A general algorithm for pade approximation to formal power series

A General Algorithm For Pade

Approximation To Formal Power Series

S.

L. Loi

No. 22

April 1982

A GENERAL ALGORITHM FOR PADE APPROXIMATION

TO FORMAL POWER SERIES.

1 INTRODUCTION

A general algorithm for constructing an interpolating

function from a linear family of functions forming a

Chebyshev system has been given by Brezinski in [l].

This algorithm was extended to generalized rational

interpolation [7] and is applied to the Pade-type

approximants for the series of functions in [l]. A

question arises as to whether this algorithm could be

extended to generalize the Pade approximation to a formal

power series f(x). Before the discussion, we recall some

.basic definitions.

The Pade approximant (m,n;x) to a function f(x)

which can be expressed as a formal power series

is the ratio of Q(m,n;x)/P(m,n;x), where (m,n;x) agrees

with the power series as far as possible. The

denominator P(m,n;x) and the numerator Q(m,n;x) are

polynomials of degree m and n respectively. These

polynomials P(m,n;x) and Q(m,n;x) are determined from

00

k

m+n+l

P(m,n;x) k ckx - Q(m,n;x) = O(x

)

k=O

by neglecting higher order terms. They can be expressed

respectively as the quotient of a determinant of order

m+l, and its minor whose elements are formed from the Ci'

n

P.o (x)

Q(m,

n

;x) =

D

_m,

_n

i

i

k

_c.

₀

_{if i}

<

₀

_a

_n

_d

_D

_{is the}

� c

k

x

,

=

where PO (x) =

J.

m,

n

k

=O

mi

n

or obtai

n

ed by elimi

n

ati

n

g the first row a

n

d the last

column of the

n

umerator.

P(m,n;x) is obtai

n

ed by replaci

n

g i

P0 (x) i

n

( 1.1) by 1 V

The Pade table of f (x) is a doubly indexed array of the

rational fu

n

ctio

n

(m,n; x). The array is followi

n

g

(O,O)

( 1, 0)

( 2, 0)

. . .

(�, 0)

(0,1)

( 1, 1)

(2,1)

. . .

.

. . . .

(0,2)

(1,2)

( 2, 2)

G e O 8 e

..

•

. . .

·-( 0, n)

The partial sums of the power series

� C

_k

x occupy the

k

k=O

first column of the table.

A Pade table is

n

ormal if

n

o

n

e of the (rn,

n

;x) is

n

il or

equal to each other. I

n

other words, the power series

f(x) is

n

ormal if, for each pair (m,

n

), (m,

n

;x) agrees

m+

n

exactly through the power x

and

n

o

n

e of the D

_m,n

= 0.

I

n

particular each coefficie

n

t c. must

n

ot va

n

ish. The

00 J.

power series � C

_k

x

k

_{is semi}

n

ormal if D

= 0 for

k

=O

m,

n

are isolated by the

n

o

n

-

n

il eleme

n

ts. Otherwise, the

series is called no

n

-

n

ormal. More detailed defi

n

itio

n

s,

properties a

n

d theorems are give

n

i

n

[6].

Differe

n

t algorithms for co

n

structi

n

g the eleme

n

ts

of the Pade table of a power series for the

n

ormal case

are fou

n

d i

n

[4]. Some of these algorithms have bee

n

modified, i

n

particular situatio

n

s for the

n

o

n

-

n

ormal

case [2] , [3], [5] .

I

n

this,paper ,the algorithm for i

n

terpolatio

n

give

n

by Brezi

n

ski i

n

[l] is exte

n

ded to compute the elements of

the Pade table of a power series i

n

the

n

ormal case.

This algorithm is used to compute separately, colum

n

by

colum

n

, the de

n

omi

n

ator P(m,

n

;x) a

n

d the

n

umerator

Q(m,n;x) of the Pade approximant of f(x). I

n

the

n

ext

sectio

n

, how this algorithm ca

n

be applied to (1.1) will

be show

n

. This algorithm is exte

n

ded to the seminormal

case in Sectio

n

3. This exte

n

sio

n

is similar to the

method used to solve the si

n

gular case i

n

1[7]. I

n

the

last sectio

n

some comme

n

ts are made about the

n

o

n

-

n

ormal

case.

