## Arnoldi versus Nonsymmetric Lanczos Algorithms for Solving Nonsymmetric Matrix Eigenvalue Problems (1996)

Venue: | BIT |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Cullum96arnoldiversus,

author = {Jane K. Cullum},

title = {Arnoldi versus Nonsymmetric Lanczos Algorithms for Solving Nonsymmetric Matrix Eigenvalue Problems},

journal = {BIT},

year = {1996},

volume = {36},

pages = {470--493}

}

### Years of Citing Articles

### OpenURL

### Abstract

We obtain several results which may be useful in determining the convergence behavior of eigenvalue algorithms based upo n Arnoldi and nonsymmetric Lanczos recursions. We derive a relationship between nonsymmetric Lanczos eigenvalue procedures and Arnoldi eigenvalue procedures. We demonstrate that the Arnoldi recursions preserve a property which characterizes normal matrices, and that if we could determine the appropriate starting vectors, we could mimic the nonsymmetric Lanczos eigenvalue convergence on a general diagonalizable matrix by its convergence on related normal matrices. Using a unitary equivalence for each of these Krylov subspace methods, we define sets of test problems where we can easily vary certain spectral properties of the matrices. We use these and other test problems to examine the behavior of an Arnoldi and of a nonsymmetric Lanczos procedure. Mathematical Sciences Department, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, NY 10598, USA, a...

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Citation Context ...o nonsymmetric Lanczos methods. Consider Equation(1) where A is a n \Theta n nonsymmetric matrix. A may be real or complex. 2.1. Arnoldi Methods The Arnoldi method is based upon the Arnoldi recursion =-=[26]-=-. Arnoldi Recursion: 1. Given v 1 with kv 1 k = 1, for j = 2; 3; : : : compute: v j+1 = Av j 2. For each j and for i = 1; : : : ; j compute: h ij = v H i v j+1 ; v j+1 = v j+1 \Gamma h ij v i 3. For e... |

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Citation Context ...er A T or A has been found. Exact breakdown is highly improbable, near breakdowns may cause numerical instabilities. To avoid such problems, various look-ahead strategies have been proposed, see e.g. =-=[25, 18]-=-. The discussions in this paper are equally applicable to the look-ahead variants of these methods. If look-ahead steps are performed, then the scalar coefficients in Equations(4) become matrices, and... |

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Citation Context ...finite precision the Lanczos vectors do not remain biorthogonal, and the basic procedure must be modified. We use modifications analogous to our modifications for the real symmetric Lanczos procedure =-=[14]-=-. We require the following assumptions. Assumption 4.1:. Lanczos Phenomenon. For large enough m, all of the desired eigenvalues of A will appear in ! (T m ). Assumption 4.2: Any spurious eigenvalues a... |

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Citation Context ...erties of A control the convergence of each of these methods'? We have not answered this question but describe several results which might be useful in such studies. See for example, the related work =-=[2, 3, 4, 5, 6, 7, 8, 27, 28, 29]-=-. In section 2 we outline briefly the Arnoldi and the nonsymmetric Lanczos eigenvalue procedures we are considering. In section 3 we exhibit a certain relationship between these two methods. We prove ... |

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Citation Context ... w 1 j v 1 . If A is a real, normal matrix, and v 1 has reasonable projections on the desired right eigenvectors of A, then setting w 1 = v 1 may be an optimal choice in terms of the mismatch theorem =-=[25]-=-. . Lemma 2.1. Let A be a real, normal matrix with n distinct eigenvalues. Let v 1 have a significant projection on each unit right eigenvector of A. Then v 1 has a significant projection on each unit... |

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Citation Context ...erties of A control the convergence of each of these methods'? We have not answered this question but describe several results which might be useful in such studies. See for example, the related work =-=[2, 3, 4, 5, 6, 7, 8, 27, 28, 29]-=-. In section 2 we outline briefly the Arnoldi and the nonsymmetric Lanczos eigenvalue procedures we are considering. In section 3 we exhibit a certain relationship between these two methods. We prove ... |

