Algebraic Continued Fractions

3.5.1 Algebraic Continued Fractions

Some functions cannot be well approximated by a power series, but can be well approximated by a quotient of power series. In order to study such approximations we first introduce continued fractions , i.e., expressions of the form

The second expression is a convenient compact notation. If the number of terms is infinite, r is called an infinite continued fraction.

Continued fractions were applied in the seventeenth century to the rational approxi- mation of various algebraic numbers. In such algebraic continued fractions r and the entries

a i ,b i are numbers. Beginning with work by Euler, analytic continued fraction expansions

r(z) =b 0 +

involving functions of a complex variable r(z) became an important tool in the approxima- tion of special classes of analytic functions of a complex variable.

322 Chapter 3. Series, Operators, and Continued Fractions We first study some algebraic properties of continued fractions. The partial fraction

is called the nth approximant of the continued fraction. There are several essentially different algorithms for evaluating a partial fraction. It can be evaluated backward in n divisions using the recurrence

y n =b n , y i −1 =b i −1 +a i /y i , i = n : −1 : 1, (3.5.4) for which r = y 0 . It can happen that in an intermediate step the denominator y i becomes

zero and y i −1 = ∞. This does no harm if in the next step when you divide by y i −1 the result is set equal to zero. If it happens in the last step, the result is ∞. 114

A drawback of evaluating an infinite continued fraction expansion by the backward recursion (3.5.4) is that you have to decide where to stop in advance. The following theorem shows how forward (or top down) evaluation can be achieved.

Theorem 3.5.1.

For the nth convergent r n =p n /q n of the continued fraction ( 3.5.1), p n and q n , n ≥ 1, satisfy the recursion formulas

p n =b n p n −1 +a n p n −2 , p −1 = 1, p 0 =b 0 , (3.5.5) q n =b n q n −1 +a n q n −2 , q −1 = 0, q 0 = 1.

(3.5.6) Another useful formula reads p

n q n −1 −p n −1 q n = (−1) −1 a 1 a 2 ···a n . (3.5.7) If we substitute a n x for a n in (3.5.5)–(3.5.6), then p n (x) and q n (x) become polynomials in

x of degree n and n − 1, respectively. Proof. We prove the recursion formulas by induction. First, for n = 1, we obtain

Next, assume that the formulas are valid up to p n −1 ,q n −1 for every continued fraction. Note that p n /q n can be obtained from p n −1 /q n −1 by the substitution of b n −1 +a n /b n for b n −1 . Hence

b n q n −1 +a n q n −2 This shows that the formulas are valid also for p n ,q n . The proof of (3.5.7) is left for Problem

114 Note that this works automatically in IEEE arithmetic, because of the rules of infinite arithmetic; see Sec. 2.2.3.

3.5. Continued Fractions and Padé Approximants 323 The evaluation of a continued fraction by forward recursion requires 4n multiplications

and one division. It is sometimes convenient to write the recursion formulas in matrix form; see Problem 3.5.2. One must also be careful about the numerical stability of these recurrence relations.

In practice the forward recursion for evaluating a continued fraction often generates very large or very small values for the numerators and denominators. There is a risk of overflow or underflow with these formulas. Since we are usually not interested in the p n ,q n themselves, but in the ratios only, we can normalize p n and q n by multiplying them by the same factor after they have been computed. If we shall go on and compute p n +1 ,q n +1 , however, we have to multiply p n −1 ,q n −1 by the same factor also . The formula

a 1 a 2 a 3 k 1 a 1 k 1 k 2 a 2 k 2 k 3 a 3 (3.5.8)

b 1 + b 2 + b ···=

where the k i are any nonzero numbers, is known as an equivalence transformation. The proof of (3.5.8) is left for Problem 3.5.6.

Suppose we are given a rational function R(z) = R 0 (z)/R 1 (z) , where R 0 (z) and R 1 (z) are polynomials. Then by the following division algorithm R(z) can be expressed

as a continued fraction that can be evaluated by backward recursion in fewer arithmetic operations; see Cheney [66, p. 151]. The degree of a polynomial R j (z) is denoted by d j . By successive divisions (of R j −1 (z) by R j (z) ) we obtain quotients Q j f (z) and remainders R j +1 (z) as follows.

For j = 1, 2, . . . , until d j +1 = 0, set R j −1 (z) =R j (z)Q j (z) +R j +1 (z)

(d j +1 <d j ). (3.5.9) Then

R 0 (z)

R(z) =

R (z) =Q 1 (z) 1 + R 1 (z)/R 2 (z) =···

Q k (z) By means of an equivalence transformation (see (3.5.8)), this fraction can be transformed

Q 2 (z) + Q

3 (z) +

into a slightly more economic form, where the polynomials in the denominators have leading coefficient unity, while the numerators are in general different from 1.

Example 3.5.1.

In the rational form

r(z) − 101z + 540z − 1204z + 958

7z 4 3 2

z 4 − 14z 3 + 72z 2 − 151z + 112

the numerator and denominator can be evaluated by Horner’s rule. Alternatively, the above algorithm can be used to convert the rational form to the finite continued fraction

r(z) =7− . z − 2− z − 7+ z − 2− z −3

324 Chapter 3. Series, Operators, and Continued Fractions To evaluate this by backward recursion requires fewer operations than the rational form, but

a division by zero occurs at the four points z = 1, 2, 3, 4. In IEEE arithmetic the continued fraction evaluates correctly also at these points because of the rules of infinite arithmetic!

