## Lanczos Algorithms for Large Symmetric Eigenvalue Computations: Vol. 1: TheoryFirst published in 1985, this book presents background material, descriptions, and supporting theory relating to practical numerical algorithms for the solution of huge eigenvalue problems. This book deals with 'symmetric' problems. However, in this book, 'symmetric' also encompasses numerical procedures for computing singular values and vectors of real rectangular matrices and numerical procedures for computing eigenelements of nondefective complex symmetric matrices. Although preserving orthogonality has been the golden rule in linear algebra, most of the algorithms in this book conform to that rule only locally, resulting in markedly reduced memory requirements. Additionally, most of the algorithms discussed separate the eigenvalue (singular value) computations from the corresponding eigenvector (singular vector) computations. This separation prevents losses in accuracy that can occur in methods which, in order to be able to compute further into the spectrum, use successive implicit deflation by computed eigenvector or singular vector approximations. |

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### Contents

CL41_ch1 | 17 |

CL41_ch2 | 32 |

CL41_ch3 | 76 |

CL41_ch4 | 92 |

CL41_ch5 | 164 |

CL41_ch6 | 194 |

CL41_ch7 | 210 |

CL41_backmatter | 252 |

### Other editions - View all

Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. II Programs Cullum,Willoughby No preview available - 2012 |

### Common terms and phrases

applied basic Lanczos recursion block Lanczos procedure Chapter column complex symmetric matrices complex symmetric tridiagonal component computed eigenvalues computer storage conjugate gradient procedure convergence corresponding eigenvectors corresponding Lanczos corresponding Ritz vectors Cullum defined denote determine diagonal entries diagonal matrix discussion eigenva eigenvalue computations eigenvalue of Tm eigenvalue problems eigenvalues and eigenvectors EISPACK equations error estimates example factorization given matrix Hermitian matrices identification test inverse iteration iterative block Lanczos Krylov subspace Lanczos algorithm Lanczos matrices Lanczos tridiagonalization Lanczos vectors Lemma linear losses in orthogonality Math maximizing nonzero norm number of eigenvalues nxn matrix obtained original matrix orthogonal matrix orthonormal Paige Parlett polynomial positive definite Proof real symmetric matrix real symmetric tridiagonal reorthogonalization residuals scalars Section single-vector Lanczos procedures singular value decomposition singular vectors sparse sparse matrix spectrum spurious eigenvalues starting vector Step storage requirements Sturm sequencing subroutine symmetric tridiagonal matrices Theorem unitary matrix zero