Pade Approximants Method and Its Applications to Mechanics (Lecture Notes in Physics, Volume 47 1976)

Lecture Notes in

Physics

Edited by J. Ehlers, MLinchen, K. Hepp, Zerich, H. A. Weidenm~ller, Heidelberg, and J. Zittartz, K61n Managing Editor: W. BeiglbSck, Heidelberg

47 Pade Approximants Method and Its Applications to Mechanics

Edited by H. Cabannes

Springer-Verlag Berlin. Heidelberg. New York 197 6

Editor Prof. Henri Cabannes Universit@ Pierre et Marie Curie Mecanique Theorique Tour 66, 4, Place Jussieu 7 5 0 0 5 Paris/France

Library of Congress Cataloging in Publication Data

Main entry under title: Fade approximants method and its applications .to mechanics. (Lecture notes in physics ; 47) Bibliography: p. Includes index. I. Pade approximant. 2. Fluid mechanics. I. Cabannes, Henri. II. Series. QC20.7.P3P35 532' .01' 515 75-46504

ISBN 3-540-07614-X Springer-Verlag Berlin - Heidelberg • New York ISBN 0-387-07614-X Springer-Verlag New York • Heidelberg • Berlin This ,,york is subject to copyright, All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re* printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1976 Printed in Germany Printing and binding: Offsetdruckerei Beltz, Hemsbach/Bergstr.

Henri Pad~

(1863 - 1953)

Henri Euggne Pad~ was born in Abbeville

(France) on Dec.

17th 1863.

Admitted in 1883 to the Ecole Normale Supgrieure, he left it in 1886 with the highest teacher's degree

(Agr~gation)

in Mathematics.

After teaching at the classi-

cal secondary school in Limoges, Carcassonne and Montpellier, in 1889 in order to study in Germany,

first in Leipzig,

he was granted a leave

then in G~ttingen.

On June

21st 1892, before the University of Paris, he defended his doctorate thesis on the approximate representation of a function by rational fractions. Henri Pad~ was appointed lecturer at the Faculty of Sciences of Lille in 1897, Professor of Rational and Applied Mechanics at the Faculty of Sciences of Poitiers

in 1902 and Professor of Mechanics at the Faculty of Bordeaux in 1903. In

1906, he was elected Dean of the Faculty of Sciences of Bordeaux and became Laureate with the major prize for mathematical on the report of Emile Picard.

sciences awarted by the Academy of Sciences

In 1908 he was named Rector of the Academy of

Besan~on, he was then the youngest rector in France.

In 1917 he became rector of the

Academy of Dijon and in 1923 rector of the Academy of Aix-Marseilles, kept until he retired in 1934. Henri Pad~ died in 1953 at the age of 89.

an office he

Henri

Pad~

(1863

-

1953)

/

•

~

i ¸f

.f~'~o~

~.-""~INISTI~ftE • . .

f

-~":,,, ii,.9'~

h,

~Z~f '

REPUBLIQUE .....

',~"4" \'~

),.~

5'~-AN GA I S E

L'INSTttUCTION PUBLIQ~f,~ e*:x~;'~Gbecause

E~N)(~)

the moments

J~> we choose,

~k are dependent

.

(11-4)

More precisely we have 4)

~N)(q~)

(11-5)

I~> In that way, we generate interesting

variational

principles

fact is that if we now increase E (N+I) I % L ~)

In other words, nal principle

Ei


and E (2) are the ground state eigenvalue O

O

and eigenvector of P2 H P2, where P2 is the projector onto the two-dimensional space ~(2) spanned by the vectors

I~> and HI~ > . This remark can be completely

generalized, and is linked to the fact that the Lanczos method fer matrices, and its generalization to Hilbert space the tri-diagonalisation Jacobi method, are deeply linked with the theory of Pad~ Approximation

6)

.

To end up this section we give the formulae generalizing (IV-2) for the N th approximation. Introducing the normalized polynomials

:

~O.....~N

~(E,N)

A(E,N)

>

0.

(iv-4)

~/GNGN_ I

~ . . . . . g2N The L th approximated eigenfunction at order N is given by : N-I

(Iv-5) L

j=O

The

I~-N$ are obtained by diagonalizing the Hermitian matrix ILl =

PN H PN

(IV-6)

25 where PN is the projector onto the N-dimensional space spanned by the vectors I~>, HI~>,..o HN'I]~>. While the

~L(N)\ar / e orthogonal but not normalized,the

of orthonormal vectors because it is known that the

A(H,j) I~> are a set

A(E,N) form a set of ortho-

gonalized polynomials with respect to the spectral measure of the operator H 5).

V - APPLICATION TO A SYSTEM OF N FERMIONS. To apply effectively the Pad4-Ritz principle (IV-5) it is necessary to find a way to compute easily the moments

~k

"

~o

(v-l)

In the Z-body problem the moments are given Z-uple integrals which are difficult or even impossible to compute if k is large. However, there are cases in which the moments can be reduced to entirely algebraic expressions. An example is given in ref. (5)~ for the case of the most general d-dimensional anharmonic oscillator, by choosing a suitable variational test vector I~>

•

Here we want to consider the case of Z fermions interacting via a two body Z potential : Z ~ 2 ~ ~pi)2 HZ

"

~ i =I

P2 ~

+ i

I i<jsZ

~ ~ V( Iri'rj ])

i-I 2M

(V-2)

where we have subtracted the center of mass energy. We shall suppose that V, the two-body potential is a superposition of Gaussian potentials : .

V( l~i-~j I)

"

OlS

2

Pp ([~i-~j ]2)

e- ~P(~i-~J)

(V-B)

p=l (where

P (x) is a polynomial in x~With such a superposition, we can build up a P large class of phenomenological potentials, The interesting case of hard cores will be postponed to a further section. We must now choose the test vector in such a way that I) The Z-uple integral reduces to an algebraic calculation. ..... ~ - - ~ - ~ - ~ 5 ~ z 9 - y ~ - ~ 5 5 ~ 5 ~ - ~ - ~ [ ~ 9 ~ 9 - ~ 5 - ~ - - 5 ~ z - f ~

illed.

~]~> is a given vector~ by using "m~trix" Pad4 Approxlment most of the results exposed here generalize to the case where ]~> is a set of vectors I~i>, ]~2> o.o l~R>, generating a "model" space.

2B We shall choose i~> to be an arbitrary sum of Slater determinants built up with the states of the harmonic oscillator. -+

-+

-~

-+

%0o(rl, t) o..go(rz, t) -+

P I

-+

o ,

be denoted by Yi"

We now describe a method

f o r computing Yi' f o r i = 1 , 2 ..... M.

Assume that Yi is Known and that the values of the derivatives exist for k > o .

Then,consider

s i [ h ] =Yi + h ' f [ x i ) Pi Let r. = - - b e z qi

f[k)[x i]

the series s. defined as follows l

h2 h3 f " [ x _ ) +~T!'f'(xi) +~!' i + ""

the Pad@ apprcximant

of s, of order [m,n]. z

[2)

This implies

that r i is an element of R m and that n

si[h).qi[h) -pi[h] =OChm+n+ki+lj

[3]

for some integer value of K i, which is as high as possible.

If qi[xi+1) # o then the value of Yi+1 is defined as follows

Yi+l =ri[xi+l]

"

(4)

Since Yo is Known and since r i exists for every i, this technique allows us to compute yl,y 2, ....YM'

The value YM can be considered

as an approximate value for y[b) or I. section that lim YM =I h~ o

It will be seen in the next

if certain conditions

ere satisfied.

73

A variant of the above technique has been used by P.J.S. Watson in [9], for the case where h = b - a

or M = I.

In the next exampie

we illustrate what Kind of formulas we get if [4) is

used

to

compute approximate vaiues for I.

Example

Let m =n = 1 then the Pad@ approximant r i of order (1,1)

of (2) is the rational function associated with the irreducible form of ~ q

where

f'(x.) p=yi.f[xi)

÷ h. [f2[xi) - Y i . T

]

and

~'[x i) q =f[x i ] -h.-----~----.

The formula [4) gives 2.f2(x.) Yi+q = Yi + h . 2 . f ( x i

) - h.f'(x 1 i)

for

i=0,1

. . . . . M-1

It is clear that YM is a nonlinear combination of values of f and its first derivative at the points x,. z As a consequence of [5) we also get the following formula for approximate integration between x i and xi+ 1 :

Xi+l

2.f2(x. )

1

fxl f[t).dt ~ h.2.f(xl ) _h.f,(xi )

(5)

74

4. Convergence

properties

In this section some convergence

properties

of the method described

in section 3 will be given. Consider any one-step method of the following form

Yi+l =yi + h'g(xi'h)

for i=0,1

for computing a solution of (I) numerically.

. . . . . M-1

(6)

Assume that g(x,h) is

defined and continuous for every (x,h) in [a,b] x [o,h o] , where h ° is some positive real number.

Under this condition

the following

result can be proved.

Theorem 1.

Let Yi be d e f i n e d by (6), then

x in

[a,~

if

and o n l y i f

lim

h~ o x=xi

Yi =y(x) f o r every

g[x,o) =f(x).

This property is a special case of a theorem about the convergence o% one-step methods for the numerical equations

solution of ordinary differential

Csee [4 ,p.?1).

Due to the definition

of r. in section 3, it is clear that l

be written in the form [6) with g{x,h) = f { x ) + h .

g(x,h) = [ri(x) -yi ] /h or

f'(x] + 3 ~~ . f"(x) .....

g[x,o) = f i x ) , consequently

[4) can

This implies that

theorem I can be applied and

~yM

=y{b)"

About the order of convergence we can prove the following result, the case where K. =o in (3) for i =0,1,2 .....M-1. l the case of normal Pad~ approximants.

for

This case is called

75

Let r. be the normal Pad@ approximant l

Theorem 2.

of s. of order &

[m,n) and Y i + l be d e f i n e d by [ 4 ) , then Y ( X i + l ) - Y i + l ~O(hm+n÷1) as h ~ o ,

f o r i = 0 , 1 , 2 . . . . . M-1.

A proof of this theorem is given in [10].

5. One-step methods w i t h o u t usin~ d e r i v a t i v e s

Consider any one-step method of the form (6) for computing of (I).

In order to find the value of g[xi,h)

that derivatives

of f must be computed.

it might be possible

This is e.g. the case if (6)

is derived by using the method in section 3. with m + n > l . compute derivatives less interesting. another expression,

a solution

of the integrend can be compllcated

The need to and numerically

Therefore one could try to replace g(xl,h) without derivatives,

the same order of convergence.

in (6) by

hoping to Keep a method with

This technique

has successfully

been

applied in some cases.

If e.g. the derivative quotient,

in (5) is replaced

by its forward difference

we get

2.f2(x.) 1 Yi+l = Yi + h • 3.f(xi ) _ f(Xi+l 3

It

can be proved (see [10]) t h a t t h i s

(7)

o n e - s t e p method has the same

o r d e r o f convergence as the method d e f i n e d by using ( 5 ) .

The f o r m u l a

(7) can a l s o be c o n s i d e r e d as a n o n l i n e a r method f o r a p p r o x i m a t e integration

of f between x i end xi+ 1 , namely as

Xi÷l I xi

f(t).dt

2.f2(Xl) ~ h.3.f(xi ) _f(xi+l

)

76

6. RemarKs

In [10] some numerical examples are given, illustrating the usefulness of the nonlinear techniques derived in this paper.

For "smooth"

integrands the classical linear methods give in general better results than the nonlinear techniques.

If the integrand has a pole near the

interval of integration than the nonlinear techniques can give better results.

Due to possible singularities in formulas of type C5) and

(7), care must be taken in applying these formulas.

Difficulties can

sometimes be avoided by a careful choice of the stepsize h.

Other nonlinear techniques for numerical integration are considered in [10 I.

Author's address Wuytack L. Department of Mathematics University of Antwerp Universiteitsplein I B-26]0 WILRJK (Belgium)

77

References

[I] CHISHOLM, J.S.R.: Applications of Pad6 approximation to numerical integration.

The Rocky Mountain Journal of Mathematics 4

C1972),159-167.

[2] CHISHOLM, J.S.R,; GENZ, A. and ROWLANOS, G.E.:

Accelerated

convergence of sequences of quadrature approximations. Journal of Computational Physics 10(1972),264-307.

[3] DAVIS, P.J.; RABINOWITZ, P.: Numerical integration. Publ. Co., London,

[41 FOX, L.:

Blaisdell

1967.

Romberg integration for a class of singular integrands.

The Computer Journal 10C1967),87-93.

[5] HENRICZ, P.:

Discrete variable methods in ordinary differential

equations.

[61 KAHANER,

O.K.:

John Wiley & Sons, New York, 1962.

Numerical quadrature by the s-algorithm.

Mathematics of Computation 26(1972),689-693.

[71LYNESS,J,N.;

NINHAM, B.W.:

expansions.

Numerical quadrature and asymptotic

Mathematics of Computation 21(1967),162-176.

[61 PIESSENS, R.; MERTENS,

I.; BRANDERS, M.:

Automatic integration

of functions having algebraic end point singularities. Angewandte InformatiK (1974), 65-66.

[91 WATSON, P.J.S.:

Algorithms for differentiation and Integration.

In "Pad~ approximants and their applications" MORRIS, P.R., editor),

[I01WUYTACK,

93-98.

(GRAVES-

Academic Press, London,1971.

L.: Numerical integration by using nonlinear techniques.

Journal of Computational and Applied Mathematics. To appear.

DETERMINATION

OF SHOCK WAVES

BY CONVERGENCE ACCELERATION by Pr.

Max BAUSSET

-

TOULON

(France)

-MAY 1 9 7 5 -

The difficulties

of the determination

of stationary detached shocks

arise from the global character of the problem to be solved as it is impossible to determine shock waves in the vicinity of a point. On the contrary for attached shocks determination These difficulties non-s~at~anary

is possible step by step from the vertex of the body. can be avoided by considering

flow. If a body situated in a motionless

so that the field of initial velocities causes a shock wave the determination

is not null,

shock problems

in a

fluid is set into motion

this motion immediately

of which in the vicinity of the initial

time is a local problem. The evolution of the shocks corresponding

to this motion in the vicinity

of the starting point of the body is presented here. On analytical of the stationary detached shocks waves related to an analytical be obtained by a process of convergence

accelaration,

representation

convex body can

then by passing on to the

limit when the permanent motion is reached. The data will be supposed to be such that operations

can be considered as possible within the fields where it is being

operated.

-

E~E~_~_~_~e_!~_~_~e~i~!~e_S!~i~

:

The space is related to orthonormated

fixed axes.

The equation of the body is : .%~ s h o c k

[Y (1)

x = f(yz) +

~(t)

oI

[.i o''x body

82

in which f is supposed to be uniform and the body occupies

the region

:

x ~

f (y z) +

~ (t).

The initial data are such that : (2)

$(O) = O

~' (0) >

0

This body is plunged into a non viscous compressible, supposedly perfect fluid. This motionless representing

respectively

the pressure,

and and

the density and ratio of specific heats.

The latter will be supposed to be constant throughout ristic quantities

non heat-conducting

is defined by quantities p, p

of this fluid (velocity,

the motion.

The characte-

pressure and density) will be desi-

gnated as V, p and p at the point of the spatio-temporal

coordinates

x, y, z,

and t. The fluid and the body are motionless before t = O. Since at the initial time

~' (0) > 0 the motion of the body immediately

propagates

(3)

itself through the fluid.

x =

F (y z t )

For reasons of calculations write (4)

causes a schock wave which

Its equation will be :

symmetry which will appear below,

let us

: ~ (x y z t) (x y

z t)

~

f (y z) +

=

F

so that when considering

(y z t)

$(t) - x - x

covector r = ] x, y, z ] the vectors normal at each

instant to any point of the body and the shock are defined by :

(5)

N,

= ~r *

=

N~

= ~r ~

=

[-1,

Fy,

F'zl

f'z In these conditions,

the normal number of Mach M at time t at any point

of the shock wave and the upstream number of Mach at infinite by :

32~ are defined

83

6t (6)

= _ _ ~'(0)

M (~) =

l -

C(~r~

c designating Let us write

x ~r~) 2

2 the speed of sound so that c p = yp at any point x, y, z, t. 2 (y + I) ~ = y - 1 and relate pressure and density to their va-

lours at infinite

introducing

P (x y z t) =

will be written

under

dr+ t

(7)

:

R (x y z t) =

the continuous

equations the form

motions

of dynamics,

of the considered

fluid are

mass and energy conservations

which

:

2 ! - ~ 2" R. ~ r P = O I +~

-2 c

dtR - R.. T r (~r v) = O

(I - 2 )

dt

dimensions

P

In these conditions, defined by the classic

without

desigmating

R. dtP

(I + 2 )

the corpuscular

The solution and the boundary

p. dt R = O

derivative

of shock problems

of limits

In the absence which

+

and Tr

the trace of matrix

~r V.

is a result of the study of this system

on the body and on the shock wave.

of viscosity,

leads to the equations

the body is necessarily

a stream surface,

:

(x y z t) = 0 (8)

6r~ x V~ = ~t ~

in which

V~

is the value of V on the body

The classic motion,

conditions

of shock phenomena

mass and energy are preserved

Theae usual conditions

~ = O.

while

can be expressed

mean that the quantities

crossing

the surface

by the equations

:

of

of the wave.

84

~ (xy

z t)

=0

(9) 6t#

-- 2

V~

~r ~ ~r ~ x ~r ~

P

+2 ~2

=

~r ~

--

'c

R~

= 21(

~2~r~-I

6t~

--

~/

2

I 7t° _ 2,)

+

2 (6 t *)

in which

V~ ) P#

and R~

are the values of V, P and R on shock ~

At any point of the spatio~temporal the shock and for t to equations

~ 0

equations

= 0.

region included between the body and

(7) are identities in x, y, z and t. Added

(8) and (9), they permit the calculation of the partial n-order

derivatives of quantities V, P, R and

~ at time t = 0.

In effect, if one place oneself at the initial time when the fluid is motionless everythere except on the body which is set into motion, the position of the shock wave coincides with the position of the body. Thus one has for t = 0 the relations

(lo)

:

~ (x y z o) =

which will be designated

~ (xy

~

=

z o)

F(y z 0) = f (y z)

~o"

O

They entail the following equations V~

= V o

: R~

$o

= o

R $o

(11) P$

o

= P

~o

~

r

~

o

= ~

r

~o

85

From the given equations (8) and (9) considered at time t = O one deduces the following relations :

6r~ °

x

= (6t~)t = O =

V~o

~'(O)

(12) 6r~ ° x V~

= - (| - 2 )

l~--]--, t - 0

o

c

0r~ ° x ~ J o

(6r¢ ° x

6r#o)

(6t~)t=O

If one notices that -.(6t$)t_O = F' (y z O) one deduces by elimination of V the t initial value of the shock velocity at any point :

(13)

F' (y z O)

t

~'(O)

+ (~

2(1_ 2 )

(O)

(1_~2)

)2

+

6r~ °

x

--

6r~ °

- Second order derivatives : . . . . . . . . . . . . . . . . . . . . . . . .

The total number of the partial n-order derivatives of a function A(x y z t) being C n n+3' one will be brought to consider the table : Jl

(14)

6n rn

6nA II = II 6xn

A ( x y z t )

~n A ;

6Y n

6nA ;

6zn

which will he a line or a matrix 3 x 3 according as A has a scalary or vectorial value. Relations (|l) then permit to calculate the spatial derivatives of (x y z t) for t = 0 in the form :

n

(15) ~r n (6r~ o) =

In

-~x - n (6r~o) ; ~ y

n

n

(6r# o) ; ~--~ (6r~ o)

1

from which the initial values of all the spatial partial derivatives of the function representing the shock wave :

(16)

6 p+q F (y z O) 6y p

6z q

=

dP+q f (~ z) 6y p

~z q

can be deduced. But concerning the temporal or spatio temporal derivatives of F it is necessary

86

to use the derivatives of the boundary of limits on the body and the shock. If one designates by ~ one differential which according to the case can be

~r'

~t' or dr, one deduces from equations (8) and (9) the following linear

equations in relation to

~(~t~), ~(~t~) and

~(~r~) :

+ ( x y z t) = 0

(17)

~r ~ x

6V#

=

6(~t ~) -

6(~r~) x

v~

(x y z t) = 0

V~ = ~. ~r ~

x

~(~t ¢)

+

~. 6r ¢

x

~r ¢ "

~(~r ¢)

+ Y "

~(~r ~)

(18)

P~ = el. ~(6t ~) +

81 •

~(~r ~)

R~ = e2" 6(6t~) +

82"

~(~r ~)

in which to following quantities which are dependent only on the first order derivatives of ~ that are known at the initial time are :

= - (I - 2),

~ 2 ~r ¢ x 6r¢ + (6t~)2 ~ #

x

8 = 2(I- 2),

~r ¢ "(6t#)2

~r ¢

y = (I- 2 ) 6r~

x

~r ~ .(6t~)2

6 ¢ t

~1 =

x 6r ~

.... -2 (6 r ~ x 6 ~¢)2 c

(19) 8L = - 2(|+ 2 ) c --2

Y2 =

2c2"

(dt~) 2 (~r ~ x ~r ~ )2

(| - 2 )

r (6t0) 2

. 6r ~

2 6r ~ x dr~ ~2 = -2 ~2(I- ~ ) (~t~)4

87

According to the choice of differential 6, equations lines and matrices

(|8) include the scaleries,

: t!

~t(~t ~) =

#t 2

~t(~r ¢)

(20) ~r co

(y Z O)

one can write : ~ ~

Ft2(Y z O) '~

f~r ~

3!a2+ 1

x ~J>

2|

(1 + p 2 ) , £ 1 ( f )

F't (y z O) ~

+ (1- !a2).~2(f )

One can then see that, for an analytical body, approximation (36) remains valid for t

> O. It is clear that such is not the case if the body's velocity is

~ndeterminate.

92

-

D~_t_ermination_of_d_et_a~_hed_s_ta_tio_~_a_~z_s_h~h~

:

It is possible to infer from the foregoing results approximate formular determining by the movements

the positions of the detached stationary shocks created

of blunt analytical bodies in translation motion.

In effect, when the second derivative

~"(t) vanishes

for t ->

the motion of the body tends to become unifo©m. It is then logical to admit that the motion of the fluid in relation to a mark linked to the body tends to become stationary. stationary

shock results from the knowledge

lary logical to admit that the stationary

Consequently

the position of a

of F(y z t) for t--~oo. It is simi-

flow is reached all the more rapidly

as the velocity limit of the body is reached more rapidly. One will place oneself in the case when is chosen so that : t

~< 0 :

$ (t) = O

t > : ~ (t) = t. ~' (0)

In any space of time when F(y z t) is an analytical variable dependent on time, one can write

: oo

(37)

F (y z t) ~

~

aj(yz).t j

j=O and one has indicated a calculation process of a. coefficients. J co ral methods allowing to replace the analytical function Ya.t j

There are seveby a sequence

L

of functions

converging towards F(yz)

in the field of

O j

convergence of the entire sequence a.t j and which are definite whatever 3 value of t is and which have a limit when time increases indefinitely. Within the framework of the

(38)

A

= n

n ~ 0

a.t j J

n ~

e-algorithm

the

theory supposes

0

is a converging sequence whose terms are dependent as parameters. Among others this theory proposes

to replace this sequence by another converging

more rapidly towards the same limit or converging

to a more extensive field

than the initial sequence. PADE'S defined by :

n-order diagonal approximation

fused in (36) for n = 17 is

93

t n 1A l ) ..............

t ° An

a 2 ) .................

an+ l

an+l

a2n

Pn

(t)

t °

Qn

(t)

tnAo

al

(39)

i

t

n

a1

an which

leads

in p a r t i c u l a r

Ql(t)

a2

................

an+ !

an+l

~ ................

a2n

~

lim

P (t) A n--5-----= (-I) n

t =o

Qn(t )

A

=

the c o f a c t o r

ao

,

ald ........

an

a2

~

a 3 ~ .......

an+ |

an

~

an+|) ......

a2n

of a

o

in

(41)

A

n

•

A.

As to SttANKS'S t r a n s f o r m a t i o n s , sequence

.....

