Chapter III: Rational Interpolants

127

CHAPTER mr Ratlonal Interpolants f l. Notations and definitions

.

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

129

52 . Fundamental properties . . . . . . . . . . . . . . . . . . . . 131 2.1. Properties of t h e rational interpolant . . . . . . . . . . . . 131 2.2. T h e table of rational interpolants . . . . . . . . . . . . . 132 2.3. Normality . . . . . . . . . . . . . . . . . . . . . . . 135 $ 3.Methods t o compute rational interpolants . . . 3.1. Interpolating continued fractions . . . . 3.2. Inverse differences . . . . . . . . . . . . 3.3. Reciprocal differences . . . . . . . . . 3.4. A generalization of the qd-algorithm . . . 3.5. A generalization of the algorithm of Gragg 3.6. The generalized &-algorithm . . . . . . . 3.7. Stoer’s recursive method . . . . . . . . 54 . Rational Hermite interpolation

4.1. 4.2. 4.3. 4.4. 4.5.

. . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

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

. 138 . 138 139

. 143 . 146

. . . . . . . . 149 . . . . . . . . 151 . . . . . . . . 153

. . . . . . . . . . . . . . . . . 156 . . . . . . . . . 156

Definition of rational llermite interpolants The table of rational Ilermite interpolants Determinant representation . . . . . . . Continued fraction representation . . . . Thiele’s continued fraction expansion . .

. . . . . . . . . 159

. . . . . . . . . 162

. . . . . . . . . 163

. . . . . . . . . 164

85 . Convergence of rational Hermite interpolants . . . . . . . . . . . 187 $ 6. Multivariate rational interpolants . . . . . . . . . . . . . . . . 169 6.2. Interpolating branched continued fractions . . . . . . . . . 169 6.2. General order Newton-Pad6 approximants . . . . . . . . . 175

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

188

Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

190

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

192

References

128

Die Lagrangesche Interpolationsformel, welche dazu dient, eine Rekhe v o n n W e r t h e n durch eine ganze F u n k t i o n { n - l l t e n Grades dnrzustellen, i s t v o n Cauchy durch eine Formel verallgemeinert morden, welche eine Rekhe v o n n f m W e r t h e n durch eine gebrochne Funktkon dnrstellt, deren Zahler u n d N e n n e r respektkve von (n-l)ten u n d mten Grade & z d . Mun k a n n die Formel dadurch deduciren, dass m a n die lineuren Gleichungen, v o n uielchen die Aujgabe ubhungt, uuflost, u n d die D e t e r m i n a n t e n , welche m a n fGr d e n Zahler u n d N e n n e r jindet, entwickelt. ”

C . JACOBI - “Uber die Darstellung ekner Rekhe yeyebner W e r t h e durch eine gebrochne rationale Funktion” (1845).

III.1. Notc~tionoand definitaorzs

129

$1. Notations and deflnitiona. Consider a function f defined on a subset G of the complex plane. Let { Z i } ; , - N be a sequence of different points belonging t o G. We still denote the exact degree of a polynomial p by a p and its order by w p . The rational interpolation n) for f cousists in finding polynomials problem of order ( m , m

and

n

with p ( z ) / q ( z )irreducible and such t h a t

e' = 0, . . m + n .)

(3.1.)

Instead of solving problem (3.1.) we consider the linear system of equations f ( z i ) q ( z i -) p ( z i )

=x

0

i=O

,...,m + n

(3.2.)

Condition (3.2.) is a homogeneous system of n-t n -+ 1 linear equatims in the + 2 unknown coefficients a; and 6; of p and q. Hence the system (3.2.) always has a t least one nontrivial solution. For different solutions of (3.2.) the following equivalence can be proved. m -t n

Theorem 3.1.

If the polynomials p i ,

q1 and p 2 , q 2 both satisfy (3.2.) then pIq2 = p z q l .

Proof For the polynomial p l q 2 - p 2 q l we can write (PI42

- P 2 q l ) f Z i ) = [(fq:!- p 2 h - ( f m - p1)q2](z;) = 0

i = 0 , . . .)rn + n

130

Since a ( p 1 ~ p2q1) m n zeros.

+

111.1.Notations and definition8

5 m +n

it must vanish identically for it has more than

I

Not all solutions of (3.2.) also satisfy (3.1.): it is very well possible t h a t the polynomials p and q satisfying (3.2.) are such t h a t p / q is reducible. Nevertheless, because of theorem 3.1., all solutions of (3.2.) have the same irreducible form. For p and q satisfying (3.2.) we shall denote by

the irreducible form of p/q, where qo(z) is normalized such that qO(x0) = 1, and we shall call tm,n(z)the rational interpolant of order ( m , n )for 1. The following result is a consequence of theorem 3.1.

Theorem 3.2. For every nonnegative m and n a unique rational interpolant of order (m, n) for j exists. Although the terminology "interpolant" is used i t may be t h a t r,,n(z) does not satisfy the interpolation conditions (3.1.) anymore [14]. A simple exampie will illustrate this. Let $0 = 0, z1 = 1, z2 = 2 and f(zo) = 0, f ( q )= 3, f(z2) = 3. Take m = n = 1. Then t h e system of interpolation conditions is

=0 3(bo 6 1 1 - (UO ~ 1 = ) 0 3(60 t 261) - (UO i-2 a l ) == O a0

+

+

A solution is p(s) = 32 and q ( z ) = 2. Thus p o ( z ) = 3 and qo(z) = 1 . Clearly

Note the similarity with the Pad6 approximation problem: the Pad6 approximant of order (m, n) did not necessarily satisfy condition (2.2.) anymore. We shall see t h a t many properties valid for Pad6 approximants can be generalized for rational interpolants [20].

III.2.I. Propertie8

of the rational interpolant

1.31

52. Fundamental properties. 2.1. Properties of the rational interpolant.

Let rm,n= p o / q o be the rational interpolant of order ( m , n ) for f . If po and yo do not satisfy the system of conditions (3.2.) themselves, it is easy t o construct polynomials p and y from po i L n d qo t ha t are a solution of (3.2.). Denote the exact degree of PO by m' and thc cxacl. degree of qo by n’.

Theorem 3.3.

If the rational interpolant of ordcr (rn,n) for f is

then a n integer a exists with 0 exist belonging t o { x u , . . ., + z},

<: H 5 min(m - m', rz - n’) and 8 points yl, . . .,y, s u c h t ha t

and

satisfy (3.2.).

Proof Let pl(z) and q ] ( z ) be a solution of ( 3 . 2 . ) . Then

(f q1)(2;) = p1 (z;)

i = 0 , . . ., m

+n

Hence if yl(zi) = 0 also p~(zi)= 0. Let {yl,. . .,ye} be the set of zeros of q1(z) belonging t o {Q,. .., z ~ + ~ We } . construct t(Z)

If

=

n;-,

(2 -?A)

132

III.2.2. The table of rational interpolanta

then a polynomial u exists with

Consequently 8

= at

5 min(m - m’,n - n’)

Since

we have

If we put p(z) = p o ( z ) t ( z ) and q ( z ) = qo(z) t ( z ) then p and q also satisfy (3.2.). I

As a conclusion we can say th at the rational interpolation problem (3.1.) has a solution if and only if p o ( z ) and qo(z) satisfy the system of equations (3.2.). 2.2. The table of rational interpolanta.

The rational interpolants of order (m,n) for f can again be ordered in a table: r0,o

r0,1

r0,2

ri ,o

71,l

r1,2

r2,o

r2,1

...

r3,0

r3,l

.. .

...

