Transpose-free multiple Lanczos and its application in Padé approximation

A transpose-free two-sided nonsymmetric Lanczos method is developed for multiple starting vectors on both the left and right. The method is mathematically equivalent to the two-sided methods without look-ahead or deflation steps, but avoids the use of the transpose of the system matrix. The method is applied to the computation of the matrix Padé approximation to a linear dynamical system. The result is a method which can be labeled transpose-free matrix Padé via Lanczos (TFMPVL). Under certain circumstances, TFMPVL will actually reduce the total number of matrix-vector products needed. It is illustrated with some numerical examples.