On preconditioned msor iterations^†

Precondition for Discretized Fractional Boundary Value Problem

“…In the finite difference form (9) and (10) for the fractional order case the behavior of the approximation of the fractional derivatives appears in the matrix structure as illustrated in figure (4), figure (5) and figure (6). The precondition introduced meets the second requirement of precondition, it is easy to construct.…”

“…technique [8][9][10][11]: precondition techniques are used to accelerate the rate of convergence . In solving linear systems iteratively varieties of preconditions.…”

“…Sometimes the preconditioned matrix is well structured than the original matrix. Figure(6) corresponding to the chequer-board ordering (odd even ordering) the numbers e B X appears in the element position (7,9) and (8,10) can be annihilated by using a precondition matrix without affecting the structure of the block sub matrices as illustrated in figure (7) and figure (8) The preconditioned system is written in the form P(AU=f) or A p U=f p (15)…”

“…For more details you can see [8] and [9]. In general, if we have a coefficient matrix of order 2ℓ with submatrices of order ℓ this precondition must be designed to annihilate the elements in positions ℓ 1, ℓ 1 and the elements in positions ℓ 2, ℓ 1 and then annihilate the element in position ℓ, ℓ 2 .…”

On preconditioned AOR method for solving linear systems

“…Let A be an irreducible Z-matrix. Assume that 0 < −α k a k,1 + β k ≤ −a k,1 , k = 2, • • • , n and A (10) has the block form…”