Thomas algorithm to solve tridiagonal matrices

Basically sets up an LU decomposition three parts 1) decomposition 2) forward substitution 3) backward substitution

1) decomposition loop from rows 2 to n ai = ai/bi-1 bi = bi-ai*ci-1 end loop 2) forward substitution loop from 2 to n ri = ri - ai*ri-1

3) back substitution xn = rn/bn loop from n-1 to 1 xi = (ri-ci*xi+1)/bi end loop

Example First decompose T

loop from rows 2 to 5 ai = ai/bi-1 bi = bi-ai*ci-1 end loop

forward substitution loop from 2 to n ri = ri - ai*ri-1 end loop

back substitution xn = rn/bn loop from n-1 to 1 xi = (ri-ci*xi+1)/bi end loop

Crout algorithm - alternate LU decomposition Instead of 1’s on diagonal of L, get 1’s on diagonal of U Operate on rows and columns sequentially, narrowing down to single element

Algorithm for j=2 to n-1 For j=n

Cholesky decomposition A decomposition for symmetric matrices Symmetric matrix e.g. a covariance matrix

For symmetric matrices, can write Can develop relationships for l

Example: 1st row Skip this equation

Row 2

Row 3

Row 4

Iterative methods for solving matrix equations 1. Jacobi 2. Gauss-Seidel 3. Successive overrelaxation (SOR) Idea behind iterative methods: Convert Ax=b into Cx=d, which has sequence of approximations x(1), x(2), … with

What are C and d? Recall from fixed-point iteration

Rewrite matrix equation in same way becomes

Then

Jacobi method is like fixed point iteration Example: Shape of a stretched membrane

Shape can be described by potential function Let us give some boundary conditions

Problem look likes this Solve for u’s

Leads to this system of equations

Choose an initial u=[1 1 1 1 …1]’ Iterate using x=Cx+d

Matlab solution, 49 iterations

Gauss-Seidel method differs from Jacobi by sequential updating - use new xi’s as they become available

Example: Jacobi Gauss-Seidel