V
CONTINUED FRACTIONS AND THEIR APPLICATION
IN THE COMPUTATION OF DEFINITE
RIEMANN INTEGRALS
by
STANLEY MAX COMPTON, B.S.
A THESIS
IN MATHEMATICS
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of
MASTER OF SCIENCE
Approved
August, 1973 ^ ^^- ^J
1973
ACKNOWLEDGMENTS
I would like to express my sincere appreciation to Professor Vadim Komkov for his guidance with respect to the development of this thesis, and to Professors Charles N. Kellogg and Robert A. Moreland for their helpful criticisms. I would also like to thank Mr. Joel R, Wilson for his inspirational interest in this paper, and Mrs. Anna F. Kirk for her fine editing and typing.
11 CONTENTS
ACK^^OWLEDGMENTS ii LIST OF TABLES iv I. INTRODUCTION AND HISTORICAL DEVELOPMENT ..... 1 Introduction 1 Historical Development 2 II. INTRODUCTORY CONCEPTS 6 Definitions and Notation 6 Successive Convergents 7 Convergence 11 Continued Fractions and Power Series 14 III. CONTINUED FRACTION EXPANSIONS 17 Introductory Examples 17 Examples of Continued Fraction Expansions for Certain Functions 23 IV. THE QUOTIENT-DIFFERENCE ALGORITHM 25 Introduction 25 Hankel Determinants 25 The Quotient-Difference Algorithm 26 The Quotient-Difference Algorithm and Continued Fractions 35 Continued Fractions and Definite Riemann Integrals 43 BIBLIOGRAPHY 51
111 LIST OF TABLES
Tanle A 30
Table B 31
Table C 34
Table D 39
Table E 42
Table F 44
Table G 46
Table H 49
IV CHAPTER I
INTRODUCTION AND HISTORICAL DEVELOPMENT
Introduction
Almost unknown to the Western student of modern mathematics, the subject of continued fractions plays an extensive, if not vital, role in analytic theory as well as arithmetic theory. Powerful applications involving the use of continued fractions exist with respect to the theory of equations, orthogonal polynomials, power series, infinite matrices and quadratic forms in infinitely many variables, definite integrals, the moment problem, analytic functions, and the summation of divergent series.
As an elementary example, let us suppose we are going to solve the quadratic equation
(1.1) x^ - 3x - 1 = 0.
We might begin by dividing equation (1.1) by x and write
(1.2) X = 3 + ^.
Then we might replace x in the right-hand side of equation
(1.2) with its equal, namely 3 + —, to obtain .TV
(1.3) x = 3 + ^. 3 + - X With repeated substitutions for x in the right-hand side of equation (1.3) we obtain
(1.4) x = 3 -»- 3 + I 1 3 +—^ 3 + - X Now it is clear that this procedure will never yield an exact solution of equation (1.1). However, if this procedure were to be truncated as in expression (1.4), and the results numerically computed, we would obtain a good approximation to the positive root of equation (1.1). In fact, the accuracy of this approximation varies as the number of terms which remain after the truncation has been performed. In Chapter II we shall see that an expansion like the one given in expression (1.4) is called a continued fraction. We shall also see that there is a very eloquent theory associated with such expansions.
Historical Development Perhaps the first major step in the creation of an analytic theory for continued fractions came with T. J. Stieltjes* [24] "Recherches sur les Fractions Continues" in 1894. In this paper Stieltjes developed the funda mental function theory and integral theory necessary for a complete treatment of an important class of continued fractions. During the years of 1889 through 1890 Stieltjes published several examples of continued fraction expansions of definite integrals of the form
^i^^t. X (z + t) where f(t) > 0. The continued fractions associated with these integrals are of the form
(1.5) ^ , z + C2 1 + — ^ z + 1 + where c. > 0, i = 1, 2, 3, ... . Stieltjes later showed that the continued fraction (1.5) could be transformed into
(1.6) ^1 z + a, T- ^ "^ ^2 " z -H a- - •. where the a. and b. are positive functions of the c. 11 1 In 1903 E. B. Van Vleck [3o] was able to connect, in certain cases, continued fractions of the form (1.6) with definite integrals of the form
f(t) ^t £-(z + t)^^' where f(t) > 0. However, H. Hamburger [6] was the first to develop a complete extension of Stieltjes' theory to continued fractions of the form (1.6) in 1920. In 1914 Hellinger and Toeplitz [7] established the fundamentals of a matrix theory for continued fractions of the form
(1.6) with b. > 0 and a. real, i=l, 2, 3, ... . In
1922 Hellinger [s] obtained a completion of this theory.
Around 1900 Pringsheim [l9, 20] obtained answers to questions about convergence of continued fractions with complex elements. These continued fractions were of the form
(1.7) ^ Zg 1 + ^ ^2 1 + 1 + '.
