A Golub–Kahan-Type Reduction Method for Matrix Pairs

@article{Hochstenbach2015AGR,
  title={A Golub–Kahan-Type Reduction Method for Matrix Pairs},
  author={Michiel E. Hochstenbach and Lothar Reichel and Xuebo Yu},
  journal={Journal of Scientific Computing},
  year={2015},
  volume={65},
  pages={767-789}
}
We describe a novel method for reducing a pair of large matrices $$\{A,B\}$${A,B} to a pair of small matrices $$\{H,K\}$${H,K}. The method is an extension of Golub–Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix A represents a discretization of a compact integral operator and B is a regularization matrix. 
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