## The Lanczos algorithm and Hankel matrix factorization (1992)

Citations: | 8 - 4 self |

### BibTeX

@MISC{Boley92thelanczos,

author = {Daniel L. Boley and Tong J. Lee and Franklin T. Luk},

title = {The Lanczos algorithm and Hankel matrix factorization},

year = {1992}

}

### OpenURL

### Abstract

We explore the connections between the Lanczos algorithm for matrix tridiagonalization and two fast algorithms for Hankel matrix factorization. We show how the asymmetric Lanczos process is related to the Berlekamp-Massey algorithm, and how the symmetrized Lanczos process is related to the Phillips algorithm. We also discuss conditions under which the analysis of Phillips applies. 1

### Citations

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(Show Context)
Citation Context ...he Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in [6] and [26]. Hankel matrices arise in various settings, ranging from system identification =-=[23]-=- to algorithmic fault tolerance [4]. In his 1977 dissertation, Kung [20] studied the Berlekamp-Massey (BM, 1967) algorithm [1], [24] for solving Hankel equations, and remarked that their algorithm is ... |

1099 | The Algebraic Eigenvalue Problem - Wilkinson - 1965 |

438 |
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Citation Context ...l matrices arise in various settings, ranging from system identification [23] to algorithmic fault tolerance [4]. In his 1977 dissertation, Kung [20] studied the Berlekamp-Massey (BM, 1967) algorithm =-=[1]-=-, [24] for solving Hankel equations, and remarked that their algorithm is related to the Lanczos process. There still exists strong interest in a simple exposition of the BM algorithm; see, e.g., [19]... |

422 |
Theory and Practice of Error Control Codes
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(Show Context)
Citation Context ...rrection schemes, the entries of H are the moments of an indefinite inner product of known knots and unknown weights that play the role of errors [4]. This corresponds to the – 13 –sReed-Solomon code =-=[2]-=- and the Berlekamp-Massey Algorithm. If the knots are unknown, they may often be computed as the roots of the n-th degree orthogonal polynomial generated by Algorithm AsymLanczos, as described in [5].... |

379 | An iteration method for the solution of the eigenvalue problem for linear differential and integral
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- 1950
(Show Context)
Citation Context ...Massey algorithm, and how the symmetrized Lanczos process is related to the Phillips algorithm. We also discuss conditions under which the analysis of Phillips applies. 1 Introduction In 1950 Lanczos =-=[22]-=- proposed a method for computing the eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce the given matrix to a tridiagonal form, from which the eigenvalues could be determined. ... |

272 |
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- 1969
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Citation Context ...rices arise in various settings, ranging from system identification [23] to algorithmic fault tolerance [4]. In his 1977 dissertation, Kung [20] studied the Berlekamp-Massey (BM, 1967) algorithm [1], =-=[24]-=- for solving Hankel equations, and remarked that their algorithm is related to the Lanczos process. There still exists strong interest in a simple exposition of the BM algorithm; see, e.g., [19] in 19... |

186 |
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- 1964
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Citation Context ...thms for factorizing a Hankel matrix appeared in [29], [30], [31], and the connection with the Lanczos algorithm either on a nonsymmetric matrix or using an indefinite inner product appeared in [14], =-=[18]-=-, [20], [28]. When the permuted Yule-Walker equations (a Hankel system with a special right hand side) are solved, the resulting method essentially computes the LU factors of the Hankel matrix [1], [2... |

150 | An implementation of the lookahead Lanczos algorithm for non-Hermitian matrices
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- 1993
(Show Context)
Citation Context ...7], [15], [27], with an approach that is similar to that proposed by Berlekamp [1] to factorize a Hankel matrix which is not strongly nonsingular. Also, there is much recent interest, e.g., [3], [5], =-=[9]-=-, [10], [12], in exploring the connections between a modified asymmetric Lanczos algorithm and orthogonal polynomials with respect to an indefinite inner product. 7 Acknowledgements The authors acknow... |

103 |
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Citation Context ... special cases of the Lanczos algorithm. For simplicity in presentation, we have avoided the breakdown problem for the asymmetric Lanczos algorithm. The problem has been considered in [5], [7], [15], =-=[27]-=-, with an approach that is similar to that proposed by Berlekamp [1] to factorize a Hankel matrix which is not strongly nonsingular. Also, there is much recent interest, e.g., [3], [5], [9], [10], [12... |

98 | On the partial stochastic realization problem
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- 1997
(Show Context)
Citation Context ...algorithms for factorizing a Hankel matrix appeared in [29], [30], [31], and the connection with the Lanczos algorithm either on a nonsymmetric matrix or using an indefinite inner product appeared in =-=[14]-=-, [18], [20], [28]. When the permuted Yule-Walker equations (a Hankel system with a special right hand side) are solved, the resulting method essentially computes the LU factors of the Hankel matrix [... |

