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 matr...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.展开更多
A generalization of the Householder transformation,renamed as elementary matrix by A.S.Householder:Unitary transformation of a nonsymmetric matrix,J.ACM,5(4),339–342,1958,was introduced by LaBudde(Math Comput 17(84):...A generalization of the Householder transformation,renamed as elementary matrix by A.S.Householder:Unitary transformation of a nonsymmetric matrix,J.ACM,5(4),339–342,1958,was introduced by LaBudde(Math Comput 17(84):433–437,1963)as a tool to obtain a tridiagonal matrix similar to a given square matrix.Some of the free parameters of the transformation can be chosen to attain better numerical properties.In this work,we study the spectral properties of the transformation.We also propose a special choice for free coefficients of that transformation to minimize its condition number.The transformation with such suitable choice of parameters is called optimal.展开更多
文摘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.
基金supported by The Natural Science Foundation of Hunan Province(No.2021JJ40708)The Natural Science Foundation of the Higher Education Institutions of Jiangsu Province(No.17KJB110008)。
基金The work of the first and third authors was partially supported by National Council for Scientific and Technological Development(CNPq),Brazil.
文摘A generalization of the Householder transformation,renamed as elementary matrix by A.S.Householder:Unitary transformation of a nonsymmetric matrix,J.ACM,5(4),339–342,1958,was introduced by LaBudde(Math Comput 17(84):433–437,1963)as a tool to obtain a tridiagonal matrix similar to a given square matrix.Some of the free parameters of the transformation can be chosen to attain better numerical properties.In this work,we study the spectral properties of the transformation.We also propose a special choice for free coefficients of that transformation to minimize its condition number.The transformation with such suitable choice of parameters is called optimal.