2 ao(ala 4 - a~) - a 1 ( a l a 4 - a 2 a 3) + a 2 ( a l a 3 - a 2) 2 a2a 4 - a 3

Q2(t)

and 6 b e i n g

~

~..•......•

to

lim t =

:

t

a2

P2(t) (4O)

n-1

~

: 2 aoa 2 - a i

P1(t) lim t ~

with

~ ...............

In, n| A n =

the sequence

B

n

=

they

are

defined

:

An+l"

An-l-

An+l + A n - I

A2 n

- 2 An

n

>

1

by associating

to

94

The process

can be iterated by considering _

C

(42)

= Bn+| Bn-I

B2 n

n

the sequence

:

~ 2

n

Bn+ 1 + Bn_ 1 - 2B n and so forth. In each of these iterations t

->

the first term B! C 2 ... has a limit for

oo

lim

2 aoa 2 - a I

B I (t)

a2 (43)

2 aoa 2 - a I

lim C2(t )

22 (ala 3 - a2) 2 a4a 2 - a 3

a2 One can then see that PADE'S

a4 2 a2

approximation

If,I] and SHANKS'S

first

order B 1 have the same limit which we shall call first order approximation. On the contrary,

second order approximation

differ.

- ! l ~ ! ~ _ ~ l ~ 2 ~ _ ~ ! _ ~ ! ~ ! ~ l _ ~ h ~

When the studied the choice between

Concerning the stationary

method

numerical

which is the case here,

and another

results

the approximation

detached wave, point

Placing

is determinate,

an approximation

a comparison with accurate

an indefinite

function

:

relative

of function

the preceding

can only be made through to know particular

cases.

x = F(y z t) representing

calculations

have been conducted

to

(x, y z) only up to n = 2.

oneself

at a mark linked

to the body,

it follows

from (35)

and : _

(44)

F't (y z O) -

~2

M 2

+

1 -

~2

1

~'(0) = c.

~Sr f x 8 rf ~'2 M

that in the vicinity

(45)

of the initial

time one can write

F (y z t) = f (y z) + ~t -~2

M2 + 1 - -~2

:

~l~ + tSF, ' t2 (y z O)

M in which

the expression

of the second derivative

$"(O) = O. In these conditions gives a position

of the detached

PADE'S

follows

(or SHANKS'S)

stationary

shock

:

from (26) with

first order approximation

95 (46)

x = f(y z) -

2( ~2 M2+ I - ~2)2.(M2 + 1).d 4 M 2. A 8 . ~

2 (f) -

(~r f x 6r f)

I

M 2. (M 2 + l)~A6.~l(f)

In this expression the velocity of the body and the conditions of the motionless fluid interven through the normal number of Mach which plays the role of an auxiliary parameter.

There results a certain complication even

for very simple analytical bodies. As shown above the permanent motion will be reached all the more rapidly as the body's velocity is high. In this hypothesis

the preceding rela-

tion is consideratly simplified and leads to the approximation

(47)

x = f (y z) + (I +

2 ~2(3 ~2 + I) ~ 2 ) . £ I ( f ) + (I- ~2).£2(f)

:

+ ~ ~22>

The field of validity of (46) remains to be specified in relation to ~4~as well as to r. As in the case of (47) it is linked to that of PADE'S approximations

for the accelerated sequence.

It can easily be verified that if analytical function f(y z) is chosen as uniform and convex everywhere,

the field of definition of (47) remains limi-

ted. The application of (46) or (47) to the two dimensional flows (or revolutions)

leads for the body's curve

(48) ~(Y) =

~(Y) -

x =

~ (y) to

2 ~2(3 ~2+ I)(I + ~,2)2y + ~ (I + ~2)y . ~,, _ 2y- ~"-0 '2 + ~ (I+ ~2).(|+ ~,2)~,

I ~

and : 2 (~2 M 2 + I -~2)2(M 2 + I),C4(I + ~,2) . y (49)

(y)

=

~(y)

-

I[M2 A8 - 2(M2 + I) A6]-y.~'2~"+ M2(M2 + l)A6(l+~'2)y~"l + ~ M 2 (M 2 + I) A 6 (I + ~ , 2 ) ~

with

M = ~ k, and k+m, for some integral m.

Additional

storage for whole sets of field data is required, and it varies from I to 4 sets, depending on the order of the transform and ways the eigen-value estimates are handled. In passing, we note that if the component,

~process

is strictly applied for each

i.e. at each grid point, not only more storage is required but the

redundant eigen-value estimates

implicit in such process

sistency and delay the approach to the limit.

We find the

would lead to incon~process

coverges

much more slowly in most cases. V.

EXAMPLE:

A DIRECHLET PROBLEM

In a study described in our report 11, we have tested this cyclic-transform technique on line-relaxation methods applied to a model Dirichlet problem, using

107

various relaxation parameters and sweep directions.

These numerical experiments

show that a reduction in iteration number by a factor of three to five is generally possible.

In the example of Fig. 2, a line Gauss-Seidel procedure is applied to

the system based on a 9-point central difference scheme, for which neither the optimum

relaxation parameter, nor the spectral

to the best of our knowledge.

radius,

A typical convergence history of the unaccelerated

results, using a 1/30th mesh, is shown as a solid curve. is the iteration number k.

is theoretically known

The abscissa of the graph

The accelerated result based on a 2nd-order transform

is shown in short dash with circles, which approaches the limit within 1% in 30 iterations~as compared to VI.

EXAMPLES:

three to four hundred for the unaccelerated one.

TRANSONIC THIN AIRFOIL PROBLEMS

We shall study below the results of application in transonic small-disturbance theory governed by the von K~rm~n equation.21 The discussion

is confined to the

flow over a symmetric circular arc airfoil, which has an embedded supersonic region.

The basic program to be accelerated is one similar to that of Murman and

Cole I, using an x-mesh of 2½% chord, and a y-mesh near the wing 2% chord.

For the

result shown in Figure 3a, the relaxation parameter is taken to be 1.4 in the subsonic region and

0.9 in the supersonic region.

This slide gives the conver-

gence history for the velocity perturbation near the mid chord.

The unaccelerated

result shown in solid curve takes 140 iterations to approach the limit within I%; cyclic acceleration using the first-order transform presented in thin solid curve takes 60 iterations for the same accuracy.

For the results using 2nd-order trans-

form, only data at the end of each cycle are shown in circles; this takes only 40 iterations to reach the limit within I%. The results in Figure 4b differ from the preceeding one in that, here, a uniform relaxation parameter~uJ= regions.

0.95, is used in the supersonic and subsonic

The convergence rate for the unaccelerated program in solid curve is

low, as expected, taking 400 iterations or more to reach the limit within I%. This is to be compared with the 65 and 30 iterations for the two accelerated solutions. We have also studied the acceleration of transonic solutions involving circulation,

i.e. airfoil at incidence.

The basic line-relaxation program is

the same as before, except for a doubling in the number of grid points to account for the asymmetry and the use of a somewhat different pair of relaxation parameters. One sees from Fig. 5 that the use of the first-order transform in solid curve achieves a convergence within I% at 150 iterations for the circulation, whereas the unaccelerated one may take more than 400. We would like to emphasize that the above examples

involve shock waves which

are "captured", so to speak, by the numerical procedure - thanks to the "numerical

108

viscosity" inherent in the computer program. near the shock is lost.

Because of this, the flow detail

A shock-fitting method, which modifies the computer

program to fit the shock as a surface of discontinuity, has been developed. 11 The natural question to be asked is whether acceleration and shock-fitting techniques can work together.

The answer is an affirmative one.

In Figure 5,

we present the shock and sonic boundaries from our iterative, shock-fitting solution, computed for a slightly supersonic Mach number.

The unaccelerated

result obtained after 240 iterations compares well with that obtained by Magnus & Yoshihara, who used a shock capturing method based on an unsteady approach.

With

acceleration based on a 2nd-order transform, the very same shock-fitting solution is recovered in 64 iterations. VII.

CONCLUDING REMARKS In summary, our study with the transonic flow and other examples show that

the cyclic acceleration techniques based on sequence transforms may effectively increase the convergence rate and the efficiency of the relaxation methods, with minimal programming and storage changes.

A reduction by a factor of three to

five in computer time is possible, with and without shock-fitting.

In fact,

where accurate description for the shock is important, the time saved by acceleration with shock fitting can be 6 to 36 fold.

One observes that the above

demonstration involves only the use of some of the most rudimentary forms of sequence transforms.

With an increase in data storage capacity (or facility),

it should be possible to employ the more sophisticated higher order transforms and their recurrence relations which are discussed in other parts of this Proceeding.

In the meantime, possibilities for reducing the data storage require-

ment for the higher-order transforms do exist.

This is supported by a study

described in Appendix A below for an iterative procedure applied to a linear system, making use of Wynn's recursive relations for the APPENDIX A.

~-algorithm.

IMPLEMENTATION OF WYNN'S ~-ALGORITHM FOR APPLICATIONS TO ITERATIVE MATRIX EQUATIONS

In Ref. 22, Wynn uses the ~-algorithm as an acceleration technique for iterative vector and matrix problems. +

The effective use of the transforms in

the cyclic iterative method discussed in the text, as well as the corresponding elements in the

~-algorithm,

are limited, in practice, by the increased storage

requirement for the higher-order transforms •

However, the possibility for using

higher-order transforms without the increas ing storage remains, and is confirmed below for a linear system.

This is accompl ished through application of Wynn's

+The symbol " ~ " employed in this Appendix, is not to be confused with the error vector " E~" used in the text.

109 rhombus rule, and other identities for a linear iteratlve equation system. The 22 result provides an alteration from Wynn's origlnal procedure with a substantial savings in data storage. Wynn's Recursive Relation Applied to Vectors and Matrices The power of the

~-algorithm lies in the fact, established through many

examples, that if the sequence

~o,

~ , ~z .....

~x ....... i.e.,

is slowly convergent, then the numerical convergence of the sequence 4

, "'" ,

to the l i m i t rapid.

~25

,

...

,

(or a n t i l i m i t ) ,

with which sequence { O x t

In the s c a l a r c a s e , the q u a n t i t i e s

$,l

~_f

is a s s o c i a t e d ,

~(k? s a t i s f y

E~÷,~ =

-z

,

_(o)

i.e.,

E(k}

E~ ,

is f a r more

the rhombus r u l e 15

I @

(A.l) which is closely related to the Shanks' transform.

As is well known, the elements

generated in this manner in the E-algorithm may be identified with those on the upper half of the Pade Table (cf. Sec.

II in text), hence, those in ShankS' e n

transform , 2n

,n (A.2)

J with

~(k)

= ~

~

E~ )

and setting

In cases in which

=

0

~:)x is a vector or a matrix, the algorithm is still mean-

ingful, provided the inverse of the entity is consistently defined.

Wynn has

considered the following alternative definitions. (i)

Primitive Inverse: independently;

In this case, each component is considered

it amounts to a simultaneous application of the scalar

~-algorithm to components of the array. (ii)

The Samelson Inverse of a Vector:

In this case, the inverse of the

~ : (X, XI,....,XN) is taken (after K. Samelson) to be

vector

N

X-'~(~ where (iii)

~

Xj X J ' ~ ' c a ' ~ ' ' ' ' ' ' ~ " ) '

is the complex conjugate of

~

(A.3) .

The Normally Defined Inverse of a Square Matrix:

This was not

recommended for large systems. Wynn discusses in Ref. 22 applications of the algorithm to numerical analyses, including boundary-value problems,

initial-value problems, Fredholm and Voltera

integral equations, and differential equations.

110

~/ynn's_Procedure for Acceleratincj Relaxation Solutions Of particular

interest are Wynn's application to the acceleration of the

Jacobi and Gauss-Seidel

relaxation methods for iterative solution of large systems 22

of linear algebraic equations.

For subsequent discussion,

it is convenient

cIk) to arrange the array of ~s into

the familiar pattern suggested by the rhombus rule, illustrated at the middle of the page, where the original column near the left.

sequence

{~kl

is given on the first non-zero

With the identification

given by Eq. (A.2) elements on each

column corresponds to those belonging to Shanks'

e - transform of the same order n The diagonal elements in the Pad~

(with the order increasing towards the right).

Table rn, n are identified with elements on the "roof top" with even subscript, i.e.,

_(o) Eis .

with

_--'-~o~c

0

o

-r~

~J

o

~2

E~

~ ~

,

~

~s

E?

In Wynn's applications,

a sequence of vectors or matrices

is obtained from

an iterative procedure for a linear system, say,

and stored as

E(~) before the acceleration

if we have three iterates be determined as

E~)

Do

J

~

procedure

, and

~2

For example,

according to the rhombus rule (which in this case is ident-

ifiable with Pad~'s rll or Shanks ~ e I { ~ l } i.e.,

is applied.

, a better estimate will then

if one wishes to obtain

E~) ~

).

with n ~

If more resolution

is needed,

4, more iterates (with k ~ 4)

have to be generated from Eq. (A.4) and stored. Underlying this procedure

is the assumption that at the end point (towards

the right) of the application of the rhombus rule, one shall arrive at (or near) the limit ~

satisfying the equation

(~) ~ (~ (~ + ~

This assumption can indeed be justified. components of ~ ,

(A.4b)

In fact,

inasmuch as the number of

say N, is finite, the exact solution ~

Do and 2N (and only 2N) successive

iterates,

i.e.,

can be predicted from

(~o ' ~ ' ~2~-'.',~,-..,

~,

111 ~A/+I ) . . . . > ~)2N-I, ~)aN, using rhombus rule.

This follows from Eq. (A.4a),

f o r whicha corollary of the Cayley-Hamilton theorem (cf. of Ref. l l )

Eq. (3.9) on p. 5,

gives ~/' N

where p j ' s

j:o /=0 (A. 5a~ in the c h a r a c t e r i s t i c polynormal of the i t e r a t i v e

are the c o e f f i c i e n t s

matrix Q N ] I ('"X'-~j) )=1

--

~ + ~,/~.'e Pz .A-='F''"

+~,V/~.N .

(A.Sb)

Now, the right hand member of Eq. (A.5a) is precisely eN(~),__ identifiable with E ZN 4o

while the

until

a matchin9

is obtained.

RESULTS A c c o r d i n 9 t o t h e method d e v e l o p e d i n S e c t i o n s 4 . ,

?ormed a n u m e r i c a l s u r v e y o£ t h e models i n t r o d u c e d The r e s u l t s

a r e summarized i n s e v e r a l t a b l e s

v e n i e n c e we have f i x e d variables

x, 4

the time

is finite

For t h e v i s c o e l a s t i c we have computed:

t=T

(O~x~cT,

5.,

in Sections 2,,

and f i g u r e s .

so t h a t

we have p e r 3.

For c o n -

t h e r a n g e oP t h e

O~4~T).

waves i n a S t a n d a r d L i n e a r S o l i d

(see (2.10)

197 (i)

the Green's Function

and t h e s o l u t i o n s (ii)

of the following

boundary v a l u e p r o b l e m s :

~ ( )

R.(t)=l

0

S

1 = ~ S

Ro(S) =

(ill)

i

I

s[i + ~ ( s ) ] ~ (iv) (v)

i

Ro(t) = e-at

Ro(s)

R o ( t ) = e "at cos ~ t

Ro(s)-

In T a b l e s

=

s

I - Vl we compare t h e s e r i e s

a=O.

In t h i s

R(x,t)=e_t/2 and a t than

(4.2)

results.

check For t h e P.A.

case t h e e x a c t s o l u t i o n

digits

accuracy

i s a c h i e v e d For ( i i i )

is explicitly

i s Found f o r

known 171:

T ~ SO

u s i n g no more

P.A,

In F i g u r e s 1-4 t h e r e s p o n s e s t o several

For t h e

i o { ~ ( t 2 _ c~ ) : } 2

least a five (12/12]

and P.A. s o l u t i o n s

i n o r d e r t o check t h e method we do a l s o

long t i m e and c o n v o l u t i o n

An even more s t r i n g e n t when

+~

(s + a ) 2 + 2

above boundary v a l u e p r o b l e m s ; quote t h e

s+~

values of

(ii)

and ( i i i )

are plotted

for

T.

For t h e t h e r m o e l a s t i c

waves t h e F o l l o w i n g

boundary v a l u e

is

considered:

go(t) = 0

~o(S) = 0

I Eo(t)=1

go(S)=~

In T a b l e s VII, VIII we e x h i b i t comparing t h e s e r i e s ,

P.A.

the thermal (separately

g+ # g - , E+ , E-) and Ion9 t i m e relevant

results

transient

and e l a s t i c

r e s p o n s e s by

computed on t h e e x p a n s i o n o£

1161 s o l u t i o n s .

In F i g u r e s 5 - 8 some

a r e shown.

From t h e p r e v i o u s examples plays a crucial

S

role

if

wave p r o b l e m s .

we are

we can i n f e r interested

that

t h e Padd method

i n a 91obal

solution

of

198 APPENDIX The r e s p o n s e o f a S . L , S .

R (x,t) = I

t o any i n p u t

For l a r g e v a l u e s o f

t

(A.i)

~ (s)

with

g i v e n by

we a p p r o x i m a t e ( A . 1 )

point method. FOr the Greenes ~unction result

i s g i v e n by

err(s) go(S) d s

f(S):S~-c--"~ Vl+~(s~ ,

where

to(t)

No(S): I

(2.10).

using the saddle

and the standard

reads

R ( x , t ) = [ 2 = t l f " ( ~ ) l ] -½ e- t l f ' ( ~ ) l where

s

i s d e f i n e d by

For i n p u t s most r e l e v a n t close to point

(ii) (iii)

f'(~)=O

( i i ) and ( i i i )

contribution

s=O.Replacin 9

contribution

and must be computed n u m e r i c a l l y ,

Ro(s)

to

has a p o l e a t t h e o r i g i n

(A.1)

f(s)

(A.2)

by

i s o b t a i n e d when t h e ~(s)= f'(O)s+

can be a n a l y t i c a l l y

f"(O)s 2

and t h e

saddle pointis the saddle

e v a l u a t e d and we g e t :

R(x,t) -~ ½Erfc[~W] R(x,t) -~ x V~ x+ct

whereW=-(1-

x

ct

V-a

Erfc[W~

)-(2

(A.4)

1,,:a.

x

a

c t 2 ~a

Ca

For t h e r m o e l a s t i c i n 1161 by t a k i n g

(A.3)

)-½

waves long t i m e a p p r o x i m a t i o n s

the

limit

as

s~O

of the transform

have been d e r i v e d solutions

and

read: O(x,t)

= - ~1

+~

1-

1+

E(x,t) = where

Z=x

V1 +e/?-

Erf

[Z]

Err [Z] ¢-t

•

(A.5) (A.6)

199 REFERENCES Ill

F. MAINARDI & G. TURCHETTI, Mechanics Res. Comm., in press.

1 2 1 F . MAINARDI & G. TURCHETTI, t o be p u b l i s h e d . 131 S. C. HUNTER, i n Progress in S o l i d Mechanics, e d i t e d by SNEDDON & HILL, Vol.

I , p. 3, N o r t h - H o l l a n d , Amsterdam, 1960.

141 R, M. CHRISTENSEN, Theory of V i s c o e l a s t i c i t y ,

Acad. Press, N.Y.,

1971. 151 J. D. ACHENBACH, "Wave Propagation in E l a s t i c S o l i d s " ,

North-

Holland, Amsterdam, 1973. 161G. DOETSCH, "Theory and A p p l i c a t i o n of the Laplace Transform", Sprlnger-Verla9,

N.Y., 1974.

171 E. M. LEE & T. KANTER, J. Appl. Phys. 24, 1115 (1956). 181 S, KALISKY, B u l l . Acad. Pol. S c i . 13, 409 (1965). 191 E. B. POPOV, J, Appl. Math. Mech. (PMM) 31, 349 (1967). I10) M. W. LORD & g. SHULMAN, J. Mech. Phys. S o l i d s 15, 299 (1967). Illl

J. D. ACHENBACH, J. Mech. Phys. S o l i d s 16, 273 (1968).

1121 H. PADE', Thesis Ann. Ecole Nor. ~, Suppl, 1 (1892). 1131 G, A. BAKER, J. Adv. Theor, Phys. ~, 1 (1965). 1141 J. NUTTAL, J. Math. Anal. and Appl. 31, 147 (1970). 1151 J, ZINN-JUSTIN, Physics Reports (Sect. C, Phys, L e t t , ) ~, 55 ( 1 9 g l ) , I161 F. R. NORWOOD & W. E. WARREN, Quart. J. Mech. Appl. Math, 22, 283

(1969).

200

TABLE CAPTIONS The meaning o f t h e symbols used i n t h e T a b l e s X :

distance

TAU :

time elapsed from the wave Front ;

SERIES :

r e s u l t s of the p a r t i a l sums

in

~

is the Following:

= t - x/c

(NS) of the series s o l u t i o n

,

NS :

number of terms in the p a r t i a l sums

ERS :

estimated accuracy of the series s o l u t i o n defined by

PADE :

diagonal

II_,(NS-I)I (NS)

[NP/NP]

Pad6 approximants

series s o l u t i o n in ~

computed on the

,

NP :

order of P . A . .

ERP:

estimated accuracy of P.A. d e f i n e d by

CONVOLUTION: c o n v o l u t i o n piecewise

LONG TIME :

NS=2NP+I I

II"

L.NP..- 1/NP- I ] I [NP/NP] I

of the input with the Green's f u n c t i o n computed using the series when ERS~ 10-5 ,

ERS > 10-5 ~ ERP

ERP> 10-5 .

Remark t h a t

the P.A. when

and the long time approximation when

A Gauss quadrature w i t h

NO

long t i m e a p p r o x i m a t i o n o f t h e s o l u t i o n

V i s c o e l a s t i c waves in a S.L.S. w i t h

a=.S

Table

I

Input ( i ) ,

Table

II

Input

(ii)

Table

III

Input

(iii)

T a b l e IV

Input

(iv)

with

a = ,1 ,

Table V

Input

(v)

with

a : . I , ~:T0,

for

points

is used.

(see A p p e n d i x ) .

T=30.

