TY - JOUR
TI - The Generalized Toda Flow, the QR Algorithm and the Center Manifold Theory
AU - Chu, Moody T.
T2 - SIAM Journal on Algebraic Discrete Methods
AB - A continuous version of the classical QR algorithm, known as the Toda flow, is generalized to complex-valued, full and nonsymmetric matrices. It is shown that this generalized Toda flow, when sampled at integer times, gives the same sequence of matrices as the OR algorithm applied to the matrix exp $(G(X_0 ))$. When $G(X) = X$, global convergence is deduced for the case of distinct real eigenvalues. This convergence property can also be understood locally by the center manifold theory. It is shown that the manifold of upper triangular matrices with decreasing main diagonal entries is the stable center manifold for the Toda flow. One interesting example is given to demonstrate geometrically the dynamical behavior of this flow.
DA - 1984/6//
PY - 1984/6//
DO - 10.1137/0605020
VL - 5
IS - 2
SP - 187-201
UR - http://dx.doi.org/10.1137/0605020
ER -
TY - JOUR
TI - On the Global Convergence of the Toda Lattice for Real Normal Matrices and Its Applications to the Eigenvalue Problem
AU - Chu, Moody T.
T2 - SIAM Journal on Mathematical Analysis
AB - The asymptotic behavior of the Toda lattice, when acting on real normal matrices, is studied. It is shown that the solution flow eventually converges to a diagonal block form where for a real eigenvalue the associated block is of size $1 \times 1$ with that eigenvalue as its element and for complex-conjugate pairs of eigenvalues the associated block is of size $2 \times 2$ with the real part as its diagonal elements and the (negative) imaginary part as its off-diagonal elements. This result generalizes the well-known asymptotic behavior of Jacobi matrices and is consistent with that from the $QR$-algorithm.
DA - 1984/1//
PY - 1984/1//
DO - 10.1137/0515004
VL - 15
IS - 1
SP - 98-104
J2 - SIAM J. Math. Anal.
LA - en
OP -
SN - 0036-1410 1095-7154
UR - http://dx.doi.org/10.1137/0515004
DB - Crossref
ER -
TY - JOUR
TI - A simple application of the homotopy method to symmetric eigenvalue problems
AU - Chu, M.T.
T2 - Linear Algebra and Its Applications
AB - The homotopy method is used to find all eigenpairs of symmetric matrices. A special homotopy is constructed for Jacobi matrices. It is shown that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. These curves are solutions of a certain ordinary differential equation with different initial values. Hence, they can be followed numerically. Incorporated with sparse matrix techniques, this method might be used to solve eigenvalue problems for large scale matrices.
DA - 1984///
PY - 1984///
DO - 10.1016/0024-3795(84)90160-5
VL - 59
IS - C
SP - 85-90
UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-0001806232&partnerID=MN8TOARS
ER -
TY - RPRT
TI - The algebraic theory of integration
AU - Kaltofen, E.
A3 - Rensselaer Polytechnic Institute, Department of Computer Science
DA - 1984///
PY - 1984///
M3 - Lecture Notes
PB - Rensselaer Polytechnic Institute, Department of Computer Science
ER -
TY - CONF
TI - The modular equation of order 11
AU - Kaltofen, E.
AU - Yui, N.
C2 - 1984///
C3 - Third Macsyma Users' Conference
DA - 1984///
SP - 472–485
PB - General Electric
ER -
TY - CHAP
TI - On a theorem by R. Dedekind
AU - Kaltofen, E.
T2 - DOPO LE PAROLE, Album in Honor of A. K. Lenstra's Doctorate
A2 - Lenstra, H.W., Jr.
A2 - Lenstra, J.K.
A2 - van Emde Boas, P.
PY - 1984/5//
ER -
TY - JOUR
TI - On using orthogonal functions with singular systems
AU - Campbell, Stephen L.
T2 - IEE Proceedings D-Control Theory & Applications
DA - 1984///
PY - 1984///
VL - 131
IS - 6
SP - 267-268
ER -
TY - CONF
TI - Explicit methods for solving singular differential equation systems
AU - Campbell, Stephen L.
T2 - Midwest Symposium on Circuits and Systems
C2 - 1984///
C3 - Proceedings of the 27th Midwest Symposium on Circuits and Systems
DA - 1984///
PY - 1984///
SP - 696–699
ER -
TY - JOUR
TI - The numerical solution of differential algebraic equation systems
AU - Campbell, Stephen L.
