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Mathematics
Numerical Methods
1. Introduction to Numerical Methods
2. Foundational Concepts in Numerical Computation
3. Root Finding Methods
4. Linear Systems of Equations
5. Interpolation and Approximation
6. Curve Fitting and Least Squares
7. Numerical Differentiation
8. Numerical Integration
9. Ordinary Differential Equations
10. Partial Differential Equations
11. Optimization Methods
12. Eigenvalue Problems
12.
Eigenvalue Problems
12.1.
Problem Formulation
12.1.1.
Standard Eigenvalue Problem
12.1.2.
Generalized Eigenvalue Problem
12.1.3.
Properties of Eigenvalues
12.2.
Power Method
12.2.1.
Basic Power Method
12.2.1.1.
Algorithm
12.2.1.2.
Convergence Analysis
12.2.1.3.
Dominant Eigenvalue
12.2.2.
Inverse Power Method
12.2.2.1.
Smallest Eigenvalue
12.2.2.2.
Shifted Inverse Iteration
12.2.2.3.
Convergence Acceleration
12.2.3.
Deflation Techniques
12.2.3.1.
Hotelling's Deflation
12.2.3.2.
Wielandt's Deflation
12.3.
QR Algorithm
12.3.1.
QR Decomposition
12.3.1.1.
Gram-Schmidt Process
12.3.1.2.
Householder Reflections
12.3.1.3.
Givens Rotations
12.3.2.
Basic QR Algorithm
12.3.2.1.
Iteration Process
12.3.2.2.
Convergence Properties
12.3.3.
Practical QR Algorithm
12.3.3.1.
Hessenberg Form
12.3.3.2.
Implicit Shifts
12.3.3.3.
Deflation Strategies
12.4.
Jacobi Method
12.4.1.
Symmetric Matrices
12.4.2.
Jacobi Rotations
12.4.3.
Convergence Theory
12.4.4.
Implementation Details
12.5.
Lanczos Method
12.5.1.
Krylov Subspaces
12.5.2.
Tridiagonalization
12.5.3.
Large Sparse Matrices
12.5.4.
Reorthogonalization
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11. Optimization Methods
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1. Introduction to Numerical Methods