Existence and Stability of Choleski Q.I.F. for Symmetric Linear Systems

Recently, Evans [1] has given a variant of the classical Choleski factorization, called Choleski Q.I.F., for the solution of symmetric linear systems. In the present paper we address the questions of existence and stability of the factorization. We show that Evans' Choleski Q.I.F. exists if the coefficient matrix is symmetric positive definite. Our proof of existence of the factorization is via obtaining the diagonal elements of the factor matrices in a closed-form in terms of certain principal minors of the coefficient matrix. By interpreting the Choleski Q.I.F. as an elimination procedure in which eliminations are carried out, alternately, in columns from the left and the right, we show that the Choleski Q.I.F. is also (numerically) strongly stable if the coefficient matrix is symmetric positive definite. We include arithmetical operations counts for the Choleski Q.I.F.; these agree with the counts for the classical Choleski factorization.

Keywords:

• Symmetric Positive Definite Linear Systems
• Choleski Q.I.F.
• Elimination Interpretation
• Operations Counts
• Existence
• Strong Stability