Pringsheim found thkt continued fractions of the form
(1.7) converge for Jz. ^ (1 - r._,)r., where
I 0 Vleck [28, 29] also found that continued fractions of the form (1.8) ^1 •" 1 ^2 ^ ^TT-^ converge when z^ / 0 and |,^(z^)| ^ k/((z^) for k > 0 and i = 1, 2, 3, ... if and only if the series ^JI^J diverge; In 1912-1915 Szasz [26, 27j found that continued fractions of the form (1.7) converge if the series 2^|z.| converges and yjlzj^l ~ R^^±^J ^ 2. Extensions of the above mentioned results, as well as many new results, within a larger analytic theory may be found due to H. S. Wall and his associates. Specifically with respect to expression (1.7) and expression (1.8), Wall and his associates found that the inequalities of Pringsheim and Van Vleck restricting the z. to lie in a neighborhood of the origin could be replaced with inequalities restricting the z. to lie in a domain bounded by certain parabolas with foci at the origin (Scott and Wall [22J and Paydon and Wall [l8J ) . Wall and his associates also extended Stieltjes* theory to a class of continued fractions of the form (1.6) with a. and b. complex, i = 1, 2, 3, ... (Hellinger and Wall [9], Wall and Wetzel [si, 32], Dennis and Wall [3]). In 1948 Wall [34] published his Analytic Theory of Continued Fractions which serves as a unified theory for continued fractions and their application. It may be noted that this excellent book is widely considered to be the authoritative text on the analytic theory of continued fractions. CHAPTER II INTRODUCTORY CONCEPTS Definitions and Notation An expression of the form bg (2.1) a^ + ^ 0 b^ 2 a, + 1 b_. a^ + "^ 2 a-3 + •.• is called a continued fraction. In general, the elements a^, a, , a^j ... and b, , b^, b^, ... may be real or complex numbers, or they may be real or complex valued functions of one or more variables. The number of terms in a continued fraction may be finite or infinite. In view of the notational difficulties involved in expression (2.1), it is more commonly accepted to repre sent a continued fraction by an expression of the form a + ^ ^ ^ 0 a, + a^ + a.^ + • • • , or by an expression of the form ,00 w V 1 • The elements a and b^, for any integer n, are called the coefficients of the continued fraction (2.1) In particular, the elements b, , b^ > b.^, ••. are called the partial numerators, a-^ is called the leading constant and the elements a-,, a2» a^> ••' are called the partial b denominators. The fraction —, for any integer n, is an called the n partial quotient. The terminating continued fraction b, ^2-^3 b P 0 a, + a^ + a- + • • • + a Q 1 Z o n n is called the n convergent or n approximation of the continued fraction (2.1). We shall assume that all of the coefficients of a continued fraction are finite and that no partial denomi nator is ever zero. Successive Convergents We note from expression (2.2) that Pi ,^ «1 ' h ^^ ^ = 8 ar0^ + a^a2 + b2 _ ^0^1^2 •" ^0^2 "• ^2^1 The last of these relations indicates, in the general case, that for all integers n^ 2, P a P T + b P ^ / 9 ON _J}. _ n n-1 n n-2 n n^n-1 n^n-2 To prove relation (2.3) we use induction and note P P , that in order to progress from -^ to ^ we must replace , ^n n+1 a^ with a_ + n+1 . Then n n a n+J 1T b ^ a P T + " P T+bP^ r,^-U P .T n n-1 a ^, n-1 n n-2 a b ^TP , n+1 _ n+1 _ n+ ^^P1 n + n+1 n-1 °n.l \ , h^ Q " ^n.A ^ Vl°„-1- n n-1 aTV^n-l n^n-2 n+1 Thus equation (2.3) is valid for all integers n ^^ 2, and hence provides a means by which successive convergents of order n ^ 2 can be computed for the continued fraction (2.1). Further, we note from the above development that P =aP T+bP ^ and Q = a Q , + b^Q ^ for all n n n-1 n n-2 n n n-1 n n-2 integers n > 2. In order that equation (2.3) be valid for n = 1, we define P_, = 1 and Q_, = 0. We now consider the difference of two successive convergents of the continued fraction (2.1). That is, consider P P , P Q , - P ,Q (-y A\ _IL n-1 _ n n-1 n-1 n K^'^) Q " Q , " Q Q • n "^n-l ^n-l^n With the help of equation (2.3) we may substitute the appropriate values for P^ and Q in the numerator of the n n right-hand side of equation (2.4) to obtain ^ ^ILzl ^^n^n-1^ ^n^n-2^Qn-l " ^^n^n-l " ^nQn-2)Pn-l ^n " ^n-l " ^n-A ,. ^n-lQn-2 - Pn-2Qn-l n Q TQ n-1 n Again with the help of equation (2.3) we may substitute appropriate values for P , and Q , in the numerator of n-1 n-1 the right-hand side of the above equation to obtain P P , n _ n-1 n n-1 (a TP ^ + b TP ^)Q ^ - (a ,Q ^ + b ,0 ^)P ^ _, n-1 n-2 n-1 n-3 n-2 n-1 n-2 n-1 n-3 n-2 ~ n Q -.Q n-1 n „ P ^Q - - P ^Q ^ f n \ 2, -. n-2 n-3 n-3 n-2 = (-1) b b , pr pr . n n-1 Q ,Q n-1 n Repeating this procedure we find that 10 P P T P^Q 1 - P .Qn ^2-^^ Q" - Q~T - ^-^^ ^n^n-1 ^1 Q~7Q ' n n-1 n-1 n Now if we recall that PQ = a^, P_, = 1, Q^. = 1, and Q_, = 0, we see that equation (2.5) may be written in the form /o r: \ P n P n-T1 / , xn+^,b1 bn n-T1 • • • b-,1 n n-1 n-1 n We now consider the difference of two convergents of the continued fraction (2.1) whose indices differ by 2. From equation (2.6) we see that P ^, P b ^^h b , ••• b, f o -7 ^ n+1 n _ / -, V n n+1 n n-1 1 ^ '^ Q ~ Q ~ ^'^ Q Q -, • • ^n+1 n "^n^n+l Adding equation (2.6) and equation (2.7) we obtain b b , bn /b , T n+1 _ n-1 _ /-i \ n n n-1 ••• 1/ n+1 n+1 n-1 Qn \Q ^T Qn-r n+1 b b T • • • ^1/ n n n-1 1 Now with the help of equation (2.3) we may substitute the appropriate value for Q ^, in the numerator of the right-hand side of the above equation to obtain /o o\ n+1 n-1 ( -, xn+1 n+1 n n-1 1 (2.8) Q = (-1) Q -Q ;^ • ^n+1 ^n-1 n-l'^n+l Replacing n with 2k - 1 in equation (2.8) we see that the relationship between successive convergents of even order 11 is given by P P ab b ...b ro ON 2k ^2k-2 . 