## A list of matrix flows with applications (1994)

Venue: | in Hamiltonian and Gradients Flows, Algorithms and Control |

Citations: | 15 - 1 self |

### BibTeX

@INPROCEEDINGS{Chu94alist,

author = {Moody T. Chu},

title = {A list of matrix flows with applications},

booktitle = {in Hamiltonian and Gradients Flows, Algorithms and Control},

year = {1994},

pages = {87--97}

}

### OpenURL

### Abstract

Many mathematical problems, such as existence questions, are studied by using an appropriate realization process, either iteratively or continuously. This article is a collection of di erential equations that have been proposed as special continuous realization processes. In some cases, there are remarkable connections betweensmooth ows and discrete numerical algorithms. In other cases, the ow approach seems advantageous in tackling very di cult problems. The ow approach has potential applications ranging from new development ofnumerical algorithms to the theoretical solution of open problems. Various aspects of the recent development and applications of the ow approach are reviewed in this article. 1

### Citations

4676 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
Citation Context ... [27], physics applications and boundary value problems[38, 44], and polynomial systems [35, 36]. 2.3 Scaled Toda Flows Original Problem: Reduce a square matrix A0 2 R n n to a certain canonical form =-=[31]-=-, e.g., triangularization, so as to solve the eigenvalue problem Discrete Method: A0x = x: (10) 1. For triangularization, use the unshifted QR algorithm: Ak = QkRk =) Ak+1 = RkQk (11) where QkRk is th... |

1087 |
The Algebraic Eigenvalue Problem
- Wilkinson
(Show Context)
Citation Context ... A0x = x: (10) 1. For triangularization, use the unshifted QR algorithm: Ak = QkRk =) Ak+1 = RkQk (11) where QkRk is the QR decomposition of Ak� orany other QR-type algorithms, e.g., the LU algorithm =-=[28, 45]-=-. 2. For a general non-zero pattern which A0 is reduced to, no discrete method is available. Flow: 4s1. The Toda ow dX dt =[X� 0(X)] (12) where [A� B] =AB ; BA, 0(X) =X ; ; X ;T and X ; is the strictl... |

510 |
Iterative Solution of Nonlinear Equations
- Ortega, Rheinboldt
- 1970
(Show Context)
Citation Context ...mes [25, 8]. 2.2 Homotopy Flows Original Problem: Solve the nonlinear equation f(x) =0 (5) where f : R n ;! R n is continuously di erentiable. Discrete Method: A classical method is the Newton method =-=[37]-=- xk+1 = xk ; k(f 0 (xk)) ;1 f(xk): (6) Motivation: At least two ways to motivate the continuous ows: 1. Think of (6) as one Euler step with step size k applied to the di erential equation [32] dx ds =... |

178 |
Applied Iterative Methods
- Hageman, Young
- 1981
(Show Context)
Citation Context ...st of Flows 2.1 Linear Stationary Flows Original Problem: Solve the linear equation where A 2 R n n and x, b 2 R n . Ax = b (1) 2swhere Discrete Method: Most linear stationary methods assume the form =-=[29]-=- xk+1 = Gxk + c� k =0� 1� 2�::: (2) G = I ; Q ;1 A c = Q ;1 b and Q is a splitting matrix of A. Motivation: Think of (2) as one Euler step with unit step size applied to a linear di erential system. F... |

99 |
Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems, Linear Algebra Appl
- Brockett
- 1991
(Show Context)
Citation Context ...ctly into a continuous model [32, 8]� Sometimes a di erential equation arises naturally from a certain physical principles [40, 44]� More often a vector eld is constructed with a speci c task in mind =-=[6, 11,15,16]-=-. We shall report the material only descriptively. For more extensive discussion, readers should refer to the bibliography. We present the ows on a case-by-case basis. For brevity, we encapsulate the ... |

