Interpolation by Rational Functions

Interpolation

2.2 Interpolation by Rational Functions

2.2.1 General Properties of Rational Interpolation

Consider again a given set of support points (Xi' };), i = 0, 1, 2, .... We will now examine the use of rational functions

p".V(x) ao

+

a1x

+ ... +

a X"

~",V(x) = - Q"'V(x) - bo =

+

b1x

+ ... +

bvxv "

for interpolating these support points. Here the integers JI. and v denote the maximum degrees of the polynomials in the numerator and denominator, respectively. We call the pair of integers (JI., v) the degree type ofthe rational interpolation problem.

The rational function ~". v is determined by its JI.

+

v

+

2 coefficients ao, at. ... , a", bo, bl, ... , bv'

On the other hand, ~", v determines these coefficients only up to a common factor P =1= 0. This suggests that ~", v is fully determined by the JI.

+

v

+

1 interpolation conditions

(2.2.1.1 ) i

=

0, 1, ... , JI.

+

^v.

We denote by A'" v the problem of calculating the rational function ~", v

from (2.2.1.1).

It is clearly necessary that the coefficients ar , bs of ~", v solve the hom- ogeneous system of linear equations

(2.2.1.2) P'" V(x;) - }; Q'" V(x;)

=

^0, i = 0, 1, ... , JI.

+

^v,

or written out in full,

ao

+

alxi

+ ... +

a"xf - };(bo

+

b1xi

+ ... +

bvxi)

=

0.

We denote the above system by S'" v.

At first glance, substituting S'" v for A'" v does not seem to present a problem. The next example will show, however, that this is not the case, and that rational interpolation is inherently more complicated than polynomial interpolation.

EXAMPLE. For support points

and Jl = v = 1:

Xi

o

2

h 1 2 2 - 1 . bo =0, ao + al - 2(bo + bd = 0, ao + 2al - 2(bo + 2bd = o.

Up to a common nonzero factor, solving the above system S1. 1 yields the coefficients

ao = 0, bo = 0, and therefore the rational expression

<1>1, 1(X)

=-,

2x

X

which for x = 0 leads to the indeterminate expression 0/0. After canceling the factor x, we arrive at the rational expression

<li1. 1(X)

=

^2.

Both expressions <1>1. ¹ and <li1. ¹ represent the same rational function, namely the constant function of value 2. This function misses the first support point (xo ,fo) =

(0, 1). Therefore it does not solve A 1, 1. Since solving S 1, 1 is necessary for any solution of A 1, 1, we conclude that no such solution exists.

The above example shows that the rational interpolation problem AI', ^v need not be solvable. Indeed, if SI', ^v has a solution which leads to a rational function that does not solve AI', v_as was the case in the example-then the rational interpolation problem is not solvable. In order to examine this situation more closely, we have to distinguish between different representa- tions of the same rational function <Ill', v, which arise from each other by canceling or by introducing a common polynomial factor in numerator and denominator. We say that two rational expressions,

are equivalent, and write

if

This is precisely when the two rational expressions represent the same rat- ional function.

A rational expression is called relatively prime if its numerator and deno- minator are relatively prime, i.e., not both divisible by the same polynomial of positive degree. If a rational expression is not relatively prime, then can- celing all common polynomial factors leads to an equivalent rational expres- sion which is.

Finally we say that a rational expression <Ill', ^v is a solution of SI', ^v if its coefficients solve SI', v. As noted before, <Ill', v solves SI', v if it solves AI', v.

Rational interpolation is complicated by the fact that the converse need not hold.

(2.2.1.3) Theorem. The homogeneous linear system of equations Sp, v always has nontrivial solutions, For each such solution

pp, V(x)

<liP' v = _---':--'..