Green's f u n c t i o n

Thermal and s t r a i n waves w i t h

NG=8

~ = .03,

NG=20

~=1.3

(NS~21, N P ~ I O ) , Table VI

T h e r m a lwaves f o r

Eo(t)=l

go(t)=O

T a b l e VII

Elastic

Eo(t )= 1

go(t) = 0

waves f o r

for

T = 5

201

TABLE I : S,LS(i ) T = 3 0

TAU

SERIES

NS

0

2.593E-03

1

2

2,123E-02

10

4

5.042E-02

6 8 I0

ERS

PADE t

NP

2.593E-03

0

1.E-06

2.124E-02

4

3.E-04

2.28E-02

12

7,E-06

5.042E-02

5

6. E-05

5.30E-02

7.222E-02

16

9.E-06

7.221E-02

7

3 • E-06

7.53 E-02

7.451 E-02

19

1,E-05

7.451E-02

9

3 • E-08

7.74E-02

5.973 E-02

23

4.E-07

5.973E-02

11

3.E-10

6.19E-02

12

3. 867E-02

25

3.E-05

3.867E-02

12

1. E-I 0

4 . OOE-02

14

2.059E-02

25

6.E-03

2,062E-02

12

I . E-O8

2.13E-02

16

9.012E-03

25

2.E-02

9.124E-03

12

6. E-07

9.45E-03

18

2.021E-02

25

3.E+O0

3.344E-03

12

I.E-06

3,47E-03

20

1.579E+00

25

2.E+O0

1.003E-03

12

1.E-05

1.04E-03

22

3.145E+02

25

2.E+O0

2.400E-04

12

3. E - 0 4

2.50E-04

24

2.965E+04

25

2.E+O0

4.911E-05

12

1,E-O1

4.57E-05

26

1,516E+06

25

2.E+O0

4.538E-06

12

2.E-01

5,75E-06

28

3.984E+07

25

2.E+O0

2.080E-07

12

7,E-01

3.85E-07

.0

ERP .0

LONG TIME --

202

TABLE fl : SLS(~ii) T = 3 0 TAU

SER IES

NS

0

5,531E-04

1

3

5,893E-02

11

6

2. 767E-01

9

£RS

NP

5,531E-04

0

9. E-06

5,893 E-02

5

2. E-05

9.27E-02

16

4.E-06

2. 767E-01

7

5,E-07

2,49 E-Ol

5. 858E-01

2o

8. E-06

5.858£-01

9

6. E-08

5,22E-01

12

8. 288E-01

25

4. E-06

8.288E-01

12

3. E-11

8,16E-01

15

9,497E-01

2.5

2. £-04

9.497E-01

12

2. E-09

9,72E-01

18

1 . 001 E+O0

25

4. E-02

9,898E-01

12

5. E-09

9• 99E-01

21

2,117 E+01

z5

2. E+O0

9.986E-01

12

7.E-08

I • OOE+O0

24

2,788E+04

25

2. £+00

9.999E-01

12

7, E-07

I , OOE+O0

27

9,103 E+06

25

2, £+00

1, O00E+O0

12

I , E-06

1 • OOE+O0

ERP

LONG TI ME

.0

ERP

LONG TI ME

PADE'

.0

TABLE Ill : SL.,S(iii) T=..~0 SER I ES

NS

o

5,531E-04

1

3

4,980E-02

12

6

2,182E-01

9

TAU

ERS ,0

PADE'

NP

5,531E-04

0

,0

9. E-07

4.980E-02

5

2,E-05

7.34E-02

15

5.E-06

2.182E-01

7

3,E-08

1.87E-01

4,402E-01

2O

8.E-06

4,402E-01

9

4,E-08

3.71E-01

12

6,031E-01

25

5,E-06

6,031E-01

12

7, E-11

5.88E-01

15

6,786E-01

25

4, E-05

6,786E-01

12

6,E-12

6.91E-01

18

7,073E-01

25

3. E-02

7,018E-01

12

8. E-09

7,07E-0l

21

-3,681E+01

25

2, E+O0

7 • 065 E-01

12

2, E-09

7,07E-01

24

-9,250E+04

25

2.E+O0

7,071E-01

12

2. £-06

7.07E-01

27

-6,870E+07

25

2.£+00

7,071E-01

12

3.£-06

7,07E-01

203

TABLE

TAU

SER I ES

NS

0

5 • 531 E-04

1

3

5,279E-02

12

6

2.224E-01

9

IV

:

ERS

SLS(iv) T=30 PADE'

NP

5,531E-04

0

7, E-07

5,279E-02

5

5.E-05

5.279E-02

14

9, E-06

2,224E-01

6

5, E-05

2,224E-01

4. 054E-01

21

3.E-06

4.054E-01

10

2.E-09

4,054E-01

12

4. 638E-01

25

I.E-05

4.63gE-01

12

2,E-lO

4,638E-01

15

3,972E-01

25

6.E-04

3,972E-01

12

1,E-08

3.972E-01

18

3.01 OE-01

25

2, E-O1

2,867E-01

12

3.E-08

2.867E-01

21

2,240E+01

25

2,E+O0

1.902E-01

12

1,E-07

1,902E-01

24

3 , 1 0 9 E+04

25

2, E+O0

1,223E-01

12

7. E-06

1,223E-01

27

1. 018E+07

25

2. E+O0

7. 808E-02

12

9. E-06

7,808E-02

PADE'

NF'

ERP

CONVOLUTI ON

5,531E-04

0

.0

TABLE

V

:

ERP .0

CONVOLUTI ON

5.531E-04

SLS(v) T=.~O

SER I ES

NS

0

5,531E-04

1

3

4,835E-02

12

4, E-07

4. $35E-02

5

2, E-04

4.835E-02

6

t . 597E-01

t6

8, E-06

1,597E-01

7

4, E-06

1,597E-01

9

1.575E-01

21

4.E-06

1,575E-01

10

5,E-08

1.575E-01

12

-1.583E-02

25

3, E-05

-1,583E-02

12

5, E-06

-1.583E-02

15

-1,400E-01

25

2. E-03

-1.399E-01

12

2. E-05

-1,399E-01

18

-6.051E-02

25

6. E-01

-7,035E-02

12

2, E-04

-7,035E-02

21

2,030E+O1

25

2,E+O0

5,592E-02

12

4.E-03

5,590E-02

24

2,749E+04

25

2, E+O0

6,375E-02

12

5, E-03

6,556E-02

27

8,939E+06

25

2, E+O0 - 2 . 870E-02

12

1, E+O0

-1.381E-02

TAtl

ERS

.0

,0

5,531 E-04.

204 TABLE Vl

: THERMAL WAVES T= PADE"

ERP

LONG TI ME

X

SERIES

5,24

-5,358E-02

O.

-5.358E-02

O.

4.81

-4.851E-02

1, E-06

- 4 . 851 E-02

3. E-06

-2,56E-02

4.18

-3,807E-02

7, E-06

- 3 . 807 E-02

I , E-05

-2.39E-02

4.18

-2,276E-02

I,E-05

-2,276E-02

2. E-05

-2,39E-02

3.54

-2,011E-02

4, E-06

-2,011E-02

8, E-06

-2.17E-02

2.90

-1,708E-02

5. E-06

- I , 708E-02

3, E-07

-1,89E-02

2,27

-1,373E-02

7.E-05

-I,373E-02

6. E-07

-1,55E-02

1.63

-1,010E-02

4. E-03

- 1 . 009 E-02

9 • E-06

-1.16E-02

.99

-6.707E-03

2, E-01

- 6 . 241 E-03

1, E-04

-7.30E-03

.35

-7.372E-03

4. E+O0

-2,263 E-03

3, E-03

-2.67E-03

ERS

w ~

TABLE VII : ELASTIC WAVES T = ~ X

SERIES

ERS

PADE'

ERP

LONG TIME

5.24

5.369E-01

O.

5.369E-01

O.

4.81

7. 709E-01

4. E-06

7. 709E-01

6. E-05

9,74E-01

4.18

9.276E-01

4. E-06

9,276E-01

1.E-05

9.76E-01

4.18

9,785E-01

4. E-06

9. 785E-01

1. E-05

9.76E-01

3.54

9,811E-01

9.E-06

9.811E-01

1.E-05

9,78E-01

2.90

9 • 840E-01

4. E-06

9. 840E-01

8. E-07

9.81E-01

2.27

9.872E-01

4;E-06

9.872E-01

7, E-09

9.84E-01

1.63

9.906E-01

3. E-04

9.906E-01

8.E-07

9.88E-01

.99

9,950E-01

1.E-02

9,942E-01

9, E-06

9.93E-01

.35

1. O08E+O0

3. E-01

9,979E-01

7, E-05

9,97E-01

205

FIGURE CAPTIONS I - 2

The response o f a S . L . S .

with

a = .5

For i n p u t ( i i )

and

T=l,3,5;10,30w50.

3- 4

The same as F i g u r e s

1 - 2 for

S- 6

Thermal and e l a s t i c

waves w i t h

Eo(t)=Z, 7- 8

go(t)=O

The same as F i g u r e s

and 5- 6

input

(iii)

s = .03 ,

~= 1 . 3

T=2.5. For

~= .03 ,

~ = .9

for

input

206

R, 1

a=O,5

,

I

i 2

SLS .

i 4

2o

Fig. l

Fig.2

a=0,5

40

R~ 1 SLS a=0,5

SLS a=0,5

t.50

2

4

Fig.3

Malnardl~

Turc

20 Fig.4

het

tl

40

207

E

=0,03 (3 = 1 , 3

= 0,03 (3 = 1,3

t=5

-0,1

I

0

4

t~5

t=2

I

6

I 4

x

Fig. 5

I 8

E

F\ = 0,03 /3=0.9

C = 0,03 /3 = 0 , 9

t=2

I 4

Fig. 7

Mainardi-Turchetti

x)~

Fig. 6

I 0

x~

t=5

,I,

4

Fig. 8

0

x

APPLICATION OF METHODS FOR ACCELERATION OF CONVERGENCE TO THE CALCULATION OF SINGULARITIES OF TRANSONIC FLOWS* Andrew H. Van Tuyl Naval Surface Weapons Center White Oak Laboratory Silver Spring, Maryland 20910 USA

i.

Introduction Initial value problems in gas dynamics which lead to transonic flows include

the inverse blunt body problem, given and the body w h i c h w o u l d

in which a bow shock wave in a uniform flow is produce it is calculated,

and the inverse calculation

of nozzle flows starting from data given on the centerline.

Each of these problems

can be expressed as an initial value problem for a second order quasi-linear differential

equation satisfied by the stream function.

When the initial curve and initial data are such that the initial curve is noncharacteristic,

it follows from the Cauchy-Kowalewskl

theorem [i, page 39] that

the initial value problem can be solved in terms of power series in the neighborhood of a given point of the initial curve.

However,

the region of convergence

series obtained may be too small for practical use, due to the occurrence singularities,

either real or complex, near the initial curve.

of the of

This was found by

Van Dyke [2] in the case of the inverse blunt body problem, where a limiting line** (envelope of characteristics) flow.

occurs in the upstream analytic continuation

of the

This limiting line lies closer to the shock than the distance between the

latter and the body, and hence, a power series solution in the neighborhood

of a

point of the shock diverges at the body and cannot be used directly to calculate the flow there. In [3], Leavitt has calculated

the shape and position of this limiting

near the axis of symmetry by a modification

line

of a method due to Domb [4], starting

* This work was supported by the Naval Surface Weapons Center Independent Fund. **Also called limit line.

Research

210

from the power solution of the inverse blunt body problem in the neighborhood the nose of the shock.

The location of the limiting line was then used to trans-

form the series so that convergence was obtained at the body. Schwartz

More recently,

[5] has used Domb's method to calculate limiting lines in the flows

produced by parabolic and paraboloidal number.

of

Various modifications

shocks in a free stream of infinite Mach

and extensions

of Domb's method have been applied to

problems of statistical mechanics by Domb, Sykes, Fisher, and others

([6], for

example). Limiting lines in solutions of the inverse blunt body problem have also been calculated by Garabedian and his students

([7] and [8]) by use of Garabedian's

method of complex characteristics. Limiting lines may also occur in nozzle flows obtained from given centerline distributions

of velocity or Math number.

region of convergence

As in the case of blunt body flows, the

of a power series solution may be restricted by a limiting

line even though the point about which the solution is obtained lles in the subsonic region.

In nozzle design,

given centerline distribution

it is of practical interest to know if a

leads to a limiting line which lies between a

desired streamline and the axis of symmetry. A procedure for calculation of limiting lines will be described, from a power series solution, are used.

in which methods for acceleration

This procedure involves the ratio of successive

power series, as in Domb's method,

of convergence

coefficients

and a necessary requirement

the extent of the region of convergence

starting

of a

is therefore that

in the direction of at least one of the

coordinate axes should be determined by a limiting line.

Sequences are constructed

which converge to points on a limiting line and to its order k ~i.

With the

assumption that the single power series used in this calculation has only one singularity on its circle of convergence, sequence transformations,

including

it is proved that certain nonlinear

the e 1(s) transformation

defined by Shanks

([9], page 39) accelerate the convergence of these sequences.

211

The results obtained hold also for analytic initial value problems for other equations or systems of equations in two independent variables, when the given equation or system of equations can be replaced by a characteristic system in two independent variables.

In particular, limiting lines can be calculated by the

present method in the one-dimensional unsteady flow produced by a given piston motion.

Finally, the present method is also applicable to some of the series

occurring in [6].

2.

Limiting Lines of Order k In both the inverse blunt body problem and the inverse calculation of nozzle

flows, the stream function @ satisfies a quasi-linear second order partial differential equation of the form a~xx + b~xy + C~yy + d = 0, where the coefficients are analytic functions of their arguments.

(2.1) The independent

variables denote cartesian coordinates in the two-dimensional case and cylindrical coordinates in the axially symmetric case.

As in [i], pp. 491-493,

(2.1) can be

replaced by the system of characteristic equations y~ = h I x Y8 = h2 x8 d pe + h2q e + ~ x

= 0

(2.2)

P8 + hlq8 + ~a x8 = 0 ~

- px

- q y~

=

0

where h I and h 2 are the roots of the equation ah 2 - bh + c = 0.

(2.3)

Real values of = and 8 correspond to values of x and y for which (2.1) is hyperbolic.

It follows from the Cauchy-Kowalewski theorem that the solution of an

analytic initial value problem for (2.2) in the real eB-plane is analytic.

212

Given a solution of (2.2) which is analytic in a domain D of the real ~B-plane, the functions x(~,B) and Y(~,B) define a mapping which is one-to-one in any portion of D in which the Jacobian J = x yB-xBy ~ does not vanish.

Let k ~ l

be an integer, and let J and its derivatives of order up to and including k-i vanish along a curve C in D.

Then the image of C in the xy-plane is defined to be

a limiting line of order k. The well-known result ([i0], for example) that a regular arc of a limiting line of the first order is an envelope of one of the families of characteristics can also be shown to hold for limiting lines of order k > l .

As in the case of

limiting lines of first order, characteristics of the second family have infinite curvature at the limiting line for k>l. Finally, we can prove also that the behavior of flow quantities in the neighborhood of a limiting line of order k ~ l Theorem i.

is given by the following theorem:

Let a solution of (2.1) have a limiting line of order k ~ l

with

the equation x = Xo(Y), where Xo(Y) is analytic for yl 2.

When bo=0 but bl~O ,

an+l/a n =

=

a - --~]{

_

l+2/m

+ j--liE e.n -~-lJ

+ O(n - N - l )

, m ~ 2

1 + E e.n-J/m-i + 0(n-N/m-l) j=l 3

m > 2.

A similar but more complicated theorem can be proved when there are several singularities on IzI=l, with a different value of m associated with each.

With

more than one singularity, however, the transformations used here become less effective.

4.

Sequences for Calculation of Limitin Z Lines Given an analytic initial value problem for an equation of the form of (2.1),

let the origin be taken at a point of the initial curve, and let (Xo,0) be a point on a limiting line of order k ~i.

Let F(x,y) be any one of the dependent

variables, such as the density or pressure in the inverse blunt body problem, and

215

as described in the introduction, let F(z,0) be analytic for Izl ~ x ° except at z = x . This assumption, that only one singularity lies on the circle of o convergence, appears to be verified in special cases which have been calculated. We have F(z,o) = ~

(4.1)

anzn , Izl < x

n= 0

o '

and by Theorem i,

(4.2)

F(z,o) = n=O~ b n ( l - xZ--) n/(k+l)o

in a cut neighborhood of z = x O .

It follows from Corollary 1 of Theorem 2

that when bo#0 in (4.2), (i

l+i/(k+l)) an = x n+l an+ I o

i + N-I ~ c.n-j-I + O(n -N-I) 1 J=l 3

k = I (4.3)

=

X

0

i + ~ c.n-j/(k+l)-I + O(n-N/(k+l)-l) , k > 2 j=l 3

From (4.3), we obtain rn

=

(Sn-l)(n+l)(n+2) = ~3

I

1

i + ~ d.n -3" + 0(n-N) j=l 3

k = i

= (l+i/(k+l)) I + N-I ~ d.n-j/(k+l) + 0 (n_N/(k+l ) )} , k ~ 2 j=l 3 2 when bo~0, where Sn=an+2an/an+ 1. N o t i n g t h a t ( n + 2 ) S n - ( n + l ) t e n d s t o u n i t y

(4.4)

as

n-~o, we see that the sequence t

= n

(Sn-l)(n+l)(n+2) (n+2)Sn_(n+l)

(4.5)

tends to the same limit as the left hand side of (4.4), and we can show that it has the same asymptotic form as (4.4) as n-~.

Denoting the coefficients of

the asymptotic expression for tn by ej, we find that e I = d I when k ~ 2 . When k = i, however, we have d I = e I + 3/2. Thus, while tn and rn have the same rate of convergence, one may be asymptotically more accurate than the other.

216

When bo=0 but bl#0 , sequences similar to the preceding can be obtained by use of Corollary 2 of Theorem 2.

By consideration of both the limit and the rate of

convergence of tn, it is possible to determine k and to decide whether or not

b

o

5.

vanishes.

Nonlinear Sequence Transformations Let A

be a given sequence, and let

n

B

= n

An+lAn 1-An

.

(5.1)

An+ l+An- i- 2An

The sequence Bn is equal to the first order transform el(An ) defined by Shanks [9] and to the sequence e~ n) o b t a i n e d by t h e e - a l g o r i t h m of Wynn [11], referred to as the Aitken 6 2 process. sequence A

n

Under certain conditions, a convergent

is transformed to a sequence B

converges more r a p i d l y .

and i s a l s o

In particular,

n

which has the same limit as A

the latter

n

and

holds when lim IAAn+I/AAnI#I. n..+~

When lim ' n'IAAn+l/AAI=l but lim AAn/ABn=S@I, the transformed sequence B converges n_>~ ~ n with the same rapidity as A . For this case, Shanks ([9], page 39) has defined n the transform

e

~S)(A n

)=

sB -A n n s-l"

(5.2)

A transformation equivalent to this, called Un, was also introduced by Lubkin [12]. Finally, Lubkin ([12], page 229) has introduced a more general transformation which is expressed in the present notation by AA BnA--~

- An n

Wn =

AA n AB n

(5.3) 1

With the preceding definitions, we can verify the following theorems by direct calculation:

217

Theorem 3.

Let a given sequence have the form

A

as n-~o for all N ~ I .

n

N-I = A + E a.n-J-~ + O(n-N-=) . j 3=o

Then

N-I e I(c~+l) (An) = A + ~ b.n -j-~-2 + 0(n-N-~-2). j=o 3

Corollary.

With the notation e~Sl)(e~S2)(An)) = e(Sl)e~S2)(An ), we have

e(~+2r+l)^(~+2r-l) ... e(e+l)(An ) = A + 0(n -=-2r-2) =i

Theorem 4.

Let a given sequence have the form

A n

as n-~o for all N ~ 2

N-I = A + ~ a.n -j/m-~ + 0(n -N/m-~) j=l 3

where m = 2,3, . . . .

,. , e(l+e+i/m) i ~'~n)

Then

. = A + N-I ~ 5.n -3/m-~ + 0(n_N/m_~) j=2 3

for N = 3,4, . . . .

Corollary.

(lq~+r/m) (l+e+(r-l))/m e (l+~+I/m) ( A ) el el "'" I n

= A + 0(n-(r+l)/m-=).

Theorems 3 and 4 have direct application to the sequences of Section 4. In particular, with a

An

(i=

3/2~

n__ n---~" an+ I

218

in the case k = i, we see that

el(2r+l)el(2r-l) ... e ~ 3 ) ( A )

= x

+ 0(n -2r-2) o

We see that the sequences A respectively,

n

in Theorems 3 and 4 must have e > 0 and e > -i/m,

in order to be convergent.

However,

the theorems do not specify the

sign of ~, and the corollaries show that the iterated transformations converge to A for sufficiently large values of r whether or not A Finally, the following theorems for W Theorem 5.

= A + n

as n ->~ for N = 1,2, . . . .

Then

W

= A + n

are nearly identical with the preceding:

N-I ~ a.n -j-~ + O(n -N-e) j=o

N-I ~ c.n -j-e-2 + O(n-N-~-2), j=o J

Let a given sequence have the form

A

= A + n

N-1 ~ a.n -j/m-~ + 0(n-N/m-a). j=l 3

as n ->~ for N = 2,3,... where m = 2,3, . . . .

W

= A + n

for N = 3,4 . . . .

converges.

Let a given sequence have the form

A

Theorem 6.

n

n

Then

N-I 0 (n-N/m-a) ~ c.n -j/m-~ + j=2 3

Corollaries which are exactly parallel to the corollaries of

Theorems 3 and 4 clearly hold.

219

6.

Numerical Examples The sequences of Section 4, together with the sequence transformations of

Section 5, have been applied to the calculation of limiting lines of first order in the inverse blunt body problem and in the calculation of nozzle flows.

The

calculation of the axial point on the upstream limiting line in the case of a paraboloidal shock at a free stream Mach number of 2 is shown in Table I, which was calculated by use of 39 coefficients of the power series for the density. Calculations were carried out on the CDC 6500 in double precision, using the method of [13]. The sequences r and t converge to 3/2 and are accelerated by n n the transformation e~ , 2) -

showing that the upstream limiting line is of first

order, and that b #0 in the expansion of the density in the neighborhood of the o limiting line.

Values of rn, tn, and e~2)(tn ) are given in Table 2, and show that

t is preferable to r in this case. n n

When the same calculations are repeated

using the power series for the stream function, it is found that rn and tn approach the limit 5/2.

Thus, it appears that bo=0 while bl#0 in the expansion of the

stream function in the neighborhood of the axial point of the limiting line. The rates of convergence shown in Section 5 cannot usually be realized beyond the transformation e1(3) when using single precision, because of loss of significance due to subtraction.

However, the accuracy obtained by use of e~ 3)- in

single precision has been found to be sufficient in all examples calculated. Agreement of successive terms of the sequence to 5 or 6 significant figures is usually obtained. Finally, calculations of limiting lines in nozzle flows have been carried out in single precision on the CDC 6500 using e~ 3).

The result of such a calculation

is shown in Figure 1 in the axially symmetric case, where u = 1 + (%/3/2)x on the centerline.

This centerline velocity distribution is the same as that of case (f)

of [14] after a change of reference quantities.

The sonic line, limiting

characteristic, and streamlines were calculated by the method of [15]. All

220

calculations

were carried out using 25 terms of the series in y, and convergence

to 5 or more figures was found except near the limiting line.

With the use of

double precision, the more rapidly convergent transformation e(5)e(3)(A ~ would i i n" give additional aceuracy. The portion of the streamline ~ = 0.2 to the left of the limiting characteristic can be taken as the wall of a nozzle in the subsonic and transonic regions, since the flow is analytic on the limiting characteristic up to and on the streamline.

However, the streamline ~ = 0.45 meets the limiting line above its

point of tangency with the limiting characteristic, and cannot be used as part of a nozzle contour.

The dashed portion of the limiting characteristic belongs to a

second solution of the flow equations having the limiting line shown.

The two

solutions correspond to the same analytic solution of equations (2.2) in the •

i

~8-plane, but on opposite sides of the curve J =, 0 which corresponds to the limiting line.

7.

References i.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York, 1962.

2.

M. D. Van Dyke, "A Model of Supersonic Flow Past Blunt Axisymmetric Bodies with Application to Chester's Solution," Journal of Fluid Mechanics, Vol. 3, (1958), pp. 515-522.

3.

J. A. Leavitt, "Computational Aspects of the Detached Shock Problem," AIAA Journal, Vol. 6 (1968), pp. 1084-1088.

4.

C. Domb, "Order-Disorder Statistics, II. A Two Dimensional Model," Proceedings of the Royal Society of London, Series A, Vol. 199 (1949), pp. 199-221.

5.

L . W . Schwartz, "Hypersonic Flows Generated by Parabolic and Paraboloidal Shock Waves," Physics of Fluids, Vol. 17 (1974), pp. 1816-1821.

6.

M. E. Fisher, "The Nature of Critical Points," Lectures in Theoretical Physics, Vol. VlIC, University of Colorado Press, Boulder, 1964, pp. 1-159.

7.

P. R. Garabedian, "Global Structure of Solutions of the Inverse Detached Shock Problem," Problems of Hydrodynamics and Continuum Mechanics, published by the Society for Industrial and Applied Mathematics, 1969, pp. 318-322.

221

8.

E. V. Swenson, "Geometry of the Complex Characteristics in Transonic Flow," Comm. Pure and App. Math., Vol. 21 (1968), pp. 175-185.

9.

D. Shanks~ "Non-Linear Transformations of Divergent and Slowly Convergent Sequences," Journal of Mathematics and Physics, Vol. 34 (1955), pp. 1-42.

10.

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948.

Ii.

P. Wynn, "On a Device for Computing the em(Sn) Transform," MTAC, Vol. i0 (1956), pp. 91-96.

12.

S. Lubkin, "A Method of Summing Infinite Series," Journal of Research of the Bureau of Standards, Vol. 48 (1952), pp. 228-254.

13.

A. H. Van Tuyl, "Use of Pad~ Fractions in the Calculation of Blunt Body Flows," AIAA Journal, Vol. 9 (1971), pp. 1431-1433.