In the f i s t column one finds the polynomial interpolants for f and in the first row the inverses of the polynomial interpolants for j. By theorem 3.3. a t least

111.2.2. The tuble of rational interpolanta

m'

+ n' + t + 1 points ( 2 0 , . . .,

with t

~ ~ t + ~ t + t }

that

2 0 exist in ( 2 0 , .. ., z,+n} such

i = 0 , . . .,m'

rm,n(z;)= f(z;)

133

+ n’ + t

On the basis of this conclusion a property comparable with the block structure of the Pad4 table can be formulatcd. Theorem 3.4. Let

and let (a)

(f qo - p o ) ( z ; ) = 0

i = 0, , . ., m' 1- n’

(b)

(f qo - p ~ ) ( z i#) 0

i = m'

(c)

for k and t satisfying m'

(d)

m
then

5

+t

+ n’ + t + 1, . . .,m +

5 k 5 m’ + t and nr 5 !? 5

n'

+ t:

n'+t

Proof For k and t satisfying m' R = min(k - m'> !?- n’) and

5k5

mr

+t

and n’

5 t 5 n’

-t t we define

For p ( z ) = p o ( z ) t ( z ) and q ( z ) = qo(z) t ( z ) we know from (a) that

Since k

+ ! 5 m' + n' + t +

8

we have

(f q - p ) ( z ; )= 0 i = 0, . . ., k + 1

134

111.2.2. The table of rational interpolants

and thus

because

Since p o / q o is the irreducible form of p / q , where p and q satisfy (jq-p)(z;)=O

i = O ,...,m+ n

it is possible, according t o theorem 3.3., t o find an integer 0 5 R 5 mdn(rn- m , n - n ) such t h a t

The upper bound on 8 implies the upper bounds m respectively. This finishes the proof.

8

with

+ t and n + t on m and n I

Remark that condition (b) in theorem 3.4. is only necessary t o assure t h a t the integer t in (a) is as large as possible. It is important t o emphasize t h a t the table of rational interpolants has the block structure described above, only when the interpolation points (20,.. ., + )z, are ordered in such a fashion t h a t the points xi in which (I4 - P ) ( Z i ) # 0 are the last points in the interpolation set. For chosen m and n such a numbering can always be arranged and then we know that all the rational interpolants lying in a square emanating rrom rrnf,=!and with rmt+t,n~+tas its furthermost corner are equal to p o / q o . Let us take a look at the following example. Let 2; = i for i = 0 , . . . , 5 with j ( z 0 ) = I, f ( x 1 ) = 1, f(x2) = 5 / 3 , j ( z 3 ) = 5/2, f(z4) = 17/5, f(z.5) = 13/5. Then

III.2,S.Normality

Po(4

r3,2(x) = -~~

4015)

-

135

1 + x2 1+ z

and (f qo - pa)(x;) = 0 for d = 0 , . . ., 4, while (f qo - po)(xs) # 0. This implies that r2,1 = r2,2 = r3,l = r3,2. If t h e interpolation points are not numbered such t h a t the conditions of theorem 8.4. are satisfied then it is possible for 2 dements in t h e table of rational interpolants t o be equal without having a square of equal elements. A disturbance in the numbering of the interpolation points in theorem 3.4. implies a disturbance in the block-structure. Let us illlistrate this. For 2; = i when i = 0 , . . . , 3 arid f ( x 0 ) = 2, f ( x 1 ) = 312, f ( ~ 2 )= 4/5, f(z3) = 112 we find

q , o ( z ) = r2,1(x) = 2 - 2 X

but r?,o(z)

=2 -

2 x - --x 1 5 10

2

and

One can also check t h a t the rational interpolation problem of order (k,!) for f as formulated in 13.1.) has a solut,iou if the integers k and f are such that

and if the conditions of theorem 3.4. are satisfied. 2.3. Normality. Again we call an entry of the table normal if it occurs only once in t h a t table. A necessary condition for the normality of the rational interpolant rm,n(x) is formulated in the following theorem.

138

KII.2.9. No rmaltty

Theorem 3.5.

If rm,,, = @

Qo

is normal and if (f qo - p~)(z;)= 0 for z' = 0 , . . ., m'

+ n’, then

m = m' and n = n’ ( j q o -PO)(%;)

f 0 for i = m + n + 1, m + n + 2

(IQo - po)(z;)

= o for i = 0,.. .,m l +

we know that rm,n= r m r , n i . Hence, if rm,n is normal, we have m = m' and n = n’. Now suppose that (f qo - p ~ ) ( z , + ~ ~ +=~ 0. ) Then

This contradicts the normality of r,,,+. If (I90 - ~ 0 ) ( ~ m + n + 2= ) 0 then

satisfy

(1q - p)(z;) = 0

for i = 0 , . , ., m

+n+2

Thus rm," = rm+l,n+l which is again a contradiction with the normality of rm,n-

I

111.2.9. Normality

137

Conclusion (b) in theorem 3.5. does not imply tha t ( f q o - p o ) ( z i ) f 0 for i

2m

+n + 1

That the conditions (a) and (b) are not sufficient to guarantee normality of r m , n ( z ) is illustrated as follows. I d z; = i for i = 0 , 1 , 2 , . . . and f(z0) = 0, f(z1) = I, f(z2) = 3, f(z3) = 4, f(zi) = i for i = 4 , 5 , 6 , .. .. For rn = 1 and n = 0 we find r,,+(z) = z with (a) and (t,) in theorem 3.5. satisfied. But rnL,n is not normal because r m , n = r k , t for k 2 3 and e 2 2. However, it is possible t o formulate a sufficient condition for the normality of rm.=.

Theorem 3.6.

If rm,n = YO

with m = m', n = n' and (fq0 - po)(z;) = 0 for a t most m + n + 1 points from the sequence {s;}~,]N, then T ~ is normal. , ~ Proof Let us suppose that rm,nis not normal, in other words that rm,n = r h , t for k >_ m, t? 2 n and with k t- !. > m + n. According t o theorem 3.3. an integer 8 exists with 0 5 8 5 min(k - m, 1 - ti) and a points {yl, . . . , y e } exist such that, the polynomials

and

satisfy

Hence ( f q o - p o ) ( Z i ) = 0 for a t least k + t? + 1 - 8 points in (z;)~&. Since 6 is bounded above by k - m and t? - n, we conclude th a t k + L + 1 - 8 > m + n 1 which contradicts the fact th at ( j q o - p o ) ( z ; ) = 0 for at most rn + n + 1 points from {zi);Em I

+

III.3.1. Interpolating continued fractions

138

$3. Methods to compute rational interpolants.

In the sequel of this chapter we suppose t h a t every rational interpolant r,,=(z) itself satisfies the interpolation conditions (3.1.). This is for instance satisfied if min(m - m', n - n’) = 0. 9.1. Interpolating continued fractions.

Theorem 3.7.

If

and rm+k,n+C

with k,C

2. 0 then

a polynomial

du

U(Z)

P2

=42

exists with

5 max(k - 1, t - I)

and

where

Proof As we assumed, the rational functions ?-m,n and rrn+k,=+( both satisfy [3.1.): for d = 0 , . . .,rn + n

(fql

-p l ) ( s i ) =0

(fq2

- p 2 ) ( ~ ;= ) 0 for i

= 0 , . . .,m

+ k + n + t?

Consequently (P1q2

- P 2 Q l ) ( Z i ) = KfQ2 - P 2 ) q l ] ( Z i ) - [ ( f q l

-p1)q2](2;)

=0

111.3.2. InvcrRp difference8

139

+

i = O , . . . , m n and thus a polynomial ~ ( z ) exists such that ) t~(z)B,+,+l(z). It is easy t o see t h a t a u 5 max(k - 1 , 1 - 1) (PlQZ' - p z q l ) ( ~ = since a ( p 1 q z - pzqi) 5 max(m + n + k , m n L). I

for

+ +

If we consider the staircase of rational interpolants ~ k = { ~ k , 0 , r k + 1 , 0 ~ r k + l , i ~ r k + 2 , i I for ~ ~ ~ }k

2 0

it is possible t o compute coefficients d;(i 2 0) such that the convcrgents of the continued fraction do + d l ( 2 - 2 0 ) dk(2 - 20). . .(z - X k - 1 )

+ + I

.

