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We introduce an eigenvalue-preserving transformation algorithm from thegeneralized eigenvalue problem by matrix pencil of the upper and the lowerbidiagonal matrices into a standard eigenvalue problem while preservingsparsity, using the theory of orthogonal polynomials. The procedure isformulated without subtraction, which causes numerical instability.Furthermore, the algorithm is discussed for the extended case where the upperbidiagonal matrix is of Hessenberg type.

An isospectral transformation between Hessenberg matrix and Hessenberg-bidiagonal matrix pencil without using subtraction