📓 CME 302
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Drawing 2023-10-04 2.excalidraw
Drawing 2023-10-04 12.13.49.excalidraw
Drawing 2023-10-04 12.26.03.excalidraw
Drawing 2023-10-15 18.35.05.excalidraw
Drawing 2023-10-15 18.36.19.excalidraw
Drawing 2023-10-18 11.37.23.excalidraw
GMRES least-squares problem 2023-11-29 10.49.27.excalidraw
Gram-Schmidt 2023-10-15 17.54.00.excalidraw
Least-squares solution using SVD 2023-10-15 20.28.15.excalidraw
MINRES 2023-12-04 08.40.09.excalidraw
QR iteration with shift 2023-10-30 11.49.00.excalidraw
QR using Givens transformations 2023-10-18 11.41.30.excalidraw
QR using Householder transformations 2023-10-18 11.37.36.excalidraw
2024 complete and painless Conjugate Gradient
Accelerating convergence using a shift
Algorithm for QR iteration with shift
Algorithm for the Arnoldi process
Algorithm for the Lanczos process
All the orthogonality relations in CG
Angle between subspaces
Applying a Householder transformation
Arnoldi process
Backward error analysis for LU
Bootcamp
Brief introduction to Conjugate Gradients
Cauchy-Schwarz
CG search directions
Chebyshev iteration
Cholesky factorization
Classical iterative methods to solve sparse linear systems
Computational cost of Arnoldi and Lanczos
Computationally efficient search directions
Computing eigenvalues
Computing eigenvectors using the Schur decomposition
Computing multiple eigenvalues
Conditioning of a linear system
Conjugate Gradients algorithm
Conjugate Gradients code
Conjugate Gradients Version 1
Connection between Arnoldi and polynomials of A
Connection between Krylov subspace and Arnoldi
Convergence of classical iterative methods
Convergence of GMRES
Convergence of Lanczos eigenvalues for symmetric matrices
Convergence of Lanczos inner eigenvalues
Convergence of the Conjugate Gradients
Convergence of the orthogonal iteration
Convergence with conjugate steps
Deflation in the QR iteration
Determinant
Diagonalizable matrices
Dot product
Eigenvalues
Eigenvalues cannot be computed exactly
Exact shift
Existence of LU
Existence of the Cholesky factorization
Flexible preconditioned conjugate gradient method
Floating point arithmetic and unit roundoff error
Floating point arithmetic is different from regular arithmetic
Floating point numbers
Forward and backward error
Gauss-Seidel iteration
Ghost eigenvalues in the Lanczos process
GMRES
GMRES algorithm
GMRES least-squares problem
Gram-Schmidt
Hermitian and symmetric matrices
Householder transformation
Invertible matrix
Iterative methods for eigenvalue computation
Jacobi iteration
Key idea of iterative methods for eigenvalue computation
Krylov iterative methods to solve sparse linear systems
Krylov methods for sparse systems
Krylov subspace
Lanczos process
Least-squares problems
Least-squares solution using QR
Least-squares solution using SVD
LU algorithm
LU and determinant
Matrix block operations
Matrix-vector and matrix-matrix product
Method of normal equation
Method of power iteration
MINRES
Motivation of iterative methods for eigenvalue computation
Operator and matrix norms
Optimal step size
Orthogonal iteration
Orthogonal iteration algorithm
Orthogonal matrix and projector
Outer form of matrix-matrix product
Preconditioned Conjugate Gradients algorithm
Preconditioned Conjugate Gradients code
Preconditioning
Preconditioning the Conjugate Gradients algorithm
Projection
Pythagorean theorem
QR factorization
QR factorization and least-squares
QR iteration
QR iteration 2x2 example
QR iteration for upper Hessenberg matrices
QR iteration with shift
QR using Givens transformations
QR using Householder transformations
Rajat's painless Conjugate Gradients
Residuals and solution increments in CG
Row pivoting
Schur decomposition
Sensitivity analysis
Sherman-Morrison-Woodbury formula
Singular value decomposition
Solving linear systems
Solving linear systems using LU
Solving triangular systems
Some orthogonality relations in CG
SOR iteration
SOR iteration as a splitting method
Space and time costs of CG and GMRES
Splitting methods
Stability of the Cholesky factorization
Stability of the LU factorization
Subspace and linear independence
Summary of convergence and cost of the QR iteration
Summary of least-squares solution methods
Symmetric and unsymmetric QR iteration
Symmetric Positive Definite Matrices
The four fundamental spaces
Three-term recurrence
Trace
Triangular factorization
Uniqueness of the QR factorization
Unitarily diagonalizable matrices
Upper Hessenberg form for the QR iteration
Vector norms
Vectors and matrices
Why eigenvalues
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Nov 19, 2024
Drawing 2023-10-15 18.35.05.excalidraw
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Nov 19, 2024
Drawing 2023-10-15 18.36.19.excalidraw
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Nov 19, 2024
Drawing 2023-10-18 11.37.23.excalidraw
excalidraw
Nov 19, 2024
GMRES least-squares problem 2023-11-29 10.49.27.excalidraw
excalidraw
Nov 19, 2024
Gram-Schmidt 2023-10-15 17.54.00.excalidraw
excalidraw
Nov 19, 2024
Least-squares solution using SVD 2023-10-15 20.28.15.excalidraw
excalidraw
Nov 19, 2024
MINRES 2023-12-04 08.40.09.excalidraw
excalidraw
Nov 19, 2024
QR iteration with shift 2023-10-30 11.49.00.excalidraw
excalidraw
Nov 19, 2024
QR using Givens transformations 2023-10-18 11.41.30.excalidraw
excalidraw
Nov 19, 2024
QR using Householder transformations 2023-10-18 11.37.36.excalidraw
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