.

are precisely the subsequent elements of

Tk.

Theorem 3.8.

If every three consecutive elements in Tk are different, then a continued fraction of the form (3.3.)exists with d k + , # 0 for i 2 1 and such t h a t the nthconvergent equals the (n + 1lth element of Tk.

The proof is left t o the reader because it is completely analogous to the one given for theorem 2.9. We shall now describe methods t h a t can be used to calculate those coefficients d;(i 2 0).

3.2.Inverse d#e re nces . Inverse differences for a function f givcn in G are defined as follows:

po[z] = f(z) for every

for every

2

in G

.

Z O , Z ~ , . .,zk

in G

140

111.3.2. Inverse daflerencea

w e call p k [ Z 0 7 . . .,zk] the kth inverse difference of f in the points 20,. . ., Z k . Usually inverse differences depend on the numbering of the points 2 0 , . . .,Z k although they are independent of the order of the last two points. If we want t o calculate an interpolating continued fraction of the form

(3.4.)

we have to compute the inverse differences in table 3.1.

Table 3.1. Po I201

Theorem 3.9.

If di = p;[so,. . ,,xi] in the continued fraction (3.4.), then the Cn of (3.4.) satisfies

ltth

Cn(z;) = f(zi) for i = 0 , . . .,n if C , ( z i ) is defined. Proof

From the definition of inverse differences we know t h a t for n

2 1:

convergent

III.8.2. Inverse d i f e r e n c e s

141

With d; = cpi[xo,.. . l z;] it follows that C,, satisfies the imposed interpolation conditions. 1 The continued fraction po[zo]

2-20 2-z1 + ( c P l [ ~ O , X l+l-JP2[2UJ21,221I+ ... --

is called a Thiele interpolating continued frsction. To illustrate this technique we give the following exarnple. Consider the data: z i = i for i = 0, . . .,3, f ( ~=)1, f ( z l ) = 3, /(Q) = 2 and f ( x 3 ) == 4 . We get

3

1/2

4

1

4

3/10

The rational function

1

I

2 2-11 I+ + ___ + 2-21

I1/2

213

13/10

= -5. x 2 - 5 x - 6 42 - 6

= r(z)

142

III.3.2. Inueree differences

indeed satisfies r(zi) = f(zi)for 4 = 0 , . . ., 3. In the previous example difficulties occured neither for the computation of the inverse differences nor for t h e evaluation of r(z;). We shall illustrate the existence of such computational difficulties by means of some examples. Consider again the data: 2 0 = 0, z1 = I, z2 = 2 with f ( s 0 ) = 0, f(z1) = 3 = f ( ~ 2 ) . Then the table of inverse differences looks like 0 3

113

3

213

Hence

3

I

I

is not defined for x = 20, and thus we cannot guarantee t h e satisfaction of the interpolation condition r(x0) = f ( 2 0 ) . If we cousider the data: zi = i for i = 0 , . . . , 4 with ~ ( Z O= ) 1, f(z1) = 0, f ( z 2 ) = 2, f(23) = -2 and f ( q ) = 5 then p2[z0,21,23] is not finite. This does not imply the nonexistence of t he ratioual interpolant in question. A simple permutation of t h e interpolation d a t a enables us t o continue the computations. For zo = 0, z1 = 2, 22 = 1, z3 = 3 and z 4 = 4 we get

I

2

2

0

-1

113

-2

-1

-113

-3

5

z

-2

-917

The rational function

7/12

III. 8.9. Reciprocal differencea

143

satisfies r(zi) = f(z;) for d = 0,.. .,4. In order t o avoid this dependence upon the numbering of the data we will introduce reciprocal differences. 9.9. Reciprocal diferences.

Reciprocal differences for a function f given in G are defined as follows :

po[z]= f ( z ) for every z in G

w e call P k ( Z 0 , . . ., z k ] the kth reciprocal difference of the function f in the points 20,.. ., Z k . There is a close relationship between inverse and reciprocal differences as stated in the next property.

Theorem 3.10.

For k

2 2 and for all

. ., Z k in (7:

20,.

Proof

The relations above are an immediate consequence of the definitions. This theorem is helpful for the proof of the following important property. Theorem 3.11. pk(201..

.,Z k ] does not depend upon the numbering

of the points

X O , . . ., Z k .

I

144

III.3.9. Reciprocal diffe T ence 8

Proof We consider the continued fraction (3.4.)and calculate the kth convergent by means of the recurrence relations (1.3.):

i = 1, ..., k

For even k = Zj,this convergent is of the form

+ a l z + . . . + 0 j Z’ bo + 6 1 2 + . . . + b j z3

a0

____I____

and for odd k = 2 j - 1, it is of the form

+ . . . + a j zi+ biz + . . . + b3-1. zi-’

ao -t- a12

_ _ I _ -

bo

In both cases we calculate the coefficients of the terms of highest degree in numerator and denominator, using the recurrence relations for the kfh convergent and the previous theorem : for k even we get

and for k odd

Since Pk[%o,. . ., zk] appears to be a quotient of coeflicients in the rational interpolant, i t is independent of the ordering of the 20, .. ., z k because the rational interpolant itself is independent of th at ordering. I

111.3.3. Reciprocal differences

145

The interpolating continued fraction of the form (3.4.) can now also be calculated as follows: compute a table o f rcciprocal differences and put do = p o [ z o ] , d l = p 1 [zo, zl]and for i 2 2: d, =; p t [ z o , . . ., z z ]- p r - 2 [ z ~ , .. ., zi-21. U p t o now we have only constructcd rational interpolants lying on the descending staircase To. To calculate a ration:tl interpolant on T k with k > 0 one proceeds as follows. Obviously it is possiblr to construct a continued fraction of the form

(3.5.) whose convergtwts are the e l e m r n ~ sof ?i, Clearly CO,. . ., c k + l are the divrdtd difrerences f[zoj,. . f[z0, . , z k + l ] since rk,O and r k + l , O , the first two coriwrgents, are the polynomial interpolants for j of degree k and k + 1 respectivc3ly If we want t o calculate for instance r k + i t we need the (2!Jth convergent of ( 3 5.) In order to compute the coefkienl,s d k + * for = 2, . . .,2e we write .j

To define

8

we proceed as follows. Thc conditions rk+l.l(z;) = j ( z ; )

imply t h a t

So

A(Z)

8

for i = 0,.. ., k

+ 2t?

must satisfy

is the ( Z t -

convergcrit of the continued fraction

III.3.4.A generalization

146

of the qd-algorithm

Hence 8(z) belongs t o t h e descending staircase To in the table of rational interpolaats for the function -.q

f-P

As soon as t h e coefficients C O , . . ., c k + l are known, the function q / ( f - p) can be constructed and inverse or reciprocal diiferences for it can be computed. The coeficients dk+i with i 2 2 are preciscly those inverse differences. So finally the computation of an clement i n T k for f is reduced to the computation of an c l e m mt in 7’0 for 4_ _ I - P 3.4. A generalization of the qd-algorzthm.

Consider continued fractions of the form gk(2)

= co

+

c k

i= 1

c;(.

- 2 " ) ( 2 - el). . .(z

- Zi-1)

(3.6.)

Theorem 3.12.