2k^2k-1^2k-2 ^1 (2.9) - - = + - - ^2k ^2k-2 ^2k-2^2k for k=l, 2, 3, '•• . Replacing n with 2k in equation (2.8) we see that the relationship between successive convergents of odd order is given by P P a bh •••"h (o in\ 2k+l ^2k-l _ '^2k+l 2k 2k-l ^1 ^2k+l ^2k-l ^2k~1^2k+l for k=l, 2, 3, '•• . Convergence Utilizing the iterative properties of equation (2.9) and equation (2.10) we have respectively P^, a-b, a.b-,b^b, /-TTX 2k ^ 21^ 4321^ ,^2k'"2k-1^2k-2 ••• '"l Q2k-2Q2k P21<-H .^ ^3^2^ _ 15^4^3^ (2.12) Qj^^, - ^0 a^ - Q^Q3 Q3Q5 ^2k+1^2k'^2k-l • " ^1 • • - Q 0 ^2k-1^2k+l We see from equation (2.3) and equation (2.11) that if all of the coefficients of a continued fraction are positive, then the convergents of even order generate a 12 monotonically increasing sequence. But from the form of the continued fraction (2.1) it is clear that all of the ^1 convergents of eve i order are less than a^ + —, Thus ^ 0 a, this monotonically increasing sequence is bounded from p , ^, ^ lim 2k . ^ above so that , ^^7:— exists. k-»<» C ^, Similarly, we see from equation (2.3) and equation (2.12) that if all of the coefficients of a continued fraction are positive, then the convergents of odd order generate a monotonically decreasing sequence which is P bounded from below by a^. Hence , ^TT exists. We note also that if all of the coefficients of a continued fraction are positive, then from the form of /P P T\ the continued fraction (2.1) it is clear that p p _2k _ 2k-l^ ^ Q ^^ ^^^ follows that fQ2k ^2k-l ro -5^ ^2k y lim^2k , lim^2k-l . ^2k-l (2.13) Q^^ < ^^^^^ . K.ooQ^^_^ - Q2^_^- .^ lim^2k lim^2k-l ., . . .^ lim n ^^-^.^ Now If ,^^«,Q— = j^^coQ -> that IS If -^ exists, 2k 2k-l n then the continued fraction is said to converge and the . ^ T , T1 • i mP n value of the continued fraction will be ^^..^OOQ"' n Thus in the case considered above where all of the coefficients of the continued fraction were positive we see from expression (2.13) that if the continued fraction 13 converges, then its value will be greater than any of its even order convergents and less than any of its odd order convergents. P P I-r^limf ._7rn - = ^+ °^o or i.^limf n n->oQQ n->«>Q = -00, then the continued n n fraction is said to be inessentially divergent. If T • P lim n does not exist, then the continued fraction is said n-><«Q n to be essentially divergent. Unlike the characteristics of a series, the conver gence of a continued fraction may be influenced by the omission of a finite number of the first few terms. If -co for all m > 1 the continued fractions V a^0;' a converge. OO Jm then the continued fraction V ^r^0J' a is said to be vj absolutely convergent. If, however, there exists some QO m > 1 such that the continued fraction V diverges, Sir^i0' a ^oo V m V then the continued fraction ^0' a is said to be vj conditionally convergent, provided it does not diverge. Suppose that the coefficients of a continued fraction are functions defined on some set, say A, and suppose that no partial denominator of this continued fraction is ever zero. Then this continued fraction is said to be uniformly convergent on A to some function, say f, if for every -6 > 0 there exists an N > 0 such that n > N 14 implies P ^ - f < € Qn on A. Now if this continued fraction is indeed uniformly convergent on A to some function, say f, then we see that the series (c.f. equation (2.6)) OO OO ^ + ) /!v Vi ^- y . ,.v.i^\-i ...\ converges uniformly on A to f. In fact, this series and the continued fraction are identically equal on A. Continued Fractions and Power Series We now turn our attention to the class of continued fractions of the form a.,z a^jZ a^z (2.14) 1 ^ ~ ^-jr ^-r ^ ••• , where z is a complex variable and a. 7^0, i = l, 2, 3, ••• . A continued fraction of the form (2.14) is a con tinued fraction corresponding to the power series i (2.15) 1 +^ Z-L^ij ^^ ^ i=l provided the Maclaurin expansion of the n convergent of the continued fraction coincides, term by term, with series (2.15) up to and including that term of series 15 (2.15) which contains z^. If the constant term in series (2.15) is c^., then the constant term in the continued fraction corresponding to series (2.15) will be CQ. It is known that to every power series of the form (2.15) there corresponds at most one continued fraction of the form (2.14) in the sense that the Maclaurin expansion of the n convergent of the continued fraction coincides, term by term, with the power series up to and including that term of the power series which contains z . (The existence of this continued fraction is dis cussed in Chapter IV of this paper.) It is a remarkable, and at the same time an extremely important fact in the theory of continued fractions, that for a given power series the corresponding continued fraction often converges when the variable z lies in certain domains for which the power series diverges. For example, the continued fraction associated with the power series 00 (2.16) YJZ-^"^ i=0 is simply (2.17) z - 1 16 which terminates abruptly with its first convergent. Series (2.16) diverges for all |z| < 1 whereas continued fraction (2.17) converges for all z except z = 1. CHAPTER III CONTINUED FRACTION EXPANSIONS Introductory Examples Let ^, where p and q are integers with q > 0, be any rational number. The division of p by q may be expressed iii the foinn where a^ is the unique integer so chosen as to make the remainder r^ greater than or equal to zero and yet less than q. Now if r^ = 0, then ^ = ^Q« However, if r^^ / 0 we write R q = ^0. ^ fr.