Padé Approximation and its Applications Amsterdam 1980: Proceedings of a Conference Held in Amsterdam, The Netherlands, October 29–31, 1980

383 Pages·1981·11.56 MB·English

by  Claude Brezinski (auth.)

THE LONG HISTORY OF CONTINUED FRACTIONS
AND
PADE APPROXIMANTS
Claude BREZINS~
U.E.R I.E.E.A - Informatique
University of Lille
58655 Villeneuve d'Ascq C@dex
ECNARF
INTRODUCTION.
Continued fractions and Pad~ approximants have played a quite important r61e in
the development of pure and applied mathematics and they are still widely used as
this congress shows. Thus I think that it is of interest for specialists in this field
to have an idea of their history. The reason is not only a cultural one but some old
works can also be the starting point of new researches.
The history of continued fractions is rather long since it begins with Euclid's
algorithm for the g.c.d. 300 years B.C. Their history also involves most of the well
known mathematicians of all ages. Thus a complete history will be too long for the-
se proceedings and I shall only give a brief account of it. The complete history with
references will be published later.
I would like to thank Bernard Rouxel from the University of Lille for his auto-
rized advice on the history of mathematics. Thanks are also due to Herman Van Rossum
and Marcel de Bruin for interesting discussions and for having accepted this long pa-
per for publication.
THE EARLY AGES.
As it is often the case in sciences, continued fractions have been used a long
time before their real discovery. It seems that their first use goes back to the
algorithm of EUCLID (c. 306 B.C. - c. 283 B.C.) for computing the .g .e d. of two
integers which leads to a terminating continued fraction. Let a and b be two positive
integers with a > b. We set r ~ = a, r I = b and we define the sequence (r k) by
rk = rk+l qk + rk+2 k = 0,i,...
with 0 ~ rk+ I < r k and qk (cid:12)9 ~" It can be proved that an index n exists such that
rn+ 2 = O. Thus rn = rn+l qn and rn+ I is the g.c.d, of a and b. Moreover we have
rk/rk+l = qk + i / (rk+i/rk+ 2)
and consequently
--+ b= qo I ql + "'" + "
Of course Euclid did not present his algorithm in that way but he used geometri-
cal considerations on the measurement of a segment by another one.
Euclid's algorithm is related to the approximate simplification of ratios as
it was practiced by ARCHIMEDES (287 B.C. - 212 B.C.) and ARISTARCHUS OF SAMOS
(c. 310 B.C. - c. 230 B.C.). Continued fractions were also implicitely used by greek
mathematicians, such as THEON OF ALEXANDRIA (c. 365), in methods for computing the
side of a square with a given area. Theon's method for extracting the square root
is in fact the Beginning of the continued fraction
rj
rl
where a is the greatest integer such that a 2 ~ A. Archimedes also gave the bounds
2165 153 _ 3 (5 + 5~ + 0~i ) < /3 < 1351 i ~ 0-- 7i 3 8 (5 0 + 5~ + + 5~ ")
Other attempts to approximate square roots have been made among the centuries.
Though none of them is directly related to continued fractions they opened the way
that will be followed later by those who really created their theory.
Another very ancient problem which also leads to the early use of continued
fractions is the problem of what is now called diophantine equations in honour to
DIOPHANTUS (c. 250 A.D.) who found a rational solution of
ax + by = e
where a, b and c are given positive integers. Such equations arose from astronomy
and their solutions are connected with theappearance of some constellations. This
pnoblem has been completely solved by the indian mathematician ARYABHATA (475-550).
Let us consider
ax + by = c
where a and b are relatively prime and a > b > O. Then by Euclid's algorithm
a
--b =qo + + "'" + "
Let us set
q--P = qo + i + ... + hn-i .
Then aq - bp : +i. If we consider the case with the positive sign then
ax + by : c(aq-bp)
and thus
(cq - x)/b : (y + cp)/a : t.
Then
x = cq - bt y = at - cp.
Giving integral values to t we obtain all the integral solutions of the diophantine
equation. This solution was also given by BRAHMAGUPTA (c. 598). In Europe the same
method will be rediscovered by Claude Gaspard BACHET DE MEZIRIAC (1581-1638) in 1612.
The same method will also be used by Nicholas SAUNDERSON (1682-1739) and by Joseph
Louis LAGRANGE (1736-1813) who wrote down explicitely in 1767 the continued fraction
for a/b but 1300 years after the indian mathematicians !
