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Deflated and restarted symmetric Lanczos methods for eigenvalues and linear equations with multiple right-hand sides. (English) Zbl 1209.65042

Summary: A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Experiments are given with large matrices from quantum chromodynamics that have many right-hand sides.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F10 Iterative numerical methods for linear systems
15A06 Linear equations (linear algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
81V05 Strong interaction, including quantum chromodynamics
65F25 Orthogonalization in numerical linear algebra

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