75 |
Pathways to solutions, fixed points, and equilibria
- Garcia, Zangwill
- 1981
(Show Context)
Citation Context ...ridge really makes the desired connection to t =0[1,2]. Example: Successful applications with specially formulated homotopyfunctions include eigenvalue problems [9, 34], nonlinear programming problem =-=[27]-=-, physics applications and boundary value problems[38, 44], and polynomial systems [35, 36]. 2.3 Scaled Toda Flows Original Problem: Reduce a square matrix A0 2 R n n to a certain canonical form [31],... |

48 | The projected gradient method for least squares matrix approximations with spectral constraints
- Chu
- 1990
(Show Context)
Citation Context ...ctly into a continuous model [32, 8]� Sometimes a di erential equation arises naturally from a certain physical principles [40, 44]� More often a vector eld is constructed with a speci c task in mind =-=[6, 11,15,16]-=-. We shall report the material only descriptively. For more extensive discussion, readers should refer to the bibliography. We present the ows on a case-by-case basis. For brevity, we encapsulate the ... |

46 | The formulation and analysis of numerical methods for inverse eigenvalue problems
- Friedland, Nocedal, et al.
- 1987
(Show Context)
Citation Context ...problem [22]. 2. Find a symmetric non-negative matrix P that has the prescribed set as its spectrum. Discrete Method: A few locally convergent Newton-like algorithms are available for the rst problem =-=[26, 33]-=-. Little is known for the non-negative matrix problem [3]. Motivation: Minimize the distance between the isospectral surface and the set of matrices of desired form. Flow: 1. Inverse eigenvalue proble... |

36 |
Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems
- Morgan
- 1987
(Show Context)
Citation Context ...s with specially formulated homotopyfunctions include eigenvalue problems [9, 34], nonlinear programming problem [27], physics applications and boundary value problems[38, 44], and polynomial systems =-=[35, 36]-=-. 2.3 Scaled Toda Flows Original Problem: Reduce a square matrix A0 2 R n n to a certain canonical form [31], e.g., triangularization, so as to solve the eigenvalue problem Discrete Method: A0x = x: (... |

28 | HOMPACK: A Suite of Codes for Globally Convergent Homotopy Algorithms - Watson, Billups, et al. - 1987 |

26 |
Least squares matching problems
- Brockett
(Show Context)
Citation Context ...the inverse eigenvalue problem and the eigenvalue problem [11]. For the latter, the ow (15) is a continuous analogue of the Jacobi method [24]. 2. The ow (15) generalizes Brockett's double bracket ow =-=[6, 7]-=- which, in turn, has been found to have other applications in sorting, linear programming and total least squares problems [4, 5]. Generalization: The idea of projected gradient ow can be generalized ... |

23 |
On the continuous realization of iterative processes
- Chu
- 1988
(Show Context)
Citation Context ...o abstract problems does exist. The construction of a bridge can be motivated in several di erent ways: Sometimes an existing discrete numerical method may beextended directly into a continuous model =-=[32, 8]-=-� Sometimes a di erential equation arises naturally from a certain physical principles [40, 44]� More often a vector eld is constructed with a speci c task in mind [6, 11,15,16]. We shall report the m... |

20 | Numerical methods for inverse singular value problems
- Chu
- 1992
(Show Context)
Citation Context ...) T ; (B0 + R(X)) T X X 2 (31) R(X) = nX k=1 <X�Bk >Bk (32) and B0�B1�:::�Bn 2 R m n are prescribed mutually orthonormal matrices. Recently insights drawn from (30) give rise to new iterative methods =-=[19]-=-. 2.7 Complex Flows Original Problem: Most of the discussion hitherto can be generalized to the complex-valued cases. One such example is the nearest normal matrix problem [30, 39]. Discrete Method: T... |

