The Conjugate Gradient Method - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 195

Problems

1.9 Write the SSOR method (1.19) in componentwise form similar to (1.14).

1.10 Verify (1.20). - Hint: Under iteration the matrix A corresponds to the
approximate inverse M − 1 . In particular, x 1

= M − 1 b for x 0

= 0.

1.11 Consider the matrices

12 2 .

2
11 1
22 1

− 11

222
−

−

1
2

A 1 = and A 2 =

1 − 12

For which of the matrices do the Jacobi method and Gauss-Seidel methods con-
verge?

1.12 Let G be an n × n matrix with lim k →∞ G k

= 0. In addition, suppose ·

n . Show that

is an arbitrary vector norm on R

∞

0 G k x

||| x ||| : =

k =

n , and that

defines a norm on ||| G ||| < 1 for the associated matrix norm. - Give

an example to show that ρ(G) < 1 does not imply

R

G < 1 for every arbitrary

norm.

2 in the SSOR method, then x k + 1

= x k . How can this be

1.13 If we choose ω =

shown without using any formulas?

1.14 Suppose the Jacobi and Gauss-Seidel methods converge for the equation
Ax = b . In addition, suppose D is a nonsingular diagonal matrix. Do we still get
convergence if AD (or DA ) is substituted for A ?

1.15 A matrix B is called nonnegative , written as B ≥ 0, if all matrix elements

are nonnegative. Let D, L, U ≥ 0, and suppose the Jacobi method converges for

A = D − L − U . Show that this implies A − 1 0.

What is the connection with the discrete maximum principle?

≥

1.16 Let M − 1 be an approximate inverse of A in the sense of (1.1). Determine
the approximate inverse that corresponds to k steps of the iteration.