- Qp, V(x)'

Qp, V(x) =1=

°

holds, i.e" all nontrivial solutions define rational expressions,

PROOF, The homogeneous linear system Sp, ^v has Ii

+

^v

+

1 equations for Ii

+

v

+

2 unknowns, As a homogeneous linear system with more unknowns than equations, sp, ^v has nontrivial solutions

(ao, at> "', all.' bo, "', bv)

=f

(0, ",,0, 0, ",,0), For any such solution, Qp, V(x) ⁼¹⁼ 0, since

Qp,V(x) == _bo

+

b1x

+," +

bvxv ==

°

would imply that the polynomial pp, V(x) == ao

+

al x

+ ", +

^{all. x p} has the zeros

Pp,V(xJ

=

0, i = 0, 1, " " Ii

+

v,

It would follow that pp, V(x) == 0, since the polynomial pp, v has at most degree Ii, and vanishes at Ii

+

v

+

1 ~ Ii

+

1 different locations, contradicting

D The following theorem shows that the rational interpolation problem has a unique solution if it has a solution at all,

(2.2.1.4) Theorem. If <II 1 and <112 are both (nontrivial) solutions of the homogen- eous linear system sp, v, then they are equivalent (<111 '" <112), that is, they deter- mine the same rational function,

PROOF, If both <IIdx) == PI (x)/Qdx) and <112(X) == P2(X)/Q2(X) solve Sp,v, then the polynomial

P(x):== P1(X)Q2(X) - P2(X)Ql(X) has Ii

+

v

+

1 different zeros

P(Xi)

=

Pl(XJQ2(Xi) - P2(Xi)Ql(XJ

=

J;Ql(Xi)Q2(Xi) - J;Q2(XJQl(XJ

=

^{0, i}

=

0, 1, ... , Ii

+

v.

Since the degree of polynomial P does not exceed Ii

+

v, it must vanish identically, and it follows that <IIl(X) '" <112(x). D Note that the converse of the above theorem does not hold: a rational expression <111 may well solve SIl, v whereas some equivalent rational expres-

sion <1>2 does not. The previously considered example furnishes a case in point. In fact, we will see that this situation is typical for unsolvable interpo- lation problems.

Combining Theorems (2.2.1.3) and (2.2.1.4), we find that there exists for each rational interpolation problem All. , a uniq!,le rational function, which is represented by any rational expression <1>1l. v that solves the corresponding linear system SIl.,. Either this rational function satisfies (2.2.1.1), thereby solving All", or All, v is not solvable at all. In the latter case, there must be some support point (Xi' J;) which is "missed" by the rational function. Such a support point is called inaccessible. Thus All. v is solvable if there are no inaccessible points.

Suppose <1>1l. '(x) == pll. V(X)/QIl, V(x) is a solution to Sil. ^v. For any i E {O, 1, ... , t1

+

v} we distinguish the two cases:

(1) QIl,V(XJ=fO, (2) Qil. '(xJ

=

0.

In the first case, clearly, <1>1l. V(Xi) = J;. In the second case, however, the sup- port point (Xi' J;) may be inaccessible. Here

pll"(XJ

= °

must hold by (2.2.1.2). Therefore, both pll., and Qil. ^v contain the factor x - Xi and are consequently not relatively prime. Thus:

(2.2.1.5). If SIl., has a solution <1>1l., which is relatively prime, then there are no inaccessible points: All. v is solvable.

Given <1>1l." let

dill.

^v be an equivalent rational expression which is relatively prime. We then have the general result:

(2.2.1.6) Theorem. Suppose <1>1l. v solves Sil. ^v. Then All. v is solvable-and <1>1l. ^v represents the solution-if and only if

dill.

^v solves Sil. v.

PROOF. If d)Il., solves Sil. ^v, then All, v is solvable by (2.2.1.5). If d)Il. v does not solve Sil. v, its corresponding rational function does not solve All. '. 0

Even ifthe linear system Sil. , has full rank t1

+

v

+

1, the rational interpola- tion problem All. , may not be solvable. However, since the solutions of Sil. , are, in this case, uniquely determined up to a common constant factor p

=f

0, we have:

(2.2.1.7) Corollary to (2.2.1.6). If Sil. v has full rank, then All. v is solvable

if

^and

only if the solution <1>1'., of Sil. v is relatively prime.

We say that the support points (Xi' J;), i

=

0, 1, ... , ^(J are in special posi- tion if they are interpolated by a rational expression of degree type (K, A)

with ^K

+

^{A. < (1.} In other words, the interpolation problem is solvable for a smaller combined degree of numerator and denominator than suggested by the number of support points. We observe that

(2.2.1.8~ The accessible support points of a nonsolvable interpolation problem AI'· v are in special position.

