TITLE:
Derivative of a Determinant with Respect to an Eigenvalue in the Modified Cholesky Decomposition of a Symmetric Matrix, with Applications to Nonlinear Analysis
AUTHORS:
Mitsuhiro Kashiwagi
KEYWORDS:
Derivative of a Determinant with Respect to an Eigenvalue; Modified Cholesky Decomposition; Symmetric Matrix; Nonlinear Finite-Element Methods; Singular Points
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.4 No.2,
March
25,
2014
ABSTRACT: In this paper, we obtain a formula for the derivative
of a determinant with respect to an eigenvalue in the modified Cholesky
decomposition of a symmetric matrix, a characteristic example of a direct
solution method in computational linear algebra. We apply our proposed formula
to a technique used in nonlinear finite-element methods and discuss methods for
determining singular points, such as bifurcation points and limit points. In
our proposed method, the increment in arc length (or other relevant quantities)
may be determined automatically, allowing a reduction in the number of basic
parameters. The method is particularly effective for banded matrices, which allow
a significant reduction in memory requirements as compared to dense matrices.
We discuss the theoretical foundations of our proposed method, present
algorithms and programs that implement it, and conduct numerical experiments to
investigate its effectiveness.