•Let v(j) be the j-th eigenvector of T(i), and V the eigenvector matrix V=[v(1),v(2),...] whose j-th column
is v(j). V is an orthogonal
matrix, so any vector x can be written
as a linear combination of the v(j).
•alpha(j) the j-th
frequency component of x,
•On the upper half of the frequency domain, multigrid acting on P(m), damps the error in the solution by the solution operator S(i). On the next coarser grid, with
half as many points,
multigrid damps the upper half of the remaining frequency components in the error. On the next
coarser grid, the upper
half of the remaining frequency components
are damped, and so on, until we solve the exact (1 unknown) problem P(1).
•