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We demonstrate that, when computing the LDU decomposition (a typical example of a direct solution method), it is possible to obtain the derivative of a determinant with respect to an eigenvalue of a non-symmetric matrix. Our proposed method augments an LDU decomposition program with an additional routine to obtain a program for easily evaluating the derivative of a determinant with respect to an eigenvalue. The proposed method follows simply from the process of solving simultaneous linear equations and is particularly effective for band matrices, for which memory requirements are significantly reduced compared to those for dense matrices. We discuss the theory underlying our proposed method and present detailed algorithms for implementing it.
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.
In recent years, many software
development organizations have been assessing and analyzing their software
product’s reliability/quality and judging whether the software product is
releasable by using Software Reliability Growth Models (SRGMs) at the final
stage of software development. The usage of SRGMs originates in the advantage that
various reliability analysis results based on the SRGMs can be acquired easily.
However, it is very difficult for general software project managers to grasp
the achievement level of reliability/quality based on its analysis results
because some sort of professional knowledge is required in order to understand
the information on the attainment progress of software product’s
reliability/quality. Moreover, it is also difficult for software project
managers and inspectors who do not deeply comprehend the details of their
project to evaluate the degree of software reliability and quality, if they
assess it without grasping the live development situation and only see the
documents submitted from their staff. In this paper, we propose a new analysis
concept for assessing the software product’s reliability/quality, and
illustrate the output results obtained by a tool, the SafeMan.