14.

G. V. R. Rao and B. Jaffe, '~ Numerical Solution of Transonic Flow in a Convergent-Divergent ~ozzle," Final Rept., Contract NAS7-635, March 1969, NASA.

15.

A. H. Van Tuyl, "Calculation of Nozzle Flows Using Pad~ Fractions," AIAA Journal, Vol. II (1973), pp. 537-541.

222

Table i.

Distance of Upstream Limiting Line from Nose of Shock for Paraboloidal Shock Wave r 2 = 2x with Free Stream Maeh Number = 2.

n

34

An = -(I

n+l'an+ 1312~an

-

0.083546753454

e~3)(An)

el(5)e(3)(An)l

0.0835451981166

0.0835451972856

35

.0835466622853

.0835451980201

.0835451972406

36

.0835465789026

.0835451979365

.0835451972874

37

.0835465024447

.0835451978019

.0835451972615

Table 2.

Sequences r

n

and t Corresponding to Axial Point on Upstream Limiting n

Line for Paraboloidal Shock Wave r 2 = 2x with Free Stream Mach Number = 2.

n

r

n

t

n

e~2)(tn )

34

1.5685409

1.5012612

1.5000020

35

1.5665483

1.5012223

1.5000018

36

1.5646684

1.5011858

1.5000017

37

1.5628918

1.5011514

1.5000016

223

1.1 = 0.45

STREAMLINES

0.9

I.

I

0,8

0.7~'~,~

~ = 0.2 LIMITING LINE

0,6

--

0,5

--

0.4--

0.3

~LIMITING CHARACTERISTIC

--

0.2--

0.1

--

0 -0.6

I

1

I

I

I

-0.5

-0.4

-0.3

-0.2

-0.1

0

I

I

0.1

0.2

0.3

X

Figure i.

Limiting line in the nozzle flow with centerline velocity distribution u = 1 +

(/'3/2)x.

THE USE OF PADE FRACTIONS IN THE CALCULATION OF NOZZLE FLOWS* Andrew H. Van Tuyl Naval Surface Weapons Center White Oak Laboratory Silver Spring, Maryland 20910 USA

i.

Introduction In nozzle design, flow calculations are usually carried out by an inverse

procedure in which either the Mach number or velocity is given on the centerline. Calculations in the supersonic region can be carried out accurately by the method of characteristics when the solution in the transonic region is known.

However, the

inverse problem is improperly-posed in the subsonic region, and it is therefore not possible to calculate the subsonic and transonic portions of the flow to arbitrary accuracy by means of finite-difference marching procedures without the use of complex extension.

Methods for calculating the flow in the transonic region

([i], for example) are mainly applicable to the design of nozzles with large throat radius of curvature, such as wind tunnel nozzles.

These methods are not

sufficiently accurate for the design of short nozzles, since rapid changes then occur near the throat.

Improved accuracy in the case of short nozzles is given by

[2], but the accuracy which can be obtained is limited by the fixed number of terms of the power series used.

Short nozzles with a uniform exit flow are of interest

in connection with gas dynamic lasers [3], where two-dimensional nozzles have been used, and with chemical lasers [4], where both two-dimensional and axially symmetric nozzles are applicable.

More accurate subsonic calculations have been carried out

in the axially symmetric case by Armitage [5] and Rao and Jaffe [6] by means of Garabedian's method of complex characteristics

[7, 8].

Calculations by the method

of complex characteristics have also been carried out by Solomon [9] in the twodimensional and axially symmetric cases.

*This work was supported by the Naval Surface Weapons Center Independent Research Fund.

226

When the flow quantity given on the centerline is analytic, the Cauchy-Kowalewski

theorem

([i0], page 39) that the inverse problem can be

solved in terms of power series in the neighborhood eenterline.

it follows from

of a given point of the

While these series are known to converge near the given point, the

region of convergence may be such that the series are either divergent or too slowly convergent at points of physical interest.

It was noted by Van Dyke [ii]

that the former occurs in the case of the inverse blunt body problem, presence of a limiting line (envelope of characteristics) continuation

of the flow.

due to the

in the upstream analytic

This limiting line lies closer to the shock than the

distance between the shock and the body, and hence, the body does not lie within the region of convergence

of the series.

However,

it was found in [12],

[13],

and [14] that Pade" fractions formed from these series give accurate results at the body, and can be used to compute the flow there.

A similar procedure has been

used in [15], where Pade" fractions are formed from certain power series in the neighborhood body problem,

of points of the centerline.

examples indicate that the use of Pade" fractions leads to convergence

when the series diverge, converge.

As in the case of the inverse blunt

and that convergence

is accelerated when the latter

The region of convergence of the series may be restricted by limiting

lines, as in the blunt body problem, calculations

or by complex singularities.

Accurate

can also be carried out in the supersonic region by use of Pade"

fractions when limiting lines do not occur near the centerline.

Near a limiting

line, convergence of both the series and the sequences of Pade" fractions

is found

to be slow. The present paper extends

[15] by investigating

fractions can be used in the calculation

additional ways in which Pade"

of nozzle flows.

In particular,

Pade"

fractions are formed from power series along given curves which intersect the centerline. expansions

By use of these results, more efficient use of the double Taylor obtained in [15] can be made.

227

2.

Power Series Solution of the Inverse Problem The coordinate system used is shown in Figure i, where x and y denote

Cartesian coordinates in the two-dimensional case and cylindrical coordinates in the axially symmetric case, and where the origin is at the sonic point on the centerline.

We will assume irrotational flow of a perfect gas with ratio of

specific heats y.

Let p* = c* = i, where p* is the critical density and c* is the

critical sound speed, and let the stream function satisfy u = (i/py°)~/~y, v = - (i/pyO)~/~x,

(2.1)

where u and v are the x and y components of velocity, respectively, p is the density, and ~ = 0 and i, respectively, in the two-dimensional and axially symmetric cases.

Substituting (2.1) into Bernoulli's equation and the condition

for irrotationality, we obtain (i/y2O)(~x 2 + ~y2) + [2/(y-1)]py+l _ [(y+l)/(y_l)]p2 = 0

(2.2)

and

P(~xx

+ ~yy _ oy-l~y) _ Px~x _ py~y = O,

(2.3)

respectively. Let the density, Mach number, and velocity on the centerline be denoted by Po(X), Mo(X) , and Uo(X) , respectively.

Then Po(X) is given in terms of Mo(X) by

the relation Po = {2/(y+l) + [ (y-i) / (y+l) ]Mo2} -I/(y-i),

(2.4)

and in terms of Uo(X), by Po = { (7+1)/2 - [(Y-I)/2]Uo2}I/(y-I)" Let M 0

(2.5)

(x) and Uo(X) have the expansions Mo(X) = ~ Mi(X-Xl)i-1 i=i

(2.6)

Uo(X ) ~ ui(x-xl )i-I i=l

(2.7)

and

228

in the neighborhood of x = x I. Given either (2.6) or (2.7), we can find the coefficients Pi of the expansion ¢o

Po (x) = .i__~lPi(X-Xl)i-I

(2.8)

by use of (2.4) or (2.5) respectively, and subroutines for power series manipulations.

We can then find the solution of the inverse problem in the

neighborhood of the point (Xl, 0) in the form 4"= ~ ~ ~ij y2i-l+~(X-Xl)J-l' i=l j=l P =

j--~lPij y 2i-2 (X-Xl)J-l" i:1 "=

(2.9)

(2.10)

We see that PlJ = PJ" To summarize the calculation of the remaining coefficients, let the left hand sides of (2.2) and (2.3) be denoted by A and B, respectively.

We have A = ~ ~ y2i-2(X-Xl )j-I i=l j=l ai~

(2.11)

B = ~ ~ bij y2i-l+O(x-xi)J-i i=l j=l

(2.12)

and

For J~2, we find that alj = 2(i+~)241141j + a~j,

(2.13)

411 = [i/(I+~)]{[ (y+l)/(y-1)]Pll2 - [2/(y-l) ]PllY+l}I/2

(2.14)

where

and where alj does not involve ~lj" For i>_2, we have aij = CI 4ij + C2 Pij + a~j,

(2.15) bij = C3 4ij + C4 Pij + b~j,

229

where C I = 2(i+0)(21-i+~)~ii C2 :

,

[2(y+l)/(y-l)](PllY-Pll

), (2.16)

C 3 = 2(2i-l+~)(i-l)Pll , C4 = -2(l+q)(i-l)~ll. and where a~j and b~j do not involve ~ij and Pij" coefficients can now be described as follows: alj = 0 in (2.13) for j = 2, 3, .... and calculating alj.

The calculation of the remaining

First, ~lj is calculated by setting

For each j, a~j is found by setting ~lj = 0

The coefficients ~iJ and Pij are then found for i ~ 2 by

setting aij and bij equal to zero in (2.15) and solving the resulting pair of equations simultaneously.

For given i and j, a~j and b~j are obtained by setting

~ij and PiJ equal to zero and calculating aij and bij.

Assuming that Pi is known

for i ~ i ~ 2K - i, we can find ~ij and Pij for i ~ i ~ K and i ~ J ~ 2K - 2i + i.

3.

Pade" Fractions Let f(z) = ~ c.z i i i=0

be a given power series with co @ O. are defined as follows:

(3.1)

Then Pade" fractions fk,n(Z), k ~ 0, n ~ O,

fk,n(Z) is a rational fraction with numerator and

denominator of degrees less than or equal to n and k, respectively, such that the Taylor expansion of fk,n(Z) in the neighborhood of the origin agrees with (3.1) to more terms than that of any other rational fraction with numerator of degree < n and denominator of degree ~ k ([16], page 377). exists and is unique.

This fraction always

Convenient methods of calculation include the QD algorithm

of Rutishauser [17] and the ~ - algorithm of Wynn [18]. The sequence fn,n(Z), which involves the first 2n + i terms of (4.1), is often found to converge much more rapidly than sequences in which either k or n

230

remains constant.

The convergence of fn,n(Z) has been proved only for special

classes of functions, and none of the available convergence results appears to be directly applicable to the series occurring in either the inverse blunt body problem or the inverse calculation of nozzle flows.

In both [14] and [15],

however, it was found that the sequence fn,n(Z) converges in most cases when the series diverges, and that it accelerates convergence when the latter converges.

4.

Power Series for Constant x Before Pade" fractions can be used, the solution of the inverse problem must

be expressed in terms of power series in one variable.

One approach is to write

(2.9) and (2.10) as single power series in y with coefficients which are functions of x.

We have @

= ~

' , , ~i~x;y

2i-i+~

(4.1)

i=l and oo

y2i-2,

(4.2)

~ij (X-Xl)J-i

(4.3)

Pi (x) = ~ Pij(X-Xl)J-i j=l

(4.4)

p =

Pi(X) i=l

where

~i (x) = ~ j=l and

when Ix - Xll and y are sufficiently small.

The coefficients ~i(x) and Pi(X) can

he calculated by use of Pade" fractions for a given value of x, after which Pade" fractions can be formed from (4.1) and (4.2).

Pade" fractions formed from (4.3)

and (4.4) are exact at x = Xl, but their accuracy decreases in general as Ix - Xll increases.

In a given case, however, it may be possible to find an

interval about x I throughout which acceptable accuracy is obtained.

It would be

more economical to use several such intervals with overlapping regions of validity than to solve the inverse problem for each value of x.

231

In nozzle calculations, expansions in powers of 4 may be more convenient than (4.1) and (4.2).

We know by the Cauchy-Kowalewski theorem that for x sufficiently

near Xl, (4.1) has a nonzero radius of convergence.

Assuming that ~l(X) is not

zero for a given x, we can invert (4.1) to obtain an expansion of the form

y = ~ Yi(X) 4 2i-1 i=l

(4.5)

in the two-dimensional case, and

Yi(x) ~i y2 = ~ i=l in the axially symmetric case for sufficiently small 4.

(4.6) By substituting (4.5) or

(4.6) in (4.2), we can obtain an expansion for 0 in powers of 4.

Similarly,

starting from the Taylor expansion of other flow quantities in the neighborhood of (Xl, 0), we can find their expansions in powers of 4 for constant x.

5.

Power Series Along Given Curves Another procedure for expressing the solution of the inverse problem in terms

of power series in one variable is to calculate ~ and 0 along given curves through (Xl, 0).

Families of such curves can be chosen which sweep out a neighborhood of

(Xl, 0).

Let the equation of a given curve be x - x I = g(y), where

g(y) = ~ gi yi i=l for sufficiently small y.

(5.1)

On substituting x - x 1 = g(y) in (2.9) and (2.10), we

obtain expansions of the forms

(5.2)

4 = ~ aiyi+° i=l and

0 = ~ biY i=l respectively.

i-i

,

(5.3)

With the equation of the given curve written in the alternate form

y = g(x - Xl) , y is replaced by x - x I in (5.1) through (5.3). case when g(y) is a function of y2, we have

In the special

232

= ~

aiy2i-l+a

(5.4)

i=l and 0 = ~ biY i=l along the given curve.

2i-2

(5.5)

We see that the coefficients a. and b. in (5.2) through I

1

(5.5) are known exactly, while the coefficients of (4.1) and (4.2) for x # x I are known only approximately.

In addition, ~i(x) and Pi(x) become less accurate for a

given x as i increases, since the number of terms of their Taylor expansions which are available then decreases.

However, the accuracy obtained in a particular

case is dependent on the choice of Xl, and can be determined only by trial. As in section 4, if a I @ 0, we can invert (5.2) to obtain y =

ci~

(5.6)

i=l in the two-dimensional case and y = ~ di~i/2 i=l in the axially symmetric case for sufficiently small ~.

(5.7) Similarly, if a I ~ 0 in

(5.4), we have y = ~

c

i=l

i*

2i-1

(5.8)

in the two-dimensional case, and y in the axially symmetric case.

2

" = ~ di~l i=l

(5.9)

By use of the preceding expansions for y, we can

obtain expansions for 0 and other flow quantities along the given curve in powers of ~ or ~i/2. A special case of an expansion along a curve of the form x - x I = g(y2) is given by the expansions at constant potential of [15].

The equation of the

equipotential through the point (Xl, 0) is of the form

x - xI =

piy i=l

2i

(5.10)

233

The calculation of the coefficients Pi' starting from the coefficients of (2.9) and (2.10), is described in detail in [15]. We see that the special choices g(y) = ~y and g(y) = ~y2 correspond to Moran's procedure in [13].

In [13], a double power series in x and y is written

so that terms of the same degree in x and y jointly are grouped together.

The

given series is then expressed as a power series in one of the variables with coefficients which are polynomials in the ratio of the variables, is held constant in a given calculation.

and this ratio

The use of the above choices for g(y),

after making the substitution y = z I/2 in the second, is seen to be equivalent to Moran's procedure. In the preceding two cases, it is possible to obtain the coefficients a i and b i explicitly as polynomials in ~.

In the case x - x I = ey (or y = a(x - Xl)),

a more symmetrical approach is to transform (2.9) and (2.10) to polar coordinates r and 8 with origin at (Xl, 0). y

2

On substituting x - x I = r cos e and

2 = r (I - cos28) in (2.9) and (2.10), we obtain power series in r for ~/yl+O

and p with coefficients which are polynomials in cos 8.

6.

The Equation of the Sonic Line The equation of the sonic line has an expansion of the form

x

=

i= I siY

(6.1)

in the neighborhood of the origin both in the two-dimensional and axially symmetric cases.

We can calculate the coefficients s. successively by substituting 1

(6.1) in the left hand side of the equation

oijy2i-2x j - l o 1

(6.2)

i=1 j=l and equating the coefficients of powers of y2 past the constant term to zero. This calculation can be carried out on a computer by use of subroutines for power

234

series manipulations.

Finally, we can obtain power series for flow quantities

along the sonic line by substituting (6.1) or its inversion in their Taylor expansions in the neighborhood of the origin.

7.

The Equation of the Limitin$ Characteristic As shown in Figure i, the limiting characteristic is the right-running

characteristic which passes through the origin.

In both the two-dimensional and

axially symmetric cases, the two families of characteristics satisfy the equations dy/dx = tan (8±~),

(7.1)

where tan8 = - ~x/~y and tans = (M2 - 1) -1/2 . Making these substitutions, we obtain [-~x(M2-1) I/2 ± ~y]dx/dy - ~y(M2-1) I/2 ~ ~x = 0.

(7.2)

The equation of the limiting characteristic has an expansion of the form x = ~

i=l

for sufficiently

s m a l l y.

r i Y 2i

Denoting the left

hand s i d e of ( 7 . 2 ) by C, we f i n d t h a t

C= £ ei y2i-l+e i=l On s e t t i n g

(7.3)

(7.4)

c 1 = 0 and p r o c e e d i n g as i n [ 1 5 ] , we o b t a i n r 1 = (~+1)P12/4

(7.5)

r I = (/5-i) (y+l)P12/8

(7.6)

i n t h e t w o - d i m e n s i o n a l c a s e , and

in the axially symmetric case. P12 is negative.

We note that both values are negative, since

We can verify that c.i = - [2i - (y+l)Pl2/2dl]ri + c~,

where cf does not involve r.. 1 1

(7.7)

Finally, in order to determine r i for i ~ 2, we

start from (7.5) in the two-dimensional case and (7.6) in the axially symmetric case, and calculate

235

r i = c~/[2i - (y+l)Pl2/2dl] , for i = 2, 3, . . . .

For each i, the calculation of cf is carried out by setting l

r° = 0 and calculating 1

c.. 1

After the coefficients

r i have been determined,

flow quantities along the limiting characteristic inversion in their Taylor expansions

8.

(7.8)

we can find power series for

by substituting

in the neighborhood

(7.1) or its

of the origin.

Numerical Results and Discussion As in [15], calculations have been carried out on the CDC 6400 in the axially

symmetric case with y = 1.4, using the velocity on the centerline given by example

(c) of [6].

Coefficients

of the Taylor expansions

computed for i ~ 25 in these calculations, equal to 49.

and hence, with the maximum value of j

The centerline velocity of example

present coordinates

(2.9) and (2.10) are

(c) is given in terms of the

and reference quantities by u = A I + A2/[A 3 + (x-A4)2]2 ,

where A 1 = 0.0657267,

A 2 = 3.1224458,

A 3 = 1.7125400,

centerline velocity is shown in Figure 2.

(8.1) and A 4 = 0.34000641.

In the present units,

This

the value of the

stream function which defines the nozzle contour in [6] becomes ~ = 0.31104. Figure 3 shows the calculated nozzle contour, characteristic

sonic line, and limiting

for the eenterline velocity of example

(c).

Equipotential ~urves

are also shown, as calculated in [15] by means of Pade" fractions formed from expansions

of x - x I and y2 in powers of 4.

calculated by use of Pade" fractions for x = x I.

Points on the nozzle contour were

formed from expansions

The sonic line was calculated

in powers of y2 or

in two ways, by solving the equation

p = 1 by Newton's method with the left hand side replaced by a Pade" fraction, and by forming a Pade" fraction from (6.1).

In the present example,

method was found to be more accurate than the latter for y > 0.6.

the former

Finally,

the

limiting characteristic was calculated in [15] by means of a Pade" fraction formed from the right hand side of (7.3).

Comparison is made with the calculations

[6] and [9] by the method of complex characteristics.

of

236

Similarly, calculations

Figure 4 compares the flow angle calculated in [15] with the

of [6] and [9].

of Pade ~ fractions

The calculations

of [15~ were carried out by means

formed from power series in y2 for x = x I.

In Tables 1 and 2, Pade" fractions of the form fn,n(Z) are compared with the corresponding partial sums of the series y.

(4.1) for x = x I = -I and for two values of

The tables indicate that the sequence of Pade ~ fractions converges

for both

values of y, while the series converges for the smaller value of y and diverges for the larger.

Tables 3 and 4 compare Pade" fractions and partial sums at the points

(-i, I.i) and (-I, 1.6) for expansions form x - x I = ~y2

with x I = -3.

of ~ in powers of y2 along parabolas of the

Finally, Tables 5 and 6 give the same comparison

for expansions of ~ in powers of y along the rays from (-3,0). along rays through hence,

The expansions

(-3, 0) contain all powers of y up to and including y48, and

49 coefficients

are found in the present calculations.

expansions use all 625 of the coefficients

We note that these

~ij obtained in the solution of the

inverse problem, while only 325 are used by the expansions

along parabolas.

The

series converges for both values of y in Tables 3 through 6, but more slowly than the sequence of Pade" fractions. Comparison through

of Tables 4 and 6 shows that the expansion along a straight line

(-3, 0) leads to more rapid convergence of the sequence of Pade" fractions

at (-i, 1.6) than the expansion along a parabola, while both tables show more rapid convergence than Table 2.

We see that the expansion along x = -i used in Tables 1

and 2 is a special case both of an expansion along a straight line through and of an expansion along a parabola. expansion

(xl, 0) in a particular

at which the flow is c~iculated.

(-i, 0)

It follows that the most suitable center of

case is not necessarily Further calculations

the one nearest the point

show that the portion of the

nozzle contour shown in Figure 3 can be calculated to 4 figures or more for x < -0.5 by means of Pade" fractions formed from power series along rays through

(-3, 0).

The remaining portion of the nozzle contour in Figure 3 can be calculated by use of Pade" fractions formed from expansions along rays through the origin and through the point

(-0.25, 0).

This method is found to be more economical than the procedure of

237

section 4, in which ~i(x) and Pi(X) are calculated by means of Pade

fractions,

since the range of x for which the latter are sufficiently accurate becomes small for i ~ 15 in the present calculations.

The time required to find expansions along

a given ray through (Xl, 0) when ~ij and PiJ are known is much less than that for solution of the inverse problem.

9.

References i.

J. R. Baron, "Analytic Design of a Family of Supersonic Nozzles by the Friedrichs Method," WADC Report 54-279, June 1954, Naval Supersonic Lab., MIT, Cambridge, Mass.

2.

D. F. Hopkins and D. E. Hill, "Effect of Small Radius of Curvature on Transonic Flow in Axisymmetric Nozzles," AIAA Journal, Vol. 4 (1966), pp. 1337-1343.

3.

J. D. Anderson, Jr. and E. L. Harris, "Modern Advances in the Physics of Gasdynamic Lasers," AIAA Paper 72-143, San Diego, Calif., 1972.

4.

T. A. Cool, "A Summary of Recent Research on Continuous Wave Chemical Lasers," Modern Optical Methods in Gas Dynamics Research, edited by D. S. Dosanjh, Plenum Press, New York, 1971, pp. 197-220.

5.

J. V. Armitage, "Flow in a deLaval Nozzle by the Garabedian Method," ARL 66-0012, Jan. 1966, Aerospace Research Labs., Dayton, Ohio.

6.

G. V. R. Rao and B. Jaffe, "A Numerical Solution of Transonic Flow in a Convergent-Divergent Nozzle," Final Report, Contract NAS7-635, March 1969, NASA.

7.

P. R. Garabedian, "Numerical Construction of Detached Shock Waves," Journal of Mathematics and Physics, Vol. 36 (1957), pp. 192-205.

8.

P. R. Garabedian, Partial Differential Equations, Wiley, New York, 1964, Chapter 16.

9.

J . M . Solomon, private communication, Jan. 1971, Naval Surface Weapons Center, White Oak, Silver Spring, Md.

i0.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York, 1962.

ii.

M. D. Van Dyke, "A Model of Supersonic Flow Past Blunt Axisymmetric Bodies with Application to Chester's Solution," Journal of Fluid Mechanics, Vol. 3 (1958), pp. 515-522.

12.

A. H. Van Tuyl, "The Use of Rational Approximations in the Calculation of Flows With Detached Shocks," Journal of the Aero/Space Sciences, Vol. 27 (1960), pp. 559-560.

238

13.

J. P. Moran, "Initial Stages of Axisymmetric Shock-on-Shock Interaction for Blunt Bodies," Physics of Fluids, Vol. 13 (1970), pp. 237-248.

14.

A. H. Van Tuyl, "Use of Pad~Fraction in the Calculation of Blunt Body Flows," AIAA Journal, Vol. 9 (1971), pp. 1431-1433.

15.

