MATLAB Mathematics

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 Gate-Based Quantum Algorithms Introduction to Quantum Computing Qubit Quantum Gate Quantum Circuit Helper Functions Types of Quantum Gates Creation Functions for SimpleGate Objects Creation Functions for CompositeGate Objects Local Quantum State Simulation Create Circuit Simulate Circuit Display State Formula Plot Possible States Query Possible States Query Qubit State Probabilities Simulate Quantum State Measurement Run Quantum Circuit on Hardware Using AWS Requirements Set Up Access to AWS Device Availability Connect to Quantum Device Create Task to Run Circuit Fetch Measurement Output Graph Coloring with Grover's Algorithm Create State Oracle for Graph Coloring Problem Create Diffuser Circuit Apply Grover's Algorithm to Graph Coloring Oracle Run Circuit on AWS Ground-State Protein Folding Using Variational Quantum Eigensolver (VQE) Model Protein Fold with Qubits Write Function to Calculate Energy of Folds Compute Minimum Energy for All Folds Write CVaR-VQE Objective Function Create Circuit Ansatz Simulate Iterations of CVaR-VQE Run Final Iteration on QPU