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Cayley Transformation Mode

  Driver dsdrv6  implements the Cayley spectral transformation. Eigenvalues closest to a shift can be obtained by computing the selected eigenvalues of

Settings of which are the same as those of the shift-invert modes. It is easy to verify that OP is symmetric with respect to the (semi) inner product defined by ${\bf M}$ so it is best to use ${\bf M}$-inner products with this transformation.   The eigenvalue of the original problem and are related by

Note that the transformation becomes ill-defined as since . The expression does have a limit as but it would be better to use regular inverse mode or to use shift-invert mode with when eigenvalues near the origin are sought.

The Cayley transformation has slightly different properties than the standard shift and invert spectral transformation used by drivers dsdrv3 and dsdrv4. To use this driver, the operations

are required.

Chao Yang