## ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi MethodsThis book is a guide to understanding and using the software package ARPACK to solve large algebraic eigenvalue problems. The software described is based on the implicitly restarted Arnoldi method, which has been heralded as one of the three most important advances in large scale eigenanalysis in the past ten years. The book explains the acquisition, installation, capabilities, and detailed use of the software for computing a desired subset of the eigenvalues and eigenvectors of large (sparse) standard or generalized eigenproblems. It also discusses the underlying theory and algorithmic background at a level that is accessible to the general practitioner. |

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### Contents

SE06_ch1 | 1 |

SE06_ch2 | 9 |

SE06_ch3 | 21 |

SE06_ch4 | 43 |

SE06_ch5 | 67 |

SE06_appendixa | 79 |

SE06_appendixb | 113 |

SE06_appendixc | 121 |

133 | |

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### Common terms and phrases

Arnoldi factorization Arnoldi iteration ARPACK array workd ipntr basis vectors BLAS Chapter columns complex arithmetic complex conjugate computational mode computed eigenvalues double-precision dsaupd eigenvalue problem bmat eigenvalues and eigenvectors eigenvalues of interest eigenvalues of largest Figure Hermitian Hessenberg matrix ido=1 Actions Required imaginary implicit restarting implicitly restarted Arnoldi implicitly shifted QR INPUT Integer invariant subspace ipntr(1 IRAM Krylov subspace Lanczos LAPACK largest magnitude matrix-vector products maximum number maxncv non-Hermitian number of eigenvalues OP+x operations original problem orthogonal output parameter pointer into WORKD polynomial positive semidefinite Postprocessing precision QR algorithm Rayleigh quotient regular inverse mode result vector returned reverse communication interface Ritz values Ritz vectors Schur decomposition Schur vectors shift strategy shift-invert mode iparam smallest specifies spectral transformation standard eigenvalue problem starting vector storage subdiagonal subroutine symmetric total number tridiagonal matrix upper Hessenberg matrix WORKL XYaupd zero znaupd