60 |
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(Show Context)
Citation Context ...ation defining the inner product may not be known. For the case of an ordinary inner product, viz., 〈f,f〉 > 0 for all nonzero f, the problem has been extensively studied in the literature; see, e.g., =-=[11]-=- and references therein. However, only recently has this problem been addressed for more general inner products; see, e.g., the modified Chebyshev algorithm in [12]. This problem was addressed in [5] ... |

55 |
Reduction to tridiagonal form and minimal realizations
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- 1992
(Show Context)
Citation Context ...which the eigenvalues could be determined. A characterization of the breakdowns in the Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in [6] and =-=[26]-=-. Hankel matrices arise in various settings, ranging from system identification [23] to algorithmic fault tolerance [4]. In his 1977 dissertation, Kung [20] studied the Berlekamp-Massey (BM, 1967) alg... |

24 | The unsymmetric Lanczos algorithms and their relations to Padé approximation, continued fractions, and the qd algorithm
- Gutknecht
- 1990
(Show Context)
Citation Context ...impulse response of a dynamical system [23]. The relation between the asymmetric Lanczos process and fast Hankel factorization and/or inversion algorithms has been explored more recently in [3], [4], =-=[16]-=-, [17], [21]. In the next two subsections, we apply the Lanczos process to define two procedures for decomposing an n × n strongly nonsingular Hankel matrix ⎛ ⎜ H = ⎜ ⎝ η0 η1 η2 · · · ⎞ ηn−1 η1 η2 η2 ... |

20 |
The nonsymmetric Lanczos algorithm and controllability
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- 1991
(Show Context)
Citation Context ...m, from which the eigenvalues could be determined. A characterization of the breakdowns in the Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in =-=[6]-=- and [26]. Hankel matrices arise in various settings, ranging from system identification [23] to algorithmic fault tolerance [4]. In his 1977 dissertation, Kung [20] studied the Berlekamp-Massey (BM, ... |

16 |
The inverses of block Hankel and block Toeplitz matrices
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- 1990
(Show Context)
Citation Context ...onse of a dynamical system [23]. The relation between the asymmetric Lanczos process and fast Hankel factorization and/or inversion algorithms has been explored more recently in [3], [4], [16], [17], =-=[21]-=-. In the next two subsections, we apply the Lanczos process to define two procedures for decomposing an n × n strongly nonsingular Hankel matrix ⎛ ⎜ H = ⎜ ⎝ η0 η1 η2 · · · ⎞ ηn−1 η1 η2 η2 η3 η3 η4 · ·... |

16 | Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials - Rissanen - 1973 |

14 |
Nonsymmetric Lanczos and Finding Orthogonal Polynomials Associated with Indefinite Weights, Numerical Analysis report NA-90-09
- Boley, Elhay, et al.
- 1990
(Show Context)
Citation Context ...oth δk and γ k are nonzero. When this assumption is relaxed, we face a breakdown problem that will be discussed in Section 5. This algorithm is the same as the nonsymmetric Lanczos algorithm found in =-=[5]-=-, except that we are using a weighted inner product to enforce W-orthogonality, as opposed to an unweighted inner product to enforce ordinary orthogonality. We refer the reader to [5] for details of t... |

12 |
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- 1989
(Show Context)
Citation Context ...udied in the literature; see, e.g., [11] and references therein. However, only recently has this problem been addressed for more general inner products; see, e.g., the modified Chebyshev algorithm in =-=[12]-=-. This problem was addressed in [5] for the case where the inner product is a discrete sum over a finite number of knots. 3.1 Asymmetric Orthogonalization Since the polynomials are of exact degrees th... |

11 |
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(Show Context)
Citation Context ... strongly nonsingular Hankel matrix have appeared in several papers under different guises, going all the way back to Chebyshev [8]. Early algorithms for factorizing a Hankel matrix appeared in [29], =-=[30]-=-, [31], and the connection with the Lanczos algorithm either on a nonsymmetric matrix or using an indefinite inner product appeared in [14], [18], [20], [28]. When the permuted Yule-Walker equations (... |

10 |
l’interpolation par la méthode des moindres carrés, Mém
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(Show Context)
Citation Context ...st Toeplitz factorizers. The resulting recursion formulae to factorize a strongly nonsingular Hankel matrix have appeared in several papers under different guises, going all the way back to Chebyshev =-=[8]-=-. Early algorithms for factorizing a Hankel matrix appeared in [29], [30], [31], and the connection with the Lanczos algorithm either on a nonsymmetric matrix or using an indefinite inner product appe... |

