Numerical Differentiation and Integration 6.1 Numerical Differentiation 6.1.1 Approximating Derivatives with Finite Differences 6.2 Numerical Integration 6.2.1 Newton-Cotes Methods 6.2.2 The Gauss Integration Method 7. Data Interpolation 5.1 Lagrange Interpolation 5.1.1 Construction of Lagrange Interpolating Polynomial 5.1.2 Uniqueness of Lagrange Interplation Polynomial 5.1.3 Lagrange Interpolation Error 5.2 Newton’s Interpolation 5.2.1 Description of the Method 5.2.2 Newton’s Divided Differences 5.3 MATLAB’s Interpolation Tools 5.3.1 Interpolation with the interp1 Function 5.3.2 Interpolation with the Spline Function 5.3.3 Interpolation with the Function pchip 5.3.4 Calling the Functions spline and pchip from interp1 5.4 Data Interpolation in Python 5.4.1 The Function interp1d 5.4.2 The Functions pchip interpolate and CubicSpline 5.4.3 The Function lagrange 6. Solving a System of Nonlinear Equations 4.1 Solving a Single Nonlinear Equation 4.1.1 The Bisection Method 4.1.2 The Newton-Raphson Method 4.1.3 The Secant Method 4.1.4 The Iterative Method Towards a Fixed Point 4.1.5 Using the MATLAB and Python solve Function 4.2 Solving a System of Nonlinear Equations Part II: Data Interpolation and Solutions of Differential Equations 5.

Ill-Conditioning and Regularization Techniques in Solutions of Linear Systems 3.1 Ill-Conditioning in Solutions of Linear Systems 3.1.1 More Examples of Ill-Posed System 3.1.2 Condition Numbers and Ill-Conditioned Matrices 3.1.3 Linking the Condition Numbers to Matrix Related Eigenvalues 3.1.4 Further Analysis on Ill-Posed Systems 3.2 Regularization of Solutions in Linear Systems 3.2.1 The Truncated SVD (TSVD) Method 3.2.2 Tikhonov Regularizaton Method 3.2.3 The L-curve Method 3.2.4 The Discrepancy Principle 4. Solving Linear Systems with Iterative and Least Squares Methods 2.1 Mathematical Backgrounds 2.1.1 Convergent Sequences and Cauchi’s Convergence 2.1.2 Vector Norm 2.1.3 Convergent Sequences of Vectors 2.2 The Iterative Methods 2.2.1 The General Idea 2.2.2 The Jacobi Method 2.2.3 The Jacobi Method in the Matrix Form 2.2.3.1 The Gauss-Seidel Iterative Method 2.2.4 The Gauss-Seidel Method in the Vector Form 2.2.5 The Relaxation Methods 2.3 The Least Squares Solutions 2.3.1 Some Applications of Least Squares Solutions 3. Solving Linear Systems Using Direct Methods 1.1 Testing the Existence of the Solution 1.2 Methods for Solving Linear Systems 1.2.1 Special Linear Systems 1.2.2 Gauss and Gauss-Jordan Elimination 1.2.3 Solving the System with the rref Function 1.3 Matrix Factorization Techniques 1.3.1 The LU Factorization 1.3.2 The QR Factorization 1.3.3 The Singular Value Decomposition (SVD) 2. Table of contents : Cover Half Title Title Page Copyright Page Dedication Contents Preface Author Part I: Solving Linear and Nonlinear Systems of Equations 1.

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