A regular splitting of a non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T. A general iterative method converges for every initial vector if T is convergent, and under certain conditions if T is semi-convergent.
Note that ρ(T) = 1/4, where ρ(T) represents the spectral radius of T, since 1/4 is the only eigenvalue of T. Let T be an n × n matrix.
The following properties are equivalent to T being a convergent matrix: A general iterative method involves a process that converts the system of linear equations into an equivalent system of the form for some matrix T and vector c. After the initial vector x(0) is selected, the sequence of approximate solution vectors is generated by computing for each k ≥ 0.
In the system of linear equations (2) above, with A non-singular, the matrix A can be split, that is, written as a difference so that (2) can be re-written as (4) above.
The expression (5) is a regular splitting of A if and only if B−1 ≥ 0 and C ≥ 0, that is, B−1 and C have only nonnegative entries.
[14] If A is possibly singular but (2) is consistent, that is, b is in the range of A, then the sequence defined by (4) converges to a solution to (2) for every x(0) ∈