18 |
A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map
- Bloch, Brockett, et al.
- 1990
(Show Context)
Citation Context ...i method [24]. 2. The ow (15) generalizes Brockett's double bracket ow [6, 7] which, in turn, has been found to have other applications in sorting, linear programming and total least squares problems =-=[4, 5]-=-. Generalization: The idea of projected gradient ow can be generalized to other types of approximation problems as will be seen below. 2.5 Simultaneous Reduction Flows Original Problem: Simultaneous r... |

18 |
Numerical linear algebra aspects of globally convergent homotopy methods
- WATSON
- 1986
(Show Context)
Citation Context ...ent ways: Sometimes an existing discrete numerical method may beextended directly into a continuous model [32, 8]� Sometimes a di erential equation arises naturally from a certain physical principles =-=[40, 44]-=-� More often a vector eld is constructed with a speci c task in mind [6, 11,15,16]. We shall report the material only descriptively. For more extensive discussion, readers should refer to the bibliogr... |

16 | A continuous Jacobi-like approach to the simultaneous reduction of real matrices
- Chu
- 1991
(Show Context)
Citation Context ...ctly into a continuous model [32, 8]� Sometimes a di erential equation arises naturally from a certain physical principles [40, 44]� More often a vector eld is constructed with a speci c task in mind =-=[6, 11,15,16]-=-. We shall report the material only descriptively. For more extensive discussion, readers should refer to the bibliography. We present the ows on a case-by-case basis. For brevity, we encapsulate the ... |

15 | Constructing symmetric nonnegative matrices with prescribed eigenvalues by differential equations
- Chu, Driessel
- 1991
(Show Context)
Citation Context |

15 |
Numerical linear algorithms and group theory
- Della-Dora
- 1975
(Show Context)
Citation Context ...e Toda ow and the QR algorithm was discovered by Symes [40]. Example: Di erent choices of the scaling matrix K give rise to di erent isospectral ows, including many already proposed in the literature =-=[12, 18,21, 41, 42]-=-. In particular, (13) can be used to generate special canonical forms that no other methods can [10]. 2.4 Projected Gradient Flows Original Problem: Let S(n) andO(n) denote, respectively, the subspace... |

14 |
Homotopy algorithm for symmetric eigenvalue problems
- Li, Rhee
- 1989
(Show Context)
Citation Context ... into account in(9) in order that the bridge really makes the desired connection to t =0[1,2]. Example: Successful applications with specially formulated homotopyfunctions include eigenvalue problems =-=[9, 34]-=-, nonlinear programming problem [27], physics applications and boundary value problems[38, 44], and polynomial systems [35, 36]. 2.3 Scaled Toda Flows Original Problem: Reduce a square matrix A0 2 R n... |

13 |
Numerical Analysis of Parameterized Nonlinear Equations
- Rheinboldt
- 1986
(Show Context)
Citation Context .... Example: Successful applications with specially formulated homotopyfunctions include eigenvalue problems [9, 34], nonlinear programming problem [27], physics applications and boundary value problems=-=[38, 44]-=-, and polynomial systems [35, 36]. 2.3 Scaled Toda Flows Original Problem: Reduce a square matrix A0 2 R n n to a certain canonical form [31], e.g., triangularization, so as to solve the eigenvalue pr... |

12 |
Matrix nearness problems and applications, in Applications of Matrix Theory
- Higham
- 1989
(Show Context)
Citation Context ...o new iterative methods [19]. 2.7 Complex Flows Original Problem: Most of the discussion hitherto can be generalized to the complex-valued cases. One such example is the nearest normal matrix problem =-=[30, 39]-=-. Discrete Method: The Jacobi algorithm [39] can be used. Motivation: The nearest normal matrix problem is equivalent to Minimize H(U) := 1 kU AU ; diag(U AU)k2 2 Subject to U 2U(n) (33) where U(n) is... |

