In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices.
Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices.
If (2) represents a regular splitting of A, then the iterative method where x(0) is an arbitrary vector, can be carried out.
Equivalently, we write (4) in the form The matrix D = B−1C has nonnegative entries if (2) represents a regular splitting of A.
[3][4] If, in addition, the splitting (2) is chosen so that the matrix B is a diagonal matrix (with the diagonal entries all non-zero, since B must be invertible), then B can be inverted in linear time (see Time complexity).
The Jacobi method can be represented in matrix form as a splitting The Gauss–Seidel method can be represented in matrix form as a splitting The method of successive over-relaxation can be represented in matrix form as a splitting In equation (1), let Let us apply the splitting (7) which is used in the Jacobi method: we split A in such a way that B consists of all of the diagonal elements of A, and C consists of all of the off-diagonal elements of A, negated.
[11] The method (5) applied to the problem (10) then takes the form The exact solution to equation (12) is The first few iterates for equation (12) are listed in the table below, beginning with x(0) = (0.0, 0.0, 0.0)T. From the table one can see that the method is evidently converging to the solution (13), albeit rather slowly.
As stated above, the Jacobi method (7) is the same as the specific regular splitting (11) demonstrated above.
Since the diagonal entries of the matrix A in problem (10) are all nonzero, we can express the matrix A as the splitting (6), where We then have The Gauss–Seidel method (8) applied to the problem (10) takes the form The first few iterates for equation (15) are listed in the table below, beginning with x(0) = (0.0, 0.0, 0.0)T. From the table one can see that the method is evidently converging to the solution (13), somewhat faster than the Jacobi method described above.