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The Minimum Degree Algorithm استاد: دکتر کاوه میثم ترکمان زاده
Introduction Consider the system A x= b , where M is a nonsingular real-valued n by n matrix, x is a one by n vector of variables, and b is a one by n vector of constants. We wish to solve this system of equations for x . In many applications, A is a large sparse matrix; that is, A has many zero elements. If the system is solved using Gaussian elimination or some other direct method, many of the zeros in A may become non-zero. To make the solution process efficient, we would like to avoid having to explicitly examine the zeros of A, and to keep the number of non-zero elements small. Therefore, the nonzero pattern of U consists of all positions which contain nonzero in the upper triangle of A, plus the fill-in, which is the set of positions of the additional nonzero generated during the elimination. • There are three strong reasons why fill-in is undesirable: • Storage must be allocated for the new nonzero • The computer time required to achieve the factorization increases rapidly when the amount of fill increases, because many more arithmetic operations are required • Error bounds increase when the amount of fill increases
Proposal • when A is positive definite pivots can be selected from the diagonal • the rows and columns of A can be permuted in any way preserves symmetry, and then elimination can be performed pivoting sequentially on the diagonal elements • that the amount of fill-in may depend drastically on the permutation selected • Band and profile methods are methods regard fill-in in a global manner, and confine it to certain areas of the matrix
The minimum degree algorithm • The algorithms which try to reduce the fill-in caused by elimination when the matrix is symmetric, positive definite and sparse, but has no particular structure. • The minimum degree algorithm (Tinney, 1969) • general purpose pivoting method • simple, cheap and effective • for problems which are not too large • he symmetric version of the Markowitz algorithm for un symmetric matrices
Central idea of minimum degree To achieve a local minimization of fill-in and number of operations by selecting, at each stage of elimination and among all possible pivots, that row and column which introduces the least number of nonzeros in the triangular factors. Consider Gauss elimination by columns, which can easily be combined with the minimum degree algorithm. At the kth stage of elimination the matrix has zeros in columns 1 to k - 1 below the diagonal All we have to do is: to examine the active submatrix J formed by rows k to n and columns k to n of , select a row (or column) with the minimum number of nonzeros, say row (or column) T 1, and interchange row 1 with row k and column 1 with column k of the complete matrix A(k) before performing step k of elimination
Implementing Total fill in16 Total fill in10