If every three consecutive elements in Tk are different, then a continued fraction of the form (3.6.) exists with c k + J # 0, q?+') # 0, e y + ' ) # 0, 1 + q y + ' ) ( z o - z k + 2 ; - , ) # 0, 1 + e[,k+')(zo- ~ k + ~ # ; ) 0 for a 2 1 and such element of Tk. t h a t the nth convergent equals the ( n

+

IIZ.3.4.A generalization

of

the qd-algorithm

147

Proof

For t h e elements in Tk we put rk+i,j

=

Pk+i,j --__

qk+i,j

for i = j , j + l a n d j = 0 , 1 , 2 , . .

and for the convergents of gk(Z) we put, prl C, = -- for n = 0 , 1 , 2 , . . . with

&O

= &I = 1

Q R

lJsing theorem I .4. a continued fraction with nth convergent equal to

I:,

(2

r,

( n 2 0)

is, alter a n equivalence transforIr\ixLron,

(3.7.)

For

we find by means of theorem 3.7. (,hat f o r

. .(.

C k + 1 ( 5 - Z ).

+

with

Ck+1

1------r

# 0 since rk,o # r k + l , o .

-

1,

Zk)l

2

1,

’2-2

ai(z - z k + i ) /

148

III.S.4.A generalization of the qd-algorithm

For (3.7.) it is even true t h a t

we find t h a t (3.7.) can be written as (3.6.).

I

To calculate the coefficients qLF+') and in (3.6.) one can use the following recurrence relations. Compute the even part of the continued fraction g k ( 2 ) and the odd part of the continued fraction gk-l(z). These contractions have the same convergents r k , O , r k + l , 1 , r k + 2 , 2 , . . . and they also have the same form. In this way one can check [2] that: for k 2 I

III.3.5. A generalization of the algorithm of Gragy

149

These coefficients are usually ordered as in the next table

Table 3.2.

Again the superscript denotes a diagonal i n the table and the subscript a column. Another qd-like algorithm exists for continued fractions of another form than the one given in (3.6.). Although it is computationally more efficient, it has less interesting properties and so we do not mention it here but refer t o [3].

3.5. A generalization of the algorithm of Gragg. The previous algorithm generalized the qd-algorithm and calculated elements on descending staircases. We can also generalize the algorithm of Gragg and calculate rational interpolants on ascending staircases [3]. To this end we assume normality of the table of rational interpolants. Consider for k 2 1 t h e staircase

and continued fractions of the form fk(2) = c g

+

c k

i= 1

C;(.

- 2 0 ) . . .(z - . & I )

-

Ck(2

- 20).

. . ( z - Zk-1)j 1 (3.8.)

150

111.8.5.A generalization of the algorithm of Gragg

Similar to theorem 3.12. one can prove that there exist coefficients f?) # 0 and 8:b) # 0 such t ha t the successive convergents of f k are the elements of s k , as soon as three consecutive elements of s k are different from each other. Making use of the relations existing between neighbouring staircases sk and S k + l we get the following recurrence relations: for k 2 1

f (1k ) - Ck-1 ck

(3.9.)

The coefficients

fr)

and

can be arranged in a two-dimensional table. Table 3.3.

f‘,“’

...

f I”

f I“’

...

Each upward sloping diagonal contains the coefficients which are necessary t o construct the continued fraction (3.8.). It is easy t o see th a t the formulas (3.9.) reduce to the corresponding algorithm of Gragg for the calculation of Pad6 approximants in case all the interpolation points coincide with the origin.

III.S.6. The generalized c-algorithm

151

3.6. The generalized c-algorithm. Let us again consider two neighbouring staircases Sm+nand Sm+n+l.Each of them can be represented by a continued fraction of the form (3.8.). The successive convergents of the continued fraction constructed from Sm+ncan be obtained by means of the forward recurrence relations (1.3.). If we write [4] rm,n=

Pm,n -~ Qm,n

then

(3.10.)

and (3.11.)

Consequently, using (3.10.),

Using (3.10.) and (3.11.) we get

152

111.3.6. The generalized

E-

algorithm

Combining these two relations, we obtain

Performing analogous operations on Sm+n+lwe obtain

Using this result it is possible to set up the following generalized E- algorithm [4], in the same way as the c-algorithm for Pad6 approximants was constructed from the star identity (2.8.) :

(")=o

6- 1

(-n-l)

€2 n

$)

=O

= rm,o(z)

m=O,l,

...

n=0,1,

...

m = 0 , 1 , . ..

111.3.7.Stoer ’5 recursive m e t h o d

153

9.7. Stoer’s recursive m e t h o d .

The use of recursive methods is especially interesting when one needs the function value of an interpolant and not the interpolant itself. Several recursive algorithms were constructed for the rational interpolation problem, one of which is the generalized ealgorithm. Other algorithms can be found in [lo, 16, 221. We shall restrict ourselves here t o t h o presentation of the algorithm described by Stoer. Let m p!&(z) =

c

ai

xi

i= 0

be defined by

i= 0

in other words, they solve the interpolation problem (3.2.) starting a t z, and let a& and b!$n indicate the coefficients of degree m and n in the polynomiais p!& and qk!n respectively. The following relations describe the successive calculation of the rational interpolants lying on the main descending staircase

with and

154

111.3.7.Stoer's recursive method

Proof We will perform the proof only for the first set of relations, because the second (i) and part is completely analogous. In case one wants to proceed from pn,n-l px:A)l to p?)- the degree of the numerator may not be raised. The coefiicient of the term of degree n + 1 in the right hand side of (3.12.) is indeed

To check the interpolation conditions in xi for i = j,. . ., j + 274 we divide the set of interpolation points into three subsets: (a) ( f q ( , i - pk),Jx,)

= -(xi - x

G+l)

3+2n) an,n-l

(b) (fq(,', - px',)(x;) = 0 for i = j + 1 , . . ., j

(d

(i)

(fqn,n--l - p n , n - - l ) ( Z i ) = 0

+ 2n - 1 since

Again these relations can easily be adapted for the calculation of rational interpolants on other descending staircases. To calculate the interpolants in

one starts with

1119.7.Stoer’8 recursive method

155

where the c; are divided differences of f. To calculate the interpolants in

one starts with

where the wi are divided differences of l / f . As for Pad4 approximants, one can also give explicit determinantal formulas for the numerator and denominator of rm,Jz). We will postpone this representation until the next section.

156

III.4.1. Definition of rational Hermite interpolants

54. Rational Hermite interpolation.

4.1. Definition of rational Hermite interpolante. Let the points ( Z , } ~ ~heNdistinct and let the numbers s;(i 2 0) belong to IN. Assume t ha t the derivatives f(')(z;) of the function f evaluated a t the point 2; are given for f = 0, . . ., 8 ; - 1. Consider fixed integers j , k, m and n with

i m+n+l=Ca; i- 0

+

k

The rational Hermite interpolation problem of order (m, n) for f consists in finding polynomials m

and

n

q(z) =

C

b; zi

with p / q irreducible and satisfying

In this interpolation problem 8i interpolation points coincide with z;, so 8; interpolation conditions must be fulfilled in 2;. Therefore this type of interpolation problem is also often referred t o a5 the osculatory rational interpolation problem 1211. In case 8; = 1 for all d 2 0 then the problem is identical t o the rational interpolation problem defined at the beginning of this chapter. In case all the interpolation conditions must be satisfied in one single point 20 then the osculatory rational interpolation problem is identical to the Pad6 approximation problem defined in the previous chapter.

