Crawfnos

Subspace algorithms for the Crawford number computation.

Description

These are MATLAB implementations of the subspace algorithms in [1] for computing the Crawford number of a matrix \(A\in \mathbb C^{n\times n}\), i.e., the distance of the numerical range \(\mathcal{F}(A)\) from zero:

\[ \gamma(A) = \min\{\, |z| \colon z\in \mathcal{F}(A)\, \},\qquad \mathcal{F}(A) = \{\, v^{*} A v \colon v\in \mathbb{C}^{n}, \|v\|_2=1\,\}. \]

To use crawfnos, the user can provide either a matrix \(A\), or a linear operator op with member functions: op.matvec, op.rmatvec, op.matmat to compute \(A\times x\), \(A^*\times x\) and \(A\times V\), for a vector \(x\in \mathbb C^{n}\) and matrix \(V\in \mathbb C^{n\times k}\), respectively.

All example data and routines used in the paper [1] are also included. For a detailed description of the usage of these routines, please check the readme files in the package.

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References

  1. Subspace acceleration for the Crawford number and related eigenvalue optimization problems
    with Daniel Kressner and Bart Vandereycken,
    SIAM J. Matrix Anal. Appl., 2018. 39(2):961–982. (preprint, MATLAB code)

Contact

Email: Ding.Lu@uky.edu
Homepage: http://www.ms.uky.edu/~dlu245/