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Citation Context ... of A onto K j (A; v 1 ) with respect to the V j . The preceding implementation is a modified Gram-Schmidt orthogonalization of the vectors fv 1 ; Av 1 ; A 2 v 1 ; : : :g. Other implementations exist =-=[31]-=-. In matrix form these recursions become AV j = V j H j + h j+1;j v j+1 e T j where H j = (h ik ); 1si; ksj: (2) Basic Arnoldi Eigenvalue Procedure: 1. Given v 1 use the Arnoldi recursion to generate ... |

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Citation Context ...vectors to convergence of Ritz vector approximations. Bai [1] considered a different nonsymmetric variant of the nonsymmetric Lanczos eigenvalue procedure and obtained extensions of theorems in Paige =-=[20, 21]-=- to the error terms F j and G j in equations (19). In his proof, Bai assumed exact local biorthogonality and normalizability. We prove a similar result for the complex symmetric variant which requires... |

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Citation Context ...vectors to convergence of Ritz vector approximations. Bai [1] considered a different nonsymmetric variant of the nonsymmetric Lanczos eigenvalue procedure and obtained extensions of theorems in Paige =-=[20, 21]-=- to the error terms F j and G j in equations (19). In his proof, Bai assumed exact local biorthogonality and normalizability. We prove a similar result for the complex symmetric variant which requires... |

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Citation Context ...roximations occurs in conjunction with losses in the biorthogonality of the Lanczos vectors. Using a symmetrized version of the nonsymmetric Lanczos procedure, we derive a variant of a theorem in Bai =-=[1]-=- connecting losses in biorthogonality to convergence of eigenvalue approximations, relaxing his assumptions of exact local biorthogonality and normalization. In section 5 we derive a simple unitary in... |

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Citation Context ...or estimates and would be excluded on that basis, along with these other good eigenvalues. Error estimates cannot distinquish between these two types of Lanczos eigenvalues. Alternatively, references =-=[24, 30]-=- track the convergence of approximations as the size of the Lanczos matrix is increased and only accept converged approximations. That approach suffers from the same difficulties as an approach which ... |

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Citation Context ...erties of A control the convergence of each of these methods'? We have not answered this question but describe several results which might be useful in such studies. See for example, the related work =-=[2, 3, 4, 5, 6, 7, 8, 27, 28, 29]-=-. In section 2 we outline briefly the Arnoldi and the nonsymmetric Lanczos eigenvalue procedures we are considering. In section 3 we exhibit a certain relationship between these two methods. We prove ... |

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Citation Context ...r only by the specified shift. Therefore , without loss of generality, we can restrict ourselves to test matrices with eigenvalues whose real parts are all positive. We have written a suite of MATLAB =-=[19]-=- codes which allow the user to generate and regenerate test matrices of the form given in equation(45). The user can also call either a Ax = b routine or a basic real or complex Arnoldi eigenvalue rou... |

2 |
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Citation Context ...ding eigenvalues of C. If A is real and diagonalizable we can replace complex V andsin equation(45) by a real orthogonal matrix and a real block diagonal matrix with 1 \Theta 1 and 2 \Theta 2 blocks, =-=[10]-=-. We can use equations(45) to specify various eigenvalue distributions and eigenvector spaces. In this paper we focus on diagonalizable test matrices. In [10] where we study the convergence of iterati... |

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Citation Context ...fferent approach. No convergence tolerances are used. Our identification test is a simple extension of the test used in our real symmetric Lanczos procedures. This extension is discussed in detail in =-=[11]-=-. The argument requires only the symmetry of the Lanczos tridiagonal matrices and is valid in finite precision arithmetic, as long as the error terms in the Lanczos recursions remain small. There is n... |

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Citation Context ... methods on nonnormal problems it is not sufficient to know the singular values of the eigenvector matrix. They also suggest a potential source of numerical difficulties for both types of methods. In =-=[9]-=- we consider similar questions for the problem Ax = b. We use the following notation. 1.1. Notation. A = (a ij ) , 1si; jsn; n \Theta n real or complex matrix A T = (a ji ) , 1si; jsn; transpose of A ... |

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Citation Context ...or estimates and would be excluded on that basis, along with these other good eigenvalues. Error estimates cannot distinquish between these two types of Lanczos eigenvalues. Alternatively, references =-=[24, 30]-=- track the convergence of approximations as the size of the Lanczos matrix is increased and only accept converged approximations. That approach suffers from the same difficulties as an approach which ... |