MATLAB Mathematics

• Publisher: MathWorks
• Year: 2022
• Tongue: English
• Leaves: 708
• Category: Library

✦ Table of Contents

Matrices and Arrays
Creating, Concatenating, and Expanding Matrices
Removing Rows or Columns from a Matrix
Reshaping and Rearranging Arrays
Multidimensional Arrays
Array Indexing
Linear Algebra
Matrices in the MATLAB Environment
Creating Matrices
Adding and Subtracting Matrices
Vector Products and Transpose
Multiplying Matrices
Identity Matrix
Matrix Inverse
Kronecker Tensor Product
Vector and Matrix Norms
Using Multithreaded Computation with Linear Algebra Functions
Systems of Linear Equations
Computational Considerations
General Solution
Square Systems
Overdetermined Systems
Underdetermined Systems
Solving for Several Right-Hand Sides
Iterative Methods
Multithreaded Computation
Factorizations
Introduction
Cholesky Factorization
LU Factorization
QR Factorization
Using Multithreaded Computation for Factorization
Powers and Exponentials
Eigenvalues
Eigenvalue Decomposition
Multiple Eigenvalues
Schur Decomposition
Singular Values
Batched SVD Calculation
Low-Rank SVD Approximations
LAPACK in MATLAB
A Brief History
Matrix Exponentials
Graphical Comparison of Exponential Functions
Basic Matrix Operations
Determine Whether Matrix Is Symmetric Positive Definite
Image Compression with Low-Rank SVD
Random Numbers
Why Do Random Numbers Repeat After Startup?
Create Arrays of Random Numbers
Random Number Functions
Random Number Generators
Random Number Data Types
Random Numbers Within a Specific Range
Random Integers
Random Numbers from Normal Distribution with Specific Mean and Variance
Random Numbers Within a Sphere
Generate Random Numbers That Are Repeatable
Specify the Seed
Save and Restore the Generator Settings
Generate Random Numbers That Are Different
Managing the Global Stream Using RandStream
Creating and Controlling a Random Number Stream
Substreams
Choosing a Random Number Generator
Configuring a Stream
Restore State of Random Number Generator to Reproduce Output
Multiple Streams
Replace Discouraged Syntaxes of rand and randn
Description of the Discouraged Syntaxes
Description of Replacement Syntaxes
Replacement Syntaxes for Initializing the Generator with an Integer Seed
Replacement Syntaxes for Initializing the Generator with a State Vector
If You Are Unable to Upgrade from Discouraged Syntax
Controlling Random Number Generation
Sparse Matrices
Computational Advantages of Sparse Matrices
Memory Management
Computational Efficiency
Constructing Sparse Matrices
Creating Sparse Matrices
Importing Sparse Matrices
Accessing Sparse Matrices
Nonzero Elements
Indices and Values
Indexing in Sparse Matrix Operations
Visualizing Sparse Matrices
Sparse Matrix Operations
Efficiency of Operations
Permutations and Reordering
Factoring Sparse Matrices
Eigenvalues and Singular Values
References
Finite Difference Laplacian
Graphical Representation of Sparse Matrices
Graphs and Matrices
Sparse Matrix Reordering
Iterative Methods for Linear Systems
Direct vs. Iterative Methods
Generic Iterative Algorithm
Summary of Iterative Methods
Choosing an Iterative Solver
Preconditioners
Equilibration and Reordering
Using Linear Operators Instead of Matrices
References
Graph and Network Algorithms
Directed and Undirected Graphs
What Is a Graph?
Creating Graphs
Graph Node IDs
Modify or Query Existing Graph
Modify Nodes and Edges of Existing Graph
Add Graph Node Names, Edge Weights, and Other Attributes
Graph Plotting and Customization
Visualize Breadth-First and Depth-First Search
Partition Graph with Laplacian Matrix
Add Node Properties to Graph Plot Data Tips
Build Watts-Strogatz Small World Graph Model
Use PageRank Algorithm to Rank Websites
Label Graph Nodes and Edges
Functions of One Variable
Create and Evaluate Polynomials
Roots of Polynomials
Numeric Roots
Roots Using Substitution
Roots in a Specific Interval
Symbolic Roots
Integrate and Differentiate Polynomials
Polynomial Curve Fitting
Predicting the US Population
Roots of Scalar Functions
Computational Geometry
Triangulation Representations
2-D and 3-D Domains
Triangulation Matrix Format
Querying Triangulations Using the triangulation Class
Working with Delaunay Triangulations
Definition of Delaunay Triangulation
Creating Delaunay Triangulations
Triangulation of Point Sets Containing Duplicate Locations
Creating and Editing Delaunay Triangulations
Spatial Searching
Voronoi Diagrams
Plot 2-D Voronoi Diagram and Delaunay Triangulation
Computing the Voronoi Diagram
Types of Region Boundaries
Convex Hulls vs. Nonconvex Polygons
Alpha Shapes
Computing the Convex Hull
Computing the Convex Hull Using convhull and convhulln
Convex Hull Computation Using the delaunayTriangulation Class
Convex Hull Computation Using alphaShape
Interpolation
Gridded and Scattered Sample Data
Interpolation versus Curve Fitting
Grid Approximation Techniques
Interpolating Gridded Data
MATLAB Gridded Interpolation Functions
