1549301742-The_Theory_of_Difference_Schemes__Samarskii

662 Methods for Solving Grid Equations

Thus, the well-known Chebyshev polynomial defined by

where

(24) ]~ 1 (t) =cos (n arccos t) for ltl < 1,

is just the solution of the original proble1n concerned. For ltl > 1 the

polyn0111ial of interest is specified by the fonnula

(25) Tn(t) = 0.5 [(t + Jt^2 - 1)" + (t - Jt^2 - 1)"], ltl > 1.

Since max ITn(t)I = 1, the relations occur:

It I:<;: 1

(26)

In an atten1pt to find the unknown parameters T 1 , T 2 , ... , T 11 by the ap-

proved rule saying that the zeroes of the sought polynomial Pn(t) should

coincide wit.h known zeroes of Chebyshev's polynomial such as

2k - 1

(27) ---7f, k = 1,2,. .. ,n,

211

we recall from calculus that the polynon1ial

has zeroes at the points xk = 1/Tk, k = 1, 2, ... , n. By formula (22),

relating x and t, we deduce that

g1vmg
2
Tk = ~( /1 +_/2+_( /-2---,-1-) -t k-) ' k = 1,2,. .. ,n.

Also, it will be sensible to introduce more co1npact notations

(28) (.,--, c - fl
/2

1-~

Po = l + ~,

1-~

1+~·

2

Ta= ---

/1 + /2