Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
Implicitly restarted Arnoldi with purification for the shift-invert transformation
Mathematics of Computation
Numerical computing with IEEE floating point arithmetic
Numerical computing with IEEE floating point arithmetic
Matrix algorithms
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
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The B-Arnoldi method is a variant of the ordinary Arnoldi method in which orthogonalization is done with respect to the inner product generated by a positive definite matrix $B$. It arises in connection with the generalized eigenvalue problem $Ax = \lambda Bx$. When $B$ is semidefinite, the algorithm can proceed formally, with “orthogonalization” taking place in the semi-inner product generated by $B$. However, it has been observed that components of the Arnoldi vectors lying in the null space of $B$ can grow rapidly. In this paper we examine the source and consequences of this growth.