RAIRO. A NALYSE NUMÉRIQUE
M ICHEL C ROUZEIX F RANÇOISE R UAMPS
On rational approximations to the exponential
RAIRO. Analyse numérique, tome 11, no3 (1977), p. 241-243
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R.A.I.R.O. Analyse Numérique/Numerieal Analysis (vol. 11, n ° 3 , 1977, p. 24ï à 243}
ON RATIONAL APPROXIMATIONS TO THE EXPONENT1AL (1) by Michel CROUZEIX and Françoise RUAMPS (2)
Communiqué par P.-A. RAVIART
Abstract. — We give in this paper a simple characterization of the A-acceptability proper ty for a family ofrational approximations to e~*. This result, which is implicitely contained in Norsett [3]
is obtained here in a direct way. As a coroUary, we obtain the results of Ehk [1] on Pade approxi- mations to e~z.
Let r(z) = — 2—, at, bi e R, be a rational approximation
W 1 + M + . . . + bn*n
to e 2. The function r (z) is said to be ^-acceptable iff
\r(z)\ < 1 for zeCRez^Q. (1) This paper is devoted to the proof of the following.
THEOREM : Assume that the rational function r (z) satisfies
r(z) = e~z + O^2""1) (z-^0) (2) Then r(z) is A-acceptable iff
kl < K (3)
or
r(z) isreducible. (4) Proof : We set p(z) = 1 + axz + . . . + anzn
q(z)= 1 +bxz+ . . . +bnzn
(1) Manuscrit reçu le 4 janvier 1977.
(2) U.E.R. Mathématiques et Informatique Université de Rennes.
R,A.I.R.O. Analyse Numérique/Numerical Analysis, vol. 11, n° 3, 1977
2 4 2 M. CROUZEIX AND F. RUAMPS 1. THE CONDITION IS NECESSARY
Assume that r(z) is ^-acceptable. Then the pôles of r(z) belong to the open half-plane R e z < 0 and we have lim \r(z)\ < 1. Moreover, if r = pjq is
|z|->oo
irreducible, we have bn ^ 0 (otherwise the polynomial q would have a positive root) and \an\ ^ bn.
2. THE CONDITION IS SUFFICIENT
Let us prove it by induction. We first notice that the resuit is clearly true for n = 1. Let us assume that the property is true for n — 1. We consider two cases :
lst case r is reducible or an = bn = 0. Then it follows from (2) that r is the (w — \jn — 1) — th Padé approximation to e~z and, from Hummel and
A (— z) Seebeck [ 2 ] and Padé [4] r(z) may be written on the form "~l V }- with
A M - "y ( 2 n - 2 - f c ) l ( i t - l ) l ,fc
An-X{z) " k2 ,o(2 w- 2 ) ! f c ! ( n ~ 1 - k)\Z *
Therefore r satisfies the theorem hypothesis for n — 1 and it follows from the induction hypothesis that r is ^-acceptable.
2nd case bn i=- 0 and |an| ^ bn. We set (a, b) = (ax, ..., afP bt, . . . , bn) and En = { (a, b);r(z) = e^ + Q(z2«~^h \an\ ^ bn and bn + 0 }.
The subset En of R2 n is convex and therefore connected.
For ail (a, b) e En we have
+ a2y4 + . . . +any2» ^ x + A / i ) 2 n_n
and then we have oq = p1 9 . . . , an^x = Pn-i- From the inequality |ûn| < bn
it follows that <xn < Pn and therefore we have for ail (a, b)e En :
| r ( i » | ^ 1 for ail y e R. (5) And so r has no pôle on the axis Re z = 0. Therefore q cannot have a root on this axis : indeed q(iy0) = 0 implies q(— iy0) = 0 and y0 =£ 0 for q(0) = 1 ; since i j0 and — Ï J0 cannot be pôles of r, r may be written r(z) = pt (z)/q1 (z) with d°p1 ^ n ~ 2 and J ° ^1 ^ M — 2, which is incompatible with (2).
Now we set
Fn = { (a, è) ; (a, b) e En and the roots of q belong to the closed half plane R ë z < 0 }
R Â I R O. Analyse Numérique/Numencal Analysis
ON RATIONAL APPROXIMATIONS TO THE EXPONENTIAL 2 4 3
Since q has no root on the imaginary axis, we have
Fn — { (a, b); (a, b) e En and the roots of q belong to the open half plane Re z < 0 }
We notice that Fn is not empty; indeed the (n — \\n - 1) - th Padé approxi- mation to e~z is irreducible and it foliows from the induction hypothesis that its denominator An_1 has no pôle in the half plane Re z ^ 0. We set
p(z) = An_l(-z)(\+z) and q(z) = A - i ^ O + &, then we have (a, b)e Fn.
From (6) and (7) it follows that Fn is an open and closed subset of En and then we have Fn = En. Therefore if (a, b) e Eni r is analytic in a neighbourhood of the half plane Re z > 0; hence from the maximum principle and from (5) it follows that r is ^4-acceptable.
COROLLARY 1 : The Padé approximations to e~z of type (njn\ {n — \jn\
(n — 2jn) are A-acceptable,
COROLLARY 2 : The pôles of Padé approximations to e~z of type (n/n), (n - l/«), {n - 2/n) belong to the open left half plane Re z < 0.
REFERENCES
1. B. L. EHLE, Astable methods and Padé approximations to the exponential. SIAM.
J. Math. Anal. 4, 1973, 671-680.
2. P. M. HUMMEL and C. L. SEEBECK, A generalization ofTaylor's expansion. Amer. Math.
Monthly, 56, 1949, p. 243-247.
3. S. P. NORSETT, C-Polynomials for rational approximation to the exponential fonction.
Numer, Math. 25, 1975, p. 39-56.
4. H. PADE, Sur la représentation approchée d'une fonction par des fractions rationnelles.
Thesis, Ann. de l'Ec. Nor. 9, 1892.
vol IL n° 3. 1977
