A Necessary and Sufficient Symbolic Condition for the Existence of Incomplete Cholesky Factorization Xiaoge Wang and Randall Bramley Department of Computer Science Indiana University - Bloomington

A Necessary and Sufficient Symbolic Condition for the Existence of Incomplete Cholesky Factorization

This paper presents a suucient condition on sparsity patterns for the existence of the incomplete Cholesky factorization. Given the sparsity pattern P(A) of a matrix A, and a target sparsity pattern P satisfying the condition, incomplete Cholesky factorization successfully completes for all symmetric positive deenite matrices with the same pattern P(A). This condition is also necessary in the s...

Incomplete Cholesky Factorization with Sparsity Pattern Modi cation

This paper proposes, analyzes, and numerically tests methods to assure the existence of incomplete Cholesky (IC) factorization preconditioners, based solely on the target sparsity pattern for the triangular factor R. If the sparsity pattern has a simple property (called property C+), then the IC factor exists in exact arithmetic. Two algorithms for modifying the target sparsity pattern to have ...

CIMGS: An Incomplete Orthogonal FactorizationPreconditioner

A new preconditioner for symmetric positive definite systems is proposed, analyzed, and tested. The preconditioner, compressed incomplete modified Gram–Schmidt (CIMGS), is based on an incomplete orthogonal factorization. CIMGS is robust both theoretically and empirically, existing (in exact arithmetic) for any full rank matrix. Numerically it is more robust than an incomplete Cholesky factoriza...

Splib: a Library of Iterative Methods for Sparse Linear Systems

Generalized Continuous Frames for Operators

In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ mathcal{H} $ wi...