2

THE ALGORITHM

The followi

n

g algorithm is based o

n

the MNA

algorithm i

n

[l]. It is co

n

ve

n

ie

n

t to cha

n

ge some of the

n

otatio

n

a

n

d i

n

itializatio

n

. P

_m

n

(x) is used to compute

P(m,

n

;x) a

n

d Q(m,

n

;x) separately by the same algorithm ..

Similarly g

_m,

n ₁

. (x) is used to compute Pg

_m,

n

. (x) a

n

d

1

3.

Qg

n

. (x) which are used for computi

n

_{g P(m,}

n

; x) a

n

d Q(m,

n

;x)

m,1

respectively.

Step 1.

Initializatio

n

For j

=

0 ,

P{

(x)

=

1

1

. . .

n

xPi-l (x)

c.

_J

c.

For i

=

2, 3 ••• m

j

g.l .

-,1 (x)

=

where P

j

_{(x) =}

0

=

j-1

XPo

c.

_J

O

j

<

0

Pi

(x)

cj+l

for computi

n

_{g the}

n

umerator P(m,

n

;x)

a

n

d Pi(x) = 1, V j

for computing the de

n

omi

n

ator Q(m,

n

;x)

Step 2.

_{For k}

=

_{2, 3}

_m·

'

Compute

Pt

(x)

If k

91_{.r..' 1} ;, j · ( _x)

j

=

O, l

n

-1

=

_{.6.pi-1 (x)}

gi-1,k(x)-.6.g�-1,k (x)

=

m stop otherwise

= .6.g f-1 , i (

x)

.6.g�-1,k(x)

g�-1,k(x)

Pi-1 (x)

j _{. )}

gk-1, i (X

i = k+l, ... m

( 2 .1)

. "( 2.2)

P� (x)

has the form (1.1) if k' j

and

xk_{Ptk (x) x}k-lpj-k+l (x)

0

cj-k+l cj-k+2

.

.,.

,,

.

..

. . .

is

• •

replaced by

. 1

xPJ- (x)₀

c.

. .

J

• •

m,n

xi_Pj-i_(x)

0-c. '+l_J-1

g:j . (x) = c.

1 cj " !..• 11;1 •_.,.�• • o •_• • cj+k-2 c. _k ' 1

k,1 _J- J+

-J.

-Dk 'J

The way of computing gk

j . (x) in (2.2) can be generalized

'1

as follows.

Instead of using gk . (x) for k= 2,3 ... m,

j

_,1

we initialize

k j-k

x Po

x

k-1 j-k+l

_Po

Hg1-k ,k (x)

=

cj-k+l cj-k+2

=

j

k-2, k-1, ... n.

cj-k+2

k j-k

x PO

x

k-1 j-k+l

_PO

j

cj-k+l cj-k+2

Vg2-k ,k

(x)

=

:i

= k-1,k ... n

cj-k+l

For

9,

=

3 , 4 •••

i = 0,1,2 ...

k = i+t ... m

For j = i+l, ... n

Compute Hg�i,k(x)

= b.H g � ( i + 1 ) , k ( x)

6Vg�i,k-l(x)

j

Vg_i,k-l(x)

- Hg�(i+l) ,k(x)

j

Vg . k(x)

-1. '

= 6Vg�i,k-l(x)

�-,--

---

���- Hg�(i+l) ,k(x)

b.Hg� (i+l) ,k (x)

- Vg�i,k-l(x)

For k

=

2, 3 •••

m

j

=

0, 1 ... n-1

Compute P�(x) 6pi_l

=

(x)

Hg6 ,k (x)

-

Pk-l(x)

j

6Hg6,k(x)

( 2. 4)

( 2. 5)

( 2 • 6)

Note: Vg_i,k(x)

=

_{0 and Hg�i k(x)}

=

_{0 when i � k}

'

Example: Pade approximants to ex. n 0 1 2 3 4 5 n

Pl (x) _(ie.P(l,n;x) Q{I,n;xl )

n 0 1 2 3 4· 5

Q n ( n

Pgl,2 x) (Hgo,2<x)l Q Pg2,3(x) (Hg0,3(x}) n n Q Pg3,4(x) (Hg0,4(x)) n n

0 0 0

...

