Method | |
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| Accelerating Convergence | |
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| Inverse Power Method | |
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| General Inverse Power Method | |
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| Convergence | |
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| QR Method | |
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| Basic QR Method | |
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| Better QR Method | |
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| Finding Eigenvectors | |
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| Accelerating Convergence | |
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| Further Topics | |
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| Singular Value Decomposition | |
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| MATLAB's Methods | |
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| Chapter Wrap-Up | |
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| Solving Linear Systems: Iterative Methods | |
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| Jacobi Method | |
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| Gauss-Seidel Method | |
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| Successive Over-Relaxation | |
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| Beyond the Basics | |
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| MATLAB's Built-in Functions | |
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| Conjugate Gradient Methods | |
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| GMRES | |
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| Simplex Method | |
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| Chapter Wrap-Up | |
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| Nonlinear Functions of Several Variables | |
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| Nonlinear Systems | |
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| Newton's Method | |
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| Secant Methods | |
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| Fixed-Point Iteration | |
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| Minimization | |
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| Descent Methods | |
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| Quasi-Newton Methods | |
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| Further Topics | |
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| Levenberg-Marquardt Method | |
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| Nelder-Mead Simplex Search | |
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| Chapter Wrap-Up | |
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| Interpolation | |
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| Polynomial Interpolation | |
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| Lagrange Form | |
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| Newton Form | |
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| Difficulties | |
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| Hermite Interpolation | |
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| Piecewise Polynomial Interpolation | |
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| Piecewise Linear Interpolation | |
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| Piecewise Quadratic Interpolation | |
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| Piecewise Cubic Hermite Interpolation | |
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| Cubic Spline Interpolation | |
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| Beyond the Basics | |
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| Rational-Function Interpolation | |
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| Using MATLAB's Functions | |
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| Chapter Wrap-Up | |
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| Approximation | |
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| Least-Squares Approximation | |
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| Approximation by a Straight Line | |
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| Approximation by a Parabola | |
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| General Least-Squares Approximation | |
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| Approximation for Other Functional Forms | |
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| Continuous Least-Squares Approximation | |
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| Approximation Using Powers of x | |
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| Orthogonal Polynomials | |
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| Legendre Polynomials | |
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| Chebyshev Polynomials | |
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| Function Approximation at a Point | |
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| Pade Approximation | |
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| Taylor Approximation | |
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| Further Topics | |
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| Bezier Curves | |
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| Using MATLAB's Functions | |
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| Chapter Wrap-Up | |
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| Fourier Methods | |
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| Fourier Approximation and Interpolation | |
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| Derivation | |
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| Data on Other Intervals | |
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| Radix-2 Fourier Transforms | |
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| Discrete Fourier Transform | |
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| Fast Fourier Transform | |
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| Matrix Form of FFT | |
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| Algebraic Form of FFT | |
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| Mixed-Radix FFT | |
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| Using MATLAB's Functions | |
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| Chapter Wrap-Up | |
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| Numerical Differentiation and Integration | |
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| Differentiation | |
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| First Derivatives | |
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| Higher Derivatives | |
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| Partial Derivatives | |
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| Richardson Extrapolation | |
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| Numerical Integration | |
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| Trapezoid Rule | |
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| Simpson's Rule | |
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| Newton-Cotes Open Formulas | |
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| Extrapolation Methods | |
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| Quadrature | |
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| Gaussian Quadrature | |
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| Other Gauss-Type Quadratures | |
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| MATLAB's Methods | |
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| Differentiation | |
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| Integration | |
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| Chapter Wrap-Up | |
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| Ordinary Differential Equations: Fundamentals | |
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| Euler's Method | |
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| Geometric Introduction | |
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| Approximating the Derivative | |
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| Approximating the Integral | |
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| Using Taylor Series | |
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| Runge-Kutta Methods | |
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| Second-Order Runge-Kutta Methods | |
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| Third-Order Runge-Kutta Methods | |
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| Classic Runge-Kutta Method | |
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| Fourth-Order Runge-Kutta Methods | |
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| Fifth-Order Runge-Kutta Methods | |
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| Runge-Kutta-Fehlberg Methods | |
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| Multistep Methods | |
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| Adams-Bashforth Methods | |
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| Adams-Moulton Methods | |
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| Adams Predictor-Corrector Methods | |
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| Other Predictor-Corrector Methods | |
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| Further Topics | |
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| MATLAB's Methods | |
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| Consistency and Convergence | |
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| Chapter Wrap-Up | |
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| ODE: Systems, Stiffness, Stability | |
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| Systems | |
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| Systems of Two ODE | |
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| Euler's Method for Systems | |
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| Runge-Kutta Methods for Systems | |
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| Multistep Methods for Systems | |
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| Second-Order ODE | |
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| Stiff ODE | |
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| BDF Methods | |
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| Implicit Runge-Kutta Methods | |
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| Stability | |
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| A-Stable and Stiffly Stable Methods | |
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| Stability in the Limit | |
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| Further Topics | |
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| MATLAB's Methods for Stiff ODE | |
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| Extrapolation Methods | |
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| Rosenbrock Methods | |
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| Multivalue Methods | |
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| Chapter Wrap-Up | |
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| ODE: Boundary-Value Problems | |
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| Shooting Method | |
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| Linear ODE | |
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| Nonlinear ODE | |
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| Finite-Difference Method | |
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| Linear ODE | |
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| Nonlinear ODE | |
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| Function Space Methods | |
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| Collocation | |
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| Rayleigh-Ritz | |
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| Chapter Wrap-Up | |
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| Partial Differential Equations | |
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| Heat Equation: Parabolic PDE | |
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| Explicit Method | |
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| Implicit Method | |
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| Crank-Nicolson Method | |
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| Insulated Boundary | |
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| Wave Equation: Hyperbolic PDE | |
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| Explicit Method | |
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| Implicit Method | |
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| Poisson Equation: Elliptic PDE | |
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| Finite-Element Method for Elliptic PDE | |
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| Defining the Subregions | |
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| Defining the Basis Functions | |
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| Computing the Coefficients | |
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| Using MATLAB | |
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| Chapter Wrap-Up | |
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| Bibliography | |
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| Answers | |
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| Index | |