III..&1. Definition of rational Hermite interpolants

157

Instead of considering problem (3.13.) we can look at the linear system of equations

(fQ- P ) ( " ( . i )

=0 (14 - P ) ( L ) ( z i + l= ) 0

for -! = 0,. . .,8 ; - 1 with i = 0 , . . . , j for C = O , ..., k - 1

(3.14.)

and this related problem always has a nontrivial solution for p ( z ) and q(z), since it is a homogeneous system of m + n 1 equations in rn + n + 2 unknowns. Again distinct solutions have the same irreducible form p o l 4 0 and we shall call

+

PO

rm,n = --

40

where 90 is normalized such that qO(z0) = 1, the rational Hermite interpolant of order (rn,n) for f. The rational Hermite interpolation problem can be reformulated as a NewtonPad6 approximation problem. We introduce the following notations: y~ = zo for -! = 0 , . . ., 80 - 1 y,qi)+l = z i for C = 0 , . . ., 8 ; c i i = 0 for i > j c i j = f[yi,. . .,t/i] for i 5 j

- 1 with d ( i ) = 8 0

+ 81 + . . . + 8i-1 (i 2. 1)

with possible coalescence of points in the divided difference f [ y i , . . .,g j ] . If we put

with

Bj(4 =

n

i

(2

- YL-1)

then formally

This series is called the Newton series for f. Problem (3.14.) is then equivalent with the computation of polynomials

158

111.4.1.Definit i o n of ratio no1 He rmit e interpol ant8

and

such that

Problem (3.15.) is called the Newton-Pad6 approximation problem of order (n, n) for f . To determine solutions p and q of (3.15.) the divided differences di

= (f q - p ) ( y o , .

..,yi]

for i = 0 , . . .,m+

must be calculated and put equal t o zero. The following lemma, which is a generalization of the Leibniz rule for differentiating a product of functions, is a useful tool.

Lemma 3.1.

For the proof we refer t o 119). Using lemma 3.1. it is now possible t o write down the linear systems of equations that must be satisfied by the coefficients a; and b; in p and q : COO COI

bo = QO

60

combo

+ c11 61 =

+

Clmbl

+ ...+

(3.16a.) cmmb,

= a,

(3.16b.)

III.d.2. The table of rational Hermite interpolant8

159

Since the problems (3.14.) and (3.15.) are equivalent, the rational function rm,= can as well be called the Newton-Pad6 approximant of order ( m ,n ) to I. In the same way as for the rational interpolation problem the following theorem can be proved. Theorem 3.14.

The rational Hermite interpolation problem (3.13.) has a solution if and only if the rational Hermite interpolant rm,== po/qo satisfies (3.14.). 4.2. The table of rational Hermite interpolants. Once again we will order the interpolants rm,nin a table with double entry: r0,o

r0,1

f0,2

...

r3.0

For a detailed study of the structure of the rational Hermite interpolation table we refer to [3].We will only summarize some results. They are based on the following property. Theorem 3.15.

If the rank of the linear system (3.18b.) is n - t then (up t o a normalization) a unique solution P(z) and q(z) of (3.16.) exists with

ap 5

m-t

aq<_n-t where at least one of the upper bounds is attained. Every other solution p ( z ) and q(z) of (3.18.) can be written in the form P(Z)

= B(z) 4 2 )

dz) = where 38 <_ t.

8(4

111.4.2. The table of rational Hermite interpolanta

160

Proof Since the rank of the linear system (3.16b.) is n - t , solutions pl, 41 and p 2 , 4 2 of (3.10.) can be constructed such that BPl dql

5m 5 n -t

and

by choosing the free parameters of the homogeneous system in an appropriate way. Then one can prove, just as in theorem 3.1., that P1Q2 = P2Ql

Since

we must have either ap1 5 m - t or 892 5 n - t. Hence there exists a solution 8ii L m - t and 8q 5 n - t and it is easy to see that it is unique (up t o a normalization). Now the polynomials P (i )(2)= Bi(z)P(z)

p, i j of (3.16.) with

with 0

5i5t

do also solve the Newton-Pad6 approximation problem of order

( m ,n) because already

(fij

-

=

c

iz m+ a+ 1

diBi(Z)

What’s more, they are a linearly independent set of solutions and hence span the

111.4.2. The table of rational Hermite interpolant8

161

solution space. Consequently, all other solutions p and q of 3.16. are a polynomial multiple of p and ij. Now if both ap < m - t and a9 < n - t we would have more than t + 1 linearly independent solutions and thus the dimension of the solution space would be greater than t + 1 which implies that the rank of the system (3.16b.) would be less t ha n n - t and this is a contradiction. I Before describing the shape of sets in the table of rational Hermite interpolants that contain equal elements, it is important t o emphasize that the structure of the table can only be studied if the ordering of the interpolation points {z;)~,=N remains fixed once it is chosen. Since the polynomials p and q constructed in the previous theorem have the property t ha t their degrees cannot be lowered simultaneously anymore unless some interpolation conditions are lost, we shall call them a minimal solution. This does not imply that p / g is irreducible. However we still have

Theorem 3.16. Let p(z) and g(z) be the minima) sohition of the Newton-Pad6 approximation problem of order ( m , n ) for f and let the rank of the linear system (3.16b.) be 92

- t.

a) If ap = m - t - tl then all the rational Hermite interpolants lying in the triangle with corner elements rm-t--tl,n-t, rm-t-tl,n+t+tl and rm+t,n--t are equal t o r,-t-tE,n-t. b) If aq = n - t - tz then all thc rational Hermite interpolants lying in the triangle with corner elements rm-t,n-tt--tl, rm-t,n+t and rm+t+tz,n-t--ta are equal t o r,-t,n-t-.ta. c)

If i > m+n-2~+ t a + l

with dm+n--2t+to+l # 0 then all the rational Hermite interpolants lying in the triangle with corner elements rm-t,n-t, r,+t+ts,n-t and rnr-t,n+t+tp are equal t o rm-t,n--t.

III.4.3. Determinant

162

representation

(m-t-t ,n-t) 1

1

(m, n ) (m+t,n-t)

(m+t,n-t) (m+t+t2,n-t-t,)

(m-t ,n-t)

(m-t,n+t) (rn-t,n+t+t3)

(rn+t+t,n-t) 3

Figure 3.1.

For the proof we refer to [5]. 4.3. Determinant repreaentation.

Theorem 3.17.

If the rank of the system of equations (3.16b.) is maximal,then (up to a normalization) rm,n = E! is given by

40

III.4.4. Continued fraction

representotion

163

where L= i

and

Fi,j(z) = 0 if i

>j

The proof is left as an exercise (see problem (5)). Again one can see that in case all the interpolation points coincide with one single point, these determinant formulas reduce to the ones given in the chapter on Pad6 approximants since the divided differences reduce to Taylor coefficients.

4.4. Continued fraction repreeentation. If one considers staircases

in the table of rational Hermite interpolants, one can again construct continued fractions of which the successive convergents equal the elements of Tk.It is easy to see that these continued fractions are of the form

+ - dk.+2(Z \

- Yk+l)I I

&+dZ

+/

- Yk+2)l

+...

The coefficients 6,. . .,dh+l are divided differences (with coalescence of points) and the other coefficients can still be obtained using the generalized qdalgorithm. The generalization of Gragg's algorithm and the generalized calgorithm also remain valid for the calculation of rational Hermite interpolants.

164

III.4.5. Thiele's continued fraction ezpansion

4.5. Thiele '8 continued fraction ezpansion. From theorem 3.9. we write formally

(3.17.)

We consider now the limiting case z ; -+ xo for i

p,(z) =

21 .

C P ~ ( ~ O ,* J. zjl

z; lim -+ 2 ...,j

a=o,

Then (3.17.) becomes I

I

I

This is a continued fraction expansion of around 5 0 . Formula (3.18.) is obtained from (3.17.) in the same way as we set u p a Taylor series development from Newton's interpolation formula. Since (3.18.)is formal, one has t o check for which values of z the righthand side really converges t o f(z). We can calculate the p j ( z ) using Thiele's method [17] :

and with

III.4.5. Thiele '8 continued fraction ezpunaion

165

we have

Now p j - l [ z o , . . . , z j - l ] does not depend upon the ordering of the points 20

,...,xj-1. s o

Consequently (3.19a.)

