ECS231 Handout
March 12, 2013
Preconditioning Techniques
1. By the convergence analysis of CG and GMRES algorithms, we learn that the convergence
rate strongly depends on the condition number of the coecient matrix A of the linear system
Ax = b, and the d
ECS 231
CG-type methods for Eigen-Computation
Generalized symmetric denite eigenvalue problem
Generalized symmetric denite eigenvalue problem
Au = Bu
where A and B are n n symmetric, and B positive denite,
All eigenvalues and eigenvectors are real
Denote
Fast Poisson Solvers II
ECS231 Handout
February 28, 2013
Block cyclic reduction
1. Block cyclic reduction (BCR) is a fast method for the Poisson model problem. Recall 2-D
Poissons model problem is given by
(IN TN + TN IN TN N ) vec(V ) = vec(h2 F ).
Write
ECS231 Handout
February 21, 2013
Fast Poisson Solvers I
Poissons equation is a partial dierential equation of elliptic type with broad utility in physical
models that include gravitation, temperature distribution, electromagnetism, elasticity and inviscid
ECS231: Spectral Partitioning
Based on Berkeleys CS267 lecture on graph partition
1
Definition of graph partitioning
Given a graph G = (N, E, WN, WE)
N = nodes (or vertices),
E = edges
WN = node weights
WE = edge weights
1 (2)
2 (2)
4
2
3
5 (1)
5
6 (2)
3
ECS231 Handout
February 14, 2013
Golub-Kahan-Lanczos algorithm
1. We have seen that the symmetric eigenvalue problem and the singular value decomposition
are closely related. The singular values of a square matrix1 A are the square roots of the
eigenvalue
ECS231 Handout
February 14, 2013
Arnoldi algorithm
1. The power method is the simplest algorithm suitable for computing just the largest eigenvalue
in absolute value, along with its eigenvector. Starting with a given x0 , k iterations of the
power method
ECS231 Handout
Rayleigh-Ritz procedure, Lanczos algorithm
February 12, 2013
1. Rayleigh-Ritz procedure is a framework of the orthogonal projection methods for solving
large scale eigenvalue problems
Let A be an n n real matrix and K be an m-dimensional su
ECS231 Handout
Eigenvalue Problem: basic theory and algorithms
February 7, 2013
Basic theory
1. Let A C nn .
(a) A scalar is an eigenvalue of an n n A and a nonzero vector x C n is a corresponding
right eigenvector if
Ax = x.
A nonzero vector y such that
ECS231 Handout
Lanczos process, Conjugate Gradient method
February 5, 2013
1. The symmetric Lanczos procedure can be regarded as a simplication of Arnoldis procedure when A is symmetric.
By an order-m Arnoldi decomposition, we know that
T
Hm = Vm AVm .
If
ECS231 Handout
January 31, 2013
Krylov subspace and GMRES method
1. Krylov subspace is dened as
Km (A, v ) = spancfw_v, Av, A2 v, . . . , Am1 v ,
where A is an n n matrix, and v is a column vector of length n.
Note that if x Km (A, v ), then x = p(A)v , w
ECS231 Handout
Large-Scale Linear Solvers - I
January 29, 2013
1. The landscape of solvers for linear systems of equations
Ax = b,
where A is an n n nonsingular matrix and b is an n-vector, x Rn is the unknown.
more robust less storage
Direct
Iterative
(u
ECS231 Handout
Floating point numbers and arithmetic
Jan.15, 2013
Part I: Floating-point numbers and representations
1. Floating-point representation of numbers (scientic notation) has four components, for example,
sign
3.1416 101 exponent
signicand base
ECS231 Handout
Frequently used matrix decompositions
January 10, 2013
1. LU decomposition (Gaussian elimination in matrix form). If A is a square nonsingular
matrix, then there exist a permutation matrix P , a unit lower triangular matrix L, and a
upper t
ECS231 Handout
Introduction
January 7, 2013
1. Scientic computing and computational science
Scientic computing is about the design and analysis of numerical algorithms and engineering software for solving mathematical problems in nite precision arithmetic