For a group of vectors which form a non-degenerate rectangular matrix
B0 with m rows and not fewer than
n (m > or = n) columns, and with maximum singular number
Lmax < or = 1.0, in the limit the iterative procedure
|
| (1) |
will give a matrix with orthogonal columns. The transformations (1) alter the singular
numbers, preserving the matrix pair P, Q, which makes up
the singular decomposition
| (2) |
and so for ideal computations
| (3) |
A detailed proof and estimates of the rate of convergence can be found in the full text
of this paper (The full text of this paper is deposited in VINITI, No 294-V93, 1993).
Formula (1) also holds for symmetric, positive-definite matrices
A0 with the restriction
Lmax < or = 1.0, when the computations are carried out by the
scheme
| (4) |
for any natural k, and not just even k.
The transformations given above can be used to solve the classical problems of linear algebra.
Of these, we can consider here the solution of system of linear algebraic equations with
a symmetric positive-definite matrix A0 :
| (5) |
where z and u are, respectively, the
required and known vector. The computations are carried out with the iterational formula
| (6) |
derived from (4). It can be shown by direct substitution that, with exact computations,
i iterations (6) are equivalent to 2i - 1 iterations by the
method of simple iterations
| (7) |
There is thus an advantage in relation to the number of arithmetic operations, despite the
extra procedure of reduction to a square matrix that is involved.
Note that, for formula (6), unlike the simple iterational method, the initial approximation
z0 = u must be in the form of the known vector.