PROOF. Let it, ... , i~ be the subscripts of the inaccessible points, and let cI>1'. ^v be a solution of SI'· ^v. The numerator and the denominator of cI>1'. ^v were seen above to have the common factors x - XiI" .. , X - Xi., whose cancellation leads to an equivalent rational expression cI>K. ). with ^K = fl - oc, A. = v - OC.

cI>K.). solves the interpolation problem A ^K.). which just consists of the fl

+

v

+

1 - oc accessible points. As

K+A.+1=fl+V+1-~<fl+V+1-~

the accessible points of AI'· ^v are clearly in special position. D The observation (2.2.1.8) makes it clear that nonsolvability of the rational interpolation problem is a degeneracy phenomenon: solvability can be restored by arbitrarily small perturbations of the support points. In what follows, we will therefore restrict our attention to fully nondegenerate prob- lems that is, problems for which no subset of the support points is in special position. Not only is AI'· ^v solvable in this case, but so are all problems A^{K .).} of

K

+

^A.

+

1 of the original support points where K

+

^{A. ~} fl

+

v. For further details see Milne (1950) and Maehly and Witzgall (1960).

Most of the following discussion will be of recursive procedures for solv- ing rational interpolation problems Am. ^n. With each step of such recursions there will be associated a rational expression CI>I" ^v of degree type (fl, v) with fl ~ m and v ~ n, and either the numerator or the denominator of CI>I" ^v will be increased by 1. Because of the availability of this choice, the recursion methods for rational interpolation are more varied than those for polyno- mial interpolation. It will be helpful to plot the sequence of degree types (fl, v) which are encountered in a particular recursion as paths in a diagram:

v fl 0 1 2 3 ...

We will distinguish two kinds of algorithms. The first kind is analogous to Newton's method of interpolation: A tableau of quantities analogous to

divided differences is generated from which coefficients are gathered for an interpolating rational expression. The second kind corresponds to the Neville-Aitken approach of generating a tableau of values of intermediate rational functions $1', v. These values relate to each other directly,

2.2.2 Inverse and Reciprocal Differences, Thiele's Continued Fraction

The algorithms to be described in this section calculate rational expressions along the main diagonal of the (11, v)-plane:

(2.2.2.1 )

v^ll 0 1 2 3 ...

o

1 2 3

Starting from the support points (Xi' /;), i

=

0, 1, ... , we build the following tableau of inverse differences:

Xi

11

0 _{Xo 10}

1 _Xl

11

^tp(xo,

xd

2 _{X2 12} tp(xo, x2 ) tp(xo, Xl> X2)

3 X3 13 tp(xo, X3) tp(xo, ^Xl' X 3 ) tp(xo, Xl' X 2 , X3)

The inverse differences are defined recursively as follows:

(2.2.2.2)

On occasion, certain inverse differences become 00 because the denomina- tors in (2.2.2.2) vanish.

Note that the inverse differences are, in general, not symmetric functions of their arguments.

Let pI', QV be polynomials whose degree is bounded by J.I. and v, respec- tively. We will now try to use inverse differences in order to find a rational expression

with

P"(x)

<I>"'"(x) = -

Q"(x)

<1>"' "(Xi) = /; for i = 0, 1, ... , 2n.

We must therefore have

P"(x) _

fc

P"(x) P"(xo) Q"(x) - 0

+

Q"(x) - Q"(xo)

p"-l(X) X-Xo

= fo

+

(x - xo) Q"(x) = fo

+

Q"(x)/P" l(X)' The rational expression Q"(X}/p"-l(X) satisfies

Q"(Xi) Xi - ^Xo

P" l(X_{i )} = /; _ fo = cp(xo, X;}

for i = 1, 2, ... , 2n. It follows that

and therefore p"-l(X;}

Q"-l(X;} ⁱ = 2, 3, ... , 2n.

Continuing in this fashion, we arrive at the following expression for $'" n(X):

pn(X) X - Xo

<Dn.n(X) = Qn(X) =fo

+

Qn(X)/pn-l(X)

x - Xo

= fo

+ ---=.---- -

X - Xl cp(XO' Xd

+

pn-l(X)/Qn-l(X)

x - Xo

= fo

+ - - - -

x - Xl

cp(XO' Xd +---~---

x - X2n-l

+

---;---''''----=---c-

cp(XO' ... , X2n)

<Dn. n(x) is thus represented by a continued fraction:

(2.2.2.3)

<Dn. n(x) = fo

+

x - Xo!cp(Xo, Xl)

+

x - xdcp(xo, Xl' X2)

+

~cp(Xo, Xl' X2, X3)

+ ...

