Computational Methods with MATLAB

Computational Methods with MATLAB
Erik Cuevas, Alberto Luque and Héctor Escobar
Contents
1 MATLAB® Fundamentals
1.1 Introduction
1.1.1 MATLAB® Environment
1.1.2 MATLAB® Languaje
1.1.3 Trigonometric Operators
1.1.4 Special Operators for Complex Numbers
1.1.5 Relational Operators
1.1.6 Logical Operators
1.1.7 Operators with Matrix and Arrays
1.1.8 Input and Output Operators
1.1.9 Operators for Data Analysis
1.1.10 MATLAB® Relationship with Matrices and
Vectors
1.1.11 Control Flow Structures in MATLAB®
1.1.12 MATLAB® Function Declarations and .m
Files
References
2 Graphing with MATLAB
2.1 Introduction
2.2 Simple Graphing
2.3 Contour of Two-Dimensional Functions
2.4 Triangular Mesh and Contours
2.5 Surface and Mesh Plotting2.6 Matlab as a Graphing Tool Implementing the
Mapping Toolbox
References
3 Linear Algebra
3.1 Introduction
3.2 Matrices and Vectors
3.3 Matrix and Vector Operations
3.4 Systems of Linear Equations
3.5 Determinant of a Square Matrix
3.6 Ill-Conditioned Problems
3.7 Gaussian Elimination
3.8 Gauss‒Jordan Elimination
3.9 LU Decomposition
3.10 Eigenvalues of Matrices
References
4 Interpolation and Polynomials
4.1 Introduction
4.2 Polynomial Expressions in Matlab
4.3 Linear Interpolation
4.4 Polynomial Interpolation with Power Series
4.5 Lagrange Interpolation
4.6 Polynomial Interpolation Error
4.7 Differentiation and Integration of the Lagrange
Interpolation Polynomial
4.8 Chebyshev and Legendre Interpolation4.9 Cubic Hermite Interpolation
4.10 Two-Dimensional Interpolation
References
5 Numerical Differentiation Methods
5.1 Introduction
5.2 Differentiation of Interpolation Polynomials
5.3 Differences Approximation
5.4 Differentiation by Taylor Expansion Method
5.5 Approximation of Differences in Partial
Derivatives
5.6 Numerical Computation of Higher-Order
Derivatives
References
6 Numerical Integration Methods
6.1 Introduction
6.1.1 Trapezoidal Rule
6.1.2 Simpson’s Integration Rule
6.1.3 Integration by Closed Newton–Cotes
Formula
6.1.4 Numerical Integration in Two-Dimensional
Domains
6.1.5 MATLAB® Commands for Numerical
Integration
References
7 Roots of Nonlinear Equations
7.1 Introduction7.2 Graphical Method
7.3 Bisection Method
7.4 Newton’s Iteration
7.5 Secant Method
7.6 Method of Successive Substitutions
7.7 Nonlinear Simultaneous Equations
References
8 Spline Interpolation
8.1 Introduction
8.2 Linear Splines
8.3 Quadratic Splines
8.4 Cubic Spline
8.4.1 Cubic Splines with Standard Form
Polynomials
8.4.2 Cubic Splines Based on Lagrange
Polynomials
8.5 Matlab Functions
References
9 Differential Equations
9.1 Introduction
9.2 Symbolic Solution of Equations
9.3 Euler’s Method
9.4 Trapezoidal Method
9.5 Runge–Kutta Method
9.6 Predictor–Corrector Method9.6.1 Functions for Generic Implementation
9.7 System of Differential Equations
9.8 Second Degree Differential Equations
9.9 Differential Equations with Boundary Value
9.10 Graphical Methods
References

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