-

... .

2 3 4

x x x

-x 2x -6x

...

-x+x_{2 - . - - - · · --}2 2x-2x_----2+x3 _{2 - - · -} -6x+6x2-3x_-z 3+x4

·-2x-x 6x+2x -24x -6x

-2x+x2 6x-4x2+x3 -24x+1Bx2-6x3 +x4

~ 3 2-3 i - - 1

-3x+2x -h 12x+6x +x · -60x-24x. -3x

...

· 2 2 3

-3x+x 12x-6x +x -60x-36x2-9x3+x4

~ l 2 3,.L.4

-4x-3x -x -.x 20x+l2x +3x..,..,. •

...

•·

... ..

-4x+rlx2

TJ-.1....4~5 -Sx-4x

-t

-,A

-i.,-x

20x-Bx2+x3

... 0.0•••••••0•0•0•••···.··· ... ···

-5x+x2

n

P2 (x) Pn

3 (x)

n

P 4 (x)

1 1 1 1

...

2 2 L.3 2 3 4

1-x 1-x++x 1-x+-h,:

-=-~

1-x+tx-f-x +2h

l+h l+tx 1+¢-x l+"t-..<:

···2···

·:;1····~···1···4···

1-t-x 1-h+tx 1-fx+h -.ix l-nc+nx 2-rtx +rrtx

l+tx+fx2 l+tx+.+x2 l+-fx+:rhc2 l~+.-tX2

···i:-hc···i:-hc~~

2

···i:-bc~~2:~

3

···i:i;~~1:~~~~

4 •••·

1+h+h2

+zihc

3

1+-bc+,:t,-2+-rtx

3 1+h+m2

+rrtx

3

... ····2· ... ·2· .. ··3··· ... .

1-tx 1-tx+:rhc 1-h:+rkx -rrfx · .

2 - 3 ~ - 4 -2 3 4

1-rh+ 1 h + rlx + n:+x 1 +h+tx + oh: +-:i-.-}x

••••••.••••••••••••• ~ ••••••••••••••••••••••••.•••• 2 •••••••••••••••••••••••••••••••••••••••••••••••••••••••.

1--}x 1-+x+rtx · •

1+

_{... i:h ... · ..}

h+tx2 +

rhc

1 + ,-tx 4~ ~5

0 • . • • • • • • • • 0 . . . .

. 3 SEMI-NORMAL CASE

The above algorithm may break down if C. = 0 in

l

(2.1) or 6gt-l,k(x)

=

0 in (2.2�2·.3) or 6Hg�i,k(x)

(or 6Vg

j

. k(x))

=

O in (2.4-2 .6). For the semi-normal

-1,

case, the extension of (2.4-2.6), allows the recursive

algorithm to carry on.

If 6Vg�i,k-l(x)

=

0 and 6Hg!i,k-l (x)

=

O (refer to

corollary 4 in [7])) in (2.4), Hg�i,k(x), Vg�(i-l),k-l(x)

and P�_1(x) cannot be computed. We have to

j

ump a step

j

-1

j j

-1

to compute Hg_ (i-l) ,k (x), Hg_ (i-l) ,k (x), Vg_ (i-l) ,k (x),

Vg�(i-1) ,k(x),

pi-

1(x) and Pa(x) by the following

a)

j

-1

Hg_ (i-1) ,k (x)

=

_{6'g·Vg_(i-1) ,k-l(x)-}

j

-1

j

-1

Hg_i,k(x)

j

-1

Vg_ (i-1) ,k (x)

=

Hg� (i-1) ,k (x)

=

j

Vg_(i-1) ,k(x)

=

1

. 1

Hg

J

-;-

(x)-6'g -1,k

6g

v

j

+l

g_ (i-1) ,k-1

1

j

+l

Hg_i,k(x)

6g

j

-1

Vg_ (i-1) ,k-1 (x)

'+l

( x)

-

Hg

-1,

J

. k(x)

j+l

_{ex) .}

₌

Vg_ (i-1) ,k-1 , 1

where 6g

=

6

1

_g

=

cj-k+l

c

j

-k+3

for k

= i+3.