To calculate

one uses the reIationship

(3.19b.)

We apply the formulas (3.19.) to construct a continued fraction expansion of f ( z ) = ez around the origin: P o ( % ) = e"

tpl(z) = e-"

iPr(z0)

=1

pl(z) = e-"

pz(z) = -2e"

'pZ(Z0)

= -2

pZ(z) = -ez

p3(z) = - 3 ~ "

'p3(ZO)

= -3

p4(z) = 2e"

'p4(ZO)

=

= 5e-"

'P5(z0)

=

p3(z)

= -2e-=

P 4 ( 4 = e2

p5(2)

188

So we get

111.4.5. Thiele’e continued fraction ezpanaion

111.5. Convergence of rational Hermite interpolanta

167

55. Convergence of rational Hermite interpolants. The theorems of chapter I can be used to investigate the convergence of interpolating continued fractions. We shall now mention some results for the convergence of columns in the table of rational Hermite interpolants. r2,o, ] . . .}, in other words The first theorem deals with the first column { r o , o ,q , ~ it is a convergence theorem for interpolating polynomials. For given complex points (20,. , z j } we define the lemniscate

.

~ ( z o , ..,z,, t) = { z E C

1

~ (z zo)(z - 2 1 ) . . .(z - z,)l = r}

Broadly speaking, the convergence of an arbitrary series of interpolation does not depend on the entire sequence of interpolation points y; (as defined in the Newton-Pad6 approximation problem) but merely on its asymptotic character, as can be seen in the next theorem.

Theorem 3.18. Let the sequence of interpolation points (yo, y1 ,~ sequence

>it

Vk(j+l)+i

. .) be

2 , .

asymptotic t o the

= zi

for i = 0,.. . , j . If the function j(z) is analytic throughout the interior of the lemniscate B(z0 , .. ., z,, r) then the rm,0 converge to f on the interior of B(z0 , . . .,z j , r). The convergence is uniform on every closed and bounded subset interior to B(z0,. . . ] z j , r).

For the proof we refer to [20 p. 611 and 19 pp. 90-911. Let us now turn t o the case of a meromorphic function f with poles 2 0 1 , . . ., wn (counted with their multiplicity). For the rational Hermite interpolant of order (m,n) we write Pm,n

rm,n = __

4m,n

and for the minimal solption of the Newton-Pad4 approximation problem of order (m,n) we write fS,,,(z) and qm,n(z).

168

111.5. Convergence of rational Hermite inderpolanta

Let the table of minimal solutions for the Newton-Pad4 approximation problem be normal. According to [6] we then have &j,,+ = n. Let wim) ( (i = 1 , . . .,n) be the zeros of qm,n for rn = 0 , 1 , 2 , . . - and let p i = I(wi - Z O ) ( W ; - 2 1 ) .. .(wi - z j ) l with 0 < p1 5 p 2 5 . . . 5 pn 5 at < r for a positive constant a.

Theorem 3.19. If the sequence of interpolation points (yo, yl, 112, . . .} is asymptotic t o the sequence { Z O , Z ~ ,...,zj,zo,z1, . . .,zj,20, 21,. . .,z j , . . .}, if f is meromorphic in the interior of B(z0,. . ., zg’, r) with poles w 1 , . . , , w , counted with their multiplicity and if the table of minimal solutions for the Newton-PadB approximation problem is normal, then

+

= w , o(am>

i = 1, ..., n

and

uniformly in every closed and bounded subset of the interior of B(z0, . . .,z j , r) not containing the points ~ 1 , ...,W n . The proof is given in [4].

III. 6.1. Interpolating branched continued fraction8

169

86. Multivariate rational interpolants. We have seen t h at univariate rational interpolants can be obtained in various equivalent ways: one can calculate the explicit solution of the system of interpolatory conditions, start a recursive algorithm, calculate the convergent of a continued fraction or solve the Newton-Pad6 approximation problem. We will generalize t h e last two techniques for niiiltivariate functions. These generalizations are written down for t he case of two variables, because the situation with more than two variables is only notationally more difficult. More details can be found in [7] and [8]. 6.1, Interpolating branched continued fractions.

Given two sequences { Z O , Z ~ , Z ? , . . .} and {yo,y1,y2 ,...} of distinct points we will interpolate the bivariate function f ( z , y ) at the points in (zo, X I , ~ 2 , . ..} X { y o , y l , y 2 , . . .}. 'TO this end we use branched continued fractions symmetric in t h e variables z a n d y and we define bivariate inverse differences as follows:

170

MI. 6.1. Interpolating branched continued fractions

Theorem 3.20. (3.20.)

with

Proof From theorem 3.9. we know that

Let us introduce the function gO(z, y) by

where

By calculating inverse differences

(0)

for go we obtain

III.6.l. Interpolating branched continued fractions

1

where ho(z,y) =

&I

yo, y]. So already

By computing inverse differences R (j 0, k) for ho we get

It is easy to see that

fl

- I/n

171

172

III.6.I . Interpolating branched continued fractione

From this we find by induction that

So we can write

where

If we introduce inverse differences which provides us with a function

C:l for hl

g 1 we can repeat the whole reasoning and inverse differences ?rj,k: (1 1

III. 6.1.Interpolating branched continued fraction8

173

In the same way as for ho we find

Finally we obtain the desired interpolatory continued fraction.

I

To obtain rational interpolants we are going to consider convergents of the branched continued fraction (3.20.). To indicate which convergent we compute we need a multi-index

The Fith convergent is then given by

with

For these rational functions the following interpolation property can be proved.

III. 6.1. Inte t p o lat ing branched c o ntinue d fr actt o )a8

174

Theorem 3.21. The convergent CK(Z,y) of (3.20.) satisfies C d Z t l , YC,) = I(ZC, t YC,)

for

( e l , &)

belonging to

Proof Let C = min (&, Cz). From theorem 3.20. we know that

where

Now f(%, yt,) = CE(ZC,, yt,) if and only if the following conditions are satisfied Cln

Q0

5 i 5 L : L1 I

mi,

and Lz 5 mi,

This is precisely guaranteed by saying ( t i , &) E I .

I

;...-

111.6.2. General order Newton-Pad6 approzimant8

175

For instance, if moz 2 m l , 2 ... 2 mnz and mOy 2 m l y 2 ... 2 mny the . as we can boundary of the set I is given by n = ( n , m o z J m o y.,.,mnr,mny), tell from the next picture which is drawn for n = 2.

m1y

mZy

-. I:.

0

-*-.

-. . 4

*

4

0

0

0

~

m2x

m1x mOx

Figure 3.8.

We illustrate this technique with a simple numerical example. Let the following d ata be given: z; = i for i = 0 , 1 , 2 , . . . and yg = j for j = O,1,2,. .. with f ( z i , Y i )= (i + j)'. Take A = ( 1 , 2 , 2 , 1 , 1 ) . Then we have to compute

The resulting convergent is

6.2. General order Newton-Pad6 approzimants.

Consider two sequences of real points { Z ; ) ~ ~ N and { Y , } ~ ~where N coalescent points get consecutive numbers. For a bivariate function f ( 2 , I/)we define the following divided differences

176

111.6.2. General order Newton-Padk approzimanta

(3.21a.) or equivalently

One can easily prove that (3.21a.) and (3.21b.) give the same result. When the interpolation points z;,. . ., zi+ri-l and yj, . . .lgj+sj-l coincide, then one must bear in mind that

We consider the following set of basis functions for the real-valued polynomials in two variables:

This basis function is a bivariate polynomial of degree i

+j .