10 |
Global homotopies and Newton methods
- Keller
- 1978
(Show Context)
Citation Context ...o abstract problems does exist. The construction of a bridge can be motivated in several di erent ways: Sometimes an existing discrete numerical method may beextended directly into a continuous model =-=[32, 8]-=-� Sometimes a di erential equation arises naturally from a certain physical principles [40, 44]� More often a vector eld is constructed with a speci c task in mind [6, 11,15,16]. We shall report the m... |

9 |
Spectral properties of nite Toeplitz matrices
- Delsarte, Genin
- 1983
(Show Context)
Citation Context ...mal, nd c =[c1�:::�cn] such that A(c) :=A0 + nX i=1 ciAi (23) has the prescribed set as its spectrum. The special case is where A(c) is aToeplitz matrix which isknown, thus far, to be an open problem =-=[22]-=-. 2. Find a symmetric non-negative matrix P that has the prescribed set as its spectrum. Discrete Method: A few locally convergent Newton-like algorithms are available for the rst problem [26, 33]. Li... |

8 | A continuous approximation to the generalized Schur decomposition
- Chu
- 1986
(Show Context)
Citation Context ...est matrices that have the desired form. 7s2. Just as the Toda lattice (12) models the QR algorithm, the system (21) models the SVD algorithm [14] for the A 2 R m n , and (22) models the QZ algorithm =-=[13]-=- for the matrix pencil (A1�A2) 2 R n n R n n . 2.6 Inverse Eigenvalue Flows Original Problem: Given a set of real numbers f 1�:::� ng, consider two kinds of inverse eigenvalue problems: 1. Given A0�::... |

8 |
Di erential equations for the symmetric eigenvalue problem
- Deift, Nanda, et al.
- 1983
(Show Context)
Citation Context ... =[X� K dt X] (13) where K is a constant matrix and represent the Hadamard product. Initial Conditions: X(0) = A0. Special Features: 1. The time-1 map of the Toda ow is equivalent to the QR algorithm =-=[20,40]-=-. 2. The time-1 map of the scaled Toda ow also enjoys a QR-like algorithm [18]. 3. For symmetric X, K is necessarily skew-symmetric. Asymptotic behavior of (13) is completely known [18]. Motivation: T... |

8 |
Nevanlinna O., What do multistep methods approximate
- Eirola
- 1988
(Show Context)
Citation Context ...r than the would-be condition for the convergence of (2). Generalization: Solving (3) by a numerical method amounts to a new iterative scheme, including highly complicated multistep iterative schemes =-=[25, 8]-=-. 2.2 Homotopy Flows Original Problem: Solve the nonlinear equation f(x) =0 (5) where f : R n ;! R n is continuously di erentiable. Discrete Method: A classical method is the Newton method [37] xk+1 =... |

7 |
Simplicial and continuation methods for approximating xed points and solutions to systems of equations
- Allgower, Georg
- 1980
(Show Context)
Citation Context ...o the di erential equation [32] dx ds = ;(f 0 (x)) ;1 f(x): (7) 3s2. Connect the system (5) to a trivial system by, for example, H(x� t) =f(x) ; tf(x0) (8) where x0 is an arbitrarily xed point in R n =-=[1, 2]-=-. Generically, the zero set H ;1 (0) is a one-dimensional smooth manifold. Flow: Either (7) or f 0 (x) ; 1 t f(x) dx ds dt ds dx ds dt ds = 0 1 : (9) Initial Conditions: For (7), x(0) can be arbitrary... |

7 |
Homotopy method for general -matrix problems
- Chu, Li, et al.
- 1988
(Show Context)
Citation Context ... into account in(9) in order that the bridge really makes the desired connection to t =0[1,2]. Example: Successful applications with specially formulated homotopyfunctions include eigenvalue problems =-=[9, 34]-=-, nonlinear programming problem [27], physics applications and boundary value problems[38, 44], and polynomial systems [35, 36]. 2.3 Scaled Toda Flows Original Problem: Reduce a square matrix A0 2 R n... |

