M. Kashiwagi, “A Method for Determining Eigensolutions of Large, Sparse, Symmetric Matrices by the Preconditioned Conjugate Gradient Method in the Generalized Eigenvalue Problem,” Journal of Structural Engineering, Vol. 73, No. 629, 2008, pp. 1209-1217.

has been cited by the following article:

• TITLE: Derivative of a Determinant with Respect to an Eigenvalue in the LDU Decomposition of a Non-Symmetric Matrix

  AUTHORS: Mitsuhiro Kashiwagi

  KEYWORDS: Derivative of Determinant; Non-Symmetric Matrix; Eigenvalue; Band Matrix; LDU Decomposition

  JOURNAL NAME: Applied Mathematics, Vol.4 No.3, March 21, 2013

  ABSTRACT: 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.