III.6.2. General order Newton-Pad6 approzimants

177

With c k ; , J j = f [ z k , . . .,z i ] [ y c , .. ., yj] we can then write in a purely formal manner [l pp. 160-1641

C

f(z,1/> =

coi,oj

B i i ( z ,YI

(i,j)EINa

The following lemmas about products of basis functions B i j ( z ,y) and about bivariate divided differences of products of functions will play an important role in the sequel of the text.

Lemma 3.2. For k

+ L 2 i + j the product B i i ( ~ , y&(z,y) ) p=o

=

u-0

Proof We write B i j ( ~v ,) = Bio(2,v ) B o j ( z ,Y). Since Bio(z,y) is a polynomial in z of degree i we can write

and

with the convention th at an empty product is equal to 1. Consequently

u=o

p=o

which gives the desired formula if we put, ,A

= a,&.

178

111.6.2. General order Newton-Pedt apptoziments

A figure in IN2 will clarify the meaning of this lemma. If we multiply B;j(z,v) by &(Z, g) and k + l >_ a + J' then the only occuring Bpu(z,v ) in the product are those with ( p , v ) lying in the shaded rectangle.

I

k

i

k+i

Figure 3.3.

Lemma 3.3.

The proof is by induction and analogous t o the proof of th e univariate case. The definition of multivariate Newton-Padh approximants which we shall give is a very general one. It includes the univariate definition and the multivariate Pad6 approximants from the previous chapter as a special case as we shall see at the end of this section. With any finite subset D of IN2 we associate a polynomial

Given the double Newton series

111.6.2. General order Newton-Pad4 approzimanta

179

with c o i , ~ = , f[z0, . . ., z ; ] [ y o , .. ., y j ] ] we choose three subsets N , D and E of IN2 and construct an “ / D I E Newton-Pad4 approximant to f ( z , y ) as follows:

C

p(z,y) =

(i,j)EN

q(z,y)=

q j ~ ; (z,y) j b;j Rij (2, y )

( i , j W

(f Q - P)(z, y ) =

C

( N from “numeratorn)

(3.22a.)

(Dfrom “denominator”)

(3.22 b.)

d i j Bij (2,

Y)

( E from “equatiom”)

(3.22c.)

(i,j)EN2\E

We select N , D and E such that

D has n + 1 elements, numbered ( i o , j o ) , . . ., (in,in)

N C E E satisfies the rectangle rule, i.e. if (i,j) E E then (k,L) E E for k 5 iand L 5 j E\N has at least n elements.

Clearly the coeBcients d;, in

(f q - P N Z , Y ) =

C

dij

Bij(z,y)

(i,i)ENa

are dij

=(~~-P)[zo,...,z~~[Yo,...,Y~~

So the conditions ( 3 . 2 2 ~are ) equivalent with

(.f q - P)[zo,. . ., ~ i ] [ y o ,. ..,i/i] = 0 for (i,j) in E

(3.23.)

The system of equations (3.23.) can be divided into a nonhomogeneous and a homogeneous part:

( I q ) [ z o , . Z i ] [ y ~ ,..., Y j ] = P [ ~ o ., .,. Z i ] [ ~ o ,...,y j ] (f q)[zo, . . ., z ; ] [ y o ,. . ., y j ] = O for (i,j ) in E\N a,

Let’s take a look a t the conditions (3.23b.).

for ( i , j ) in N

(3.23a.) (3.23b.)

111.6.2.General order Newton-Pad6 approximants

180

Suppose that E is such t h a t exactly n of the homogeneous equations (3.23b.) are linearly independent. We number the respective n elements in E\N with ( h l ,k l ) , . . ., (hR?k,) and define the set = { ( h l i I),

.‘

. J (hnl

‘n)}

c E\N

( H from “homogeneous equations”)

By means of lemma 3.3. we have

Since the only nontrivial q [ z o , . . zP][yo,. . .,yu] are the ones with ( p , v) in D we can write .J

. = 0 if p > i or v > j. So the Remember that f [ z C c J . ,. z i ] [ y U ,..,yj] homogeneous system of n equations in n + 1 unknowns looks like

because

D

= {(iO, ~ O ) J * . * J (in?jn))

As we suppose the rank of the coefficient matrix t o be maximal, a solution q(z, y) is given by

111.6.2. General order N e w t o n - P a d 6 approximanta

181

By the conditions (3.23a.) and lemma 3.3. we find

Consequently a determinant representation for p(z y) is given by

(3.25b.)

If for all k, L 2 0 we have Q(ZkJYc)# 0 then

$(z,g) can be written as

with e i i = +[so,.. ., z i ] [ y o , .. yj]. Hence by the use of lemma 3.2. and since E satisfies the inclusion property .)

The following theorem describes which interpolation conditions are now satisfied by P / 9 .

182

111.6.2. General order Newton-Pad4 approzimanta

Theorem 3.22.

If

q ( Z k , yt)

# 0 for

( k , I!) in E then

where

If

r k = 1 = 6~

this reduces t o

Proof Given rk and 6~ for fixed (zk,yc), consider the following situation for the interpolation points, with respect to E

(3.26.)

I

I

, I

k Figure 3.4.

I

I 1

k+fk.1

I

*

111.6.2. Gener af order Nerut o n - f ad6 approzimant 8

and define

T

Using these d e h i t i o n s we rewrite I as

with

Because q ( z t , yt)

# 0 for ( k , l ) in R we have

183

IIl.S.2. General order Newton-Pad4 approximonte

184

To check the interpolation conditions we write

apf” R ‘J. . = p t - u ( B ~Boj) o

-

ax@ dy”

If we cover N2\E with three regions

because B;a(z,y) contains a factor (x - zk)pE+l, and

au

B~~

ay”

l(zkrYl)

= 0 for (i,j ) in B and

(I(,

v) in I

because Boj(x, y) contains a factor (y - Y $ ‘ E + ~ . Analogously

a’ B~~ axp

l(x*iYl)

= 0 for (i,j)in

C and

( p , v ) in I2

The most general situation for the interpolation points with respect to E is slightly more complicated but completely analogous to the one given in (3.26.). We illustrate this remark by means of the following figure:

111.6.2. General order Newton-Pad6 approzirnants

185

I

I

I

I

The proof in this case is performed in the same way as above.

I

We will now obtain the determinant representation given in theorem 3.17. for univariate Newton-Pad6 approximants, from the determinant representations (3.25a.) and (3.25b.). Consider the Newton interpolating series for f ( x , 0) and choose

+

If the points {(k,O) I m + 1 5 k 5 m n } supply linearly independent equations, then the determinant representations for p(z, 0) and q(z, 0) are

186

111.6.2.General order Newton-Pad6 approzimunts

We c a n also obtain the multivariate Pad6 approximants defined in the previous chapter as a special case. One only has t o choose

D = { ( i , j )I m n 5 i + j 5 mn+n} N={(i,j)Imn~%’+jImsa+m} b’= { ( i , j ) 1 mn 5 i t j _< mn + m t n }

because when all the interpolation points coincide with the origin, then

Bij

( 2 , ~= ) Z ’

yi

Let us now illustrate the multivariate setting by calculating a Newton-Pad4 approximant for 2 f ( z , ? / )= 1 + ___ sin(xy) 0.1 - y

+

with y j = ( j- 1)fi

i = 0 , 1 , 2 ' ... j = 0,1,2,. . .