A. H. Van Tuyl, "Calculation of Nozzle Flows Using Pad~ Fractions," AIAA Journal, Vol. Ii (1973), pp. 537-541.

16.

H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand, New York, 1948.

17.

P. Henrici, "The Quotient-Difference Algorithm," Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eisenvalues, National Bureau of Standards Applied Mathematics Series, No. 49, 1958, pp. 23-46.

18.

P. Wynn, "On a Device for Computing the em(Sn) Transform," Math. Tables and Other Aids to Computation, Vol. I0 (1956), pp. 91-96.

Table I.

Pade" fractions for stream function at (-i, i.i) from power series on the line x = -i.

No. of terms

ii 13 15 17 19 21 23 25

Table 2.

Series

Pade" fractions

0.22136847 0.22139025 0.22137986 0.22138595 0.22138235 0.22138439 0.22138328 0.22138386

0.22138164 0.22138385 0.22138366 0.22138367 0.22138366 0.22138367 0.22138367 0.22138367

Pade" fractions for stream function at (-I, 1.6) from power series on the line x = -i.

No. of terms

ii 13 15 17 19 21 23 25

Series

0.86601815 0.72548523 -0.44164134 1.9872070 -3.3585190 8.5084977 -17.484076 38.076299

Pade" fractions

0.32739899 0.32960800 0.32908139 0.32913508 0.32907155 0.32911728 0.32912434 0.32911631

239

Table 3.

Pade ~ fractions for stream function at (-i, i.i) from power series along parabola through (-3, 0).

No. of terms

ii 13 15 17 19 21 23 25

Table 4.

Pade" fractions

0.22082834 0.22121951 0.22135508 0.22138460 0.22138868 0.22138819 0.22138670 0.22138537

0.22139959 0.22139261 0.22140720 0.22138378 0.22138298 0.22138362 0.22138367 0.22138366

Pade" fractions for stream function at (-i. 1.6) from power series along parabola through (-3, 0).

No. of terms

ii 13 15 17 19 21 23 25

Table 5.

Series

Series

Pade ~ fractions

0.33051685 0.32947504 0.32910476 0.32904116 0.32906254 0.32909083 0.32910815 0.32911519

0.32899479 0.32910273 0.32913379 0.32911671 0.32911675 0.32912158 0.32911528 0.32911678

Pade" fractions for stream function at (-i, i.i) from power series along straight line through (-3, 0).

No. of terms

35 37 39 41 43 45 47 49

Series

Pade" fractions

0.22138349 0.22138358 0.22138358 0.22138363 0.22138366 0.22138366 0.22138366 0.22138367

0.22138363 0.22138366 0.22138367 0.22138367 0.22138367 0.22138367 0.22138367 0.22138367

240 Table 6.

Pade" fractions for stream function at (-i, 1.6) from power series along straight line through (-3, 0).

No. of terms

35 37 39 41 43 45 47 49

"~

Series

Pade" fractions

0.32910043 0.32910691 0.32910959 0.32911685 0.32911775 0.32911594 0.32911575 0.32911506

0.32913355 0.32911442 0.32911469 0.32911464 0.32911467 0.32911474 0.32911480 0.32911470

"~k

/

~1/

SONlCUne~

LIMITING

J

\,{,' 0

Figure I.

\

/ -/~

/ ', I CHARACTERISTIC

~/ X

Schematic diagram of nozzle flow.

241

U 0.5

I

I

-3.0

Figure 2.

-2.0

I

I

-I. 0

X

0

0.5

Centerline velocity distribution of example (c) of [6].

2.5

o,

~ REF. 9

NOZZLE CONTOUR

11"0'I041

Y

-3.0

Figure 3.

-2.5

-2.0

-1.5 x

-1.0

-0.5

Subsonic and transonic portions of nozzle in example (c).

242

/

20

e,-

elm

PRESENTMETHOD I; 15 ......... REF.6 I/ o REF. 9 //~i 10

I

,

,~.~>....7.,I"

0.2 Figure 4.

0.4

y

,

,

,

0.6 Ii 0.8 !

Flow angle e along sonic line in example (c).

A B I B L I O G R A P H Y ON PADE A P P R O X I M A T I O N AND SOME RELATED MATTERS Claude BREZINSKI University

of

Lille

The aim of this paper is to give a bibliography on Pad6 approximants, related matters and applications.

some

For several years, Pad6 approximants had become more and more important in mathematics, numerical analysis and various fields in physics and engineering. They are closely related to many subjects in mathematics as analytic function theory, difference equations, the theory of moments, approximation, analytic continuation, continued fractions, etc. Then a whole bibliography should be a huge one to include the corresponding references of these disciplines. I have divided the references given in this paper into three sections. The first one deals with Pad6 approximation and I hope it is quite complete. The second one is devoted to continued fractions and includes only some historical references and most of the recent papers on this subject. The thrid section contains some applications of Pad6 approximants with a special emphasis on mechanics ; I have also included references on numerical analysis and methods to accelerate the convergence of sequences. Miscellaneous references end the paper. It is obvious that this bibliography is far to be complete because of the limited number of pages of this volume. It is, in fact, less than half of a bigger bibliography on this subject and on all the related matters that I hope to publish in the future. I apologize in advance for any errors and omissions and I thank everybody who would send me any new reference on this subject.

I - PADE approximants i - R.J. ARMS, A. EDREI - The Pad6 tables and continued fractions generated by totally positive sequences - in "Mathematical essays dedicated to A.J. Macintyre" (1970) 1-21. 2 - G.A. BAKER Jr.- The Pad6 approximant method and some related generalizations in "The Pad6 approximant in theoretical physics", G.A. Baker Jr. and J.L. Gammel eds., Academic Press, New York, 1970. 3 - G.A. BAKER Jr.- The theory and application of the Pad~ approximant method J. Adv. Theor. Phys., i (1965) 1-56. 4 - G.A. BAKER Jr.- Reeursive calculation of Pad6 approximants - in "Pad~ approximants and their applications", P.R. Graves - Morris ed., Academic Press, New-York, 1973. 5 - G.A. BAKER Jr.- Best error bounds for Pad~ approximants to convergent series of Stieltjes - J. Math. Phys., i0 (1969) 814-820. 6 - G.A. BAKER Jr.- Certain invariance and convergence properties of Pad~ approximants - Rocky Mountains J. Math., 4 (1974) 141-150. 7 - G.A. BAKER Jr.- The existence and convergence of subsequences of Pad6 approximants - J. Math. Anal. Appl., 43 (1973) 498-528.

246

8

- G.A. 1975.

BAKER Jr.- E s s e n t i a l

of Pads a p p r o x i m a n t s

9 - G.A. B A K E R Jr., J.L. G A M M E L eds. A c a d e m i c Press, New York, 1972.

- Academic

- The Pads a p p r o x i m a n t

Press,

New-York,

in t h e o r e t i c a l

physics-

i0 - G.A. BAKER Jr., J.L. GAMMER, J.G. WILLS - An i n v e s t i g a t i o n of the applicability of the Pads a p p r o x i m a n t m e t h o d - J. Math. Anal. Appl., 2 (1961) 405-418. ii - M. B A R N S L E Y - The b o u n d i n g p r o p e r t i e s of the m u l t i p o i n t Pads a p p r o x i m a n t series of Stieltjes - Rocky M o u n t a i n s J. Math., 4 (1974) 331-334.

to.a

12 - M. B A R N S L E Y - The b o u n d i n g p r o p e r t i e s of the m u l t i p o i n t Pads a p p r o x i m a n t series of Stieltjes on the r e a l line - J. Math. Phys., 14 (1973) 299-313.

to a

13 - M. BARNSLEY, P.D. R O B I N S O N - Dual v a r i a t i o n a l p r i n c i p l e s x i m a n t s - J. Inst. Math. Appl., 14 (1974) 229-250. 14

-

M.

BARNSLEY,

P.D.

tions for Kirkwood (1974) 251-265.

ROBINSON

- Riseman

and P a d S - t y p e

- Pads a p p r o x i m a n t b o u n d s and a p p r o x i m a t e integral e q u a t i o n s - J. Inst. Math. Appl.,

15 - J.L. BASDEVANT - Pads a p p r o x i m a n t s - in "Methods VoI. IV, Gordon and Breach, London, 1970. 16 - J.L. B A S D E V A N T - The Pads a p p r o x i m a t i o n der Physik, 20 (1972) 283-331. 17 - A.F. B E A R D O N - On the c o n v e r g e n c e Appl., 21 (1968) 344-346.

in s u b n u c l e a r

and its p h y s i c a l

of Pads a p p r o x i m a n t s

appro-

solu14

physics",

applications

- J. Math.

- Fort.

Anal.

18 - D° BESSIS - Topics in the theory of Pads a p p r o x i m a n t s - in "Pads approximants", P.R. G r a v e s - M o r r i s ed., The institute of physics, London, 1973. 19 - D. BESSIS, J.D. T A L M A N - V a r i a t i o n a l a p p r o a c h to the t h e o r y of o p e r a t o r Pads a p p r o x i m a n t s - Rocky M o u n t a i n s J. Math., 4 (1974) 151-158. 20 - C. BREZINSKI - C o n v e r g e n c e of Pads a p p r o x i m a n t s for some special sequences Thrid C o l l o q u i u m on a d v a n c e d c o m p u t i n g m e t h o d s in t h e o r e t i c a l physics, Marseille, 1973. 21 - C. BREZINSKI - Rhombus f r a c t i o n s - to appear. 22 - C. BREZINSKI to appear.

algorithms

- Computation

of Pads

23 - C. BREZINSKI - SSries de Stieltjes m e c h 58, Toulon, 12-14 m a i 1975.

connected

-

to the Pads table and c o n t i n u e d

approxlmants

and c o n t i n u e d

et a p p r o x i m a n t s

de Pads

24 - C. BREZINSKI Linear Algebra,

- Some r e s u l t s in the t h e o r y of the v e c t o r 8 (1974) 77-86.

25 - C. BREZINSKI Rocky M o u n t a i n s

- Some r e s u l t s and a p p l i c a t i o n s J. Math., 4 (1974) 335-338.

about

26 ~ C. BREZINSKI - G ~ n S r a l i s a t i o n s de la t r a n s f o r m a t i o n de Pads et de l ' e - a l g o r i t h m e - C a l c o l o (to appear).

fractions

- Colloque

~-algorithm

the vector

de Shanks,

Euro-

-

~-algorlthm

de la table

-

247

27 - J.S.R. ximants",

CHISHOLM - Mathematical P.R. G r a v e s - M o r r i s ed.,

t h e o r y of P a d 6 a p p r o x i m a n t s - in "Pad6 a p p r o The i n s t i t u t e of P h y s i c s , L o n d o n , 1973.

28 - J.S.R. C H I S H O L M - R a t i o n a l a p p r o x i m a n t s Math. Comp., 27 (1973) 841-848.

defined

29 - J.S.R. C H I S H O L M - A p p r o x i m a t i o n by sequences of m e r o m o r p h y - J. Math. Phys., 7 (1966) 39-44,

from

of P a d 6

double

power

approximants

series

-

in r e g i o n s

30 - J.S.R. C H I S H O L M - C o n v e r g e n c e p r o p e r t i e s of P a d 6 a p p r o x i m a n t s - in "Pad6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973. 31 - C.K. CHUI, 0. S H I S H A , P.W. S M I T H - P a d 6 rational approximants - J. A p p r o x . T h e o r y ,

approximants as l i m i t s 12 (1974) 2 0 1 - 2 0 4 .

32 - A.K. C O M M O N - P a d 6 a p p r o x i m a n t s P h y s . , 9 (1968) 32-38.

and bounds

33 - A.K. C O M M O N , P.R. G R A V E S - M O R R I S J. Inst. Math. A p p l i c s . , 13 (1974)

- Some properties 229-232.

34 - J.D.P. D O N N E L L Y - The P a d 6 t a b l e D.C. H a n d s c o m b ed., P e r g a m o n P r e s s ,

to s e r i e s

of b e s t

of S t i e l ~ e s

of C h i s h o l m

- in " M e t h o d s of n u m e r i c a l N e w - Y o r k , 1966.

- J. Math.

approximants

-

approximation",

35 - A. E D R E I - The P a d 6 t a b l e of m e r o m o r p h i c f u n c t i o n s of s m a l l o r d e r w i t h n e g a t i v e z e r o s a n d p o s i t i v e p o l e s - R o c k y M o u n t a i n s J. M a t h . , 4 (1974) 1 7 5 - 1 8 0 . 36 - A. E D R E I - C o n v e r g e n c e de la m 6 t h o d e m o r p h e s - C o l l o q u e E u r o m e c h 58, T o u l o n ,

de P a d 6 a p p l i q u 6 e s 1 2 - 1 4 m a i 1975.

aux fonctions

37 - D. E L L I O T - T r u n c a t i o n e r r o r s in P a d 6 a p p r o x i m a t i o n s to c e r t a i n an a l t e r n a t i v e a p p r o a c h - Math. Comp., 21 (1967) 3 9 8 - 4 0 6 .

functions

38 - S.T. E P S T E I N , M. B A R N S L E Y - A v a r i a t i o n a l a p p r o a c h to the t h e o r y point Pad6 approximants - J. Math. P h y s . , 14 (1973) 3 1 4 - 3 2 5 . 39 - W. F A I R - P a d 6 a p p r o x i m a t i o n Math. Comp., 18 (1964) 627-634.

to the

solution

40 - W. F A I R , Y.L. L U K E - P a d 6 a p p r o x i m a t i o n s Numer. M a t h . , 14 (1970) 3 7 9 - 3 8 2 . 41 - J. F L E I S C H E R - N o n l i n e a r P h y s . , 14 (1973) 246-248.

Pad6

of the R i c c a t i

to t h e o p e r a t o r

approximants

for

Legendre

m6ro-

of m u l t i -

equation

-

exponential

series

:

-

- J. Math.

42 - J. F L E I S C H E R - N o n l i n e a r P a d 6 a p p r o x i m a n t s f o r L e g e n d r e s e r i e s - in " P a d 6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973. 43 - J. F L E I S C H E R - G e n e r a l i z a t i o n s of P a d 6 a p p r o x i m a n t s - in " P a d 6 P.R. G r a v e s - M o r r i s ed., The i n s t i t u t e of p h y s i c s , L o n d o n , 1973. 44 - N.R. Theory,

F R A N Z E N - Some c o n v e r g e n c e 6 (1972) 254-263.

results

for Pad6

approximants

45 - N.R. F R A N Z E N - C o n v e r g e n c e o f P a d 6 a p p r o x i m a n t s for a certain m e r o m o r p h i c f u n c t i o n s - J. A p p r o x . T h e o r y , 6 (1972) 264-271.

approximants",

- J. A p p r o x .

class

of

248

46 - J.L. GAMMEL - Review of two recent g e n e r a l i z a t i o n s of the Pad6 approximant in "Pad6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., Academic Press, New-York, 1973. 47 - J.L. GAMMEL - Effect of r a n d o m errors (noise) in the terms of a power series on the convergence of the Pad~ a p p r o x i m a n t s - in "Pad~ approximants", P.R. G r a v e s - M o r r i s ed., The institute o f physics, London, 1973. 48 - J.L. GAMMEL, J. N U T T A L L - C o n v e r g e n c e of Pad~ approximants to quasi-analytic functions beyond n a t u r a l b o u n d a r i e s - J. Math. Anal. Appl. 49 - J. G I L E W I C Z - N u m e r i c a l detection of the best Pad~ a p p r o x i m a n t and determination of the Fourier coefficients of the i n s u f f i c i e n t l y sampled functions - in "Pad~ approximants and their applications", P.R. G r a v e s - M o r r i s ed., Academic Press, New-York, 1973. 50 - J. GILEWICZ - Totally monotonic and totally positive sequences for the Pad~ approximants m e t h o d - Colloque E u r o m e c h 58, Toulon, 12-14 mai 1975. 51 - J.E. GOLDEN, J.H. McGUIRE, J. N U T T A L L - C a l c u l a t i n g Bessel functions w i t h Pad~ approximants - J. Math. Anal. Appl., 43 (1973) 754-767. 52 - W.B. GRAGG - The Pad~ table and its r e l a t i o n to certain algorithms of numer i c a l analysis - SIAM Rev., 14 (1972) 1-62. 53 - W.B. GRAGG - On Hadamar's theory of polar singularities - in "Pad~ approxim a n t s and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 54 - W.B. GRAGG, G.D. J O H N S O N - The Laurent - Pad~ table - P r o c e e d i n g s IFIP Congress, North-Holland, 1974. 55 ~ P.R. G R A V E S - M O R R I S ed. - Pad~ approximants and their applicationsA c a d e m i c Press, New-York, 1973. 56 - P.R. G R A V E S - M O R R I S ed. - Pad~ approximants - The institute of physics, London, 1973. 57 - T.N.E. GREVILLE - On some conjectures of P. Wynn concerning the E-algorithm MRC Technical summary report 877, Madison, 1968. 58 - A.S. H O U S E H O L D E R - The Pad~ table, the Frobenius identities and the q-d a l g o r i t h m - Linear Algebra, 4 (1971) 161-174. 59 - A.S. HOUSEHOLDER, G.W. STEWART - Bigradients, Hankel determinants and the Pad~ table - in "Constructive aspects of the f u n d a m e n t a l t h e o r e m of algebra", B. D e j o n and P. Henrici eds., A c a d e m i c Press, New-York, 1969. 60 - R.C. JOHNSON - A l t e r n a t i v e a p p r o a c h to Pad~ approximants - in "Pad~ appreximants and their application", P.R. G r a v e s - M o r r i s ed., Academic Press, NewYork, 1973. 61 - W.B. JONES - T r u n c a t i o n error b o u n d for continued fractions and Pad~ approximants - in "Pad~ approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 62 - W.B. JONES - Analysis of t r u n c a t i o n error of a p p r o x i m a t i o n s b a s e d on the Pad~ table and continued fractions - R o c k y Mountains J. Math., 4 (1974) 241-250.

249

63 - W.B. JONES, W.J. THRON - On c o n v e r g e n c e Anal., 6 (1975) 9-16. 64 - I.M. LONGMAN (1971) 53-64.

- Computation

Of Pad6 a p p r o x i m a n t s

of the Pad6 table

65 - Y.L. LUKE - E v a l u a t i o n of the g a m m a f u n c t i o n SIAM J. Math. Anal., i (1970) 266-281. 66 - Y.L. LUKE - The Pad6 table and the T - m e t h o d 110-127. 67 - A. MAGNUS - Certain c o n t i n u e d Math. Z., 78 (1960) 361-374. 68 - A. MAGNUS - P - f r a c t i o n s 4 (1974) 257-260.

fractions

- Intern.

by means

and the Pad6 table

J. comp.

Math.,

3B

of Pad6 a p p r o x i m a t i o n s -

- J. Math.

associated

- SIAM J. Math.

Phys.,

37 (1958)

w i t h the Pad6 table -

- Rocky M o u n t a i n s

J. Math.,

69 - D. M A S S O N - Hilbert space and Pad~ a p p r o x i m a n t - in "The Pad~ a p p r o x i m a n t in t h e o r e t i c a l physics", G.A. Baker Jr. and J.L. G a m m e l eds., A c a d e m i c Press, New-York, 1970. 70 - D. M A S S O N - Pad6 a p p r o x i m a n t s and Hilbert spaces - in "Pad6 a p p r o x i m a n t s and their a p p l i c a t i o n s " , P.R. G r a v e s - M o r r l s ed., A c a d e m i c Press, 1973. 71 - J.H. M c C A B E Pad6 q u o t i e n t s

- A formal e x t e n s i o n of the Pad6 table to include - J. Inst. Math. Applies., 15 (1975) 363-372.

two point

72 - J.B.

- A note on the e - a l g o r i t h m

17-24.

McLEOD

- Computing,

7 (1971)

73 - G. MERZ - Pad6sche N [ h e r u n g s b r [ c k e und I t e r a t i o n s v e r f a h r e n h ~ h e r e n D o c t o r a l thesis, T e c h n i s c h e Hochschule Clausthal, Germany, 1967.

Ordnung

-

74 - J. N U T T A L L - The c o n n e c t i o n of Pad6 a p p r o x i m a n t s w i t h s t a t i o n a r y v a r i a t i o n a l p r i n c i p l e s and the c o n v e r g e n c e of c e r t a i n Pad6 a p p r o x i m a n t s - in "The Pad6 a p p r o x i m a n t in t h e o r e t i c a l physics", G.A. Baker Jr. and J.L. G a m m e l eds., A c a d e m i c Press, New-York, 1970. 75 - J. N U T T A L L - The c o n v e r g e n c e J. Math. Anal. Appl., 31 (1970)

of Pad6 a p p r o x i m a n t s 147-153.

of m e r o m o r p h i c

functions

-

76 - J. N U T T A L L - V a r i a t i o n a l p r i n c i p l e s and Pad6 a p p r o x i m a n t s - in "Pad6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 77 - J. N U T T A L L - The c o n v e r g e n c e J. Math., 4 (1974) 269-272.

of c e r t a i n

Pad6 a p p r o x i m a n t s

78 - H. PADE - Sur la r e p r 6 s e n t a t i o n a p p r o c h 6 e r a t i o n n e l l e s - Ann. Ec. Norm. Sup., 9 (1892) 79 - C. P O M M E R E N K E - Pad6 a p p r o x i m a n t s Anal. Appl., 41 (1973) 775-780.

d'une f o n c t i o n 1-93.

and c o n v e r g e n c e

- Rocky Mountains

par des f r a c t i o n s

in c a p a c i t y

- J. Math.

80 - L.D. PYLE - A g e n e r a l i z e d inverse c - a l g o r i t h m for c o n s t r u c t i n g i n t e r s e c t i o n p r o j e c t i o n matrices, with a p p l i c a t i o n s - Numer. Math., i0 (1967) 86-102. 81 - J. R I S S A N E N

- Recursive

evaluation

of Pad6 a p p r o x i m a n t s

for m a t r i x

sequences-

250

IBM J.

Res.

develop.,

(july

1972)

401-406.

82 - E.B. S A F F - An e x t e n s i o n of M o n t e s s u s de B a l l o r e ' s t h e o r e m on t h e c o n v e r g e n c e of i n t e r p o l a t i n g r a t i o n a l f u n c t i o n s - J. Approx. T h e o r y , 6 (1972) 63-67. 83 - W.F. T R E N C H - An a l g o r i t h m f o r the i n v e r s i o n S I A M J. Appl. M a t h . , 13 (1965) 1 1 0 2 - 1 1 0 7 . 84 - R.P. W A N DE R I E T - On c e r t a i n P a d ~ f u n c t i o n - r e p o r t TN 32, M a t h e m a t i c a l 85 - H. V A N R O S S U M - A t h e o r y V a n G o r a i m , A s s e n , 1953.

of f i n i t e

Hankel

polynomials in c o n n e c t i o n c e n t e r , A m s t e r d a m , 1963.

of o r t h o g o n a l

matrices

with

the b l o c k

polynomials

based

on the P a d 6

86 - H.S. W A L L - On the r e l a t i o n s h i p a m o n g the d i a g o n a l Bull. Amer. Math. Soc., 38 (1932) 7 5 2 - 7 6 0 .

files

of a P a d 6

87 - H.S. W A L L - On the P a d 6 a p p r o x i m a n t s associated a n d s e r i e s of S t i e l t j e s - Trans. Amer. Math. Soc., 88 - H.S. W A L L - On the P a d ~ w e r s e r i e s - Trans. Amer. 89 - H.S. ximants

WALL - General - Trans. Amer.

w i t h the c o n t i n u e d 31 (1929) 91-116.

approximants associated with a positive Math. Soc., 33 (1931) 511-532.

t h e o r e m s on t h e c o n v e r g e n c e of s e q u e n c e s Math. Soc., 34 (1932) 4 0 9 - 4 1 6 .