6g

=

-61. 62. 63 for k

=

i+4, i+S,

. . .

v

j

+l

g-1,k-2 (x)

where 61

=

6Vg� (i-1) ,k-2 (x)

1, 2 ...

6H g � (

i

_{+ 1 ) , k ( x) 6Hg� (i+l) ,k-1 (x)}

6H g � ( i + 1) , k-1 ( x)

'+1

62

=

H

_{g_ (i+l) ,k-1. ,}

j

+l

CXJ

I

6vg!i ,k-2 (x)

6v

j

+l

_�><>

'+2

6

Hg� ( i+l) ,k-1 (x)

·+2

6

_{Vg�i ,k-2 (x)}

'+l

6

_{Vg�i,k-l(x)}

'+2

Hg�(i+l)k-l(x)

and by the same method, we have

6

'g = -

6

_{1 .}

6

_{2 . 63}

j-1

6H g -( i + 1 ) , k -1 ( x)

6

_{Vg_i,k-l (x)}

j-1

v j+2

_g-i,k-2(x)

6

_{Hg_i,k-l(x)}

j+l

Vg�i,k-2(x)

j-1

6

_{Vg_i,k-2(x)}

J '

-Hg_(i+l) ,k-1 ( ) 6

x

Hg_i,k-1 (x)

j-1

b) Pi-l(x)

Pi (x)

where

6

_p

6

_p

and

6

'p

I

where

6

₁

A more

-·

=

=

=

=

=

6 I

j-1

j-1

p HgO,k(x) - Pk-1 (x)

6

p

_HgO,k(x)

j+ 1

Pi�i (x)

6

_p

'

=

cj+l for k

=

2.

c. 1

J--

6

'

1

6

_{2 ·43}

-

6

₁

'

1

6

_{2 .}

6

₃

6

Pf_2(x)

'+l

fork

=

3,4

Vg6,k-2(x)

. . .

detailed theoretical derivation

[ 7] . The idea of this extension is similar

of

[ 6]

is given in

to Theorem 7.4

4

NON-NORMAL CASE

For the non-normal case, the nil elements appear in

the Pade table more than once. Hence it will be more

complicated to generalize. An algorithm and some

techniques for computing the non-normal staircase Pade

table can be found in [3], [5]. For the more general

scheme, perhaps we can apply the general Neville-Aitken

algorithm

p

a+m (x)

in [l]

Pj _m (x)

gtm+l {x)

gtm+k (x) 1

gt�+l (x)

j _{( '} gm,m+k x.

That is

. . . .

pt+k (x) Pi {x) pa+m(x)

'+k

g�,m+l {x) gtk+l (x) gk,k+l j+m (x)

. . . .

'+k

g�,m+k{x)

₌

ga,k+m(x) gk,k+m(x)j+m

1

1

. . .

j+m 1 I '+k

gj·,k+l {x)

g�,m+l (x) gk,k+l (x) I

'+k

g�,m+k{x) gf,k+m(x) gk,k+mj+m (x)

This relation holds for gk

j

+ . (x) if we replace the first

m,

J.

row in the numerators by (g

j

_{. (x)}

m,

J.

g

m, J

'+k

J.

.

(x) )

and

( gk

j

_.

_(x)

_{... gk . (xJ;respect1ve y where ,m = , ...}

j

_{+m .,}

_{· 1}

_k

_{1 2}

.

'

].

'

].

It

can be used in the semi-normal case when m = 2 in the

second relation, but for m> 2, we have to evaluate a

[image:11.595.73.537.297.741.2]

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Math.

6 ( 19 8 0) 2 5 9-2 6 6.

3. A. BULTHEEL, Recursive algorithms for non-normal

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SIAM Appl. Math.,

39 (1980) 106-118.

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J.

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Appl. Math

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(1972) 1-62.

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