Matrix factorization method

matrix forward-backward substitution method

A method for solving finite-difference systems that approximate boundary value problems for systems of ordinary differential equations in one-dimensional problems, and for elliptic equations in two-dimensional problems.

The solution of the three-point difference scheme

where is an unknown grid vector, is the right-hand side vector and are given square matrices, under the boundary conditions

is sought for, as in the scalar case, in the form

(*)

The coefficients (the matrix and the vector ) are determined from the recurrence relations ( "forward substitution" )

while and are given by the left boundary condition:

The are calculated by formula (*) ( "backward substitution" ), and

There is stability of this method to rounding errors under the conditions

which implies that , (see [1]). A different form of the stability conditions is also available (see [2], [3]). The matrix factorization method is applied also to two-point difference schemes (see [3]). A variant in which inversion of matrices is replaced by orthogonalization is also used (see [4]).

References

[1]	A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)
[2]	V.V. Ogneva, "The "sweep" method for the solution of difference equations" USSR Comp. Math. Math. Phys. , 7 : 4 (1967) pp. 113–126 Zh. Vychisl. Mat. i Mat. Fiz. , 7 : 4 (1967) pp. 803–812
[3]	A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)
[4]	S.K. Godunov, "A method of orthogonal successive substitution for the solution of systems of difference equations" USSR Comp. Math. Math. Phys. , 2 : 6 (1962) pp. 1151–1165 Zh. Vychisl. Mat. i Mat. Fiz. , 2 : 6 (1962) pp. 972–982
[5]	E.L. Wachspress, "Iterative solution of elliptic systems and applications to the neutron diffusion equations of reactor physics" , Prentice-Hall (1966)

References

[a1]	I. [I. Babushka] Babuška, M. Práger, E. Vitásek, "Numerical processes in differential equations" , Interscience (1966)