7 |
Can real symmetric Toeplitz matrices have arbitrary spectra
- Driessel, Chu
(Show Context)
Citation Context ...negative matrix. Generalization: 8s1. For the inverse Toeplitz eigenvalue problem, the descent ow (24) may converge to a stationary point thatisnotToeplitz. Anew ow that seems to converge globally is =-=[23]-=-. where kij(X) := 8< dX dt =[X� k(X)] (28) : xi+1�j ; xi�j;1 if 1 i<j n 0 if 1 i = j n xi�j;1 ; xi+1�j if 1 j<i n 2. The idea of (24) can be generalized to inverse singular value problem: where (29) d... |

7 |
Anumerical approach to the inverse Toeplitz eigenproblem.SIAM
- Laurie
- 1988
(Show Context)
Citation Context ...problem [22]. 2. Find a symmetric non-negative matrix P that has the prescribed set as its spectrum. Discrete Method: A few locally convergent Newton-like algorithms are available for the rst problem =-=[26, 33]-=-. Little is known for the non-negative matrix problem [3]. Motivation: Minimize the distance between the isospectral surface and the set of matrices of desired form. Flow: 1. Inverse eigenvalue proble... |

5 |
Isospectral ows and abstract matrix factorizations
- Chu, Norris
(Show Context)
Citation Context ... rise to di erent isospectral ows, including many already proposed in the literature [12, 18,21, 41, 42]. In particular, (13) can be used to generate special canonical forms that no other methods can =-=[10]-=-. 2.4 Projected Gradient Flows Original Problem: Let S(n) andO(n) denote, respectively, the subspace of all symmetric matrices and the group of all orthogonal matrices in R n n . Let P (X) denote the ... |

4 |
Steepest descent, linear programming
- Bloch
(Show Context)
Citation Context ...i method [24]. 2. The ow (15) generalizes Brockett's double bracket ow [6, 7] which, in turn, has been found to have other applications in sorting, linear programming and total least squares problems =-=[4, 5]-=-. Generalization: The idea of projected gradient ow can be generalized to other types of approximation problems as will be seen below. 2.5 Simultaneous Reduction Flows Original Problem: Simultaneous r... |

4 |
The generalized Toda ow, the QR algorithm and the center manifold theorem
- Chu
(Show Context)
Citation Context ...e Toda ow and the QR algorithm was discovered by Symes [40]. Example: Di erent choices of the scaling matrix K give rise to di erent isospectral ows, including many already proposed in the literature =-=[12, 18,21, 41, 42]-=-. In particular, (13) can be used to generate special canonical forms that no other methods can [10]. 2.4 Projected Gradient Flows Original Problem: Let S(n) andO(n) denote, respectively, the subspace... |

4 |
Closest normal matrix nally found
- Ruhe
(Show Context)
Citation Context ...o new iterative methods [19]. 2.7 Complex Flows Original Problem: Most of the discussion hitherto can be generalized to the complex-valued cases. One such example is the nearest normal matrix problem =-=[30, 39]-=-. Discrete Method: The Jacobi algorithm [39] can be used. Motivation: The nearest normal matrix problem is equivalent to Minimize H(U) := 1 kU AU ; diag(U AU)k2 2 Subject to U 2U(n) (33) where U(n) is... |

4 |
The QR algorithm and scattering for the nite non-periodic Toda lattice
- Symes
(Show Context)
Citation Context ...ent ways: Sometimes an existing discrete numerical method may beextended directly into a continuous model [32, 8]� Sometimes a di erential equation arises naturally from a certain physical principles =-=[40, 44]-=-� More often a vector eld is constructed with a speci c task in mind [6, 11,15,16]. We shall report the material only descriptively. For more extensive discussion, readers should refer to the bibliogr... |

