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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 | |