90 - H. W A L L I N - On the c o n v e r g e n c e t h e o r y of P a d 6 a p p r o x i m a n t s the O b e r w o l f a c h c o n f e r e n c e on a p p r o x i m a t i o n t h e o r y , 1971. 91 - H. W A L L I N - The c o n v e r g e n c e of P a d ~ a p p r o x i m a n t s a n d t h e s e r i e s c o e f f i c i e n t s - A p p l i c a b l e A n a l . , 4 (1974) 2 3 5 - 2 5 2 . 92 - J.L~ W A L S H approximation

- P a d ~ a p p r o x i m a n t s as l i m i t of r a t i o n a l - J. Math. M e c h . , 13 ( 1 9 6 4 ) 305-312.

-

table

table

-

fraction

definite

of P a d 6

po-

appro-

- Proceedings

size

-

of

of the p o w e r

functions

of b e s t

93 - J.L. W A L S H - P a d 6 a p p r o x i m a n t s as l i m i t s of r a t i o n a l f u n c t i o n s of b e s t approximation, r e & l d o m a i n - J. A p p r o x . T h e o r y , ii (1974) 2 2 5 - 2 3 0 . 94 - P. W Y N N - U p o n t h e i n v e r s e of f o r m a l p o w e r s e r i e s o v e r c e r t a i n C e n t r e de r e c h e r c h e s m a t h ~ m a t i q u e s , U n i v e r s i t ~ de M o n t r 6 a l , 1970. 95 - P. W Y N N - U p o n t h e g e n e r a l i z e d i n v e r s e of a f o r m a l p o w e r v a l u e d c o e f f i c i e n t s - C o m p o s i t i o M a t h . , 23 (1971) 4 5 3 - 4 6 0 . 96 - P. W Y N N C.R. Acad.

- Sur l ' 6 q u a t i o n a u x d 6 r i v 6 e s p a r t i e l l e s Sc. P a r i s , 278 A (1974) 8 4 7 - 8 5 0 .

99 - P. W Y N N - A g e n e r a l s y s t e m 18 ser. 2 (1967) 69-81. i00 - P. W Y N N

- Some recent

recursions

of orthogonal

developments

series

de la s u r f a c e

97 - P. W Y N N - U b e r f i n e n i n t e r p o l a t i o n s - algorithmus d i e in d e r t h e o r i e d e r i n t e r p o l a t i o n d u t c h r a t i o n a l e Numer. M a t h . , 2 (1961) 1 5 1 - 1 8 2 . 98 - P. W Y N N - D i f f e r e n c e - d i f f e r e n t i a l L o n d o n Math. Soc., 23 (1971) 2 8 3 - 3 0 0 .

algebras

und gewise funktionen

for Pad6

polynomials

in the t h e o r i e s

with vector

de Pad6

-

andere formeln, bestehen -

quotients

~ Quart.

-

- Proc.

J. M a t h . ,

of c o n t i n u e d

fractions

251

a n d the P a d 6 t a b l e

- Rocky Mountains

J. M a t h . ,

4 (1974)

297-324.

i01 - P. W Y N N - U p o n the d i a g o n a l s e q u e n c e s s u m m a r y r e p o r t 660, M a d i s o n , 1966.

of t h e P a d 6 t a b l e

102 - P. W Y N N - E x t r e m a l p r o p e r t i e s H u n g a r i c a e , 25 (1974) 2 9 1 - 2 9 8 .

quotients

of P a d 6

- MRC technical

- A c t a Math.

Acad.

103 - P. W Y N N - U p o n a c o n v e r g e n c e r e s u l t in the t h e o r y Trans. Amer. Math. Soc., 165 (1972) 2 3 9 - 2 4 9 .

of t h e P a d 6

104 - P. W Y N N - Z u r t h e o r i e d e r m i t g e w i s s e n s p e z i e l l e n P a d ~ s c h e n t a f e l n - Math. Z., 109 (1969) 66-70.

funktionen

105 - P. W Y N N - U p o n the P a d 6 t a b l e d e r i v e d J. Numer. A n a l . , 5 (1968) 8 0 5 - 8 3 4 .

from a Stieltjes

106 - P. W Y N N - U p o n s y s t e m s of r e c u r s i o n s w h i c h o b t a i n the P a d 6 t a b l e - N u m e r . M a t h . , 8 (1966) 2 6 4 - 2 6 9 . 107 - P. W Y N N - L ' s - a l g o r i t m o (1961) 4 0 3 - 4 0 8 .

e la t a v o l a

di P a d 6

108 - P. W Y N N - U p o n a c o n j e c t u r e c o n c e r n i n g equations, and certain other matters - MRC M a d i s o n , 1966. 109 - P. W Y N N - G e n e r a l p u r p o s e M a t h . , 6 (1964) 22-36.

vector

for

II - C o n t i n u e d

algorithm

iterated

di Mat.

114 - D. B E R N O U L L I N o v i comm. Acad.

- D i s q u i s i t i o n e s u l t e r i o r e s de i n d o l e Sc. Imp. St P 6 t e r s b o u r g , 20 (1775)

20

problems

related

to t h e m

de f r a c t i o n i b u s

of c o n t i n u e d

fractionum

fractions

continues

fractions

- Nouvelles

CHEBYCHEV

- Sur les f r a c t i o n s

continues

- Jour.

-

continuarum

-

- SIAM Rev.,

Ann.

de M a t h . ,

-

continuis

Irra~ionalit[t unendlieher Kettenbr~che Z qV x v - M a t h . A n n . , 76 ( 1 9 1 5 ) 2 9 5 - 3 0 0 .

evaluation

-

continued

de M a t h . ,

(3) 6 (1887) 118 - P.

of

- Numer.

and matrix

and algorithms

- Adversaria analytica miscellanea Sc. Imp. St P 6 t e r s b o u r g , 20 (1775)

- Sur q u e l q u e s

Roma,

procedures

vector

i13 - D. B E R N O U L L I N o v i comm. Acad.

117 - E. C E S A R O

- SIAM

the q u o t i e n t s

112 : F.L. B A U E R , E. F R A N K - N o t e on f o r m a l p r o p e r t i e s o f c e r t a i n f r a c t i o n s - Proc. Amer. Math. Soc., 9 (1958) 3 4 0 - 3 4 7 .

116 - G. B L A N C H - N u m e r i c a l 7 (1964) 3 8 3 - 4 2 1 .

verkn~pften

fractions

iii - F.L. B A U E R - Use of c o n t i n u e d f r a c t i o n s C I M E s u m m e r s c h o o l l e c t u r e s , P e r i g i a , 1964.

115 - S. B E R N S T E I N , 0. S Z A S Z - U b e r m i t e i n e r A n w e n d u n g a u f die R e i h e

-

a method for solving linear t e c h n i c a l s u m m a r y r e p o r t 626,

epsilon

ii0 - P. W Y N N - A c c e l e r a t i o n t e c h n i q u e s Math. C o m p . , 16 (1962) 3 0 1 - 3 2 2 .

- Rend.

table

series

among

Sci.

set.

11,3

252

(1858)

289-323.

119 - V.F. C O W L I N G , general continued

W. L E I G H T O N , W.J. T H R O N - T w i n c o n v e r g e n c e r e g i o n s f o r t h e f r a c t i o n - Bull. Amer. Math. Soc., 49 (1943) 913-916.

120 - I.V. C Y G A N K O V - S o l u t i o n o f R i c c a t i e q u a t i o n s by c o n t i n u e d Perm. Gos. Univ. Ucen. Zap. M a t . , 17 (1960) 99-107. 121 - I.V. C Y G A N K O V f r a c t i o n s - Perm.

- Solution Gos. Univ.

fractions

of a s p e c i a l R i c c a t i e q u a t i o n b y c o n t i n u e d Ucen. Zap. M a t . , 17 (1960) 109-113.

122 - J.D.P. D O N N E L L Y - C o n t i n u e d f r a c t i o n s - in " M e t h o d s o f n u m e r i c a l m a t i o n " , D.C. H a n d s c o m b ed., P e r g a m o n P r e s s , O x f o r d , 1965. 123 - H.G. E L L I S - C o n t i n u e d f r a c t i o n s s o l u t i o n s o f the g e n e r a l t i a l e q u a t i o n - R o c k y M o u n t a i n s J. M a t h . , 4 (1974) 3 5 3 - 3 5 6 . 124 - L. E U L E R - De f r a c t i o n i b u s St P ~ t e r s b o u r g , ii (1739).

continuis

125 - L. E U L E R - De f o r m a t i o n e St P ~ t e r s b o u r g , 1779.

fractionum

126 - W. (1971)

continued

FAIR - Noncommutative 226-232.

127 - W. F A I R - A c o n v e r g e n c e J. A p p r o x . T h e o r y , 5 (1972)

theorem 74-76.

observationes

continuarum

fractions

Acad.

Acad.

Sc.

- S I A M J. Math.

129 - W. F A I R - C o n t i n u e d f r a c t i o n s o l u t i o n to F r e d h o l m R o c k y M o u n t a i n s J. M a t h . , 4 (1974) 3 5 7 - 3 6 0 .

FRANK - Corresponding 89-108.

133 - E. diques

GALOIS - Ann.

type

continued

fractions

Imp.

2

equations

131 - D.A. F I E L D , W.B. J O N E S - A p r i o r i e s t i m a t e s for t r u n c a t i o n n u e d f r a c t i o n s K ( i / b n) - Numer. M a t h . , 19 (1972) 2 8 3 - 3 0 2 .

Imp.

-

in a B a n a c h

truncation error estimates for continued J. M a t h . , 4 (1974) 3 6 1 - 3 6 2 .

132 - E. (1946)

Sc.

fractions

equation

integral

differen-

Anal.,

continued

to the R i c c a t i 318-323.

approxi-

Riccati

- Comm.

- Acta

for noncommutative

128 - W. F A I R - C o n t i n u e d f r a c t i o n s o l u t i o n a l g e b r a - J. Math. Anal. A p p l . , 39 (1972)

130 - D.A. F I E L D - A p r i o r i K ( i / b n) - R o c k y M o u n t a i n s

-

-

fractions

error

of c o n t i -

- Amer.

J. M a t h . ,

68

- D~monstration d ' u n t h ~ o r ~ m e sur les f r a c t i o n s Math. P u r e s et A p p l . , 19 ( 1 8 2 8 - 1 8 2 9 )

continues

p~rio-

134 - H.L. G A R A B E D I A N , H.S. W A L L - T o p i c s in c o n t i n u e d f r a c t i o n s a n d s u m m a b i l i t y N o r t h w e s t e r n Univ. S t u d i e s in Math. a n d phys. s c i e n c e s , Vol. i, E v a n s t o n a n d C h i c a g o , (1941) 89-132. 135 - I. G A R G A N T I N I , P. H E N R I C I - A c o n t i n u e d f r a c t i o n a l g o r i t h m f o r the c o m p u t a t i o n of h i g h e r t r a n s c e n d e n t a l f u n c t i o n in t h e c o m p l e x p l a n e - Math. C o m p . , 21 (1967) 18-29. 136 - T.F. G L A S S , W. L E I G H T O N - On the c o n v e r g e n c e Bull. Amer. Math. Soc., 49 (1943) 133-135. 137 - W.B.

GRAGG

- Truncation

error

bounds

of a c o n t i n u e d

for ~fraetions

- Bull.

fraction

Amer.

-

Math.

253

Soc.,

76 (1970)

1091-1094.

138 - W.B. G R A G G - M a t r i x i n t e r p r e t a t i o n s and a p p l i c a t i o n s f r a c t i o n s a l g o r i t h m - R o c k y M o u n t a i n s J. M a t h . , 4 (1974)

of the c o n t i n u e d 213-226.

139 - W.B. G R A G G - T r u n c a t i o n ii (1968) 3 7 0 - 3 7 9 .

- Numer.

error

bounds

for g-fractions

140 - M. H A M B U R G E R - U e b e r di K o n v e r g e n z e i n e s K e t t e n b r u c h s - Math. Ann., 81 (1920) 31-45.

mit

141 - H. H A M B U R G E R - B e i t r [ g e K e t t e n b r [ c h e - Math. Z e i t . ,

zur K o n v e r g e n z t h e o r i e 4 (1919) 1 8 6 - 2 2 2 .

142 - T.L. 7 (1965)

fraction

HAYDEN - Continued 292-309.

143 - T.L. H A Y D E N M a t h . , 4 (1974)

- Continued 367-370.

145 - E. (1922)

HELLINGER 18-29.

- Zur

der

approximation

fractions

144 - T.L. H A Y D E N - A c o n v e r g e n c e Math. Soc., 14 (1963) 546-552.

einer

in B a n a c h

problem

Stieltjesschen

for

Potenzreihe

Stieltjes

spaces

147 - P. H E N R I C I , P. P F L U G E R - T r u n c a t i o n e r r o r f r a c t i o n s - Numer. M a t h . , 9 (1966) 120-138.

J.

fractions

- Proe.

Amer.

- Math.

continued 1965. for

continued

Ann.,

- 0ber Math.,

eine besondere Art der 12 (1889) 3 6 7 - 4 0 5 .

152 - T.H. J E F F E R S O N - T r u n c a t i o n Numer. A n a l . , 6 (1969) 3 5 9 - 3 6 4 . 153 - W.B. J O N E S Ph.D. thesis,

error

- Contributions to the Vanderbilt University,

estimates

fractions

- Doctoral

continued

reeller

i n t e g r a l e f.~ e -x2 dx 12 ( 1 8 3 4 ) , A 346-347.

for

T-fractions

- SIAM

J.

t h e o r y of T h r o n c o n t i n u e d f r a c t i o n s N a s h v i l l e , T e n n e s s e e , 1963.

154 - W.B. J O N E S , R.I. S N E L L - T r u n c a t i o n S I A M J. Numer. A n a l . , 6 (1969) 2 1 0 - 2 2 1 . 155 - W.B. J O N E S , R.I. S N E L L f r a c t i o n s K ( a n / l ) - Trans,

-

Stieltjes

Kettenbruchentwicklung

151 - C.G.J. J A C O B I - De f r a c t i o n e c o n t i n u e , in q u a m e v o l d e r e l i c e t - J. f i r die r e i n e u. a n g e w , m a t h . ,

86

fractions

149 - K.L. H I L L A M , W.J. T H R O N - A g e n e r a l c o n v e r g e n c e c r i t e r i o n f o r f r a c t i o n s K ( a n / b n) - Proc. Amer. Math. Soc., 16 (1965) 1 2 5 6 - 1 2 6 2 . 150 - A. H U R W I T Z GrSssen - Acta

Math.,

Mountains

estimates

148 - K.L. H I L L A M - S o m e c o n v e r g e n c e c r i t e r i a f o r t h e s i s , U n i v e r s i t y o f C o l o r a d o , B o u l d e r , 1962.

- Numer.

- Rocky

Kettenbruchtheorie

146 - P. H E N R I C I - E r r o r b o u n d s f o r c o m p u t a t i o n s w i t h in " E r r o r in d i g i t a l c o m p u t a t i o n " , Wiley, New-York,

assoziierten

' sehen

to f u n c t i o n s

continued

Math.,

- Sequences Amer. Math.

156 - W.B. J O N E S , W.J. T H R O N - C o n v e r g e n c e M a t h . , 20 (1968) 1 0 3 7 - 1 0 5 5 .

error

bounds

for

continued

-

fractions

of c o n v e r g e n c e r e g i o n s f o r c o n t i n u e d Soc., 1 7 0 (1972) 4 8 3 ~ 4 9 7 . of c o n t i n u e d

fractions

- Canad.

J.

-

254

157 - W.B. K(an/l)

J O N E S , W.J. T H R O N - T w i n c o n v e r g e n c e r e g i o n s - Trans. Amer. Math. Soc., 150 (1970) 9 3 - 1 1 9

158 - W.B. J O N E S , W.J. continued fractions

T H R O N - A p o s t e r i o r i b o u n d s f o r the t r u n c a t i o n - S I A M J. N u m e r . A n a l . , 8 ( 1 9 7 1 ) 6 9 3 - 7 0 5 .

159 - W.B. J O N E S , W.J. T H R O N - N u m e r i c a l s t a b i l i t y f r a c t i o n s - Math. C o m p . , 28 (1974) 7 9 5 - 8 1 0 . 160 - W.B. J O N E S , W.J. 166 (1966) i 0 6 - i 1 8 . 161 - J.Q. J O R D A N , W. continued fraction (1938) 8 7 2 - 8 8 2 . 162 - A. Ya.

for continued

THRON

- Further

properties

in e v a l u a t i n g

fractions

error

of

continued

of T - f r a c t i o n s

- Math.

Ann.,

L E I G H T O N - On the p e r m u t a t i o n of the c o n v e r g e n t s of a a n d r e l a t e d c o n v e r g e n c e c r i t e r i a - Ann. M a t h . , (2) 39

KHINTCHINE

- Continued

fractions

- P. N o o r d h o f f ,

Groningen,

1963.

163 - A.N. K H O V A N S K Z I - The a p p l i c a t i o n o f c o n t i n u e d f r a c t i o n s a n d t h e i r g e n e r a l i z a t i o n s to p r o b l e m s in a p p r o x i m a t i o n t h e o r y - P. N o o r d h o f f , G r o n i n g e n , 1963. 164 - E. L A G U E R R E - Sur la r @ d u c t i o n & t e n d u e de f o n c t i o n s - C.R. Acad.

en f r a c t i o n s c o n t i n u e s Sc. P a r i s , 87A (1878).

d'une

classe

assez

165 - E. L A G U E R R E - Sur la r @ d u c t i o n en f r a c t i o n s c o n t i n u e s de e F ( x ) , F(x) d @ s i g n a n t u n p o l y n S m e e n t i e r - J. Math. p u r e s et a p p l . , (3) 6 ( 1 8 8 0 ) . 166 - R.E. L A N E - The v a l u e J., 12 (1946) 2 0 7 - 2 1 6 .

for continued

fractions

- D u k e Math.

167 - R.E. L A N E - The c o n v e r g e n c e a n d v a l u e s of p e r i o d i c Bull. Amer. Math. Soc., 51 (1945) 2 4 6 - 2 5 0 .

continued

fractions

168

region

problem

- L.J. L A N G E - On a f a m i l y of t w i n c o n v e r g e n c e f r a c t i o n s - Iii. J. of M a t h . , i0 (1966) 97-108.

169 - L.J. regions 170 - W. Soc.,

regions

-

for c o n t i n u e d

L A N G E , W.J. T H R O N - A t w o - p a r a m e t e r f a m i l y of b e s t t w i n c o n v e r g e n c e f o r c o n t i n u e d f r a c t i o n s - Math. Z e i t . , 73 (1969) 2 7 7 - 2 8 2 .

LEIGHTON - A test-ratio 45 (1939) 97-100.

171 - W. L E I G H T O N - C o n v e r g e n c e 5 (1939) 2 9 8 - 3 0 8 .

test for

theorems

continued

fractions

for continued

- Bull.

fractions

172 - W. L E I G H T O N - S u f f i c i e n t c o n d i t i o n s f o r t h e c o n v e r g e n c e f r a c t i o n - D u k e Math. J., 4 (1938) 7 7 5 - 7 7 8 . 173 - W. L E I G H T O N , W.T. S C O T T - A g e n e r a l Amer. Math. Soc., 45 (1939) 5 9 6 - 6 0 5 . 174 - W. L E I G H T O N , W.J. T H R O N - On v a l u e Amer. Math. Soc., 48 (1942) 9 1 7 - 9 2 0 . 175 - W. L E I G H T O N , W.J. T H R O N - C o n t i n u e d Math. J., 9 (1942) 7 6 3 - 7 7 2 . 176 - W. L E I G H T O N ,

W.J.

THRON

continued

regions

fraction

for

fractions

- On the c o n v e r g e n c e

Math.

- D u k e Math.

J.,

of a c o n t i n u e d

expansion

continued

with

Amer.

fractions

complex

of continued

- Bull.

elements

fractions

- Bull.

- Duke

to

255

meromorphic

functions

- Ann.

Math.,

(2)

44

(1943)

177 - J.S. M A C N E R M E Y - I n v e s t i g a t i o n s c o n c e r n i n g f r a c t i o n s - D u k e Math. J., 26 (1959) 6 6 3 - 6 7 8 . 178 - A. M A G N U S - E x p a n s i o n (1962) 2 0 9 - 2 1 6 .

of power

179 - A. M A G N U S - On P - e x p a n s i o n s S e l s k a b s S k r i f t e r (1964), n°3.

series

of p o w e r

80-89.

positive

into P-fractions

series

- Det.

180 - A. M A G N U S - The c o n n e c t i o n b e t w e e n P - f r a c t i o n s Prec. Amer. Math. $oc., 25 (1970) 6 7 6 - 6 7 9 .

- Math.

Kgl.

Norske

and associated

181 - A. M A R K O V - D e u x d ~ m o n s t r a t i o n s de la c o n v e r g e n c e c o n t i n u e s - A c t a M a t h . , 19 (1895) 93 - 104.

de c e r t a i n e s

182 - A. M A R K O V - N o t e sur les f r a c t i o n s c o n t i n u e s - Bull. Acad. Imp. Sc. St P ~ t e r s b o u r g , 5 (2) (1895) 9-13. 183 - D.F. M A Y E R S - E c o n o m i z a t i o n n u m e r i c a l a p p r o x i m a t i o n " , D.C.

definite

classe

continued

Z.,

Videnskabers

fractions

- On t r u n c a t i o n A n a l . , 3 (1966)

-

fractions

Physico-Math.

of c o n t i n u e d f r a c t i o n s - in " M e t h o d s of H a n d s c o m b ed., P e r g a m o n P r e s s , O x f o r d , 1965.

1 8 4 - J.H. Mc C A B E - A c o n t i n u e d f r a c t i o n e x p a n s i o n , w i t h a t r u n c a t i o n e s t i m a t e , f o r D a w s o n ' s i n t e g r a l - M a t h . C o m p . , 28 (1974) 8 1 1 - 8 1 6 . 185 - E.P. M E R K E S S I A M J. N u m e r .

80

errors for 486-496.

continued

fraction

186 - E.P. M E R K E S , W.T. S C O T T - C o n t i n u e d f r a c t i o n s o l u t i o n s e q u a t i o n - J. Math. Anal. A p p l . , 4 (1962) 3 0 9 - 3 2 7 .

error

computations

of the Riccati

187 - R. DE M O N T E S S U S DE B A L L O R E - Sur les f r a c t i o n s Bull. Soc. Math. de F r a n c e , 30 (1902) 28-36.

continues

alg~briques

-

188 - R. DE M O N T E S S U S DE B A L L O R E - Sur les f r a c t i o n s C.R. Acad. Sc. P a r i s , 134 (1902) 1 4 8 9 - 1 4 9 1 .

continues

alg~briques

-

189 - R. DE M O N T E S S U S DE B A L L O R E - Sur les f r a c t i o n s c o n t i n u e s L a g u e r r e - C.R. Acad. Sc. P a r i s , 140 (1905) 1 4 3 8 - 1 4 4 0 .

alg~briques

de

190 - R. DE M O N T E S S U S DE B A L L O R E - Les f r a c t i o n s A c t a M a t h . , 32 (1909) 2 5 7 - 2 8 1 . 191 - T. M U I R - A t h e o r e m in c o n t i n u a n t s w i t h an i m p o r t a n t a p p l i c a t i o n - Phil.

continues

-

alg~briques

- E x t e n s i o n of a t h e o r e m H a g . , (5) 3 (1877).

-

in c o n t i n u a n t s

192 - T. M U I R - On the p h e n o m e n o n of g r e a t e s t m i d d l e in the c y c l e of a c l a s s of a c l a s s of p e r i o d i c c o n t i n u e d f r a c t i o n - Prec. Roy. Soc. E d i n b u r g h , 12 (1884) 578-592. 193

- H. P A D E - M 6 m o i r e sur les d 6 v e l o p p e m e n t s en f r a c t i o n s c o n t i n u e s de la f o n c t i e n e x p o n e n t i e l l e p o u v a n t s e r v i r d ' i n t r o d u c t i o n ~ la t h 6 o r i e d e s f r a c t i o n s c o n t i n u e s a l g 6 b r i q u e s - Ann. Fac. Sci. de l'Ec. Norm. Sup., 16 ( 1 8 9 9 ) 395-436.

194 - H. P A D E C.R. Acad.