^' 0 < "^1 < and then proceed to express the division of q by r^ in the form where a, is the unique integer so chosen as to make the remainder r^ greater than or equal to zero and yet less than r, . Now if r2 = 0, then q = ^Q + —. However, if r^ / 0 we write 17 18 1 -r,^ = a,1 + —r^, 0 < r^ 2^< r1, , and repeat the above procedure with respect to the division of r, by r2» etc. Now the remainders r^, r^, r^, ••• in the above development form a decreasing sequence of non-negative integers, q > r-, ^ r2 > • • • • Since q is finite, there exists an integer n > 0 such that r _^, =0, and hence ^ ^ n+1 q can be represented by a finite continued fraction, namely (3.1) ^=^n+ — ^ — ^ ^—• q^ 0 a,1 + 2a ^ + ••• + a n Due to the manner in which the coefficients a^, a, , a-i^:* •••, an where chosen, we note that the continued fraction (3.1) is a unique representation of the rational number ^ with the one exception that we can always modify the last term, a , so as to make the number of terms in n the expansion either even or odd. That is, if a / 1, then £ _. ^ + — — — -y q 0 a^ + a2 + • • • + (a^ " •'"^ •*" T = a + -1- -^ 1 1 i, - 0 aj^ + a2 + • • • + a^_^ + a^ - 1 + 1 and if an =1, then 19 ^ = a« + — ^0 " aT1 + a-2^ + • • • + n-a^1 1 + 1' Thus any rational number can be expressed as a finite continued fraction in which the last term can be modified so as to make the number of terms in the con tinued fraction either even or odd. As an example, consider the continued fraction expansion of the rational number -^29. To obtain this expansion we perform the following operations: 67)29(0 = a^ 0 29)67(2 = a, 58 ^ 9)29(3 = a^ 27 2)9(4 = a.. 8 1)2(2 = a. 1 0 Thus 67~^2+3+4+2' or 67 - °% ^ 3 + 4+ 1 +^ 2 + 3 + 4+ 1 + r Similarly, we can also represent irrational numbers by continued fraction expansions. The concept of successive division as presented in 20 the above example is the concept which is most commonly used in obtaining the continued fraction expansions for elementary functions. For example consider the complex- valued polynomial P(z) = 1 + 3z + z^. Factoring z out of the last two terms of P(z) we obtain P(z) = 1 + z(3 + z) which we rewrite as P(z) = 1 + 1 • 3 + z Now performing the division ^5—-— we obtain P(z) = 1 + 1 3^ 3 3 + z which we rewrite as 3z P(z) = 1 + 1 - ^ 3 + z That is, P(z) = 1 + — 1 ' 1-^ ^^f which is valid for all z / -3. 21 As another example, consider the complex-valued function e^. The Taylor series for e about z = 0 is given by 2 3 4 ®e^ = -l+-^ 1^! + ^2 ! + -3^! + 4^+.-1 - . To obtain the first few terms of the continued fraction expansion for e we truncate the above series, say to the fourth term, and then factor z out of the last three of these terms to obtain 2 e^ f^ 1 + z(l +§ + -%-) which we rewrite as e^ ^^^ 1 + 1 + ^ + ^ •^2 6 Now performing the division ^ we obtain •^2^6 e^ ^ 1 + ^ _ 2 6_ 2 1 -h ^2 + 6^ which we rewrite as 22 e^ «v 1 + 1 - ^ •^ 2 1 + ^ + ^ •^2 6 1 + ^ 2 6 2 1 + ^ + ^ 9 6 Now performing the division —r we obtain e^ ^ 1 + 1 - 2 ^ + z_ 2 + 16 + ^6 2 6 which we rewrite as ^ .i3«< 1 4. . , Z ^9 1 1 Z 1 - 2 + ^ 1 + ^ 2 6 1 + ^ 6 6 2 6 Now performing the division ^^ we obtain i + — 6 6 e^ ^ 1 + 1 - 2 + z^ 3 - 3 1 + ^ 6 6 which we rewrite as 23 e^ ^^ 1 + 1 - 2 + 1 + ^ 2 2 which is valid for all z / -1, In general, e^ = 1 + f _ f ^ f _ f ^ ... for |z| < OO. Examples of Continued Fraction Expansions for Certain Functions Utilizing a procedure similar to that described in the previous section, the following continued fraction expansions may be obtained. e^ - 1 Z Z Z Z , I I y e^ + 1 ^ + 6 + To + 14 + .-. ^^^^ 1^1 ^ "^ 2 2 2 tanz=-|_-^_-^_^_ ... when (z) < «, 1 M+^-Z^ Z^ 2^ ^ ^ ^ 9Z logu + 2^-1 + 2+3 + 4 + 5 + 6 + 7 +••• when (zj < 1, ,T ^ sk _ 1 kz (1 + k)z (1 - k)z 2(2 + k)z (1 + z) -Y_ — + "2 + (2)(3) + (3)(4) 2(2 - k)z 3(3 + k)z + (4)(5) + (5)(6) + ... when )zI < 1, z zi. (2)(3)z^ (4)(5)z^ sin Z = -:r , :5' 9 n ^ ^ (2)(3) - z^ + (4)(5) - z^ + (6)(7) - z^ + •'• when jzj <. «>, 24 2 .2 Q 2 ,^2 z z 4z 9z 16z arctan z = 1+3+5 + 7+ 9 +••• when)zj < 1, ^2 3 2 2 2 -tV^. x^ 9xf. 13x 275x s ... 0® ^^ " ^ " 3 + 10 - 21 + 78 + for all real x > 0, ®^-dt = ^ 1 1 2 2 3 i^z+; t Z+1+Z+1+Z+1+'** for all z not lying on the negative real axis. CHAPTER IV THE QUOTIENT-DIFFERENCE ALGORITHM Introduction From its initial announcement by E. Stiefel [23] in 1953, the Quotient-Difference algorithm has been success fully applie''! to nvmerous problems of numerical analysis. These problems include the determination of the poles of a meromorphic function from the coefficients of one of its Taylor expansions, the computation of eigenvalues and eigenvectors of a matrix, the approximation of the zeros of a polynomial and the determination of the continued fraction expansion of a function represented by a power series. We are concerned here with the last application mentioned above. First, however, we shall consider the Quotient-Difference algorithm in general to permit a complete understanding of the scheme and its limitations. This study requires knowledge of the Hankel Determinants which is where we shall begin. Hankel Determinants For n = 0, 1, 2, ... and k = 1, 2, 3, ••• the determinants 25 26 n ^n+1 • • • ^n+k-1 (k) ^n+1 ^n+2 • • • ^n+k (4.1) H n ^n+k-1 ^n+k * * ' ^n+2k-2 with H^ ^ = 1 for all values of n, are called the Hankel Determinants. One of the basic properties of Hankel Determinants is that (4.2) LMY _ n^yuM , ^(k+l)^(k-l) ^ Q \^ n J n-1 