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Preface | |

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Introduction | |

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Background | |

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Representation of Numbers on a Computer | |

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Errors in Numerical Solutions | |

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Round-Off Errors | |

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Truncation Errors | |

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Total Error | |

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Computers and Programming | |

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Problems | |

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Mathematical Background | |

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Background | |

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Concepts from Pre-Calculus and Calculus | |

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Vectors | |

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Operations with Vectors | |

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Matrices and Linear Algebra | |

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Operations with Matrices | |

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Special Matrices | |

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Inverse of a Matrix | |

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Properties of Matrices | |

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Determinant of a Matrix | |

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Cramer's Rule and Solution of a System of Simultaneous Linear Equations | |

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Norms | |

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Ordinary Differential Equations (ODE) | |

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Functions of Two or More Independent Variables | |

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Definition of the Partial Derivative | |

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Chain Rules | |

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The Jacobian | |

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Taylor Series Expansion of Functions | |

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Taylor Series for a Function of One Variable | |

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Taylor Series for a Function of Two Variables | |

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Problems | |

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Solving Nonlinear Equations | |

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Background | |

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Estimation of Errors in Numerical Solutions | |

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Bisection Method | |

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Regula Falsi Method | |

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Newton's Method | |

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Secant Method | |

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Fixed-Point Iteration Method | |

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Use of MATLAB Built-In Functions for Solving Nonlinear Equations | |

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The fzero Command | |

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The roots Command | |

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Equations with Multiple Solutions | |

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Systems of Nonlinear Equations | |

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Newton's Method for Solving a System of Nonlinear Equations | |

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Fixed-Point Iteration Method for Solving a System of Nonlinear Equations | |

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Problems | |

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Solving a System of Linear Equations | |

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Background | |

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Overview of Numerical Methods for Solving a System of Linear Algebraic Equations | |

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Gauss Elimination Method | |

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Potential Difficulties When Applying the Gauss Elimination Method | |

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Gauss Elimination with Pivoting | |

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Gauss-Jordan Elimination Method | |

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LU Decomposition Method | |

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LU Decomposition Using the Gauss Elimination Procedure | |

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LU Decomposition Using Crout's Method | |

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LU Decomposition with Pivoting | |

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Inverse of a Matrix | |

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Calculating the Inverse with the LU Decomposition Method | |

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Calculating the Inverse Using the Gauss-Jordan Method | |

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Iterative Methods | |

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Jacobi Iterative Method | |

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Gauss-Seidel Iterative Method | |

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Use of MATLAB Built-In Functions for Solving a System of Linear Equations | |

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Solving a System of Equations Using MATLAB's Left and Right Division | |

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Solving a System of Equations Using MATLAB's Inverse Operation | |

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MATLAB's Built-In Function for LU Decomposition | |

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Additional MATLAB Built-In Functions | |

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Tridiagonal Systems of Equations | |

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Error, Residual, Norms, and Condition Number | |

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Error and Residual | |

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Norms and Condition Number | |

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Ill-Conditioned Systems | |

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Eigenvalues and Eigenvectors | |

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The Basic Power Method | |

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The Inverse Power Method | |

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The Shifted Power Method | |

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The QR Factorization and Iteration Method | |

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Use of MATLAB Built-In Functions for Determining Eigenvalues and Eigenvectors | |

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Problems | |

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Curve Fitting and Interpolation | |

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Background | |

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Curve Fitting with a Linear Equation | |

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Measuring How Good Is a Fit | |

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Linear Least-Squares Regression | |

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Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form | |

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Curve Fitting with Quadratic and Higher-Order Polynomials | |

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Interpolation Using a Single Polynomial | |

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Lagrange Interpolating Polynomials | |

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Newton's Interpolating Polynomials | |

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Piecewise (Spline) Interpolation | |

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Linear Splines | |

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Quadratic Splines | |

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Cubic Splines | |

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Use of MATLAB Built-In Functions for Curve Fitting and Interpolation | |

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Curve Fitting with a Linear Combination of Nonlinear Functions | |

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Problems | |

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Numerical Differentiation | |

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Background | |

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Finite Difference Approximation of the Derivative | |

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Finite Difference Formulas Using Taylor Series Expansion | |

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Finite Difference Formulas of First Derivative | |

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Finite Difference Formulas for the Second Derivative | |

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Summary of Finite Difference Formulas for Numerical Differentiation | |

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Differentiation Formulas Using Lagrange Polynomials | |

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Differentiation Using Curve Fitting | |

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Use of MATLAB Built-In Functions for Numerical Differentiation | |

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Richardson's Extrapolation | |

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Error in Numerical Differentiation | |

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Numerical Partial Differentiation | |

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Problems | |

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Numerical Integration | |

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Background | |

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Overview of Approaches in Numerical Integration | |

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Rectangle and Midpoint Methods | |

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Trapezoidal Method | |

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Composite Trapezoidal Method | |

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Simpson's Methods | |

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Simpson's 1/3 Method | |

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Simpson's 3/8 Method | |

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Gauss Quadrature | |

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Evaluation of Multiple Integrals | |

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Use of MATLAB Built-In Functions for Integration | |

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Estimation of Error in Numerical Integration | |

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Richardson's Extrapolation | |

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Romberg Integration | |

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Improper Integrals | |

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Integrals with Singularities | |

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Integrals with Unbounded Limits | |

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Problems | |

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Ordinary Differential Equations: Initial- Value Problems | |

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Background | |

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Euler's Methods | |

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Euler's Explicit Method | |

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Analysis of Truncation Error in Euler's Explicit Method | |

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Euler's Implicit Method | |

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Modified Euler's Method | |

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Midpoint Method | |

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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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Fourth-Order Runge-Kutta Methods | |

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Multistep Methods | |

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Adams-Bashforth Method | |

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Adams-Moulton Method | |

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Predictor-Corrector Methods | |

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System of First-Order Ordinary Differential Equations | |

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Solving a System of First-Order ODEs Using Euler's Explicit Method | |

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Solving a System of First-Order ODEs Using Second-Order Runge-Kutta Method (Modified Euler Version) | |

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Solving a System of First-Order ODEs Using the Classical Fourth-Order Runge-Kutta Method | |

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Solving a Higher-Order Initial Value Problem | |

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Use of MATLAB Built-In Functions for Solving Initial-Value Problems | |

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Solving a Single First-Order ODE Using MATLAB | |

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Solving a System of First-Order ODEs Using MATLAB | |

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Local Truncation Error in Second-Order Range-Kutta Method | |

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Step Size For Desired Accuracy | |

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Stability | |

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Stiff Ordinary Differential Equations | |

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Problems | |

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Ordinary Differential Equations: Boundary-Value Problems | |

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Background | |

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The Shooting Method | |

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Finite Difference Method | |

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Use of MATLAB Built-In Functions for Solving Boundary Value Problems | |

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Error and Stability in Numerical Solution of Boundary Value Problems | |

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Problems | |

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Introductory MATLAB | |

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Background | |

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Starting with MATLAB | |

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Arrays | |

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Mathematical Operations with Arrays | |

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Script Files | |

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Function Files | |

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Programming in MATLAB | |

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Relational and Logical Operators | |

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Conditional Statements, if-else Structures | |

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Loops | |

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Plotting | |

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Problems | |

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MATLAB Programs | |

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Index | |