- Sur la d i s t r i b u t i o n des r 6 d u i t e s Sc. P a r i s , 130 (1900).

anormales

d'une

fonction

-

256

195 - H. P A D E - Sur le d 6 v e l o p p e m e n t en f r a c t i o n c o n t i n u e de la f o n c t i o n F ( h , l , h ' , u ) et la g 6 n 6 r a l i s a t i o n de la t h ~ o r i e des f o n c t i o n s s p h 6 r i q u e s C.R. Acad. Sc. P a r i s , 141 ( 1 9 0 5 ) 8 1 9 - 8 2 1 . 196 - O. P E R R O N - 0 h e r z w e i K e t t e n b r [ c h e n Math. - Nat. KI. S. - B., (1957) 1-13. 197 - O. P E R R O N 1950.

- Die

Lehre yon dem

198 - O. P E R R O N - U b e r P a l e r m o , 29 (1910). 199 - S. P I N C H E R L E Norm. Sup., (3)

eine

Kettenbr[chen

spezielle

Klasse

- Sur les f r a c t i o n s 6 (18-89) 1 4 5 - 1 5 2 .

200 - A. P R I N G S H E I M - U b e r G l i e d e r n - Sb. M ~ n c h e n ,

v o n H.S.

von

continues

Wall

- Bayer,

- Chelsea

Pub.

Kettenbr~chen

alg~briques

ein K o n v e r g e n z k r i t e r i u m 29 ( 1 8 9 9 ) .

fir

Akad.

Co.,

-

Wiss.

New-York,

- Rend. Circ. Mat.

- Ann.

Sci.

Kettenbr[che

Ec.

mit positiven

201 - E. R O U C H E - M 6 m o i r e sur le d 6 v e l o p p e m e n t d e s f o n c t i o n s en s ~ r i e s O r d o n n ~ e s s u i v a n t les d ~ n o m i n a t e u r s des r ~ d u i t e s d ' u n e f r a c t i o n c o n t i n u e - J. Ee. P o l y t e c h n i q u e , 37 (1858). 202 - H. R U T I S H A U S E R - 0 b e r e i n e V e r a l l g e m e i n e r u n g Math. M e c h . , 38 (1958) 2 7 8 - 2 7 9 . 203 - H. R U T I S H A U S E R - B e s o h l e u n i g u n g d e r K e t t e n b r [ c h e n - ZAMM, 38 (1958) 187.

der

Konvergenz

Kettenbr[che

einer gewisse

204 - F.T. S C H U B E R T - De t r a n s f o r m a t i o n e s e r i e s in f r a c t i o n e m Acad. So. Imp. St P ~ t e r s b o u r g , 7 ( 1 8 1 5 - 1 8 1 6 ) . 205 - W.T. S C O T T - The c o r r e s p o n d i n g M a t h . , (2) 51 (1950) 56-67.

continued

206 - H. S I E B E C K - 0 b e r p e r i o d i s c h e r m a t h . , 33 (1846) 71-77.

Kettenbr6che

207 - V. S I N G H , W . J . T H R O N n u e d f r a c t i o n s - Proc. 208

- J. S H E R M A N nued fraction

fraction

Klasse

von

continuam

- Mem.

of a J - f r a c t i o n

- Ann.

- J. f ~ r die r e i n e

- A f a m i l y of b e s t t w i n c o n v e r g e n c e Amer. M a t h . Soc., 7 (1956) 2 7 7 - 2 8 2 .

- On t h e n u m e r a t o r s - Tran. Amer. Math.

- Z. A n g e w .

regions

u. a n g e w .

for conti-

of the c o n v e r g e n t s of the S t i e l t j e s S o c . , 35 (1933) 64-87.

conti-

209 - T.J. S T I E L T J E S - Sur la r ~ d u c t i o n en f r a c t i o n c o n t i n u e d ' u n e s ~ r i e p r ~ c ~ d a n t s u i v a n t les p u i s s a n c e s d e s c e n d a n t e s de la v a r i a b l e - Ann. Fac. Sci. T o u l o u s e , 3 (1889) 1-17. 210 - T.J. S T I E L T J E S - N o t e appl. m a t h . , 25 (1891).

sur q u e l q u e s

fractions

211 - T.J. S T I E L T J E S - R e c h e r c h e s sur les f r a c t i o n s U n i v . T o u l o u s e , 8 (1894) 1-122. 212 - T.J. S T I E L T J E S - S u r un d ~ v e l o p p e m e n t Sc. P a r i s , 99 (1884) 5 0 8 - 5 0 9 . 213 - T.J.

STIELTJES

- Recherches

continues

continues

en f r a c t i o n s

sur les f r a c t i o n s

- Quart.

- Ann.

continues

continues

J. p u r e

Fac.

- C.R.

- M6moires

and

Sci.

Acad.

pr6sen-

257

t6s p a r divers savants ~ l'acad6mie des Sciences Sciences et M a t h ~ m a t i q u e s , (2) 32 (2) (1892).

de l ' i n s t i t u t

de France,

214 - W.B. SWEEZY, W.J. THRON - E s t i m a t e s of the speed of c o n v e r g e n c e c o n t i n u e d fractions - SIAM J. Numer. Anal., 4 (1967) 254-270. 215 - W.J. THRON - On p a r a b o l i c c o n v e r g e n c e Math. Zeit., 69 (1958) 172-182.

regions

for c o n t i n u e d

of certain

fractions

-

216 - W.J. THRON - Recent approaches to c o n v e r g e n c e t h e o r y of c o n t i n u e d f r a c t i o n s in "Pad~ a p p r o x i m a n t s and their a p p l i c a t i o n s " , P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 217 - W.J. THRON - A survey of recent Rocky M o u n t a i n s J. Math., 4 (1974)

convergence 273-282.

results

218 - W.J. THRON - Two families of twin c o n v e r g e n c e f r a c t i o n s - Duke Math. J., i0 (1943) 677-685. 219 - W.J. THRON - Twin convergence Math., 66 (1944) 428-439. 220 - W.J. THRON - A family Duke Math. J., ii (1944)

regions

of simple 779-791.

for c o n t i n u e d

regions

for c o n t i n u e d

convergence

221 - W.J. THRON - C o n v e r g e n c e regions for the g e n e r a l Amer. Math. Soc., 49 (1943) 913-916. 222 - W.J. THRON - Some p r o p e r t i e s of the c o n t i n u e d Bull. Amer. Math. Soc., 54 (1948) 206-218.

for c o n t i n u e d

fractions

regions

fractions-

- Amer.

for c o n t i n u e d

continued

fraction

J.

fractions-

fraction

- Bull.

(l+doz)+K(z/(l+dnZ))-

223 - W.J. THRON - Z w i l l i n ~ s k o n v e r g e n z g e b l e t e f~r K e t t e n b r ~ c h e l+K(a /i), deren eines die K r e i s s c h e i b e |a2n_l 1 < 02 ist. - Math. Zeit., 70 (1959) n 310-344. 224 - W.J. THRON - Convergence regions for c o n t i n u e d f r a c t i o n s processes - Amer. Math. Monthly, 68 (1961) 734-750.

and other infinite

225 - W.J. THRON - Convergence of sequences of linear f r a c t i o n a l t r a n s f o r m a t i o n s and of c o n t i n u e d fractions - J. Indian Math. Soc., 27 (1963) 103-127. 226 - W.J. THRON - Some results and p r o b l e m s in the a n a l y t i c fractions - Math. Student, 32 (1964) 61-73. 227 - W.J. THRON - On the c o n v e r g e n c e fractions - Math. Zeit., 85 (1964)

t h e o r y of c o n t i n u e d

of the even part of c e r t a i n 268-273.

228 - J. T R E M B L E Y - Recherches sur les f r a c t i o n s Belles-Let., Berlin (1794-1795)

continues

229 - K.T. V A H L E N - 0her N [ h e r u n g s w e r t e angew, math,, 115 (1895) 221-233.

und K e t t e n b r [ c h e

230 - E.B. VAN V L E C K - On an e x t e n s i o n Amer. Math. Soc., 4 (1908) 297-332.

Of the 1894 m e m o i r

- Mem.

V A N V L E C K - On the c o n v e r g e n c e

and c h a r a c t e r

Acad.

Roy.

Sc.

- J. fur die reine u.

of Stieltjes

231 - E.B. VAN V L E C K - On the c o n v e r g e n c e of c o n t i n u e d f r a c t i o n s elements - Trans. Amer. Math. Soc., 2 (1901) 215-233. 232 - E.B.

continued

- Trans.

with c o m p l e x

of the c o n t i n u e d

fraction

258

alZ/l+... 233

- Trans.

- E.B. V A N V L E C K coefficients have

Amer.

Math.

Soc.,

476-483.

- On t h e c o n v e r g e n c e of a l g e b r a i c c o n t i n u e d f r a c t i o n s w h o s e l i m i t i n g v a l u e s - Trans. Amer. Math. Soc., 5 (1904) 253-262.

234 - E.B. V A N V L E C K - S e l e c t e d t o p i c s t i n u e d f r a c t i o n s - Amer. Math. Soc. 1903. 235

2 (1901)

in t h e t h e o r y of d i v e r g e n t C o l l o q u i u m Pub., i, B o s t o n

- E.B. V A N V L E C K - On t h e c o n v e r g e n c e of t h e c o n t i n u e d o t h e r c o n t i n u e d f r a c t i o n s - Ann. M a t h . , 3 (1901) 1-18.

s e r i e s and conColloquium,

fraction

236 - H. V O N K O C H - Q w e l q u e s t h & o r ~ m e s c o n c e r n a n t la t h ~ o r i e t i o n s c o n t i n u e s - O f v e r s i g t af. Kongl. V e t e n s k a p s - Akad.

of G a u s s

and

g ~ n ~ r a l e des f r a c F6rhandlingen, 52

(1895). 237 - B. V I S C O V A T O F F - De la m ~ t h o d e g ~ n ~ r a l e p o u r r ~ d u i r e t o u t e s s o r t e s de q u a n t i t ~ s en f r a c t i o n s c o n t i n u e s - Mem. Acad. Sc. Imp. St P & t e r s b o u r g , 1 (1803-1804). 238 - H. W A A D E L A N D - T - f r a c t i o n s J. M a t h . , 4 (1974) 3 9 1 - 3 9 4 .

from

a different

point

of v i e w

- Rocky

Mountains

239 - H. W A A D E L A N D - On T - f r a c t i o n s of f u n c t i o n s h o l o m o r p h i c a n d b o u n d e d c i r c u l a r d i s c - N o r s k e Vid. Selsk. S k r ( T r o n d h e i m ) 1964, n°8. 240 - H. W A A D E L A N D - A c o n v e r g e n c e p r o p e r t y N o r s k e Vid. Selsk. Skr ( T r o n d h e i m ) 1966,

of certain n°9.

T-fraction

241 - H.S, W A L L - On some c r i t e r i a of C a r l e m a n for t h e c o m p l e t e J - f r a c t i o n - Bull. Amer. Math. Soc., 54 (1948) 528-532. 242 - H.S. W A L L - C o n v e r g e n c e of c o n t i n u e d f r a c t i o n s Bull. Amer. Math. Soc., 55 (1949) 3 9 1 - 3 9 4 . 243

- H.S. W A L L - N o t e 56 (1949) 96-97.

244 - H.S. W A L L of S t i e l t j e s 73-84.

on a p e r i o d i c

expansions

convergence

in p a r a b o l i c

fraction

- Amer.

domains

Math.

-

of a

-

Monthly,

- Concerning continuous continued fractions and certain systems i n t e g r a l e q u a t i o n s - Rend. Circ. Mat. di P a l e r m o , 11,2 (1953)

245 - H.S. W A L L - P a r t i a l l y 7 (1956) 1 0 9 0 - 1 0 9 3 .

bounded

246 - H.S. W A L L - S o m e c o n v e r g e n c e M o n t h l y , 54 (1957) 95-103. 247 - H.S. W A L L - N o t e 51 (1945) 930-934. 248 - H.S. W A L L t i o n - Bull.

continued

in a

on a c e r t a i n

continued

problems

continued

fractions

for

- Proc.

continued

fraction

fractions

- Bull.

- N o t e on the e x p a n s i o n of a p o w e r s e r i e s Amer. Math. Soc., 51 (1945) 97-105.

on a r b i t r a r y

J-fractions

Soc.,

- Amer.

Math.

Soc.,

into a continued

frac-

- Bull,

Amer.

Math.

Math.

249 - H.S. W A L L - C o n t i n u e d f r a c t i o n e x p a n s i o n s f o r f u n c t i o n s p a r t s - Bull. Amer. Math. Soc., 52 (1946) 138-143. 250 - H.S. W A L L - T h e o r e m s (1946) 671-679.

Amer.

with positive

Amer.

Math.

real

Soc.,

52

259

251 - H.S.

WALL - Bounded

J-fractions

- Bull.

Amer.

Math.

252 - H.S. W A L L - S o m e r e c e n t d e v e l o p m e n t s in the t h e o r y Bull. Amer. Math. Soc., 47 (1941) 4 0 5 - 4 2 3 . 253

Soc.,

52

of continued

- H.S. W A L L - A c o n t i n u e d f r a c t i o n r e l a t e d to s o m e p a r t i t i o n E u l e r - Amer. Math. M o n t h l y , 48 (1941) 1 0 2 - 1 0 8 .

254 - H.S. W A L L - The b e h a v i o u r of c e r t a i n s i n g u l a r l i n e - Bull. Amer. Math. S o c . , 255 - H.S. W A L L - C o n t i n u e d f r a c t i o n s Math. Soc., 50 (1944) 1 1 0 - 1 1 9 . 256 - H.S. W A L L - C o n v e r g e n c e Soc., 17 (1931) 5 7 5 - 5 7 9 .

criteria

257 - H.S. W A L L - On the c o n t i n u e d t i o n s - Bull. Amer. Math. Soc., 258 - H.S. W A L L - C o n t i n u e d f o r m a t i o n s - Bull. Amer.

Stieltjes 48 (1942)

and bounded

for

fractions 39 (1933)

fractions and cross-ratios groups Math. Soc., 40 (1934) 5 7 8 - 5 9 2 . of the form

260 - H.S. W A L L - On c o n t i n u e d f r a c t i o n s Math. Soc., 44 (1938) 94-99.

representing

261 - H.S. W A L L - C o n t i n u e d f r a c t i o n s a n d t o t a l l y Amer. Math. Soc., 48 (1940) 1 6 5 - 1 8 4 . - The a n a l y t i c

268 - H.S. W A L L , J.J. D E N N I S f r a c t i o n - D u k e Math. J., 264 - H.S. W A L L , H.L. f r a c t i o n s - Trans.

theory

- The l i m i t c i r c l e 12 (1945) 2 5 5 - 2 7 3 .

Amer.

meromorphic

of C r e m o n a

Math.

func-

trans-

Amer.

functions

- Bull.

Amer.

sequences

- Trans.

- Van Nostrand,

for a positive

G A R A B E D I A N - H a u s d o r f f m e t h o d s of s u m m a t i o n Amer. Math. Soc., 48 ( 1 9 4 0 ) 1 8 5 - 2 0 7 .

the

Amer.

- Bull.

fractions

case

near

- Bull.

- Bull.

-

of

l+Kl(bZ/l)

monotone

of continued

fractions

fractions

686-693.

fractions

formulas

functions

which represent 942-952.

259 - H.S. W A L L - On c o n t i n u e d f r a c t i o n s Math. Soc., 41 ( 1 9 3 5 ) 7 9 7 - 7 3 6 .

262 - H.S. W A L L Y o r k , 1948.

continued 427-431.

analytic

continued

(1946)

New-

definite

J-

and continued

265 - H.S. W A L L , E, H E L L I N G E R - C o n t r i b u t i o n s to the a n a l y t i c t h e o r y f r a c t i o n s a n d i n f i n i t e m a t r i c e s - Ann. M a t h . , 44 ( 1 9 4 3 ) 1 0 3 - 1 2 7 .

of c o n t i n u e d

266 - H.S. W A L L , W. L E I G H T O N - On the t r a n s f o r m a t i o n f r a c t i o n s - Amer. J. M a t h . , 58 (1936) 2 6 7 - 2 8 1 .

of c o n t i n u e d

267 - H.S. W A L L , J.F. P A Y D O N - The c o n t i n u e d t r a n s f o r m a t i o n s - D u k e Math. J., 9 (1942) 268 - H.S. W A L L , and odd parts

fraction 360-372.

R.E. L A N E - C o n t i n u e d f r a c t i o n s w i t h - Trans. Amer. Math. S o c . , 67 (1949)

and

convergence

as a s e q u e n c e

absolutely 368-380.

269 - H.S. W A L L , W.T. S C O T T - On the c o n v e r g e n c e a n d d i v e r g e n c e f r a c t i o n s - Amer. J. M a t h . , 69 (1947) 5 5 1 - 5 6 1 . 270 - H,S. W A L L , s e r i e s - Ann.

W.T. S C O T T - C o n t i n u e d f r a c t i o n M a t h . , 41 (1940) 3 2 5 - 3 4 9 .

expansion

of linear

convergent

even

of c o n t i n u e d

for arbitrary

power

260

271 - H.S. W A L L , W.T. S C O T T - V a l u e r e g i o n s Math. Soc., 47 (1941) 5 8 0 - 5 8 5 . 272 - H.S. 1-18.

WALL,

for c o n t i n u e d

W.T.

SCOTT

- Continued

fractions

273 - H.S. W A L L , W.T. Trans. Amer. Math.

SCOTT Soc.,

- A convergence theorem 47 (1940) 1 5 5 - 1 7 2 .

274 - H.S. W A L L , M. W E T Z E L - Q u a d r a t i c f o r m s n u e d f r a c t i o n s - D u k e M a t h . J.~ ii ( 1 9 4 4 )

- Natl.

fractions

Math.

Mag.,

for continued

and convergence 89-102.

275 - H.S. W A L L , M. W E T Z E L - C o n t r i b u t i o n s to the a n a l y t i c T r a n s . Amer. Math. Soc., 55 (1944) 3 7 3 - 3 9 7 .

- Bull.

13

Amer.

(1939)

fractions

regions

theory

for

-

conti-

of J - f r a c t i o n s

276 - R. W I L S O N - D i v e r g e n t c o n t i n u e d f r a c t i o n s a n d p o l a r s i n g u l a r i t i e s - Proc. Lond. Math. Soc., 26 (1927) 1 5 9 - 1 6 8 / 27 (1928) 4 9 7 - 5 1 2 / 28 (1928) 1 2 8 - 1 4 4 . 277 - R. W I L S O N - D i v e r g e n t c o n t i n u e d f r a c t i o n s Proc. Lond. Math. Soc., 30 (1928) 38-57.

and nonpolar

singularities

-

278 - A. W I N T N E R - E i n q u a l i t a t i v e s K r i t e r i u m f i r dis K o n v e r g e n z K e t t e n h r ~ c h e - M a t h . Z e i t . , 30 (1929) 2 8 5 - 2 8 9 .

des a s s o z i i e r t e n

279 - L. W U Y T A C K - E x t r a p o l a t i o n to t h e p o l a t i o n - R o c k y M o u n t a i n s J. M a t h . ,

fraction

limit by using continued 4 (1974) 3 9 5 - 3 9 7 .

inter-

280 - P. W Y N N - C o n t i n u e d f r a c t i o n s w h o s e c o e f f i c i e n t s o b e y a n o n c o m m u t a t i v e of m u l t i p l i c a t i o n - Arch. Rat. M e c h . A n a l . , 12 ( 1 9 6 3 ) 2 7 3 - 3 1 2 . 281 - P. W Y N N - A n o t e on t h e c o n v e r g e n c e of c e r t a i n n o n c o m m u t a t i v e f r a c t i o n s - M R C t e c h n i c a l s u m m a r y r e p o r t 750, M a d i s o n , 1967. 282 - P. W Y N N - V e c t o r

continued

fractions

- Linear

Algehra,

283 - P. W Y N N - U p o n t h e d e f i n i t i o n o f an i n t e g r a l as t h e f r a c t i o n - Arch. Rat. M e c h . A n a l . , 28 (1968) 8 3 - 1 4 8 .

continued

i (1968)

limit

law

357-395.

of a c o n t i n u e d

284 - P. W Y N N - An a r s e n a l of A l g o l p r o c e d u r e s for the e v a l u a t i o n of c o n t i n u e d f r a c t i o n s a n d f o r e f f e c t i n g the e p s i l o n a l g o r i t h m - C h i f f r e s , 9 (1966) 327-362. 285 - P. W Y N N - F o u r l e c t u r e s tions - CIME summer school

on t h e n u m e r i c a l l e c t u r e s , 1965.

286 - P. W Y N N - C o n v e r g i n g f a c t o r s i (1959) 2 7 2 - 3 0 7 a n d 3 0 8 - 3 2 0 .

for

application

continued

fractions

of continued

- Numer.

frac-

Math.,

287 - P. W Y N N - A n o t e on a m e t h o d of B r a d s h a w f o r t r a n s f o r m i n g s l o w l y c o n v e r g e n t s e r i e s a n d c o n t i n u e d f r a c t i o n s - Amer. M a t h . M o n t h l y , 69 (1962) 8 8 3 - 8 8 9 . 288 - P. W Y N N - A n u m e r i c a l 175-196.

s t u d y of a r e s u l t

of S t i e l t j e s

- Chiffres,

6 (1963)

289 - P. W Y N N - On s o m e r e c e n t d e v e l o p m e n t s in the t h e o r y a n d a p p l i c a t i o n c o n t i n u e d f r a c t i o n s - S I A M J. N u m e r . A n a l . , i (1964) 1 7 7 - 1 9 7 , 290 - P. W Y N N - N o t e on a c o n v e r g i n g N u m e r , M a t h . , 5 (1963] 3 3 2 - 3 5 2 .

factor

for a certain

continued

of

fraction

-

-

261

291 - P. W Y N N - The n u m e r i c a l e f f i c i e n c y o f c e r t a i n c o n t i n u e d K o n i n k l . N e d e r l . Akad. Wet., 65 A (1962) 1 2 7 - 1 4 8 . 292 - P. W Y N N - C o m p l e x n u m b e r s on a p p l i c a t i o n to the t h e o r y r e p o r t 646, M a d i s o n , 1966.

fraction

expansions

a n d o t h e r e x t e n s i o n s to the C l i f f o r d a l g e b r a w i t h of c o n t i n u e d f r a c t i o n s , M R C t e c h n i c a l s u m m a r y

llI-

Applications

293 - A.C. A I T K E N - On B e r n o u l l i ' s n u m e r i c a l s o l u t i o n Proc. Roy. Soc. E d i n b u r g , 46 (1926) 2 8 9 - 3 0 5 .

of a l g e b r a i c

294 - A.C. A I T K E N - On i n t e r p o l a t i o n b y p r o p o r t i o n a l parts, d i f f e r e n c e s - Proc. Edin. Math. Soc., 3 (1932) 56-76. 295 - R. A L T - M 6 t h o d e s A - s t a b l e s p o u r l ' i n t 6 g r a t i o n mal conditionn6s - T h ~ s e 3~me c y c l e , P a r i s , 1971.

des

296 - F.L. B A U E R - C o n n e c t i o n s b e t w e e n t h e q - d a l g o r i t h m e - a l g o r i t h m of W y n n - D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t

equations

without

syst~mes

use

of

diff6rentiels

of R u t i s h a u s e r a n d the Teeh. Rep. B a / 1 0 6 , 1957.

297 - F.L. B A U E R - N o n l i n e a r s e q u e n c e t r a n s f o r m a t i o n s - in " A p p r o x i m a t i o n f u n c t i o n s " , G a r a b e d i a n ed., E l s e v i e r , N e w - Y o r k , 1965. 298 - F.L.

BAUER

- The g - a l g o r i t h m

- SIAM

J.,

8 (1960)

-

of

1-17.