n+1 n-1 n+1 for all positive values of n and k. We note that expression (4.2) can be rewritten as ^(k)^(k) _ / (k)) (4.3) (k+1) n-1 n+1 V n / Hn- 1 (k-1) Hn+ 1 to provide a formula which can be used in a recursive manner to generate the H '^ determinant. That is, knowing HJ^ = 1 and H = a , we can compute H^ from expression n n n n ^ (4,3). Then, knowing H and H , we can compute H from expression (4.3), etc. The Quotient-Difference Algorithm Consider the problem of determining the zeros of a polynomial of the form 27 (4.4) P(z) = a^z^ + a.z'^ + ••• + a .z + a , 0 1 n-1 n where P(z) has n distinct zeros, say z , z ,,•••, z, . ^ n n-1 1 A method due to D. Bernoulli for finding the n zeros of P(z) considers the related difference equation (4.5) D^(u) = a^u^ + a^u,^_^ + ... + a„u,^_„. Note that if P(ZQ) = 0 and ^ = ZQ, then k-nT n . n-1 , j. ^ I = 0 SO that u. = ZQ is a solution of DJ^(ZQ) = 0. Thus if u^"^^, u^^\ ••• , u^'^^ are solutions of D (u) = 0 k and c, , c^, ••• , c are constants, then c^u^"*"^ + c^u^^^ + ... + c u^^' is also a solution of D, (u) = 0 so that k (4.6) u^ = c^z^ + c^z^ + ••• + c^z|^ is a general solution of Dj^(u) = 0* With these results in mind, we continue with a description of Bernoulli's method by assuming that c-j^ 7^ 0 and that P(z) has a single dominant zero, say z^^; 28 i.e. |z^| > [zjl for j = 2, 3, 4, ••• , n. Then expression (4.6) can be rewritten as ki (4.7) u^ = c,z^ l.^/^^ .fi ilv =iW/ + ... + It follows that in any sequence, {u. }, generated recursivly by setting Dj^(u) = 0 in expression (4.5), the th lim lim^i+1 i " term is approximated by ^^^c^z^. In fact ^^-^ = z. "^+1 To see this, let q, = so that from expression (4.6) we have k+1 ^ k+1 ^ + c zk+ 1 (4.8) ^+1 ^1^1 "" ^2^2 "• n n ^k = k ^ k _^ k ^1^1 ^ ^2^2 "• n n Now dividing the numerator and denominator of expression (4,8) by c,z, we obtain k+1 , , k+1 )c+l 1 + + •. • + =ii^i/ c^iz^ z . Now —1 < 1 for j=2, 3, 4, ••• ,n. Thus i^^t = ° lim for j = 2, 3, 4, ••• , n. Hence j^->«,q,^ ~ ^1* ^° ^^ ^^^^ has a dominant zero, then the sequence (^yA will converge to it. 29 It must be noted that in order to generate the sequence ^\JL\ from the recursive formula (4.5) with D. (u) = 0, starting values for UQ, u_,, ••• » u_ +1 must be chosen. Further, in order to obtain the pre- ceeding results, these starting values must be chosen such that the dominant zero of P(z), z,, is represented by expression (4.6). This is to insure that c, / 0. Now it has been shown that if UQ = 1 and u_, = u_^ = • • • = u_ _j^, = 0 are chosen as the starting values for expression (4.5), then c, ?^ 0. As an elementary example, let us apply Bernoulli's method to determine the dominant root of the real-valued polynomial represented in the equation (4.9) x"^ - 5x^ + 9x^ - 7x + 2 = 0. The difference equation associated with equation (4.9) is (4.10) u^ - 5u^_^ + 9u^_2 - 7u^_3 + 2u^_4 = 0. Taking UQ = 1 and u_^ = u_2 = ^,3 = 0 as the starting values for the recursive formula (4.10), we compute u^. Then knowing u,, UQ, U_^ and u_2 we compute U2, etc. The resulting sequence, fu^}» as well as the ratio - ,!Vi ^k u. , are given in Table A. 30 Ic 1 2 3 4 5 6 7 "^ 5 16 42 99 219 466 968 3.2000 2.6250 2.3571 2.2121 2.1279 2.0773 2.0465 "^k k 8 9 10 11 12 13 14 "k 1,981 4,017 8,100 16,278 32,647 65,399 130,918 . ^k 2.0278 2.0164 2.0096 2.0056 2.0032 2.0018 TABLE A Thus the sequence -Tq^^) appears to be converging to the number 2 as the single dominant zero of equation (4.9). Now Bernoulli's method gives only the dominant zero of a polynomial. A more powerful scheme, known as the Quotient-Difference algorithm, exists as an extension of Bernoulli's method, and gives all of the zeros of a polynomial, including complex conjugate pairs, simultan eously. This method involves computing a table of quotients and differences (resembling a difference table) from which the zeros are deduced. To extend Bernoulli's method to form the Quotient- Difference algorithm we define ^(i) (i+1) Wl (i) (4.11a) (Tf^k+l ^ 31 and (4.11b) 4^^ - ,u) - ,(i) . .a-1) for i=l, 2, 3, ... , n-1 and k=0, 1, 2, ••• , where ^k ^ ~u^ ^^^ d^ ^ = 0 for all k > 0. The various k quotients and differences formed from (4.11) may be arranged according to the format of Table B. a^^'> q^2) (3) ^2 ^1 ^0 0 4^) 42) d(3) 4'^ 4'' .p^ ,(4) 0 41^ 42) d{3) .i^^ 4^' 4'' 4^^ 0 41) 42) 43) (1) . (2) . (3) . (4) : •'s : "^4 : "^a : "^2 TABLE B Relations (4.11) can be remembered by observing the rhombus-shaped parts of Table B. In a rhombus centered in a quotient column the sum of the SW pair will equal 32 the sum of the NE pair. In a rhombus centered in a difference column the product of the SW pair will be equal to the product of the NE pair. These are called the rhombus rules. Now the number of quotient columns in Table B will be equal to the degree of the polynomial which is being considered, and each quotient column will, with increasing values of k, converge to a root of the polynomial. If the polynomial does not possess a pair of complex con jugate roots, then the difference columns will converge to zero. The presence of a pair of complex conjugate roots is indicated whenever a difference column does not converge to zero. In this case the polynomial P. = z^ - A.z + B. is formed where Aj = ^i^(q,^+i + "^k"*^^^) ^^^ B. = , III, q, ^ . The roots of this polynomial will J k-^^^'k k be complex conjugate roots of the original polynomial which is being considered. It must be noted that there are two procedures which may be followed in order to produce Table B. Of course the most obvious