Grid Representations
Example: Temperature Interpolation on 2-D Grid
Gridded Interpolation Methods
Interpolation of Multiple 1-D Value Sets
Interpolation of 2-D Selections in 3-D Grids
Interpolating Scattered Data
Scattered Data
Interpolating Scattered Data Using griddata and griddatan
scatteredInterpolant Class
Interpolating Scattered Data Using the scatteredInterpolant Class
Interpolation of Complex Scattered Data
Addressing Problems in Scattered Data Interpolation
Interpolation Using a Specific Delaunay Triangulation
Nearest-Neighbor Interpolation Using a delaunayTriangulation Query
Linear Interpolation Using a delaunayTriangulation Query
Extrapolating Scattered Data
Factors That Affect the Accuracy of Extrapolation
Compare Extrapolation of Coarsely and Finely Sampled Scattered Data
Extrapolation of 3-D Data
Normalize Data with Differing Magnitudes
Resample Image with Gridded Interpolation
Optimization
Optimizing Nonlinear Functions
Minimizing Functions of One Variable
Minimizing Functions of Several Variables
Maximizing Functions
fminsearch Algorithm
Reference
Curve Fitting via Optimization
Set Optimization Options
How to Set Options
Options Table
Tolerances and Stopping Criteria
Output Structure
Optimization Solver Iterative Display
Optimization Solver Output Functions
What Is an Output Function?
Creating and Using an Output Function
Structure of the Output Function
Example of a Nested Output Function
Fields in optimValues
States of the Algorithm
Stop Flag
Optimization Solver Plot Functions
What Is a Plot Function?
Example: Plot Function
Optimize Live Editor Task
What Is the Optimize Live Editor Task?
Minimize a Nonlinear Function of Several Variables
Solve a Scalar Equation
Optimization Troubleshooting and Tips
Function Handles
Parameterizing Functions
Overview
Parameterizing Using Nested Functions
Parameterizing Using Anonymous Functions
Ordinary Differential Equations (ODEs)
Choose an ODE Solver
Ordinary Differential Equations
Types of ODEs
Systems of ODEs
Higher-Order ODEs
Complex ODEs
Basic Solver Selection
Summary of ODE Examples and Files
Summary of ODE Options
Options Syntax
Compatibility of Options with Each Solver
Usage Examples
ODE Event Location
What is Event Location?
Writing an Event Function
Event Information
Limitations
Simple Event Location: A Bouncing Ball
Advanced Event Location: Restricted Three Body Problem
Solve Nonstiff ODEs
Solve Stiff ODEs
Solve Differential Algebraic Equations (DAEs)
What is a Differential Algebraic Equation?
Consistent Initial Conditions
Differential Index
Imposing Nonnegativity
Solve Robertson Problem as Semi-Explicit Differential Algebraic Equations (DAEs)
Nonnegative ODE Solution
Troubleshoot Common ODE Problems
Error Tolerances
Problem Scale
Solution Components
Problem Type
Differential Equations
Solve Predator-Prey Equations
Solve Equations of Motion for Baton Thrown into Air
Solve ODE with Strongly State-Dependent Mass Matrix
Solve Stiff Transistor Differential Algebraic Equation
Solve System of ODEs with Multiple Initial Conditions
Solve Celestial Mechanics Problem with High-Order Solvers
Boundary Value Problems (BVPs)
Solving Boundary Value Problems
Boundary Conditions
Initial Guess of Solution
Finding Unknown Parameters
Singular BVPs
BVP Solver Selection
Evaluating the Solution
BVP Examples and Files
Solve BVP with Two Solutions
Solve BVP with Unknown Parameter
Solve BVP Using Continuation
Verify BVP Consistency Using Continuation
Solve BVP with Singular Term
Solve BVP with Multiple Boundary Conditions
Partial Differential Equations (PDEs)
Solving Partial Differential Equations
What Types of PDEs Can You Solve with MATLAB?
Solving 1-D PDEs
Example: The Heat Equation
PDE Examples and Files
Solve Single PDE
Solve PDE with Discontinuity
Solve PDE and Compute Partial Derivatives
Solve System of PDEs
Solve System of PDEs with Initial Condition Step Functions
Delay Differential Equations (DDEs)
Solving Delay Differential Equations
Constant Delay DDEs
Time-Dependent and State-Dependent DDEs
DDEs of Neutral Type
Evaluating the Solution at Specific Points
History and Initial Values
Discontinuities in DDEs
Propagation of Discontinuities
DDE Examples and Files
DDE with Constant Delays
DDE with State-Dependent Delays
Cardiovascular Model DDE with Discontinuities
DDE of Neutral Type
Initial Value DDE of Neutral Type
Numerical Integration
Integration to Find Arc Length
Complex Line Integrals
Singularity on Interior of Integration Domain
Analytic Solution to Integral of Polynomial
Integration of Numeric Data
Calculate Tangent Plane to Surface
Fourier Transforms
Fourier Transforms
Basic Spectral Analysis
Spectral Analysis Quantities
Noisy Signal
Audio Signal
Polynomial Interpolation Using FFT
FFT in Mathematics
Interpolate Asteroid Data
2-D Fourier Transforms
Two-Dimensional Fourier Transform
2-D Diffraction Pattern
Square Wave from Sine Waves
Analyzing Cyclical Data with FFT