1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular n1ethod 679

As shown above, Seidel method is quite applicable for any operator

A = A• > 0. However, the extra restriction 0 < w < 2 is necessary for

the convergence of the upper relaxation method. This is certainly true

under condition (8) with a known operator Bo. Along these lines, it is

straightforward to verify that B = w A-+ D, Tk = w and

Bo = t ((w A-+ D) + (w A++ D)) = t (w A+ (2 - w) D).

This supports the view that B 0 > 0 for 0 < w < 2 and condition (8) 1s

fulfilled for w < 2, since

T W (2 - w)

B 0 - - A = B 0 - - A = D > 0 for w < 2.

2 2 2

But the convergence rate depends on the para111eter w. However, there are

analytical estin1ates for w and the convergence rate when the subsidiary

information on the spectral bounds of the operator D-^1 (A-+A+) is avail-

able, but their determination is some problem in itself. Just for this reason

the parameter w is so chosen as to minimize the total number of iterations.

In dealing with numerous similar problems this approach is more convenient

and effective.

By appeal once again to the model problem of interest it is plain to

show that the upper relaxation method is a perfect tool in such n1atters,

since the work and storage require

3. Implicit iteration schemes. Convergence of implicit iteration schemes
   was the subject of investigation in Section 2 for the special case

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when the iterations Yk+i can be immediately found by the formula

As a matter of fact, the upper relaxation method and Seidel method
are nothing more than the implicit scheme (6) with B f:. E incorporated.
Still using the trnual framework of i1nplicit iterative methods, the value lh+i
is determined from the equation

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