procedure is to produce the table column by column from left to right making use of the recursive formulas of (4.11). However, this method is very sensitive to roundoff error and is therefore considered 33 to be unstable. A second procedure is to produce the table row by row from top to bottom utilizing fictitious entries for the top two rows as given by (c.f. (4.4)) 0 0 0 , ^0 0 n ^n-1 and then applying the rhombus rules to fill each new row in its turn. This method is less sensitive to roundoff error than is the first procedure mentioned and is therefore considered to be the superior of the two. Finally, we note that the first expression of (4.11) implies that the Quotient-Difference algorithm fails to exist whenever one of the d/^ , 0 < k < n, becomes zero. Again, however, this condition will only occur whenever one of the u. becomes zero. As an example of the application of the Quotient- Difference algorithm, let us consider the problem of determining the roots of the real-valued polynomial represented in the equation (4.12) x'^ - lOx^ + 35x^ - 50x + 24 = 0. Utilizing the row by row procedure the quotient- difference table for (4.12) may be given as 34 d q d q d q d q 10 0 0 0 0 -3.5000 -1.4286 -.4800 6.5000 2.0714 .9486 .4800 0 -1.1154 -.6542 -.2429 5.3846 2.5326 1.3599 .7229 0 -.5246 -.3513 -.1291 4.8600 2.7059 1.5821 .8520 0 -.2921 -.2054 -.0695 4.5679 2.7926 1.7180 .9215 0 -.1786 -.1264 -.0373 4.3893 2.8448 1.8071 .9588 0 -.1158 -.0803 -.0198 4.2735 2.8803 1.8676 .9786 0 -.0780 -.0521 -.0104 4.1955 2.9062 1.9093 .9890 TABLE C Now the zeros of equation (4,12) are 4, 3, 2 and 1, which are the numbers that the quotient columns of Table C appear to be converging to. Note that the difference coliimns all appear to be converging to zero. It has been shown (Peter Henrici [lOJ) that a necessary and sufficient condition for the Quotient- 35 Difference algorithm to be valid is that the Hankel Determinants, H^"'" , composed of the elements a., always be nonzero for i=l, 2, 3, ••• ,n and k = 0, 1, 2, •• Further, if the Quotient-Difference algorithm is valid, then (i)„(i-l) (4.13a) q (i) "k + l"k k (i) (i-1) X K+1 and (i+1) (i-1) (A ^•:i>.\ ^(i) K n<+l ^^•^'"^^ ^k = „(i)„(i) ^k ^+1 for i=l, 2, 3, ••• ,n and k=0, 1, 2, ••• . The Quotient-Difference Algorithm and Continued Fractions Recall that to every formal power series OO (4.14) 1 + / c.z^ i=1 '• there corresponds at most one continued fraction of the form a-iZ a,^z a.^z (4.15) i+-L2J^ 1 + 1 + 1 + • • • where a. ?^0, i = l, 2, 3, ••• . Recall also that this continued fraction corresponds to power series (4.14) in 36 the sense that the Maclaurin expansion of the n conver gent of the continued fraction coincides, term by term, with power series (4.14) up to and including that term of the power series which contains z^. It has been shown by H. S. Wall [34] that such a continued fraction exists for power series (4.14) if and only if the determinants ^2 • • • ^j ^2 ^3 ••• ^j + 1 A . = , j = 1, 2, 3, J j+l*•• ^2j-l and c -3 ... ^ -1 •^ J ^4 • • • ^j+l B . = , j = 2, 3, 4 J j + l •2j-2 are all non-zero. H. S. Wall [34] has also shown that this is equivalent to saying that an infinite continued fraction of the form (4.15) cannot correspond to a power series of the form (4.14) which represents a rational 37 function of z. Now if the determinants A., j = 1, 2, 3, "•• and Bj> j = 2, 3, 4, ... , are all non-zero, then it has been shown (H. S. Wall [34]) that the coefficients a., i=l, 2, 3, ••• ,of the continued fraction which corresponds to power series (4.14) will be given by • (4.16a) ^1 "^ '^1 "" ^1' (4.16b) a^. = Jt\.-^ 2j A .B . J J and A.^,B . (4.16c) a :. __J±L_I 2j + l B.^TA. J+l J for j=l, 2, 3, ••• , and where AQ and B, are both defined to be one. We note that in terms of Hankel Determinants, A. = H|J"^ for j = 0, 1, 2, ••• and B. = H2"'"''•' for j=l, 2, 3, ••• . Thus from equations (4.16b) and (4.16c) we see that ^^•''^^ ^2j =- (j)Jj-l) "1 ^2 and 38 (j+l) (j-1) (4.17b) ^ - __L__2___ ^2j+l -- HH^J^H^J2 H^ ) for j=l, 2, 3, ••• . Further, we see from equations (4.13) and equations (4.17) that a^ . = -q)"' and a^2j+ ..l , = -d)-1 ^ for Jj = l', 2», 3', ... . That is, the coefficients of the continued fraction which corresponds to power series (4.14) may be obtained from a quotient- difference table composed of the coefficients of the power series. For example, consider again the power series (4.18) 1 ^ iT^ fr^fr" ••• • We note that this series obviously does not represent a rational function of z. Thus there exists a continued fraction of the form (4.15) which corresponds to power series (4.18), and the coefficients of this continued fraction are given by (4.19) aQ = CQ, a^ = c^, ^^. = -q^^ and a2j^i = -d^ ^ for j=l, 2, 3, ••• . Recalling that d](0^ )^ = 0r. and^ q^(1 ) = ^i+-^—1» 39 i=0, 1, 2, ••• ,we form the quotient-difference table with respect to the relations of (4.11). This quotient- difference table appears as Table D below (c.f. Table B). 1 0 2 1 1 2 6 0 _1 _1 6 6 3 6 10 0 -i _-!_ 12 10 1 _3_ 4 20 _1_ 0 20 1. 5 TABLE D We then obtain the coefficients of the continued fraction which corresponds to series (4.18) from Table D by utilizing the relations of (4,19). The resulting continued fraction is 1 1 i _1_ 1 + z 2I §1 §1 1^ ^ 1-1+1-1+ 1+--- which we rewrite as 40 (4.20) 1 + II ?L ?L ?L "L 1-2 + 3-2 + 5 + which is valid for all z in the complex plane. To see that continued fraction (4.20) does indeed correspond to series (4.18) we note that (c.f. relation 2.3)) PQ^^) 1 Qo(z) ~ 1' ^l^'^^ = 1(1) + z(l) _ 1 + z Q^(z) 1(1) + z(0) 1 ' ^2^^^ _ 2(1 + z) - z(l) _ 2 + z Q2(z) 2(1) - z(l) - 2 - z' ^3^^^ ^ 3(2 + z) + z(l + z) ^ 6 + 4z + z^ Q3(z) 3(2 - z) + z(l) 6 - 2z Further, the Maclaurin expansions for these convergents of continued fraction (4.20) are given by Po(z) Qo(z) = 1 + 0 + 0 + PT(Z) J- i V =l + z + 0 + 0 + . . . ^ Q^(Z) P^(z) 2 3 4 2 ^ 1,,,^,^,^, ... 