Numerical Evaluation of a Slowly Convergent Series: Problem 89-15Author(s): Alan GibbsSource: SIAM Review, Vol. 32, No. 3 (Sep., 1990), pp. 481-483Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2031625 .

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PROBLEMS AND SOLUTIONS 481

Therefore, the final result is

1/2(a - 1)i3((a - 1)/2) (In x): owX ~ r8F(a/2) xa

X{+Ln2 a- ,1 2A 2 )2 (2 -Inx

+[(ln 2 -2(ln 21t)( l A1 3 41( )- I (I))

4~(3 ( 21) (2))2 1el +(3(2 1) (2)) 4+ (I 2 )4(2]2(Int) 2 - nx)3)4

+L1+l1x+(ln2-l - a+1--2V/\2!+2f\a 2 21lXJ

( 1)2( ) (X2j X)2)} x? 1, Re(2 )> 1

and, if need be, it can easily be extended to higher orders.

Also solved by W. B. JORDAN (Scotia, New York)

Numerical Evaluation of a Slowly Convergent Series

Problem 89-15*, by R. E. SHAFER (Berkeley, California). Evaluate

oo (-I)n

n=21n (n)

to within an error e, where lIl < 5 X 106.

Solution by ALAN GIBBS (Private consultant, Richland, Washington). This would be a good classroom problem, since several methods can be illustrated.

The five methods used below were implemented in simple FORTRAN programs written by the author and run in double precision on a CDC Cyber 920 UNIX workstation computer. The first four methods all yield (to ten figures)

S= 0.9242998972.

1. Hutton's method [1, p. 159] averages successive partial sums Sn = Zk=2 (-1)k/ln k to obtain a new sequence that can itself be averaged, i.e., for n=2,3, N,

S(n, 1)=Sn,

S(n+ 1,k)= 0.5x(S(n+ 1,k - 1)+S(n,k - 1)), k=2,3 ... K.

This method yields the stated result for K = 10 and N _ 25.

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482 PROBLEMS AND SOLUTIONS

2. The Euler-van Wijngaarden transform [2] writes a general alternating sum of f(n) as

S= X, (-1) f(n)+ E (-1 (-f(n). n=2 n=M+ 1 2

Here, Av is the forward difference operator, and the number of directly summed terms M is selected adaptively to maximize convergence of the second sum. This method yields the stated result for M = 8 and N ' 20.

3. The nonlinear Wynn-Shanks epsilon algorithm [3] constructs a table T(n,k),n=2,3 ... N, from

T(n, 1)= 0.0,

T(n,2)=Sn,

T(n,k+ 1)=T(n+ I,k- 1) + 1.0 k=2,3* N T(n +1, k) -T(n, k)' k23. ,

Under certain conditions [4], T(n, 2k) converges to the sum. The stated result is obtained from T(2, 16).

4. A semi-analytical method given by Hildebrand [1, p. 157] can be used to sum an alternating series from the Mth term on:

M_-I

S= Ei ( 1) f(n)+( 1)m0.5 n=2

(1) (1I fl () 17 {

- 2 (24)f ( )2 1ff "I(M) + M

2440320 7)(M)- ..

To use the method here, define f(x) by

(2) f(x) ln (x + M) = 1.

Differentiate this expression n times, solve for Dnf(x) and use

Dm ln (x+M)=(-1)m(m- 1)!/(x+M)m, m' 1

to obtain

(3) D nf(x) = ln (X +M) 2E (- O)M m(m - 1)! D(n - m)f(x)/(x + M) m.

Numerical values of f(x),f'(x), ... can be obtained from (2) and, sequentially, (3). Using them in (1) and retaining terms through f(7) yields the stated result for M_ 16.

In contrast to the above methods, Aitken extrapolation [5] of the partial sums Sn yields

S = 0.924295

after 44 terms, which is within the requested accuracy. But additional improvement is slow, e.g., seven-figure accuracy requires 303 terms.

The methods above all eclipse direct summation. There, achieving a truncation error (smaller than the last term) of e requires - exp (1 /e) terms. Thus for one significant figure approximately 4.85 x 108 terms are needed, and obtaining the

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PROBLEMS AND SOLUTIONS 483

requested accuracy, roundoff errors notwithstanding, would require an astronomical 10868519 terms!

REFERENCES

[1] F. B. HILDEBRAND, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956. [2a] E. T. GOODWIN, Modern Computing Methods, Second edition, Philosophical Library, New York,

1961, Chap. 13. [2b] W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY, AND W. T. VETTERLING, Numerical Recipes, the

Art of Scientific Computing, Cambridge University Press, Cambridge, U.K., 1986, p. 134. [3a] D. SHANKS, Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys.,

34 (1955), pp. 1-42. [3b] P. WYNN, On a device for computing the em(Sn) transformation, Math. Tables Aids Comput., 10

(1956), pp. 91-96. [4] , On the convergence and stability of the epsilon algorithm, SIAM J. Numer. Anal., 3 (1966),

pp. 92-122. [5] K. E. ATKINSON, An Introduction to Numerical Analysis, John Wiley, New York, 1978, p. 65.

Also solved by JOHN P. BOYD (University of Michigan), STEPHEN P. CHICATELLI

(University of Akron), JOHN A. CROW (Oregon State University), CLARK DORMAN and MURLI GUPTA (George Washington University), ALAN GENZ (Washington State University), C. C. GROSJEAN (State University of Ghent, Ghent, Belgium), H. GUGGENHEIMER (West Hempstead, New York), MICHAEL HAAG, S. HABER (Temple University), KAYA IMRE and HAROLD WEITZNER (Courant Institute of the Mathematical Sciences), W. B. JORDAN (Scotia, New York), JOHN F. KINKEL (Irvine, California), DOUGLAS A. KURTZE (Clarkson University), JESPER K. LARSEN (Math-Tech, Charlottenlund, Denmark), KEE-WAI LAU (Hong Kong), PATRICK DALE MCCRAY (Searle Research and Development, Illinois), A. J. O'CONNOR (Griffith University, Nathan, Australia), MICHAEL RENARDY (Virginia Institute of Technology), RENAN S. SEZGINER (Bridgeport Engineering Institute, Fairfield, Connecticut), EUGENE L. SMITH (Skokie, Illinois), H. A. STEINBERG (CADAM, Inc., Tarrytown, New York), ROBERT TERRELL (Cornell University), PETER WAGNER (University of Innsbruck), NANCY WALLER (Portland State University, Portland, Oregon), J. ERNEST WILKINS, JR. (Chicago, Illinois), G. WILSON (Annandale, Virginia), JINCHAO Xu (Pennsylvania State University), and the proposer.

A Conjectured Matrix Inequality

Problem 89-14*, by W. H. STEEB and C. M. VILLET (Rand Afnikaans University, Johannesburg, South Africa). Let A and B be two n x n symmetric matrices. Then it can be shown [1], [2] that

tr e tr eAeB < 2 re (e2A + e2s)

tr eA+B ' tr eAeB e (tr ePA) l /P(tr eP2B) l/P2

where P1 > 1 and P2 > 1 with

- += 1 PI P2

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