Mathematics

Matrices and Linear Algebra

Function Summary

Matrices in MATLAB

Creating Matrices

Adding and Subtracting Matrices

Vector Products and Transpose

Multiplying Matrices

The Identity Matrix

The Kronecker Tensor Product

Vector and Matrix Norms

Solving Linear Systems of Equations

Computational Considerations

General Solution

Square Systems

Overdetermined Systems

Underdetermined Systems

Inverses and Determinants

Overview

Pseudoinverses

Cholesky, LU, and QR Factorizations

Cholesky Factorization

LU Factorization

QR Factorization

Matrix Powers and Exponentials

Eigenvalues

Singular Value Decomposition

Polynomials and Interpolation

Polynomials

Polynomial Function Summary

Representing Polynomials

Polynomial Roots

Characteristic Polynomials

Polynomial Evaluation

Convolution and Deconvolution

Polynomial Derivatives

Polynomial Curve Fitting

Partial Fraction Expansion

Interpolation

Interpolation Function Summary

One-Dimensional Interpolation

Two-Dimensional Interpolation

Comparing Interpolation Methods

Interpolation and Multidimensional Arrays

Triangulation and Interpolation of Scattered Data

Tessellation and Interpolation of Scattered Data in Higher Dimensions

Selected Bibliography

Data Analysis and Statistics

Column-Oriented Data Sets

Basic Data Analysis Functions

Function Summary

Covariance and Correlation Coefficients

Finite Differences

Data Preprocessing

Missing Values

Removing Outliers

Regression and Curve Fitting

Polynomial Regression

Linear-in-the-Parameters Regression

Multiple Regression

Case Study: Curve Fitting

Polynomial Fit

Analyzing Residuals

Exponential Fit

Error Bounds

The Basic Fitting Interface

Difference Equations and Filtering

Fourier Analysis and the Fast Fourier Transform (FFT)

Function Summary

Introduction

Magnitude and Phase of Transformed Data

FFT Length Versus Speed

Function Functions

Function Summary

Representing Functions in MATLAB

Plotting Mathematical Functions

Minimizing Functions and Finding Zeros

Minimizing Functions of One Variable

Minimizing Functions of Several Variables

Fitting a Curve to Data

Setting Minimization Options

Output Functions

Finding Zeros of Functions

Tips

Troubleshooting

Numerical Integration (Quadrature)

Example: Computing the Length of a Curve

Example: Double Integration

Parameterizing Functions Called by Function Functions

Providing Parameter Values Using Nested Functions

Providing Parameter Values to Anonymous Functions

Differential Equations

Initial Value Problems for ODEs and DAEs

ODE Function Summary

Introduction to Initial Value ODE Problems

Solvers for Explicit and Linearly Implicit ODEs

Examples: Solving Explicit ODE Problems

Solver for Fully Implicit ODEs

Example: Solving a Fully Implicit ODE Problem

Changing ODE Integration Properties

Examples: Applying the ODE Initial Value Problem Solvers

Questions and Answers, and Troubleshooting

Initial Value Problems for DDEs

DDE Function Summary

Introduction to Initial Value DDE Problems

DDE Solver

Solving DDE Problems

Discontinuities

Changing DDE Integration Properties

Boundary Value Problems for ODEs

BVP Function Summary

Introduction to Boundary Value ODE Problems

Boundary Value Problem Solver

Solving BVP Problems

Using Continuation to Make a Good Initial Guess

Solving Singular BVPs

Solving Multi-Point BVPs

Changing BVP Integration Properties

Partial Differential Equations

PDE Function Summary

Introduction to PDE Problems

MATLAB Partial Differential Equation Solver

Solving PDE Problems

Evaluating the Solution at Specific Points

Changing PDE Integration Properties

Example: Electrodynamics Problem

Selected Bibliography

Sparse Matrices

Function Summary

Introduction

Sparse Matrix Storage

General Storage Information

Creating Sparse Matrices

Importing Sparse Matrices from Outside MATLAB

Viewing Sparse Matrices

Information About Nonzero Elements

Viewing Sparse Matrices Graphically

The find Function and Sparse Matrices

Adjacency Matrices and Graphs

Introduction to Adjacency Matrices

Graphing Using Adjacency Matrices

The Bucky Ball

An Airflow Model

Sparse Matrix Operations

Computational Considerations

Standard Mathematical Operations

Permutation and Reordering

Factorization

Simultaneous Linear Equations

Eigenvalues and Singular Values

Selected Bibliography

Nondouble Data Types

Introduction

Integer Mathematics

Integer Data Types

Largest and Smallest Values for Integer Data Types

Integer Arithmetic

Example -- Digitized Signals

Warnings for Integer Data Types

Single-Precision Mathematics

Data Type single

Single-Precision Arithmetic

The Function eps

Example -- Writing M-Files for Different Data Types

Largest and Smallest Numbers of Type double and single

References

Mathematics

Matrices and Linear Algebra

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