41 P3(z) 2 3 4 5 • Note that the Maclaurin expansion of the i convergent of continued fraction (4.20) coincides, term by term, with series (4.18) up to and including that term of series (4.18) which contains z"^, i = 0, 1, 2 and 3. Further, these results can be extended for i = 4, 5, 6, ••• so that we conclude that continued fraction (4.20) does indeed correspond to series (4.18). As a second example, consider the power series (4.21) 1 - z + z^ - z^ + z"^ - z^ + z^ + ... The coefficients of the continued fraction correspond ing to series (4.21) are obtained from a quotient- difference table formed from the coefficients of series (4.21). Recalling that df°^ = 0 and qf-*"^ = -~^, 1 1 c^ i=0, 1, 2, '•• ,we form Table E with respect to the relations of (4.11). 42 -1 0 0 -1 0 0 0 0 -1 0 0 0 -1 0 TABLE E We then obtain the coefficients of the continued fraction corresponding to series (4.21) from Table E with respect to the relations a2 • = -q!"' c^nd a2.j^, = -d)-' , j = l, 2, 3, ••• , where a^ = CQ = 1 and a, = c, = -1. This continued fraction is 1 _ -5. — = 1 ^ l+l"l+z* which is valid for all z ;«^ -1. It may be noted that if a power series converges, then the continued fraction corresponding to this power series will also converge. That is, ^^I3q- = "^^«~^^ - -^ since any convergent series satisfies the Cauchy criterion. Thus i^Sd.-" = 0 and 43 j^^q^^ = 0 for j = 1, 2, 3, ••• ; i.e. the continued fraction converges. (The preceeding example provides a good demonstration of these results.) Continued Fractions and Definite Riemann Integrals It is now apparent that continued fractions may be successfully used with respect to many problems in mathematics. Specifically, we note that definite Riemann integrals may be evaluated from a continued fraction corresponding to the infinite series representation of the integral. To see how continued fractions are obtained for definite Riemann integrals, consider the following three examples. Example 1 X ^ I -z—TTf* where x is real, as a continued fraction. We first express T—•;^^—r as an infinite series. That IS 1 2 3 4 5 1 •*+• t = 1 - t + t"^ - f^ + t^ - t"" + • •• , and then integrate term by term to obtain 3 4 5 ^6 K_ x_ . 2i_ 21- ... (4-22) JQT^ = ^ - ¥ 3 4 5 ~ 6 44 Now the coefficients of the continued fraction corresponding to the series in relation (4.22) are determined from a quotient-difference table formed from the coefficients of this series. Recalling that d^°^ = 0 and = -^^, ^ 1 q^-^^^1 c. 1 i=l, 2, 3, ••• ,we form Table F with respect to the relations of (4.11). 0 1 2 1 0 6 1 _i 3 3 0 ~\2 5 1 __9_ __3_ 4 20 10 0 1 .2- '20 15 1 __8_ 5 15 __1_ 0 30 6 0 TABLE F 45 The coefficients of the continued fraction correspond ing to the series in relation (4.21) are then obtained from Table F with respect to the relations a.^ • ~ ~^i and a2.j + 2. ~ ~^i *j = l»2, 3, ••• , where a^ = CQ = 0 and a, = c-. =1. This continued fraction is 1 1 i 1 _!. ^ 2^ 6^ ^ 5^ 10^ 1+1+ 1 +1 +1 + 1 +... which we rewrite as X X X x X 3x 1+2+3+1+5+ 2 J.+•• ..•. which is valid for all [x| < 1. Example 2 Express j^r^e -t^dt , where x is real, as a continued fraction. _t2 We first express e as an infinite series. That IS o 9 2 2 ^ 2 "* e =1 '^ ^ ^ 2! 31 4! _,_,2,tl_t!,|!_.. , - -^ ^ 21 3! 41 ' and then integrate term by term to obtain 46 2 3 5 7 (4.23) J^Ooe^ ^ dt = X - ^X X + X 3 (5)2! (7)31 9 X X , 2 3 = X _ iiiii + (x^) _ (x^) 3 10 42 216 J* Now the coefficients of the continued fraction corresponding to the series enclosed within brackets in relation (4.23) are determined from a quotient-difference table formed from the coefficients of this series. Recalling that d^'^^ = 0 and qf^^ = -ii^, 1 1 c. 1 i=0, 1, 2, ••• ,we form Table G with respect to the relations of (4.11). 0 1 3 1 0 30 3 39 10 70 13 739 0 210 1638 5 275 1638 21 11 0 252 7 36 0 TABLE G 47 The coefficients of the continued fraction corre sponding to the series enclosed within brackets in relation (4.23) are then obtained from Table G with respect to the relations a2 . = -q!'' and SL^.^^ = -d^ » j = l, 2, 3, ••• , where aQ = CQ = 1 and ^j. ~ ^1 ~ ""3* This continued fraction is 1^ 2 _3_ 2 13 2 275 2 _ i!L 10^ ^10^ 1638^ ^ 1 + 1 -1 + 1 +••• which we rewrite as ^ x^ 9x2 13x1 275x2 3 + 10 - 21 + 78 + which is valid for all real x > 0. Finally, we have r^ -t2^^ _ x^ 9x^ 13x2 275x2 ^^e at - X - 3 ^ -LO - 21 + 78 + ••• , which is valid for all real x> 0. Example 3 Express L^^!? ^dt, where x is real, as a continued fraction. We first express ^^" ^ as an infinite series. That is, 48 sin t _ 1, i! ^ t^ t"^ ^ t^ ^ , t ~ t^^ ~ 3! •" 5T " TT "" "gT "*• '"^ -1 t!^ tl t!^ t!, 31 51 71 9! ' and then integrate term by term to obtain X ^ R 7 J Sin t. _ X X X 9 J. X . (9)91 ^ 2 3 = X 1 -ii^ + (x^) _ (x2) 18 600 35280 + -iii^3265920 + ...1J .* Now the coefficients of the continued fraction corresponding to the series enclosed within brackets in relation (4.24) are determined from a quotient-difference table formed from the coefficients of this series. Recalling that df°^ 0 and = -~^, ^1 = q^-^^^1 c. i=0, 1, 2, ••• ,we form Table H with respect to the relations of (4.11). 