Download Computer Solutions of Large Sparse Positive Definite Systems by Alan George PDF

By Alan George

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Extra resources for Computer Solutions of Large Sparse Positive Definite Systems

Sample text

Show that a symmetric matrix A is irreducible if and only if its associated graph GA is connected. Let A be a symmetric matrix. 3) if and only if there exists the path (XI, X2, •••• XN) in the lUlliociuted gntph G I ' . Chap. 3: Some Graph Theory Notation 41 Characterize the graphs associated with the following matrices. 2 Computer Representation of Graphs In general, the performances of graph algorithms are quite sensitive to the way the graphs are represented. For our purposes, the basic operation used is that of retrieving adjacency relations between nodes.

2 The number of operations required to compute the triangular factor L of the matrix A is given by I N- 1 - ~ 71(Vi)[71(Vi)+3] 2 i =1 I =- N - 1 ~ [71(L. i )-I][71(L. i )+2]. 7) Proof The three formulations of the factorization differ only in the order in which operations are performed. 2) is used. 71(Vi)[71(Vi) + 1] operations are needed to form the symmetric matrix 2 _ The result follows from summing over all the steps. H) Chap. lN 2 _ 2N. 9) 3 Consider also the Cholesky factorization of a symmetric positive definite tridiagonal matrix, an example of a sparse matrix.

We will often use the phrase "the component Chap. 1 An example showing how the array prescribe a subgraph of G. 1. i MASK(i) 1 1 2 1 3 0 4 1 5 1 6 0 MASK 45 can be used to ® Subgraph of G prescribed by MASK. prescribed by ROOT and MASK" to refer to this connected subgraph. 1 would specify the graph 1~-----{ To summarize, some frequently used parameters in our subroutines, along with their contents are listed as follows: (XADJ, ADJNCY) PERM INVP MASK ROOT the array pair which stores the graph in its original ordering.