1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction inethods^715

The norm II Tn II can be expressed through the eigenvalues of the operators

A~ and A~ such as

As far as the operators A~ and A~ are comrnuting, other operators A 1 ,

A2, A and Tn possess con1n1on systems of eigenfunctions. By utilizing this

fact we denote by Ak(Tn) the eigenvalues of the operator Tn and take into

account that

with missing subscripts j and k. All this enables us to find that

n

( 11) A(Tn) =II

j=l

with

As known, the norm of the operator T, 1 is equal to thP greatest eigen-
value maxk Ak(T, 1 ). Further replacements of Cl'k 1 and f3ko in (11) by contin-
uous variables a and j3 lead to increased n1aximum of the right-hand side
of (11); meaning

(12) II Tn II< max

CYE(ry,l]

n

IT

j =1

Also, we may accept w(l) = wl^2 J and ct :::: (3, since the argun1ents o: and

/3 run over one and the same segment [17, l] and their positions in formula

(12) are certainly symmetric. The problem statement here is to find the

parameters w 1 , w 2 , ... , wn for which

(13) n ( )2

1 - W CY

min n1ax J

{~: 1 } crE[17,l] l1 l +w 1 Cl

A solution of such a problem is already known and so it remaius only to
write the final expressions for optin1al parameters r}1l and r§') in question.

Having stipulated the conditions CY = /3 = 17 for .>-(A1) = 01 , A(A2) = (^02)