One of the most important indian mathematicians is probably BHASCARA who is born
in Bidur in 1115 and worked at Ujjayni. Around 1150 he wrote a book "Lilav~ti" wheme
he treated the equation ax - by : c. He proved that the solution can he obtained from
the continued fraction for a/b. He showed that the convergents C k = Ak/B k of this con-
tinued fraction satisfy
Ak : qk Ak-i + Ak-2
Bk : qk Bk-i + Bk-2
AkBk_ 1 - ~_iBk : (-i) k-l.
Then the solution is given by
x = $ c Bn_ I + bt y = + c An_ I + at
according as a Bn_ I - b An_ I = + .i
We must anticipate to say that the recurrence relationship of continued fractions
will only be known in Europ 500 years later. Thus it is important to notice that the
first english translation of Bhascara's book appeared in 1816 and that his work was
probably not known earlier in Eurepe.(except, maybe, by some latin translation of its
arabic translation).
.krameR The notation
b + + + ...
o
has been introduced in 1898 by Alfred PRINGSHEIM (1850-1941) while the notation
a I a 2
b+ ...
o bl+ b2+
has been introduced in 1820 by Sir John William HERSCHEL 61792-1871).
THE FIRST STEPS.
In Europe the birth place of continued fractions is obviously the north of Italy.
The first attempt for a general definition of a continued fraction was made by
Leonardo FIBONACCI .c( 1170 - .c 1250) also called Leonardo of Pisa. He was a merchant
who traveled quite widely in the East and was in contact with the Arabic mathematical
writings. In his very celebrated book "Lib~ Abaci" (written in 1202, revised in 1228
but only published in 1857) he introduced a kind of ascending continued fraction which
is not of very great interest.
The first mathematician who really used our modern infinite continued fractions
was Rafael BOMBELLI (1526 - 1572) the discoverer of imaginary numbers. Little is
"L'Algebra paste eroiggam
known about his life and career but he published a book
d~Z' ~%im~tica divisa in tAeZibri" in Bologna in 1572 followed by a second edition
in 1579 with the title "L'Algebra Op~za". In this second edition he gave a recursive
algorithm for extracting the square root of 13 which is completely equivalent to the
infinite continued fraction
Bombelli gives not hint for the success of his method nor how he discovered it
but, of course, it consists in writing
A : a2 + r
where a is the greatest integer such that a 2 < A. Otherwise we have
A-/( + a)(-A - a) = r
and thus
~).
= a + r/(a +
Replacing in the denominator, /Aby its expression and repeating indefinitely this
process we get
a+S+S+
Bombelli admits that the first version of his book was based on AL-KHOWARIZMI,
the great muslim mathematician who lived in Bagdad around 830.
This method could also be attributed to AL-HAYYAM (e. 1048 - c. 1131).
The next and most important contribution to the theory of continued fractions is
by Pietro Antonio CATALDI (1548-1626) who can be considered as the real founder of
the theory. In his book "Trattato del odom brev~simo ditrova~e la radice aAdauq
d@~/ n~mer/ ..." published in Bologna in 1613, he followed the same method as
Bombelli for extracting the square root but he was the first to introduce a symbo-
lism for continued fractions. He computed the continued fraction for i~ up to the
15 th convergent and proved that the convergents are alternately greater and smaller
than I~ and that they converge to it.
At the same period continued fractions (or, more precisely, a device related to
them) were still used to find approximate values for ratios and to simplify fractions.
About this problem we must mention the contributions of Daniel SCHWENTER (1585-1636),
Frans von SCHOOTEN (1615-16609 and Albert GIRARD (1595-1632).
THE BEGINNING OF THE THEORY.
We must now emigrate to England for the next major step in the development of
the theory.
In 1655, John WALLIS (1616-1703) in his book "AY~xL~@~ca Infi~orum" obtained
an infinite product for 4/~
4 2.3.5.5.7.7 ....
2.4.4.6.6.8 ....
Then he says about Lord William BROUNCKER (1620-1684) one of the founders and
the first President of the Royal Society
"that Most Noble Man, after having considered
this matter, saw fit to bring this quantity
by a method of infi~imals peculiar to him,
to a form which can thus be conve~e~tly writ-
ten (in our modern notation)
4
--: i+
T7
and Wallis continues
epmeN" si u~L~ti adjunga~ fra~o, quae deno-
minatorem habe~ continue fracture,,
Thus the words " continued fractions" were invented.
