The varieties in the above Theorem are called the "'irreducible components "'of and can be regarded as a natural output for a decomposition algorithm, or, in other words, for an algorithm solving a system of equations in.
12.
By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix " L " and an upper triangular matrix " U " if and only if all its leading principal minors are non-zero.
13.
Custom hybrid and decomposition algorithms can be constructed using the GMP system library which makes available at the modeling level many of the basic building blocks used internally by the higher level solution methods present in AIMMS, matrix modification methods, as well as specialized steps for customizing solution algorithms for specific problem types.
14.
In its original formulation, BDD performs well only for 2nd order problems, such elasticity in 2D and 3D . For 4th order problems, such as plate bending, it needs to be modified by adding to the coarse problem special basis functions that enforce continuity of the solution at subdomain corners, P . Le Tallec, J . Mandel, and M . Vidrascu, " A Neumann & ndash; Neumann domain decomposition algorithm for solving plate and shell problems ", SIAM Journal on Numerical Analysis, 35 ( 1998 ), pp . 836 & ndash; 867.