Indeed, the continued fraction form can be shown to have smaller errors for z ∈ [0, 4] and to be immune to overflow; see Higham [199, Sec. 27.1].

Every positive number x can be expanded into a regular continued fraction with integer coefficients of the form

Set x 0 = x, p −1 = 1, q −1 = 0. For n = 0, 1, 2, . . . we construct a sequence of numbers,

Evidently b n = ⌊x n ⌋, the integer part of x n , and x n +1 = 1/(x n −b n ) . Compute p n ,q n , according to the recursion formulas of Theorem 3.5.1, which can be written in vector form,

(p n ,q n ) = (p n −2 ,q n −2 ) +b n (p n −1 ,q n −1 )

(since a n = 1). Stop when |x − p n /q n | < tol or n > nmax. If the number x is rational this expansion is finite. The details are left for Problem 3.5.1. Note that the algorithm is related to the Euclidean algorithm; see Problem 1.2.6.

The above algorithm has been used several times in the previous sections, where some coefficients, known to be rational, have been computed in floating-point. It is also useful for finding near commensurabilities between events with different periods; 115 see Problem 3.5.1 (c).

The German mathematician Felix Klein [228] 116 gave the following illuminating description of the sequence {(p n ,q n ) } obtained by this algorithm (adapted to our notation):

Imagine pegs or needles affixed at all the integral points (p n ,q n ) , and wrap a tightly drawn string about the sets of pegs to the right and to the left of the ray,

p = xq. Then the vertices of the two convex string-polygons which bound our two point sets will be precisely the points (p n ,q n )..., the left polygon having the even convergents, the right one the odd.

Klein also points out that “such a ray makes a cut in the set of integral points” and thus makes Dedekind’s definition of irrational numbers very concrete. This construction, shown in Figure 3.5.1, illustrates in a concrete way that the successive convergents are closer to x than any numbers with smaller denominators, and that the errors alternate in sign. We omit the details of the proof that this description is correct.

Note that, since a j = 1 for all j, (3.5.7) reads

p n q n −1 −p n −1 q n

= (−1) n −1 .

115 One of the convergents for log 2/ log 3 reads 12/19. This is, in a way, basic for Western music, where 13

quints make 7 octaves, i.e., (3/2) 12 ≈2 7 .

116 Felix Christian Klein (1849–1925), a German mathematician, was born 4/25. He delighted in pointing out that the day (5 2 ), month (2 2 ), and year (43 2 ) of his birth was the square of a prime number.

3.5. Continued Fractions and Padé Approximants 325 6

p Figure 3.5.1. Best rational approximations {(p, q)} to the “golden ratio.”

This implies that the triangle with vertices at the points (0, 0), (q n ,p n ) , (q n −1 ,p n −1 ) has the smallest possible area among triangles with integer coordinates, and hence there can be no integer points inside or on the sides of this triangle.

Comment: If we know or guess that a result x of a computation is a rational number with a reasonably sized denominator, although it was practical to compute it in floating-point arithmetic (afflicted by errors of various types), we have a good chance of reconstructing the exact result by applying the above algorithm as a postprocessing.

If we just know that the exact x is rational, without any bounds for the number of digits in the denominator and numerator, we must be conservative in claiming that the last fraction that came out of the above algorithm is the exact value of x, even if |x − p n /q n | is very small. In fact, the fraction may depend on tol that is to be chosen with respect to the expected order of magnitude of the error of x. If tol has been chosen smaller than the error of x, it may happen that the last fraction obtained at the termination is wrong, while the correct fraction (with smaller numerator and denominator) may have appeared earlier in the sequence (or it may not be there at all).

So a certain judgment is needed in the application of this algorithm. The smaller the denominator and numerator are, the more likely it is that the fraction is correct. In a serious context, it is advisable to check the result(s) by using exact arithmetic. If x is the root of an equation (or a component of the solution of a system of equations), it is typically much easier to check afterward that a suggested result is correct than to perform the whole solution process in exact arithmetic.

The following theorem due to Seidel 117 gives a necessary and sufficient condition for convergence of a continued fraction of the form (3.5.12).

117 Philipp Ludwig von Seidel (1821–1896), German mathematician and astronomer. In 1846 he submitted his habilitation dissertation entitled “Untersuchungen über die Konvergenz and Divergenz der Kettenbrüche”

(Investigations of the Convergence and Divergence of Continued Fractions).

326Chapter 3. Series, Operators, and Continued Fractions

Theorem 3.5.2.

Let all b n

be positive in the continued fraction

Then this converges if and only if the series

b n diverges.

Proof. See Cheney [66, p. 184].

Example 3.5.2.

The following are continued fraction expansions of some important irrational num- bers:

e =2+ 1+ ···. 2+ 1+ 1+ 4+ 1+ 1+ 6+ For e there is a regular pattern in the expansion, but for π a general formula for the expansion

is not known. The partial fractions for π converge rapidly. For example, the error in the third convergent π ≈ 355/113 is 0.266 · 10 −6 .

Figure 3.5.1 corresponds to the expansion √

5 + 1)/2, which is the “golden section ratio” (see also Problem 3.5.3). Note also that, by (3.5.7) with a j = 1,

Then, note that x = 1 + 1/x, x > 0, hence x = (