By L. Bieberbach

Translated from the fourth German variation by way of F. Steinhardt, with an extended Bibliography.

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**Example text**

In fact any arbitrary particular solution, u , of the homogeneous equation, taking on at n = 0 and n = 1 any prescribed values uo and ul, may be written in this form. , t o set In particular, Yn and ,Z, defined in $1 as the solutions of the homogeneous equation satisfying the conditions Y 0 zo =1, = 0, Y 1 z1 = o = 1, have the form 1 'n=-- From Eqs. ( 6 ) we see that these equations are inapplicable in the case of multiple roots q1 = q2. Let us now consider this case. When q1 = q 2 one particular solution can again be written in the form = To find a second, let us make, in Eq.

M = CN or M = CN2. Our definition of good conditioning is equivalent to one which is customary in the theory of systems of linear equations, where the measure of conditioning of a system of equations Ax = g with matrix A is taken to 44 Good-Conditioning Criteria 33 IIA-ll), the product of the norms of the matrices A be the quantity IIAII and A-I. *** Fulfillment of inequality ( 5 ) indicates that the sensitivity of the solution [un] to errors (for example measurement or rounding errors) occurring in the given right-hand sides 6 , $ or [fn}, does not grow with increasing N.

Find a solution of the equation u n-1 - -25 un + un+l = ‘a which is bounded for n + -14s and takes on the value uo = 1. 3 . Write out the thousandth term of the sequence uo’ ul, U 2 ’ first two terms of which are equal to one, uo = 1, u1 = 1, while the following terms are defined by the recurrence relation Y the Difference Equation of Second Order 93 n = 1, 2, u = u + u n+l n-1 n' 29 ... 4 . , will have at least one nontrivial bounded solution. ) 5. , will be bounded. 6 . What must be true of the roots of the characteristic equation if all solutions of the equation aun-l + bun + cun+l = 0 are to tend to zero as n + m ?