300 - M. B A U S S E T - U n e d 6 t e r m i n a t i o n des s u r f a c e s de c h o c p a r a e c 6 1 6 r a t i o n c o n v e r g e n c e - C o l l o q u e E u r o m e e h 58, T o u l o n , 1 2 - 1 4 m a i 1975.

de

301 - L.C. B R E A U X - A n u m e r i c a l s t u d y o f t h e a p p l i c a t i o n of a c c e l e r a t i o n techniq u e s a n d p r e d i c t i o n a l g o r i t h m s to n u m e r i c a l i n t e g r a t i o n - M. Sc. T h e s i s , Louisiana State Univ., New-Orleans, 1971. 302 - C. B R E Z I N S K I - C o n v e r g e n c e d ' u n e f o r m e e o n f l u e n t e C.R. Acad. Sc. Paris, 273 A (1971) 5 8 2 - 5 8 5 .

de l ' E - a l g o r i t h m e

303 - C. B R E Z I N S K I - L ' E - a l g o r i t h m e et les s u i t e s t o t a l e m e n t o s c i l l a n t e s - C.R. Acad. Sc. P a r i s , 276 A (1973) 3 0 5 - 3 0 8 . 304 - C. rique

BREZINSKI - M6thodes d'acc616ration - T h ~ s e , Univ. de G r e n o b l e , 1971.

de la c o n v e r g e n c e

305 - C. B R E Z I N S K I - A p p l i c a t i o n du o - a l g o r i t h m e C.R. Aead. Sc. P a r i s , 270 A (1970) 1 2 5 2 - 1 2 5 3 . 306 - C. B R E Z I N S K I (1971) 1 5 3 - 1 6 2 .

- Etudes

sur les

307 - C. B R E Z I N S K I RIRO, R3 (1970)

- RSsultats 147-153.

get

309 - C. B R E Z I N S K I - R e v i e w of m e t h o d s Rend. Mat. Roma, 7 (1974) 3 0 3 - 3 1 6 .

- Numer.

de s o m m a t i o n

de l ' s - a l g o r i t h m e

to a c c e l e r a t e

the

et

en a n a l y s e

~ la q u a d r a t u r e

p-algorithmes

s u r les p r o c 6 d 6 s

308 - C. B R E Z I N S K I - F o r m e c o n f l u e n t e M a t h . , 23 (1975) 3 6 3 - 3 7 0 .

monotones

-

num6rique

Math.,

num6-

-

17

et l ' c - a l g o r i t h m e

topologique

convergence

-

- Numer.

of s e q u e n c e s -

-

262

310 - C. BREZINSKI - Acc~l~ration de la c o n v e r g e n c e en analyse num~rique P u b l i c a t i o n 41, labo. de Calcul, Univ. de Lille, 1973. 311 - C. BREZINSKI - Conditions d ' a p p l i c a t i o n et de convergence de p r o c ~ d ~ s d ' e x t r a p o l a t i o n - Numer. Math., 20 (1972) 64-79. 312 - C. BREZINSKI - A p p l i c a t i o n de l'e-algorithme ~ la r ~ s o l u t i o n des syst~mes non lin~aires - C.R. Acad. Sc. Paris, 271 A (1970) 1174-1177. 313 - C. BREZINSKI - Integration des syst~mes d i f f ~ r e n t i e l s ~ l'aide du 0-algorithme - C.R. Acad. Sc. Paris, 278 A (1974) 875-878. 314 - C. BREZINSKI - Sur un algorithme de r ~ s o l u t i o n des syst~mes non lin~aires C.R. Acad. Sc. Paris, 272 A (1971) 145-148. 315 - C. BREZINSKI - Numerical stability of a quadratic m e t h o d for solving systems of non linear equations - Computing, 14 (1975) 205-211. 316 - C. BREZINSKI - C o m p u t a t i o n of the eigenelements of a m a t r i x by the e-algorithm - Linear Algebra, ii (1975) 7-20. 317 - C. BREZINSKI, M. CROUZEIX - Remarques sur le proc~d~ A 2 d'Aitken - C.R. Acad. Sc. Paris, 270 A (1970) 896-898. 318 - C. BREZINSKI, A.C. RIEU - The solution of systems of equations using the e-algorithm, and an a p p l i c a t i o n to b o u n d a r y value p r o b l e m s - Math. Comp., 28 (1974) 731-741. 319 - H. CABANNES, M. BAUSSET - A p p l i c a t i o n of the m e t h o d of Pad~ to the determ i n a t i o n of shock wavres - in "Problems of h y d r o d y n a m i c s and continuum mechanics ", in honor of L.I. Sedov, English ed. publ. by SIAM (1968) 95-114. 320 - H.K. CHENG, M. HAFEZ - Convergence a c c e l e r a t i o n of iterative solutions and transonic flow computations - Colloque Euromech 58, Toulon, 12-14 mai 1975. 321 - J.S.R. CHISHOLM - Pad~ a p p r o x i m a n t s and linear integral equations - in "The Pad~ approximant in t h e o r e t i c a l physics", G.A. Baker Jr. and J.L. Gammel eds., Academic Press, New-York, 1970. 322 - J.S.R. CHISHOLM - A p p l i c a t i o n of Pad~ a p p r o x i m a t i o n to n u m e r i c a l integration - Rocky Mountains J. Math., 4 (1974) 159-168. 323 - J.S.R. CHISHOLM - Pad~ a p p r o x i m a t i o n of single v a r i a b l e integrals Colloquium on c o m p u t a t i o n a l methods in t h e o r e t i c a l physics, Marseille, 1970. 324 - J.S.R. CHISHOLM - A c c e l e r a t e d convergence of sequences of quadratur e approximants - second c o l l o q u i u m on c o m p u t a t i o n a l m e t h o d s in t h e o r e t i c a l physics, Marseille, 1971. 325 - J.S.R. CHISHOLM, A.C. GENZ, G.E. ROWLANDS - A c c e l e r a t e d convergence of sequences of quadrature a p p r o x i m a t i o n - J. Comp. Phys., i0 (1972) 284-307. 326 - J. COUNTS, J.E. AKIN - The a p p l i c a t i o n of continued fractions to wave p r o p a g a t i o n problems in a v i s c o e l a s t i c rod - DEMVPI R e s e a r c h Rep. i-i, Dept. of Eng. Mech., V i r g i n i a p o l y t e c h n i c institute, Blacksburg, 1968. 327 - J.D.P. D O N N E L L Y - Applications of the q-d and e-algorithms - in "Methods of numerical approximation", D.C. Handscomb ed., P e r g a m o n Press, Oxford, 1965.

263

328 - B.L. EHLE - A - s t a b l e m e t h o d s and Pad~ a p p r o x i m a t i o n s SIAM J. Math. Anal., 4 (1973) 671-680.

to the e x p o n e n t i a l

-

329 - B.L. EHLE - On Pad~ a p p r o x i m a t i o n s to the e x p o n e n t i a l f u n c t i o n and A - s t a b l e m e t h o d s for the n u m e r i c a l s o l u t i o n of initial value p r o b l e m s - R e s e a r c h rep. CSRR 2010, dept. of AACS, Univ. of Waterloo, Ontario, 1969. 330 - M. F R O I S S A R T - A p p l i c a t i o n s of the Pad~ m e t h o d to n u m e r i c a l analysis C o l l o q u i u m on c o m p u t a t i o n a l m e t h o d s in t h e o r e t i c a l physics, Marseille, 1970. 331 - E. G E K E L E R - U b e r den e - a l g o r i t h m u s 51 (1971) 53-54.

von W y n n - Z. Angew.

332 - E. G E K E L E R - On the s o l u t i o n of systems of e q u a t i o n s r i t h m of Wynn - Math. Comp., 26 (1972) 427-436.

Math.

Mech.,

by the e p s i l o n

algo-

333 - A. GENZ - The e - a l g o r i t h m and some other a p p l i c a t i o n s of Pad~ a p p r o x i m a n t s in n u m e r i c a l a n a l y s i s - in "Pad~ a p p r o x i m a n t s " , P.R. Graves- Morris ed., The institute of physics, London, 1973. 334 - A. G E N Z - A p p l i c a t i o n s of the e - a l g o r i t h m to q u a d r a t u r e p r o b l e m s - in "Pad~ a p p r o x i m a n t s and their a p p l i c a t i o n s ", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 335 - B. G E R M A I N

BONNE - T r a n s f o r m a t i o n s

de suites

- RAIRO,

R1

(1973)

84-90.

336 - M. HAFEZ, H.K. CHENG - On a c c e l e r a t i o n of c o n v e r g e n c e and s h o c k - f i t t i n g t r a n s o n i c f l o w c o m p u t a t i o n s - Univ. So. Calif. Memo., 1973. 337 - M, HAFEZ, H.K. CHENG - C o n v e r g e n c e a c c e l e r a t i o n and shock f i t t i n g t r a n s o n i c a e r o d y n a m i c s c o m p u t a t i o n s - AIAA paper, No. 75-51. 338 - P. HENRICI - Some a p p l i c a t i o n s of the q u o t i e n t - d i f f e r e n c e Symp. Appl. Math., 20 (1963) 159-183. algorithm

- NBS appl.

340 - D . C . ' J O Y C E - Survey of e x t r a p o l a t i o n Rev., 13 (1971) 435-490.

processes

in n u m e r i c a l

341 - D.K. K A H A N E R - N u m e r i c a l (1972) 689-694.

quadrature

by the e - a l g o r i t h m

for

algorithm

339 - P. HENRICI - The q u o t i e n t - d i f f e r e n c e 49 (1958) 23-46.

Math.

- Proc.

series,

analysis

- Math.

in

Comp.,

342 - D. LEVIN - Development of n o n - l i n e a r t r a n s f o r m a t i o n s for i m p r o v i n g gence of sequences - Intern. J. Comp. Math., B3 (1973) 371-388.

- SIAM

26

conver-

343 - I.M. LONGMAN - Use of Pad~ table for a p p r o x i m a t e Laplace t r a n s f o r m inversion - in "Pad~ a p p r o x i m a n t s and their a p p l i c a t i o n s " , P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 344 - E.D. MARTIN, H. LOMAX - Rapid finite d i f f e r e n c e c o m p u t a t i o n and t r a n s o n i c a e r o d y n a m i c flows - AIAA paper, No. 74-11. 345 - I. MARX - Remark c o n c e r n i n g a n o n l i n e a r tion - J. Math. Phys., 42 (1963) 834-335. 346 - K.J.

OVERHOLT

- Extended

sequence

Aitken acceleration

to sequence

- BIT,

5 (1965)

of subsonic

transforma-

122-132.

264

347

- R. P E N N A C C H I - S o m m a di s e r i e n u m e r i c h e t i c a T2, 2 - C a l c o l o , 5 (1968) 51-61.

348

- R. P E N N A C C H I 5 (1968) 37-50.

- La t r a s f o r m a z i o n i

mediante

razionali

la t r a s f o r m a z i o n e

di u n a

349 - A. R O N V E A U X - P a d 6 a p p r o x i m a n t a n d h o m o g r a p h i c p h a s e e q u a t i o n s - in " P a d 6 a p p r o x i m a n t s and thein M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973.

successione

- Calcolo,

transformation applications",

350 - H. R U T I S H A U S E R - On a m o d i f i c a t i o n of the q - d a l g o r i t h m c o n v e r g e n c e - ZAMP, 13 (1962) 4 9 3 - 4 9 6 .

with

quadra-

of Riccati's P.R. G r a v e s -

Graeffe

type

351 - H. R U T I S H A U S E R - B e s t i m m u n g d e r E i g e n w e r t e u n d E i g e n v e k t o r e n einer Matrix m i t H i l f e des Q u o t i e n t e n - D i f f e r e n z e n A l g o r i t h m u s - ZAMP, 6 (1955) 387-401. 352

- H. R U T I S H A U S E R - E i n e F o r m e l y o n W r o n s k i u n d i h r e B e d e n t u n g t i e n t e n - D i f f e r e n z e n A l g o r i t h m u s - ZAMP, 7 (1956) 164-169.

353 - H. R U T I S H A U S E R - S t a b i l e S o n d e r f [ l l e des Q u e t i e n t e n r i t h m u s - Numer. M a t h . , 5 (1963) 95-112. 354 - H. R U T I S H A U S E R 233-251.

- Der Quotienten

355 - H. R U T I S H A U S E R - A n w e n d u n g e n ZAMP, 5 (1954) 496-508. 356 - J.R. S C H M I D T b y an i t e r a t i v e 357

des

Quotienten

- Differenzen

Algorithmus

SCHWARTZ - Series - Phys. Fluids.

solution

structure

- Differenzen

asymmetric

361 - R.E. S H A F E R - On q u a d r a t i c (1974) 447e460.

approximation

-

equations

conical

blunt-body

359 - L.W. S C H W A R T Z - S o l u t i o n s to t h e a s y m m e t r i c b l u n t - b o d y p r o b l e m approximants - C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975. 360 - L.W. S C H W A R T Z - H y p e r s o n i c f l o w s g e n e r a t e d b y p a r a b o l i c s h o c k w a v e s - Phys. F l u i d s , 17 (1974) 1 8 1 6 - 1 8 2 1 .

5 (1954)

Algorithmus

of t h e T a y l o r - M a c c a l l

for the planar

Algo-

- ZAMP,

- On the n u m e r i c a l s o l u t i o n of l i n e a r s i m u l t a n e o u s m e t h o d - Phil. M a g . , 7 (1951) 3 6 9 - 3 8 3 .

- L.W. S C H W A R T Z - On t h e a n a l y t i c f l o w s o l u t i o n - ZAMP.

358 - L.W. problem

- Differenzen

fir den Quo-

using

Pad6

and paraboloidal

- S I A M J. Numer.

Anal.,

ii

362 - D. S H A N K S - An a n a l o g y b e t w e e n t r a n s i e n t s a n d m a t h e m a t i c a l sequences and s o m e n o n l i n e a r s e q u e n c e to s e q u e n c e t r a n s f o r m s s u g g e s t e d by it - N a v a l O r d n a n c e Lab. Mem. 9994, W h i t e Oak, M d . , 1949. 363 - D. S H A N K S - N o n l i n e a r t r a n s f o r m a t i o n s s e r i e s - J. Math. P h y s . , 34 (1955) 1-42. 364 - M.D. V A N D Y K E N e w - Y o r k , 1964.

- Perturbation

365 - M.D. V A N D y K E - A n a l y s i s Mech. Appl. Math.

and

methods

of d i v e r g e n t

in f l u i d

improvement

and slowly

mechanics

of p e r b u r t a t i o n

convergent

- Academic

series

Press,

- Quart.

J.

265

366 - A.H. V A N T U Y L - C a l c u l a t i o n ii (1973) 537-541.

of n o z z l e

using

Pad~

fractions

- AIAA

J.,

367 - A.H. V A N T U Y L - A p p l i c a t i o n of m e t h o d s f o r a c c e l e r a t i o n of c o n v e r g e n c e to the c a l c u l a t i o n of s i n g u l a r i t i e s of t r a n s o n i c f l o w s - C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975. 368

- A.H. V A N T U Y L flows - Colloque

- The u s e of P a d ~ f r a c t i o n s in the c a l c u l a t i o n E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975.

369 - H.S. WALL, W.T. S C O T T - The t r a n s f o r m a t i o n Amer. Math. Soc., 51 (1942) 255-279.

of s e r i e s

and

of n o z z l e

sequences

- Trans.

370 - P.J.S. W A T S O N - A l g o r i t h m s f o r d i f f e r e n t i a t i o n a n d i n t e g r a t i o n - in " P a d s approx~mants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973. 371 - L. W U Y T A C K - The u s e of P a d ~ a p p r o x i m a t i o n C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a l 1975.

in n u m e r i c a l

integration

372 - P. W Y N N - On a c o n n e c t i o n b e t w e e n t h e f i r s t a n d the s e c o n d of the e - a l g o r i t h m - Nieuw. Arch. W i s k . , ii (1963) 19-21. 373 - P. W Y N N - U p o n a s e c o n d c o n f l u e n t Math. Soc., 5 (1962) 160-165.

form

of the

e-algorithm

-

confluent

- Proc.

form

Glasgow

374 - P. W Y N N - A c o m p a r i s o n b e t w e e n t h e n u m e r i c a l p e r f o r m a n c e s of the E u l e r transformation a n d the E - a l g o r i t h m - C h i f f r e s , i (1961) 23-29. 375 - P. W Y N N 19-22.

- On r e p e a t e d

application

o f the

e-algorithm

376 - P. w Y N N - The e p s i l o n a l g o r i t h m a n d o p e r a t i o n a l a n a l y s i s - Math. C o m p . , 15 (1961) 1 5 1 - 1 5 8 . 377

- P. W Y N N - A c c e l e r a t i o n t e c h n i q u e s in n u m e r i c a l r e f e r e n c ~ to p r o b l e m s in one i n d e p e n d a n t v a r i a b l e N o r t h H o l l a n d , (1962) 149-156.

378 - P. W Y N N - A n o t e on p r o g r a m m i n g C h i f f r e s , 8 (1965) 23-62. 379 - P. W Y N N - The r a t i o n a l defined by a power series 380 - P. W Y N N recherches

repeated

formulas

of n u m e r i c a l

analysis with particular - Proc. I F I P C o n g r e s s ,

application

of the

- The a b s t r a c t t h e o r y of t h e e p s i l o n a l g o r i t h m math~matiques n ° 74, Univ. de M o n t r e a l , 1971.

e-algorithm

arrays

- Centre

- Louisiana

State

- P. W Y N N - I n v a r i a n t s a s s o c i a t e d w i t h t h e e p s i l o n a l g o r i t h m a n d c o n f l u e n t f o r m - Rend. Circ. Mat. P a l e r m o , (2) 21 (1972) 31-41.

383 - P. W Y N N 175-195.

- Singular

384 - P. W Y N N

- A sufficient

4 (1961)

-

approximation of f u n c t i o n s w h i c h a r e f o r m a l l y e x p a n s i o n - Math. C o m p . , 14 (1960) 1 4 7 - 1 8 6 .

381 - P. W Y N N - U p o n a h i e r a r c h y of e p s i l o n N e w - O r l e a n s , techn, rep. 46, 1970. 382

- Chiffres,

rules

for certain

condition

nonlinear

for the

algorithms

instability

de

Univ.,

its f i r s t

- BIT,

of the

3 (1963)

e-algorithm

-

266

Nieuw.

Arch.

Wisk.,

3 (1961)

117-119.

385 - P. WYNN - On the p r o p a g a t i o n Numer. Math., i (1959) 142-149,

of error in certain n o n l i n e a r

386 - P. WYNN - On the c o n v e r g e n c e and s t a b i l i t y SIAM J. Numer. Anal., 3 (1966) 91-122.

of the e p s i l o n

algorithms

algorithm

-

-

387 - P. W Y N N - H i e r a r c h i e s of arrays and f u n c t i o n sequences a s s o c i a t e d w i t h the epsilon a l g o r i t h m and its first c o n f l u e n t f o r m - Rend. Mat. Roma, 5 (1972) 819-852. 388 - P. WYNN - A c c 6 1 6 r a t i o n de la c o n v e r g e n c e de s6ries d ' o p 6 r a t e u r s n u m 6 r i q u e - C.R. Acad. Se. Paris, 276 A (1973) 803-806. 389 - P. W Y N N - T r a n s f o r m a t i o n s de s6ries Sc. Paris, 275 A (1972) 1351-1353. 390 - P. WYNN - U p o n some c o n t i n u o u s 197-234 and 235-278. 391 - P. WYNN - Confluent ii (1960) 223-234.

forms

~ l'aide de l ' s - a l g o r i t h m e

prediction

of certain

392 - P. W Y N N - A note on a confluent (1960) 237-240.

en analyse

algorithms

nonlinear

- C.R. Aead.

- Calcolo,

algorithms

f o r m of the s - a l g o r i t h m

9 (1972)

- Arch.

- Arch.

Math.,

Math.,

ii

393 - P. WYNN - On a p r o c r u s t e a n t e c h n i q u e for the n u m e r i c a l t r a n s f o r m a t i o n of slowly convergent sequences and series - Proc. Camb. Phil. Soc., 52 (1956) 663-671. 394 - P. W Y N N - T r a n s f o r m a t i o n s to accelerate the c o n v e r g e n c e of F o u r i e r series G e r t r u d e Blanch a n n i v e r s a r y vol., Wright P a t t e r s o n Air Force Base, (1967) 339-379. 395 - P. WYNN - P a r t i a l d i f f e r e n t i a l equations algorithms - ZAMP, 15 (1964) 273-289.

associated

with certain nonlinear

396 - P. W Y N N - A n u m e r i c a l m e t h o d for e s t i m a t i n g p a r a m e t e r s in m a t h e m a t i c a l m o d e l s - Centre de r e c h e r c h e s m a t h 6 m a t i q u e s , Univ. de Montr6al, rep. CRM 443, 1974. 397 - P. WYNN - On a device i0 (1956) 91-96.

for c o m p u t i n g

the em(S n) t r a n s f o r m a t i o n

398 - P. W Y N N - A c o n v e r g e n c e t h e o r y of some m e t h o d s r e c h e r c h e s m a t h ~ m a t i q u e s , Univ. de Montr6al, rep. 399 - P. W Y N N - A sufficient c o n d i t i o n Numer. Math., i (1959) 203-207.

of i n t e g r a t i o n CRM-193, 1972.

for the i n s t a b i l i t y

400 - P. W Y N N - A c o m p a r i s o n t e c h n i q u e for the n u m e r i c a l slowly c o n v e r g e n t series b a s e d on the use of r a t i o n a l Math., 4 (1962) 8-14.

- MTAC,

- Centre de

of the q-d a l g o r i t h m

t r a n s f o r m a t i o n of f u n c t i o n s - Numer.

-

267

IV - M I S C E L L A N E O U S

401 ~ G.D. A L L E N , C.K. CHUI, W.R. M A D Y C H , Pad~ approximation of S t i e l t j e s s e r i e s

F.J. N A R C O W I C H , P.W. S M I T H - J. Appr. T h e o r y , 14 (1975) 3 0 2 - 3 1 6 .

402 - G.D. A L L E N , C.K. CHUI, W.R. M A D Y C H , F.J. N A R C O W I C H , P.W. S M I T H Pad~ approximant~, N u t t a l l ' s f o r m u l a a n d a m a x i m u m p r i n c i p l e - to a p p e a r . 403 - M.F. B A R N S L E Y - C o r r e c t i o n 16 (1975) 9 1 8 - 9 2 8 4 0 4 - C. B R E Z I N S K I to appear.

- Matrices

terms

for Pad~

semi-d~finies

approximants

positives

et s u i t e s

405 - F. C O R D E L L I E R - U n e n o u v e l l e f o r m e de l ' e - a l g o r i t h m e C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975. 406 - F. C O R D E L L I E R - R ~ g l e s p a r t i c u l i ~ r e s C o l l o q u e d ' A n a l y s e N u m ~ r i q u e , La G r a n d e

- J.

de m o m e n t s

~tape

vectoriel

407

- F. Publ.

408

~ F. C O R D E L L I E R - L ' e - a l g o r i t h m e vectoriel : interpretation r~gles singuli~res - Colloque d'analyse num~rique, Gourette,

409

- A.C. G E N Z - The a p p r o x i m a t e extrapolation methods - Ph.D.

- S.T. PENt, S I A M J. Math.

412 - J. T O D D

lemniscate

-

g~om~trique m a i 1974.

-

et

c a l c u l a t i o n of m u l t i d i m e n s i o n a l integrals using T h e s i s , Univ. of K e n t at C a n t e r b u r y , 1975.

A. H E S S E L - C o n v e r g e n c e o f n o n c o m m u t a t i v e A n a l . , 6 (1975) 7 2 4 - 7 2 7 .

- The

-

de l ' c - a l g o r i t h m e

410 - P. H I L L I O N - M ~ t h o d e d ' A i t k e n i t ~ r ~ e p o u r les s u i t e s o s c i l l a n t e s x i m a t i o n s - C.R. Acad. Sc. P a r i s , 280 A (1975) 1 7 0 1 - 1 7 0 4 . 411

Phys.,

et ses v a r i a n t e s

pour l'c-algorithme M o t t e , m a i 1975.

CORDELLIER - Interpretation g~om~trique d'une 40, Labo. de C a l c u l , Univ. de L i l l e , 1973.

Math.

constants

- Comm.

ACM,

18

continued

(1975)

d'appro-

fractions

14-19.

UER d'IEEA - Informatique BP 36 59650 - Villeneuve d'Ascq FRANCE

-