Wallis' most important contribution arises ten pages farther when he writes
"Nos inde hanc col~igimus regulam, cujus ope a
principio reductionem inchoemur quo~q ; lib~
oc ntinuandam
P Q P Q Q
NI N2 N3
N3 (cid:141) :+ D3 (cid:141) =: .
DI D2 D3
This is our modern recurrence relationship for the convergents of a continued frac-
tion. Wallis also made the first step to a proof of convergence when he pointed out
that the convergents of Brouneker's continued fraction are successively larger and
smaller than 4/~ and when he claimed that the process converges : "ad numeyzu~
justum acceditu~".
We must also mention the Dutch mathematician and astronomer Christiaan HUYGENS
(1629-1695) who built, in 1682, an automatic planetarium. He used continued fractions
for this purpose as described in his book "De~ipt~o automat/ p/aneta~'published
after his death. In one year Saturn covers 12 ~ 13' 3~" 18"' and the earth 359 ~ 45'
40" 30'" which gives the ratio 2640858/77706431. For finding the smallest integers
whose ratio is close to this ratio (which will give him the number of teeth of the
wheels of his planetarium) he divided the greastest number by the smallest one, then
the smallest by the remainder and so on. He thus obtained, for his ratio, the con-
tinued fraction
Huygens was also interested by the solution of the diophantine equation
py - qx = (cid:127) .i He developed p/q into a continued fraction and noted that the conver-
gents converge to p/q and are alternately smaller and greater than p/q. x and y are
respectively given by the numerator and the denominator of the convergent immedia-
tely preceding p/q. Then py - qx = *i or -i according as x/y is smaller or greater
than p/q. This method was in fact Aryabhata's method and it was used by the
Englishman Nicholas SAUNDERSON to solve ax - by = c where c is the g.c.d, of a and
b. Saunderson proved some additional results on the method such as optimality pro-
perties for the convergents of the continued fraction for a/b.
This subject was also studied by Roger COTES (1682-1716), Gottfried Wilhelm
LEIBNIZ (1646-1716), Robert SIMSON (1687-1768) and some others.
To end this section we must not forget Johann BERNOULLI I (1667-1748) who used
continued fractions to simplify the ratio of large integers and who was Euler's
professor.
THE GOLDEN AGE.
The eighteenth century is the golden age of continued fractions. It has been
marked by three outstanding mathematicians : Euler, Lambert and Lagrange all of whom
belonging to the Academy of Sciences in Berlin.
Obviously the major contribution to the theory of continued fractions is due
to Leonhard EULER (1707-1783). In his first paper on the subject, dated 1737, he
proved that every rational number can be developed into a terminating continued frac-
tion, that an irrational number gives rise to an infinite continued fraction and
that a periodic continued fraction is the root of a quadratic equation. He also gave
the continued fractions for e, (e+l)/(e-l) and (e-l)/2 by integrating the Riccati's
equation by two different methods. It must be noticed that, apart from the conver-
gence of these continued fractions which he did not treated, Euler proved the irra-
tionality of e and e .2
The first extensive and systematic exposition of the theory of continued frac-
tions was given by Euler in 1748 in his celebrated book "I~toduc~o in an~ys~
nfi~os He first gives the recurrence relationship for the convergents
a_,
Ck: JBk of the continued fraction bo + i~-i + ~--bi + ... and then shows how to
transform a continued fraction into a series
al...a n
C - : (-l)n-i B
n Cn-i Bn-i n
which leads to the relation
al...a
C : b + (-i) n-I n
o n= 1 Bn_ 1 B n
Reciprocally Euler shows that an infinite series can be transformed into a continued
fraction
+ ...
(-l)n-ic = + + ... + ~n_l-Cn
n=l n
After some examples he treats the case of a power series.
Then he comes to the problem of convergence showing how to compute the value of
- i
the periodic continued fraction C : ~ + o~ + "'" by writing C = I/(2+C) which gives
C 2 + 2C = i and thus C = ~ - .i From this example he derives Bombelli's method for the
continued fraction expansion of the square root and a general method for the solution
of a quadratic equation. The chapter ends with Euclid's algorithm and the simplifi-
cation of fractions with examples.
Euler published some papers where he applied continued fractions to the solution
of Riccati's differential equation and to the calculation of integrals. He also
showed that certain continued fractions derived from power series can converge outsi-
de the domain of convergence of the series. This is, in particular, the case for the
divergent series x - l!x 2 + 2Ix 3 - 3!x 4 + .... We proved that this series formally
satisfies the differential equation x2y '' + y = x and he got the solution
~o xe-t
y(x) = f + i7 dt.