T2 - All About Simulators
DA - 1984///
PY - 1984///
VL - 14
IS - 1
SP - 7–12
ER -
TY - CONF
TI - Non-BDF methods for the solution of time varying implicit differential equations
AU - Campbell, Stephen L.
T2 - American Control Conference
C2 - 1984///
C3 - Proceedings of the 1984 American Control Conference
DA - 1984///
PY - 1984///
DO - 10.1109/ACC.1984.4171500
VL - 21
SP - 1315–1318
ER -
TY - CONF
TI - Nonlinear singular systems and contraction mappings
AU - Campbell, Stephen L.
AU - Rodriguez, J.
T2 - American Control Conference
C2 - 1984///
C3 - Proceedings of the 1984 American Control Conference
DA - 1984///
PY - 1984///
DO - 10.1109/ACC.1984.4171540
VL - 21
SP - 1513–1519
ER -
TY - RPRT
TI - A parallel QR method using fast Givens’ rotations
AU - Ipsen, I.C.F.
A3 - Department of Computer Science, Yale University
DA - 1984///
PY - 1984///
M1 - 299
M3 - Research Report
PB - Department of Computer Science, Yale University
SN - 299
ER -
TY - CHAP
TI - A note on the Risch differential equation
AU - Kaltofen, Erich
T2 - EUROSAM 84
A2 - Fitch, J.
T3 - Lecture Notes in Computer Science
PY - 1984/12/1/
DO - 10.1007/bfb0032858
SP - 359–366
PB - Springer-Verlag
SN - 354013350X
SV - 174
UR - http://dx.doi.org/10.1007/bfb0032858
ER -
TY - CHAP
TI - Explicit construction of the hilbert class fields of imaginary quadratic fields with class numbers 7 and 11
AU - Kaltofen, Erich
AU - Yui, Noriko
T2 - EUROSAM 84
A2 - Fitch, J.
T3 - Lecture Notes in Computer Science
PY - 1984///
DO - 10.1007/bfb0032853
SP - 310–320
PB - Springer-Verlag
SN - 354013350X
SV - 174
UR - http://dx.doi.org/10.1007/bfb0032853
ER -
TY - CHAP
TI - Effective Hilbert irreducibility
AU - Kaltofen, Erich
T2 - EUROSAM 84
PY - 1984///
DO - 10.1007/bfb0032850
SP - 277–284
PB - Springer-Verlag
SN - 354013350X
UR - http://dx.doi.org/10.1007/bfb0032850
ER -
TY - JOUR
TI - Comments on a result of Yin, Bai, and Krishnaiah for large dimensional multivariate F matrices
AU - Silverstein, Jack W
T2 - Journal of Multivariate Analysis
AB - A theorem in Yin, Bai, and Krishnaiah ( J. Multivariate Anal. 13 (1983), 508–516) shows that the smallest eigenvalue of a class of large dimensional sample covariance matrices stays almost surely bounded away from zero. The theorem assumes a certain restriction on the class of matrices. With slight modifications of the proof in op cit, it is shown here that the theorem is true for all relevant matrices.
DA - 1984/12//
PY - 1984/12//
DO - 10.1016/0047-259x(84)90059-9
VL - 15
IS - 3
SP - 408-409
J2 - Journal of Multivariate Analysis
LA - en
OP -
SN - 0047-259X
UR - http://dx.doi.org/10.1016/0047-259x(84)90059-9
DB - Crossref
ER -
TY - RPRT
TI - On Limit of the Largest Eigenvalue of the Large Dimensional Sample Covariance Matrix.
AU - Yin, Y. Q.
AU - Bai, Z. D.
AU - Krishnaiah, P. R.
A3 - Defense Technical Information Center
AB - Abstract : The authors showed that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity. The above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products (SP) matrix exist. Key words include: Largest eigenvalue, Sample covariance matrix, Large dimensional random matrices, Limit.
DA - 1984/10/1/
PY - 1984/10/1/
DO - 10.21236/ada150589
PB - Defense Technical Information Center
UR - http://dx.doi.org/10.21236/ada150589
DB - Crossref
ER -
TY - JOUR
TI - Some limit theorems on the eigenvectors of large dimensional sample covariance matrices
AU - Silverstein, Jack W
T2 - Journal of Multivariate Analysis
AB - Let { v ij } i , j = 1, 2,…, be i.i.d. standardized random variables. For each n , let V n = ( v ij ) i = 1, 2,…, n ; j = 1, 2,…, s = s ( n ), where ( n s ) → y > 0 as n → ∞, and let M n = ( 1 s )V n V n T . Previous results [7, 8] have shown the eigenvectors of M n to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process X n on [0, 1], constructed from the eigenvectors of M n , is known to converge weakly, as n → ∞, on D [0, 1] to Brownian bridge when v 11 is N (0, 1), but it is not known whether this property holds for any other distribution. The present paper provides evidence that this property may hold in the non-Wishart case in the form of limit theorems on the convergence in distribution of random variables constructed from integrating analytic function w.r.t. X n ( F n ( x )), where F n is the empirical distribution function of the eigenvalues of M n . The theorems assume certain conditions on the moments of v 11 including E ( v 11 4 ) = 3, the latter being necessary for the theorems to hold.
