THE QD-ALGORITHM AS A METHOD FOR FINDING THE ROOTS OF A POLYNOMIAL EQUATION WHEN ALL ROOTS ARE POSITIVE.
STANFORD UNIV CALIF SCHOOL OF HUMANITIES AND SCIENCES
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The Quotient-Difference QD-scheme, symmetric functions and some results from the theory of Hankel determinants are treated. Some well known relations expressing the elements of the QD-scheme by means of the Hankel determinants are presented. The question of convergence of the columns of the QD-scheme is treated. An exact expression for q sub nk is developed for the case of different roots. It is proved that the columns of the QD-scheme will converge not only in the well known case of different roots, but in all cases where the roots are positive. A detailed examination of the convergence to the smallest root is presented. An exact expression for q sub nN is developed. This expression, is correct in all cases of multiple positive roots. It is shown that the progressive form of the QD-algorithm is only mildly unstable. Finally, some ALGOL programs and some results obtained by means of these, are given.