The Newton interpolating series looks like 10 f(Z,YI = 1 + x+ zb + d3 O.l+fi O.l+&

Choose

111.6.2. General order Newton-Pad4 approzimants

Writing down the system of equations (3.23b.), it is easy t o check that

LI

= {(2? ')I

('1

2))

The determinant formulas for p(z, V ) and q(z, v ) yield 1 q ( z , f l )=

-

V-tJ;;

c02,01

c12,11

c01,02

L'11,02

c02,11

c01,12

100 1 0.01 - T (1 - 0.1 fi(Y

+

with

1

1

Finally we obtain

- 0.1+2-y 0.1 - y

+ d4)

187

IH. Prob lema

188

Problems. Let

o rm,= = P~40

be the rational interpolant of order (m,n) for f ( z ) with m' = apo and fir = r3g.o. Prove th at there exist at least m' + nr + 1 points {yl , .. .,ye} from the points ( 2 0 , . . ., z ,+,} such th a t rm,,(vi) = f(vi) for i = 1 , . . ., 5 . Formulate and prove the reciprocal and homografic covariance of rational inter pol ants. Prove theorem 3.8. Which interpolation conditions are satisfied by the nth convergent of the continued fraction (3.4) if a) d, = 0 b) d, = (x, Prove theorem 3.17 Prove the following result for the error function (f - rm,,)(z). Let I be an interval containing all the interpolation points 5 0 , . . ., z ~ + Then ~ . 'dz E I ,

3y, E I :

Compute rZ,z(z) satisfying rZ,z(si) = f(q) for i = 0 , . . ., 4 and r1,3(z) satisfying rl,s(zi) = f(zi) for i = 0 , . . ., 4 with z; = d (i = 0 , . . .,4) and f(z0) = 4, f ( z 1 ) = 2, f ( z 2 )= 1, f(z3) = -1, f ( Z 4 ) = -4If f ( z ) is a rational function

with at 5 m and 8 s 5 n then k with 0 5 k 5 2 max(m, n).

'pk[Zo,.

. .,X k - 1 ,

21

is constant for a certain

111. P r o ble m8

(9)

If for some k, p k [ Z O , . . tion.

(10)

Compute

r2,1(2)

(i = 0 , . . ., 3) and

of

.)

zk-1,2]

189

is constant then f(z) is a rational func-

r 2 , 1 ( z ; ) = f(z;) for i = 0 , . . . , 3 with z i = i = I , f(z1)= 3, f(z2) = 2, f(z3) = 4 J by means

satisfying f(z0)

a) the generalized qd-algorithm

b) the algorithm of Gragg (11)

Construct a continued fraction expansion using Thiele's method for f ( z ) = l n ( l + z) around 3: = 0.

(12)

Calculate the inverse differences for gl(z,y) and 7r!.li for h l ( ~y), and perform one more step in the proof of theorem 3.20. in order to obtain the contribution b---zl)(Y - Y l ) B 2 ( 2 , Y)

(!:i

with B 2 ( Z 1 Y ) = Pz[% 2 1 , Z2l[Yo,

in the continued fraction (3.20.).

Y l , Y21

III. Remarka

190

Remarks. (1)

Instead of polynomials

m

i=O

and

n i-0

one could also use linear combinations m

i= 0

and

of basis functions (g;}iEN, which we call generalized polynomials, and study the generalized rational interpolation problem

A unique solution of this interpolation problem exists provided ( g i ( Z ) ) ; ~ N satisfies the Haar condition, i.e. for every k 2 0 and for every set of distinct points (20,. . .,zk} the generalieed Vandermonde determinant

Examples of such interpolation problems can be found in [lS]. A recursive algorithm for the calculation of these generalized rational interpolants is given in [HI.

III. Re m ar ka

(2)

19 1

The Newton-Pad6 approximation problem (3.15.) is a linear problem in t ha t sense t hat rm,ncan he considered as the root of the linear equation

where p and q are determined by the following interpolation conditions (q f - p ) ( z j ) = 0

+

j = 0 , . . .,m n

Instead of such linear equations one can also consider algebraic equations

where the polynomials pi of degree m;are determined by

More generally we can consider for different functions fo(z), . . ., fk(Z) the interpolation conditions

An extensive study of this type of problems is made in [14] and [lo]. (3)

Rational interpolants have also been defined for vector valued functions [ l l , 251 using generalized vector inverses. For other definitions of multivariate rational interpolants we refer to [17] and [S]: Siemaszko uses nonsymmetric branched continued fractions while in [8] Stoer's recursive scheme for the calculation of univariate rational interpolants is generalized t o the multivariate case.

111.Reference8

192

References. Computing methods I. Addison Wesley,

[

11 Berezin J. and Zhidkov N . New York, 1965.

[

A generalization of the qd-algorithm. J. Comput. Appl. 21 Clsessens G. Math. 7, 1981, 237-247.

[

A new algorithm for osculatory rational interpolation. 31 Clsessens G . Numer. Math. 27, 1976, 77-83.

[

41 Claessens G.

[

A useful identity for the rational Hermite interpolation 51 Claessens G. table. Numer. Math. 29, 1978, 227-231.

[

61 Claessens G. On the Newton-Pad4 approximation problem. J. Approx. Theory 22, 1978, 150-260.

[

71 Cuyt A. and Verdonk B. General order Newton-Pad4 approximants for multivariate functions. Numer. Math. 43, 1984, 293-307.

[

81 Cuyt A. and Verdonk B. Computing 34, 1985, 41-61.

[

91 Davis Ph.

Some aspects of the rational Hermite interpolation table and its applications. Ph. D., University of Antwerp, 1976.

Multivariate rational interpolation.

Intcrpolation and approximation. Blaisdell, New York, 1965.

Contribution B l’approximation de fonctions de la variable complexe au sens Hermite-Pad6 et de Hardy. Ph. D., University of Grenoble, 1980.

[ 101 Della Dora J .

[ 111 Graves-Morris P. and Jenkins C . Generalised inverse vector valued rational interpolation. In [22], 144-156. ( 121 Larkin F.

Some techniques for rational interpolation. Comput. J. 10, 1967, 178-187.

An algorithm for generalized rational inter[ 131 Loi S. and Me Innes A. polation. BIT 23, 1983, 105-117.

[ 141 Liibbe W.

Ueber ein allgemeines Interpolationsprobiem und lineare Identitaten zwischen benachbarten Losungssystemen. Ph. D., University of Hannover, 1983.

III. Reference8

193

151 MuhIbach G . The general Neville-Aitken algorithm and some applications. Numer. Math. 31, 3978, 97-110.

[ 161 Saleer H .

Note on osculatory rational interpolation. Math. Comp. 16, 1962, 486-491.

[ 171 Siemaszko W.

Thielp-type branched continued fractions for twovariable functions. J. Corn put. Appl. Math. 9, 1983, 137-153.

[ 181 Stoer J.

Ueber zwei Algorithmen zur Interpolation mit rationalen Funktionen. Numer. Math 3, 1961, 285-304.

[ 191 Thiele T.

Interpolationsrechnung. Teubner, Leipzig, 1909.

[ 201 Walsh J.

Interpolation and approximation by rational functions in the complex domain. Amer. Math. SOC.,Providence Rhode Island, 1909.

[ 211 Warner D.

Hermite interpolation with rational functions. Ph. D., University of California, 1974.

[ 221 Werner H . and Biinger H .

Pad6 approximation and its applications. Lecture Notes in Mathematics 1071, Springer, Berlin, 1984.

[ 231 Wuytack L. On some aspects of the rational interpolation problem. SIAM. J. Numer. Anal. 1 1 , 1974, 52-80. [ 241 Wuytack L.

On the osculatory rational interpolation problem. Math. Comp. 29, 1975, 837 - 843.

[ 251 Wynn P.

Continued fractions whose coefficients obey a noncommutative law of multplication. Arch. Rational Mech. Anal. 12, 1963, 273-31 2. Ueber einen Interpolations-Algorithmusund gewisse andere Formeln, die in der Theorie der Interpolation durch rationale Funktionen bestehen. Numer. Math. 2, 1960, 151-182.

[ 261 Wynn P .