The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive de...The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.展开更多
To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a spe...To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a special form (e.g. the dissection form [11]). The main tool for deriving our methods is the diagonally compensated reduction (cf. [1]). The convergence of such methods is also discussed by using this tool. [WT5,5”HZ]展开更多
For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 30...For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).展开更多
It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negati...It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negative) definiteness, and Hurwitz(Shur) stability, as well as the aperiodic property of polytopes of symmetric matrices are generalized, and a series of insightful necessary and sufficient conditions for some general set of symmetric matrices contained in Sn(α,β) are presented,which are directly available for analysis of the positive(negative) definiteness, Hurwitz(Shur) stability and the aperiodic property of a wide class of sets of symmetric matrices.展开更多
A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The conv...A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.展开更多
文摘The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.
文摘To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a special form (e.g. the dissection form [11]). The main tool for deriving our methods is the diagonally compensated reduction (cf. [1]). The convergence of such methods is also discussed by using this tool. [WT5,5”HZ]
文摘For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).
文摘It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negative) definiteness, and Hurwitz(Shur) stability, as well as the aperiodic property of polytopes of symmetric matrices are generalized, and a series of insightful necessary and sufficient conditions for some general set of symmetric matrices contained in Sn(α,β) are presented,which are directly available for analysis of the positive(negative) definiteness, Hurwitz(Shur) stability and the aperiodic property of a wide class of sets of symmetric matrices.
基金Subsidized by The Special Funds For Major State Basic Research Projects G1999032803.
文摘A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.