+

X - X2n- t!cP(Xo, Xl' ... , X2n)'

It is readily seen that the partial fractions of this continued fraction are nothing but the rational expressions <DI'. I'(x) and <DI' + 1. I'(x), J.l = 0, 1, ... , n - 1, which satisfy (2.2.1.1) and which are indicated in the diagram (2.2.2.1):

EXAMPLE

<Do. O(x) = fo,

<Dl. O(x) = fo

+

X - xo!cp(xo, xd,

<Dc l(X)

=

^fo

+

X - xo!cp{xo, Xl)

+

X - xdcp(xo, Xl' X2),

o

1 2 3

o

1 2 3

o

-1

-,

2

9

-1

-3

t ^-t

^{Z 3}

<Il2.1(X) = 0 + ~ - 1 + x - 1/- 1/2 + ~ = (4x2 - 9x)/( -2x + 7).

Because the inverse differences lack symmetry, the so-called reciprocal differences

are often preferred. They are defined by the recursions

(2.2.2.4 )

+

P(Xi+ 1, ... , Xi+k-

d·

For a proof that the reciprocal differences are indeed symmetrical, see Milne-Thompson (1951).

The reciprocal differences are closely related to the inverse differences.

(2.2.2.5) Theorem. ^{For p}

=

1,2, ... [letting p(xo, ... , Xp-2)

= o

^{for P}

=

1], q>(xo, Xl, •.. , xp)

=

p(xo, ... , xp) - p(xo, ... , Xp_ 2).

PROOF. The proposition is correct for p

=

1. Assuming it true for p, we conclude from

that

By (2.2.2.4),

( ) ( ) Xp+ 1 - xp

P x p+ b XO, .•. , xp - P ^{Xo, ... ,} Xp-1 = ( ) ( ) P xo, ... , xp - P Xp^+1, XO, ... , xp Since the p( ... ) are symmetric,

whence (2.2.2.5) has been established for p

+

^1.

o

The reciprocal differences can be arranged in the tableau Xo fo

p(xo, xd

Xl fl p(xo, Xl' X2)

(2.2.2.6) P(Xl' X2) p(xo, Xl' X2' X3)

X2 f2 P(Xl' X2, X3)

P(X2' X3) X3 f3

Using (2.2.2.5) to substitute reciprocal differences for inverse differences in (2.2.2.3) yields Thiele's continued fraction:

(2.2.2.7)

<1>'" n(x)

=

fo

+

^{X - Xo/p(Xo, xd}

+

~p(xo, Xl' X2) - p(Xo)

+ ... +

^{X -} X2n-l!P(XO, ... , X2n ) - p(Xo, ... , X2n-2)'

2.2.3 Algorithms of the Neville Type

We proceed to derive an algorithm for rational interpolation which is analo- gous to Neville's algorithm for polynomial interpolation.

A quick reminder that, after discussing possible degeneracy effects in rational interpolation problems (Section 2.2.1), we have assumed that such effects are absent in the problems whose solution we are discussing. Indeed, such degeneracies are.not likely to occur in numerical problems.

We use

<1>1'. V(x) == p~. V(x)

s Q~' V(x) to denote the rational expression with

<I>~' V(x;)

= /;

for i

=

s, s

+

1, ... , s

+

J,l

+

v,

p~. v, Q~' v being polynomials of degrees not exceeding J,l and v, respectively.

Let p~ .• and q~' v be the leading coefficients of these polynomials:

P~' V(x)

=

p~. vxl'

+ ... ,

^{Q~' V(x)} = q~' VXV

+ ....

For brevity we put

rJ.i:= X - Xi and T~' V(x, y):= p~. V(x) - yQ~' V(x), noting that

T~' V(Xi' /;) = 0, ⁱ = s, s

+

1, ... , s

+

J,l

+

v.