He thus obtained a method for summing a divergent series. Then he converted the pre-
ceding series into the continued fraction
and he used it to compute the "u~s of the divergent series i! - 2! + 3! - ...
Euler's ideas on the subject will be extended later by Laguerre and Stieltjes.
In a letter dated 1743 and in a paper published in 1762, Euler investigated
the problem of finding the integers a for which a 2 + i is divisible by a given
prime of the form 4n + i = p2 + qZ. Its solution involves the penultimate conver-
gent of the continued fraction for p/q.
In 1765, Euler studied the Pellian equation x 2 : Dy 2 + i. He developed /D
into a continued fraction ~ : v + a~ + b~ + ~cl + .... He denoted the successive
convergents of this continued fraction by
i (v) (v,a) (v,a,b) (v,a,b,c)
1 (a) (~,ZS-- (a,b~c,)
He stated several equalities and proved that
(v) 2 - D.I 2 : -e ; (v,a) 2 - D(a) 2 : 8 ; (v,a,b) 2 - D(a,b) 2 = -Y ; ...
where e, B, Y, ... can be obtained from v, a, b, c, ....
The study of the numerators and denominators of the convergents as functions
of the partial denominators was first seriously undertaken by Euler around the same
time. Denoting by (a), (a,b)/b, (a,b,c)/(h,c) .... the convergents of a + b~ + c~+ ...
he proved a long list of identities such as
(a,b,c,d .... ) : a(b,c,d,...) + (c,d .... )
(a,b,c ..... q) : (q,...,c,b,a)
(a,b)(b,c) - (b)(a,b,c) : i
(a,b,c)(d,e,f) - (a,b ..... f) : -(a,b)(e,f), etc ...
In 1771, Euler applied continued fractions to the approximate determination of
the geometric mean of two numbers whose ratio is as i/x. The method can be used to
get approximate values of x p/q.
In 1773, Euler used continued fractions to find x and y making mx 2 - ny 2 minimum
and in 1780 for seeking f and g such that fr 2 - gs e = x.
In 1783, Euler proved that the value of the continued fraction m~2'" + ~ + ...
is a rational number when m is an integer not smaller than 2. ~ p
Thus Euler was the first mathematician not only to give a clear exposition of
10
continued fractions but also to use them extensively to solve various problems. It
is obvious that his influence is prominent in the development of the subject.
In 1775, Daniel BERNOULLI (1700-1782) solved the problem of finding a continued
fraction with a given sequence of convergents.
By 1750, the number ~ had been expressed as infinite series, infinite products
and infinite continued fractions but the problem of the quadrature of the circle
still remained unsolved. A first step to the negative answer to this problem was
done by Johan Heinrich LAMBERT (1728-1777). Using Euler's work on continued fractions
he got in 1766 the development
x I ~ 12x
tan x : ~i- - - I 5 -...
Then he proved that tan x cannot be rational if x is a rational non zero number.
Since tan z/4 : i it follows that neither ~/4 nor ~ are rational. Then, from an
analogy between hyperbolic and trigonometric functions, he proved, from the conti-
nued fraction for eX+l, that en(n (cid:12)9 ~) is irrational and that all the rational num-
bers have irrational natural logarithms. Lambert also proved the convergence of the
continued fraction for tan x and he ended his work with the conjecture that on"
circ~a~ or logarithmic t~anscendental quantity into which no other tra~cende~tal
quantity enters can be expressed by any l~ra~ional radical quantity".
Lambert gave some examples of divergent series whose continued fraction conver-
ges and he obtained the continued fractions for Arctan x, Log(l+x), (eX-l)/(eX+l)
and ~. It is very much remarkable for that time that Lambert gave a complete theore-
tical justification of these expansions, although a little bit complicated but per-
fectly rigourous.
The next fundamental contributions to the theory of continued fractions are
due to Joseph Louis LAGRANGE (1736-1813). In 1766 he gave the first proof that
x 2 = Dy 2 + i has integral solutions with y ~ 0 if D is a given integer not a square.
The proof makes use of the continued fraction for .D-~
In 1767, Lagrange published a "M~moire surla r~sol~on sed ~quations nm~ri-
qua" where he gave a method for approximating the real roots of an equation by con-
tinued fractions. One year later he wrote an "Add~Qn" to the preceding "M~moire"
where he proved the converse of Euler's result :