4 |
Isospectral ows
- Watkins
- 1984
(Show Context)
Citation Context ...e Toda ow and the QR algorithm was discovered by Symes [40]. Example: Di erent choices of the scaling matrix K give rise to di erent isospectral ows, including many already proposed in the literature =-=[12, 18,21, 41, 42]-=-. In particular, (13) can be used to generate special canonical forms that no other methods can [10]. 2.4 Projected Gradient Flows Original Problem: Let S(n) andO(n) denote, respectively, the subspace... |

3 |
A di erential equation approach to the singular value decomposition of bidiagonal matrices
- Chu
- 1986
(Show Context)
Citation Context ...ng the distance from the best reduced matrices to the nearest matrices that have the desired form. 7s2. Just as the Toda lattice (12) models the QR algorithm, the system (21) models the SVD algorithm =-=[14]-=- for the A 2 R m n , and (22) models the QZ algorithm [13] for the matrix pencil (A1�A2) 2 R n n R n n . 2.6 Inverse Eigenvalue Flows Original Problem: Given a set of real numbers f 1�:::� ng, conside... |

3 | Least squares approximation by real normal matrices with speci ed spectrum
- Chu
(Show Context)
Citation Context ...ures: The putative nearest normal matrix to A is given by Z := U(1)diag(W(1))U(1) [15]. Generalization: Least square approximation by real normal matrices can also be done by a method described by Chu=-=[17]-=- 3 Conclusion dX dt = X� [X� AT ] ; [X� A T ] T 2 : (36) Most matrix di erential equations by nature are complicated, since the components are coupled into nonlinear terms. Nonetheless, as we have dem... |

3 |
On isospectral gradient ows | solving matrix eigenproblems using di erential equations
- Driessel
- 1986
(Show Context)
Citation Context ...quares approximation problem subject to spectral constraints, the inverse eigenvalue problem and the eigenvalue problem [11]. For the latter, the ow (15) is a continuous analogue of the Jacobi method =-=[24]-=-. 2. The ow (15) generalizes Brockett's double bracket ow [6, 7] which, in turn, has been found to have other applications in sorting, linear programming and total least squares problems [4, 5]. Gener... |

2 |
A survey of homotopy methods for smooth mappings
- Allgower
- 1980
(Show Context)
Citation Context ...o the di erential equation [32] dx ds = ;(f 0 (x)) ;1 f(x): (7) 3s2. Connect the system (5) to a trivial system by, for example, H(x� t) =f(x) ; tf(x0) (8) where x0 is an arbitrarily xed point in R n =-=[1, 2]-=-. Generically, the zero set H ;1 (0) is a one-dimensional smooth manifold. Flow: Either (7) or f 0 (x) ; 1 t f(x) dx ds dt ds dx ds dt ds = 0 1 : (9) Initial Conditions: For (7), x(0) can be arbitrary... |

2 |
Numerical solution of a class of de cient polynomial systems
- Li, Sauer, et al.
(Show Context)
Citation Context ...s with specially formulated homotopyfunctions include eigenvalue problems [9, 34], nonlinear programming problem [27], physics applications and boundary value problems[38, 44], and polynomial systems =-=[35, 36]-=-. 2.3 Scaled Toda Flows Original Problem: Reduce a square matrix A0 2 R n n to a certain canonical form [31], e.g., triangularization, so as to solve the eigenvalue problem Discrete Method: A0x = x: (... |

2 |
Self-similar ows
- Watkins, Elsner
(Show Context)
Citation Context |

1 |
Scaled Toda-like ows
- Chu
- 1992
(Show Context)
Citation Context ...t. Initial Conditions: X(0) = A0. Special Features: 1. The time-1 map of the Toda ow is equivalent to the QR algorithm [20,40]. 2. The time-1 map of the scaled Toda ow also enjoys a QR-like algorithm =-=[18]-=-. 3. For symmetric X, K is necessarily skew-symmetric. Asymptotic behavior of (13) is completely known [18]. Motivation: The Toda lattice originates as a description of a one-dimensional lattice of pa... |