Direct 111ethocls^651where 'Pk ( ih 1 ) is the Fourier coefficient of the function 'P( x):
N2-1
'Pk(ih1) = ~ 'PCih1,Jh2)μk(jh2)h2.
j=1
Due to the problen1 statement (14) and the orthogonality of the funcLions
{tk we derive fron1 ( 16) Lhe problem statement for delerrnination of the
numbers ck for all k = 1, 2, ... , N 2 - 1:Because of this form, the applications of the elimination method for
N 2 - 1 times to ck(ih 1 ) as a function of the argument x 1 = ih 1 for fixed k
permit us to find a solution of problem ( 13) by means of formula ( 15 ). As
can readily be observed, the calculations of the Fourier coefficients 'Pk and
solutions Yij can be carried out by the satne formulas related to connnon
sums of the special type
j = 1, 2, ... , N - 1.
0111itting n10re detail8 on this point, we refer the readers to the well-
developed algorithn1 of the fa.st Fourier transfonn, in the framework of
which Q arithmetic operations, Q ~ 2N log 0 1V, N = 2", are necessary in
connection with computations of these sutns-(instead of O(N^2 ) in Lhe case
of the usual summation), thus causing 0( 111 JV 2 log 0 N 2 ) arithmetic opera-
tions performed in the numerical solution of the Di~·ichlet problem (2) in a
rectangle.
In such tnatters some progress can be achieved by combinations of
Lhe decomposition method and the method of separation of variables. For
exatnple, this can be done using the method of separation of variables
for the "reduced" system ( 6) upon elin1inating the unknown vectors with
odd subscripts j. This trick allows one to solve problen1 (2): here the
expenditures of time are Q ~ 2111112 log 2 N 2 arithmetic operation, half as
much than required before in the tnethod of separation of variables.
- The inethod of matrix eliniination. The system of equations (3) is one
particular case of the following problem:
j = l, 2, ... , N - 1,
CuYo-BoY1 =Fo,