Hence,
<-1 + Ah II A-BI +, IIA-I(AA AA)||oo
21 IIA-IBII 1 IIA-IB|lo
whenever /,lA-1'Boo < 1, and where A = max A,. Our goal in this section is to
show that IC-Dlloo = 1 + O(At).
5.3.1 Max-Norm for Matrix C-1D in 1-Dimensional Space
The preliminary results from Section 5.2 are used to show that IIC-1DIk =
1 + O(At) in one dimensional space.
Lemma 5.3.1 Let Ax be the matrix described in Lemma 5.2.1. Let B1 be the n x n
tridiagonal matrix with O's on the diagonal, with did's on the superdiagonal, and with
-di's on the subdiagonal. Then
h|1Al-'Blloo = O(At).
Proof: The matrix B1 has the form
0 di
-d2 0 d2
-d3 0 d3
-d4 0 d4
where each d, = O(At). We can write the matrix B1 as