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Subroutine XYapps  applies the shifts using an implicitly shifted QR   mechanism on the projected matrix If a shift is complex, then [s,d]napps employs a double implicit shift application so that the complex conjugate of the shift is applied simultaneously. This allows the implicit shift application to take place in real arithmetic. Finally, XYapps updates the current Arnoldi/Lanczos basis to obtain the new length k factorization and Subdiagonal elements of the Hessenberg matrices are set to zero if they satisfy the same criterion as used in a standard implementation of the QR algorithm. (This amounts to checking if any subdiagonal of is less than the product of machine precision and the sum of the two adjacent diagonal elements of )

Chao Yang