DA - 1984/12//
PY - 1984/12//
DO - 10.1016/0047-259x(84)90054-x
VL - 15
IS - 3
SP - 295-324
J2 - Journal of Multivariate Analysis
LA - en
OP -
SN - 0047-259X
UR - http://dx.doi.org/10.1016/0047-259x(84)90054-x
DB - Crossref
ER -
TY - JOUR
TI - Applications of the method to transport calculations, Trans
AU - Kelley, C.T.
T2 - Transport Theory and Statistical Physics
DA - 1984///
PY - 1984///
VL - 13
SP - 85–96
ER -
TY - JOUR
TI - Applications of the {F_N} method to transport calculations
AU - Kelley, C T
T2 - Trans. Th. Stat. Phys.
DA - 1984///
PY - 1984///
VL - 13
SP - 85-96
ER -
TY - JOUR
TI - Regularizations of linear time varying singular systems
AU - Campbell, Stephen L.
T2 - Automatica
AB - Singular systems of differential equations arising in reduced-order models are often regularized by the introduction of small parameters. The types of regularization used for linear time invariant systems need not work on linear time varying systems. This paper gives examples and discusses this behavior. Conditions under which the standard type of regularization works are developed.
DA - 1984/5//
PY - 1984/5//
DO - 10.1016/0005-1098(84)90052-9
VL - 20
IS - 3
SP - 365-370
J2 - Automatica
LA - en
OP -
SN - 0005-1098
UR - http://dx.doi.org/10.1016/0005-1098(84)90052-9
DB - Crossref
ER -
TY - CONF
TI - Nonlinear time-varying generalized state-space systems: An overview
AU - Campbell, Stephen
T2 - The 23rd IEEE Conference on Decision and Control
AB - This paper will review the current literature on nonlinear and time varying generalized state-space systems of the form F(t,y,y') = O.
C2 - 1984/12//
C3 - The 23rd IEEE Conference on Decision and Control
DA - 1984/12//
DO - 10.1109/cdc.1984.272332
PB - IEEE
UR - http://dx.doi.org/10.1109/cdc.1984.272332
DB - Crossref
ER -
TY - JOUR
TI - Review of polynomials and linear control systems, by Stephen Barnett
AU - Campbell, Stephen L.
T2 - Linear Algebra and its Applications
DA - 1984/9//
PY - 1984/9//
DO - 10.1016/0024-3795(84)90038-7
VL - 61
SP - 289-290
ER -
TY - CONF
TI - Singular Value Decomposition With Systolic Arrays
AU - Ipsen, Ilse
T2 - 28th Annual Technical Symposium
A2 - Bromley, Keith
AB - Systolic arrays for determining the singular value decomposition of a mxn, m n, matrix A of bandwidth w are presented. After A has been reduced to bidiagonal form B by means of Givens plane rotations, the singular values of B are computed by the Golub-Reinsch iteration. The products of plane rotations form the matrices of left and right singular vectors. Assuming each processor can compute or supply a plane rotation, O(wn) processors accomplish the reduction to bidiagonal form in O(np) steps, where p is the number of superdiagonals. A constant number of processors then determines each singular value in about 6n steps. The singular vectors are computed by rerouting the rotations through the arrays used for the reduction to bidiagonal form, or else along the way by employing another rectangular array of O(wm) processors.
C2 - 1984/11/28/
C3 - Real-Time Signal Processing VII
DA - 1984/11/28/
DO - 10.1117/12.944004
PB - SPIE
UR - http://dx.doi.org/10.1117/12.944004
DB - Crossref
ER -
TY - JOUR
TI - Dynamical behavior of differential equation models of frequency and density dependent populations
AU - Selgrade, J. F.
AU - Namkoong, G.
T2 - Journal of Mathematical Biology
DA - 1984///
PY - 1984///
VL - 19
IS - 1
SP - 133
ER -