10 |
Multivariable and Multidimensional Systems: Analysis and Design
- Kung
- 1977
(Show Context)
Citation Context ... and observability was addressed in [6] and [26]. Hankel matrices arise in various settings, ranging from system identification [23] to algorithmic fault tolerance [4]. In his 1977 dissertation, Kung =-=[20]-=- studied the Berlekamp-Massey (BM, 1967) algorithm [1], [24] for solving Hankel equations, and remarked that their algorithm is related to the Lanczos process. There still exists strong interest in a ... |

9 |
Algorithmic fault tolerance using the Lanczos method
- Boley, Brent, et al.
- 1992
(Show Context)
Citation Context ...ebraic conditions of controllability and observability was addressed in [6] and [26]. Hankel matrices arise in various settings, ranging from system identification [23] to algorithmic fault tolerance =-=[4]-=-. In his 1977 dissertation, Kung [20] studied the Berlekamp-Massey (BM, 1967) algorithm [1], [24] for solving Hankel equations, and remarked that their algorithm is related to the Lanczos process. The... |

9 |
A simple Hankel interpretation of the Berlekamp-Massey algorithm. Linear Algebra and its Applications
- Jonckheere, Ma
- 1989
(Show Context)
Citation Context ... [1], [24] for solving Hankel equations, and remarked that their algorithm is related to the Lanczos process. There still exists strong interest in a simple exposition of the BM algorithm; see, e.g., =-=[19]-=- in 1989. In 1971 Phillips [28] proposed a Hankel triangularization scheme, and presented a derivation of his method using a special symmetrized Lanczos process with a weighted and possibly indefinite... |

7 |
The triangular decomposition of Hankel matrices
- Phillips
- 1971
(Show Context)
Citation Context ...quations, and remarked that their algorithm is related to the Lanczos process. There still exists strong interest in a simple exposition of the BM algorithm; see, e.g., [19] in 1989. In 1971 Phillips =-=[28]-=- proposed a Hankel triangularization scheme, and presented a derivation of his method using a special symmetrized Lanczos process with a weighted and possibly indefinite inner product. In this paper, ... |

6 |
An algorithm for the inversion of finite Hankel matrices
- Trench
- 1965
(Show Context)
Citation Context ...gly nonsingular Hankel matrix have appeared in several papers under different guises, going all the way back to Chebyshev [8]. Early algorithms for factorizing a Hankel matrix appeared in [29], [30], =-=[31]-=-, and the connection with the Lanczos algorithm either on a nonsymmetric matrix or using an indefinite inner product appeared in [14], [18], [20], [28]. When the permuted Yule-Walker equations (a Hank... |

3 |
Parallel and superfast algorithms for hankel systems of equations
- Heinig, Jankowski
- 1990
(Show Context)
Citation Context ...e response of a dynamical system [23]. The relation between the asymmetric Lanczos process and fast Hankel factorization and/or inversion algorithms has been explored more recently in [3], [4], [16], =-=[17]-=-, [21]. In the next two subsections, we apply the Lanczos process to define two procedures for decomposing an n × n strongly nonsingular Hankel matrix ⎛ ⎜ H = ⎜ ⎝ η0 η1 η2 · · · ⎞ ηn−1 η1 η2 η2 η3 η3 ... |

1 |
Error correction, orthogonal polynomials, Hankel matrices and the Lanczos algorithm
- Boley
- 1991
(Show Context)
Citation Context ... from the impulse response of a dynamical system [23]. The relation between the asymmetric Lanczos process and fast Hankel factorization and/or inversion algorithms has been explored more recently in =-=[3]-=-, [4], [16], [17], [21]. In the next two subsections, we apply the Lanczos process to define two procedures for decomposing an n × n strongly nonsingular Hankel matrix ⎛ ⎜ H = ⎜ ⎝ η0 η1 η2 · · · ⎞ ηn−... |

1 |
A completed theory for the Lanczos algorithm
- Gutknecht
- 1989
(Show Context)
Citation Context ...wed as special cases of the Lanczos algorithm. For simplicity in presentation, we have avoided the breakdown problem for the asymmetric Lanczos algorithm. The problem has been considered in [5], [7], =-=[15]-=-, [27], with an approach that is similar to that proposed by Berlekamp [1] to factorize a Hankel matrix which is not strongly nonsingular. Also, there is much recent interest, e.g., [3], [5], [9], [10... |

1 |
Fast triangular factorization of matrices with arbitrary rank profile
- Pal
- 1990
(Show Context)
Citation Context ...0], [28]. When the permuted Yule-Walker equations (a Hankel system with a special right hand side) are solved, the resulting method essentially computes the LU factors of the Hankel matrix [1], [24], =-=[25]-=-. The same sets of equations arise in identification problems, where we would like to construct the transfer function from the impulse response of a dynamical system [23]. The relation between the asy... |