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Simultaneous multidiagonalization for the CS decomposition

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Abstract

When an orthogonal matrix is partitioned into a two-by-two block structure, its four blocks can be simultaneously bidiagonalized. This observation underlies numerically stable algorithms for the CS decomposition and the existence of CMV matrices for orthogonal polynomial recurrences. We discover a new matrix decomposition for simultaneous multidiagonalization, which reduces the blocks to any desired bandwidth. Its existence is proved, and a backward stable algorithm is developed. The resulting matrix with banded blocks is parameterized by a product of Givens rotations, guaranteeing orthogonality even on a finite-precision computer. The algorithm relies heavily on Level 3 BLAS routines and supports parallel computation.

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Authors and Affiliations

  1. Department of Mathematics, Randolph-Macon College, PO Box 5005, Ashland, VA, 23005, USA

    Kingston Kang, William Lothian, Jessica Sears & Brian D. Sutton

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  1. Kingston Kang
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  2. William Lothian
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  3. Jessica Sears
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  4. Brian D. Sutton
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Correspondence to Brian D. Sutton.

Additional information

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0914559.

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Kang, K., Lothian, W., Sears, J. et al. Simultaneous multidiagonalization for the CS decomposition. Numer Algor 66, 479–493 (2014). https://doi.org/10.1007/s11075-013-9744-5

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