REMARKS ON THE CONTINUED FRACTION CALCULATION OF EIGENVALUES AND EIGENVECTORS

Journal Article · Fri Sep 01 04:00:00 EDT 1961 · J. Math. Phys.

DOI:https://doi.org/10.1063/1.1703766· OSTI ID:4837338

Swalen, J D; Pierce, L

For eigenvalue problems in which the secular determinant has tridiagonal form, e.g., the rigid asymmetric rotor; the secular equation may be writien in the form f( lambda ') = 0, where f( lambda ') is a continued fraction and lambda ' an eigenvalue. Furthermore, if the secular problem is of nth order, then the continued fraction f( lambda ') may be developed in n different ways. Since the eigenvalues are roots of a function f( lambda ), it is convenient to find the eigenvalues by means of the Newton-Raphson iterative procedure This requires that the derivative of f( lambda ) with respect to lambda (f'( lambda )) be determined. An exact expression for f'( lambda ) is derived and it is shown that f'( lambda ') is in fact the form of the eigenvector belonging to the eigenvalue lambda '. A simple recursion formula, in continued fraction form, for the eigenvector elements is also derived. The Newton-Raphson procedure is further shown to be equivalent to the variational method for iterative calculation of eigenvalues. The former procedure has, however, the advantage of bypassing the necessity of solving a set of simultaneous equations. Advantage is taken of the relation between f'( lambda ') and the eigenvector of lambda ' to formulate a reasonable criterion for choosing the best possible development of f( lambda ) in order to avoid convergence to an undesired root of f( lambda ). (auth)

• Research Organization: Harvard Univ., Cambridge, Mass.; and Shell Development Co., Emeryville, Calif.
• Sponsoring Organization: USDOE
• NSA Number: NSA-15-032893
• OSTI ID: 4837338
• Journal Information: J. Math. Phys., Journal Name: J. Math. Phys. Vol. Vol: 2
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

DIFFERENTIAL EQUATIONS
EIGENVALUES
MATHEMATICS
PHYSICS
VECTORS