Page 184 - Matrices theory and applications

9.6. Exercises
167
ii. Deduce that m = n, hence that the SOR converges for every
ω ∈ (0, 2).
7. (Extension of Theorem 9.4.1 to complex values of ω). We still assume
that A is tridiagonal, that the Jacobi method converges, and that the
spectrum of J is real. We retain the notation of Section 9.4.
(a) Given an index a such that λ a > 0, verify that ∆(λ a ) vanishes for
two real values of ω, of which only one, denoted by ω a, belongs
to the open disk D = D(1; 1). Show that 1 <ω a < 2.
2
(b) Show that if ω ∈ D\[ω a , 2), then the roots of X +ω−1−ωλ aX
have distinct moduli, with one and only one of them, denoted
1/2
by µ a (ω), of modulus larger than |ω − 1| .
(c) Show that ω → µ a is holomorphic on its domain, and that
lim |µ a (ω)| 2 = 1,
|ω−1|→1
2
lim |µ a (ω)| = γ − 1 if γ ∈ [ω a , 2).
ω→γ
(d) Deduce that |µ a (ω)| < 1 (use the maximum principle), then that
the relaxation method converges for every ω ∈ D.
(e) Show, finally, that the spectral radius of L ω is minimal for ω =
ω r , which previously was denoted by ω J .
8. Let B be a cyclic matrix of order three. With square diagonal blocks,
it reads blockwise as
 
0 0 M 1
B =  M 2 0 0  .
0 M 3 0
We wish to compare the Jacobi and Gauss–Seidel methods for the
3
matrix A := I −B. Compute the matrix G. Show that ρ(G)= ρ(J) .
Deduce that both methods converge or diverge simultaneously and
that, in case of convergence, Gauss–Seidel is three times faster than
T
Jacobi. Show that for A , the convergence or the divergence still
holds simultaneously, but that Gauss–Seidel is only one and a half
times faster. Generalize to cyclic matrices of any order p.