49 0 1 18 0 23 900 3 100 • • '" 191 0 14700 5 24625 294 3032316 0 197 31752 7 648 0 TABLE H The coefficients of the continued fraction corre sponding to the series enclosed within brackets in relation (4.24) are then obtained from Table H with respect to the relations a^ . = -qj"' and ^2.^-1 = ~^i > j = 1, 2, 3, ••• , where ag = CQ = 1 and ^1 = ^]_ - ~Y&' This continued fraction is 1 ,,2 3 „2 191 „2 24625 ..2 TS1 ^ TOO^ 14700^ 3032316^ 1 - 1 + which we rewrite as 50 1 _ 2ii 27x2 191x2 24625x2 18 + 50 - 294 + 10314 + ••• , which is valid for all real x > 0. Finally, we have r^sin t^^ = X - — 27x2 i9ix2 24625x2 0 t 18 + 50 - 294 + 10314 + ••• , which is valid for all real x > 0. BIBLIOGRAPHY 1. Ahlfors, Lars V., Complex Analysis, New York: McGraw-Hill Book Company, 1966. 2. Aitken, "On Bernoulli's Numerical Solution of Algebraic Equations," Proc. Roy. Soc. Edinburgh, vol. 46 (1925-6), pp. 289-305, 3. Dennis, J. J. and Wall, H. S., "The Limit-Circle Case for a Positive Definite J-Fraction," Duke Math. Jour., vol, 12 (1945), pp. 255-273. 4. Frank, E., "Corresponding Type Continued Fractions," Amer. Jour, of Math., vol. 68 (1946), pp. 89-108. 5. Gargantim, I. and Henrici, Peter, "A Continued Fraction Algorithm for the Computation of Higher Trancendental Functions in the Complex Plane," Math. Comp., vol. 21 (1967), pp. 18-29. 6. Hamburger, H., "Uber eine Erweiterung des Stieltjes- schen Momentenproblems," Parts I, II, III, Math. Ann., vol. 81 (1920), pp. 235-319; vol. 82 (1920), pp. 120-164 and pp. 168-187. 7. Hellinger, E. and Toeplitz, O., "Zur Einordnung der Kettenbriichtheorie in die Theorie der QuadratisChen Formen von Unendlichvielen Veranderlichen," Jour, fiir Math., vol. 144 (1914), pp. 212-238. 8. Hellinger, E., "Zur Stieltjesschen Kettenbruchtheorie, Math. Ann., vol. 86 (1922), pp. 18-29. 9. Hellinger, E. and Wall, H, S,, "Contributions to the Analytic Theory of Continued Fractions and Infinite Matrices," Ann. of Math., 2, vol. 44 (1943), pp. 103-127. 10. Henrici, Peter, "The Quotient-Difference Algorithm," Amer. Math. Soc, vol. 49 (1958), pp. 23-46, 11. Henrici, Peter, Elements of Numerical Analysis, New York: J. Wiley, 1964. 51 52 12. Henrici, Peter, "An Algorithm for Analytic Continua tion," SIAM Jour. Numer. Anal., vol. 3 (1966), pp. 67-78. 13. Hildebrand, F. B., Introduction to Numerical Analysis, New York: McGraw-Hill Book Company, 1956. 14. Jones, W. B. and Snell, R. I., "Truncation Error Bounds for Continued Fractions," SIAM Jour. Numer. Anal., vol. 6 (1969), pp. 210-220. 15. Khinchin, A. Y., Continued Fractions, Chicago: University of Chicago Press, 1964. 16. Khovanskii, Alexey N., The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory, Groningen: P. Noordhoff, 1963. 17. Olds, C. D., Continued Fractions, New York: Random House, 1963. 18. Paydon, J. F. and Wall, H. S., "The Continued Fraction as a Sequence of Linear Transforma tions," Duke Math. Jour., vol. 9 (1942), pp. 360-372. 19. Pringsheim, A,, "Uber die Konvergenz Unendlich Kettenbrliche," Sb. Miinchen, vol. 28 (1898), pp. 295-324. 20. Pringsheim, A., "Uber einige Konvergenzkreiterien fiir Kettenbrliche mit Komplexen Gliedern," Sb. Miinchen, vol. 35 (1905). 21. Rudin, Walter, Principles of Mathematical Analysis, New York: McGraw-Hill Book Company, 1964. 22. Scott, W. T. and Wall, H. S,, "A Convergence Theorem for Continued Fractions," Trans. Amer. Math. Soc, vol. 47 (1940), pp. 155-172. 23. Stiefel, E., "Zur Interpolation von Tabellienten Funktionen durch Exponentialsummen und zur Berechnung von Eigenwerten aus den Schwarzschen Konstanten," Z. Angew. Math. Mech., vol. 33 (1953), pp. 260-262. 24. Stieltjes, T. J., "Recherches sur les Fractions Continues," Ann. Fac Sci. Toulouse, vol. 8 53 (1894), J, pp. 1-122; vol. 9 (1894), A, pp. 1-47; Oeuvres, vol. 2, pp. 402-566. 25. Sweezy, W. B. and Thron, W. J., "Estimates of the Speed of Convergence of Certain Continued Fractions," SIAM Jour. Numer. Anal., vol. 4 (1967), pp. 254-270. 26. Sz^sz, O. , "Uber Gewisse Unendliche Kettenbriiche- Determinants und Kettenbrliche mit Komplexen Elementen," Sb. Munchen, 1912. 27. Szasz, O., "Uber eine Besondere Klasse Unendlicher Kettenbrliche mit Komplexen Elementen," Sb. Miinchen, 1915. 28. Van Vleck, E. V., "On the Convergence of Continued Fractions with Complex Elements," Trans. Amer. Math. Soc, vol. 2 (1901), pp. 215-233. 29. Van Vleck, E. V., "On the Convergence and Character of the Continued Fraction . . .," Trans. Amer. Math. Soc, vol. 2 (1901), pp. 476-483. 30. Van Vleck, E. V,, "On an Extension of the 1894 Memoir of Stieltjes," Trans. Amer. Math. Soc, vol. 4 (1903), pp. 297-332. 31. Wall, H. S. and Wetzel, Marion, "Quadratic Forms and Convergence Regions for Continued Fractions," Duke Math, Jour., vol. 11 (1944), pp. 89-102. 32. Wall, H. S. and Wetzel, Marion, "Contributions to the Analytic Theory of J-Fractions," Trans. Amer. Math. Soc, vol. 55 (1944), pp. 373-397. 33. Wall, H. S., Analytic Continuation of Continued Fractions, New Yt)rk: D. Van Nostrand Company, Inc., 1946. 34. Wall, H. S., Analytic Theory of Continued Fractions, New York: D. Van Nostrand Company, Inc., 1948. 35. Wynn, Peter, "Converging Factors of Continued Fractions," Num. Math., vol. 1 (1959), pp. 272- 307. 36. Wynn, Peter, "The Numerical Efficiency of Certain Continued Fraction Expansions," Proc Kon. Akad. Wet,, Ser. A, vol. 65 (1962), pp. 127-148. 54 37. Wynn, Peter, "On the Convergence and Stability of the €-Algorithm," SIAM Jour. Numer. Anal., vol. 3 (1966), pp. 91-122. 38. Wynn, Peter, "Upon the Definition of an Integral as the Limit of a Continued Fraction," Arch. Rational Mech. Anal., vol. 28 (1967), pp. 83- 148.