Beiträge zur Kenntnis des Biorthogonalisierungs-Algorithmus von Lanczos

Summary

The paper begins with a review of the essential points ofLanczos's orthogonalization procedure, which is of great importance for the determination of the eigenvalues of a real, but otherwise general matrix. Then several properties useful for numerical computation are proved:

If the degree of the reduced characteristic polynomial of the matrix ism, then it is possible to choose trial vectorsx, y such that the iteration may be continued for exactlym steps. At that point the process must stop because the iterated vectorsx _m+1,y _m+1 vanish (theorem 1).

If all the eigenvalues are real, then the co-diagonal elementsβ _i [see equation (12)] can be made arbitrarily small by proper choice of the trial vectorsx, y (theorem 5), which considerably simplifies the evaluation of the eigenvalues and eigenvectors. Should, in addition, all theβ _i still be positive (see theorem 4), then bounds for the eigenvalues may readily be given (theorem 3). Further remarks are made concerning matrices with complex eigenvalues.

Finally it is shown that by starting from a certain one-parameter family of trial vectors (11), the diagonal and co-diagonal elementsα(t),β(t) are solutions of a system of differential equations (25).