Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Numerical Analysis ↓


Numerical Linear Algebra


Numerical linear algebra is an essential field in numerical analysis that focuses on solving linear algebra problems using numerical methods. These problems include solving systems of linear equations, finding eigenvalues and eigenvectors, and performing matrix factorization. Numerical linear algebra is widely used in scientific computing, engineering, physics, and data analysis.

Basic concepts of linear algebra

Before delving deeper into numerical methods, it is important to understand some basic concepts of linear algebra:

Matrix: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, a matrix with m rows and n columns is written as:

        
A = [a 11 a 12 ... a 1n
     a 21 a 22 ... a 2n
     ... a m1 a m2 ... a mn ]
One 11 A 21 One M1 a 1n a 2n a mn

Vector: A vector is a special type of matrix that has only one column. It represents a point in a multi-dimensional space.

        
v = [v 1 v 2 ... v n ]

Solving systems of linear equations

One of the fundamental tasks in numerical linear algebra is solving systems of linear equations. A system of linear equations can be written in matrix form as follows:

        Ax = b

Where:

  • A is a known matrix.
  • x is the unknown vector that contains the variables of the system.
  • b is a known vector.

Direct methods

Direct methods solve linear systems in a finite number of steps, which are calculated directly from the matrix A and the vector b.

Gaussian elimination

Gaussian elimination is the process of transforming the system into an upper triangular matrix, which can then be solved by back-substitution.

LU decomposition

In LU decomposition, the matrix A is decomposed into the product of lower triangular matrix L and upper triangular matrix U

        A = LU

Iterative methods

Iterative methods produce a sequence of approximate solutions that converge on the exact solution. These are particularly useful for solving large linear systems.

Jacobi method: Each diagonal element is solved using an initial guess, until convergence is achieved.

Gauss-Seidel method: This is similar to the Jacobi method, but it uses the latest values as soon as they are available.

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are key concepts in linear algebra, which have many applications in numerical simulations and scientific calculations.

The problem of finding the eigenvalues and eigenvectors of a square matrix A can be represented as:

        Av = λv
  • v is the eigenvector.
  • λ is the eigenvalue.

Power iteration method

The power iteration method is an iterative technique designed to find the dominant (largest) eigenvalue and its corresponding eigenvector.

QR algorithms

The QR algorithm decomposes a matrix into orthogonal matrices, and through iteration, estimates its eigenvalues and eigenvectors.

Matrix factorization

Matrices can be split or decomposed into simpler matrices, making many matrix problems easier to solve.

Singular value decomposition (SVD)

SVD decomposes a matrix A into three matrices U, Σ, and V* such that:

        A = UΣV*
A , U Σ V*

Cholesky decomposition

The Cholesky decomposition applies to Hermitian, positive-definite matrices and decomposes a matrix into a lower triangular matrix and its conjugate transpose.

        A = LL*

Applications of numerical linear algebra

Numerical linear algebra algorithms have many applications in modern computing and science:

  • Computer graphics: transformation and manipulation of 3D objects.
  • Data analysis: Principal Component Analysis (PCA) uses SVD to reduce the dimensionality of data.
  • Engineering simulation: solving partial differential equations and structural analysis.

Mastering numerical linear algebra helps practitioners efficiently tackle large-scale problems across multiple domains. It combines mathematical ingenuity with computational techniques to achieve accurate, scalable results.


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Numerical Linear Algebra

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