claim: John S. McCain Jr. went to school.
evidence: In numerical linear algebra , an incomplete LU factorization of a matrix is a sparse approximation of the LU factorization often used as a preconditioner .. numerical linear algebra. numerical linear algebra. matrix. matrix ( mathematics ). sparse. sparse matrix. LU factorization. LU factorization. preconditioner. preconditioner. Consider a sparse linear system .. sparse. sparse matrix. These are often solved by computing the factorization , with L unit-lower triangular and U upper triangular .. One then solves , , which can be done efficiently because the matrices are triangular .. For a typical sparse matrix , the LU factors can be much less sparse than the original matrix .. matrix. matrix ( mathematics ). sparse. sparse matrix. The memory requirements for using a direct solver can then become a bottleneck in solving linear systems .. One can combat this problem by using fill-reducing reorderings of the matrix 's unknowns , such as the Cuthill-McKee ordering .. matrix. matrix ( mathematics ). Cuthill-McKee ordering. Cuthill-McKee algorithm. An incomplete factorization instead seeks triangular matrices L , U such that rather than .. Solving for can be done quickly but does not yield the exact solution to .. So , we instead use the matrix as a preconditioner in another iterative solution algorithm such as the conjugate gradient method or GMRES .. matrix. matrix ( mathematics ). preconditioner. preconditioner. conjugate gradient method. conjugate gradient method. GMRES. GMRES. The sparsity pattern of L and U is often chosen to be the same as the sparsity pattern of the original matrix A .. matrix. matrix ( mathematics ). If the underlying matrix structure can be referenced by pointers instead of copied , the only extra memory required is for the entries of L and U .. matrix. matrix ( mathematics ). This preconditioner is called ILU -LRB- 0 -RRB- .. preconditioner. preconditioner. One can obtain a more accurate preconditioner by allowing some level of extra fill in the factorization .. preconditioner. preconditioner. A common choice is to use the sparsity pattern of A2 instead of A ; this matrix is appreciably more dense than A , but still sparse over all .. matrix. matrix ( mathematics ). sparse. sparse matrix. This preconditioner is called ILU -LRB- 1 -RRB- .. preconditioner. preconditioner. One can then generalize this procedure ; the ILU -LRB- k -RRB- preconditioner of a matrix A is the incomplete LU factorization with the sparsity pattern of the matrix Ak +1 .. matrix. matrix ( mathematics ). LU factorization. LU factorization. preconditioner. preconditioner. More accurate ILU preconditioners require more memory , to such an extent that eventually the running time of the algorithm increases even though the total number of iterations decreases .. Consequently , there is a cost / accuracy trade-off that users must evaluate , typically on a case-by-case basis depending on the family of linear systems to be solved .. The ILU factorization can be performed as a fixed-point iteration in a highly parallel way .. fixed-point iteration. fixed-point iteration