(2.2.3.1) Theorem. Starting with P~' O(x)

=!s,

the following recursions hold,' (a) Transition (/l - 1, v) ~ (/l, v):

P~' V(x) = (Xsq~-1, v p~;

t,

^{V(x) - (Xs+ !,+vq~;}

t,

^{v p~-1, V(x),}

Q !" _s V(x) = (X q!'-l, _ss vQ!'-I, _s+1 v(x) _ (X _s+!'+vs+l q!'-I, vQ!'-l, _s V(x) _' (b) Transition (/l, v - 1) ~ (/l, v):

P~' V(x)

=

^{(Xsp~' v-I} P~+vl-l(X) - (Xs+!'+vP~+vl-l P~' v-l(X),

Q !" _s V(x)

=

(X p!" _ss v-IQ!" v-l(X) _ s+1 (X s+!'+vs+l p!" v-IQ!" v-l(X) s '

PROOF, We show only (a), the proof of (b) being analogous, Suppose the rational expressions <I>~-I, v and <I>~;

t,

^v meet the interpolation requirements

T~-I,v(Xi,j;)

=

0 for i

=

s, "" S

+

/l

+

v - 1, (2.2,3.2)

T !'-I, _{s+ 1} V( Xi oJi 1') -- 0 ^f" lor 1= S

+

¹ , "" s

+ +

/l v,

If we define P~' V(x), Q~' V(x) by (a), then the degree of P~' ^v clearly does not exceed /l, The polynomial expression for Q~' v contains formally a term with x^v+ 1, whose coefficient, however, vanishes, The polynomial Q~' v is therefore of degree at most v, Finally,

T~' V(x, y) = (Xsq~-^1, vy~;

f'

V(x, y) - (Xs+!'+ v q~;

t,

^{vy~-l, V(x, y),}

From this and (2.2.3.2),

T~' V(Xi' J;)

=

0 for i

=

s, ' ", s

+

/l

+

v,

Under the general hypothesis that no combination (/l, v, s) has inacces- sible points, the above result shows that (a) indeed defines the numerator

and denominator of <I>~' v, D

Unfortunately, the recursions (2.2.3.1) still contain the coefficients P~' v-t,

q~-l, v, The formulas are therefore not yet suitable for the calculation of

cI>:'

n(x) for a prescribed value of x, However, we can eliminate these coefficients on the basis of the following theorem,

PROOF. The numerator polynomial of the rational expression

PI'-1. V(X)QI'-l. v-1(X) _ pl'-l. v-1(X)QI'-1. V(x)

<l>1'-l.v() <l>1'-1,v-1()_ s s+l s+l s

s X - s+l ^{X -} Q~-l,V(x)Q~+f'v l(X) is at most of degree J1 - 1

+

v and has J1

+

v - 1 different zeros

Xi' i

=

s

+

1, s

+

2, ... , s

+

J1

+

v - 1

by definition of <I>~-1. v and <I>~;

f'

v-i. It must therefore be of the form k1 . (x - _Xs+

d ...

(x - xS+I'+v-1) with k1 = - p~; f' v-1q~-1, v This proves (a). (b) is shown analogously.

(2.2.3.4) Theorem. For J1 ~ 1, v ~ 1,

PROOF. By Theorem (2.2.3.1a),

IJ( ql'-l, v pl'-l, V(x) _ ^IJ( ql'-l, v pl'-l, V(x)

<l>1'V() s s s+ 1 '+I'+v .+ 1 s

s' X = ^IJ( ql' 1.vQI' 1,v(X)_1J( ql' 1,vQI' 1,v(X)'

s s s+l '+I'+v s+l s

o

We now assume that p~;

f,

v-1 =1= 0, and multiply numerator and denomina- tor of the above fraction by

-P~;f,v-1(X - xs+1)(x - Xs+2)'" (x - x^S+I'+v-1)

Q~+ f' V(x )Q~ 1. V(x)Q~; f' v- l(X) Taking Theorem (2.2.3.3) into account, we arrive at (2.2.3.5)

where

[ ]1 = <I>~-l, V(x) - <I>~;

f'

v-1(X), [ ]2 = <I>~;f'V(x) ^- <I>~;f'v-1tX).

(a) follows by a straightforward transformation. (b) is derived analogously.

o

The formulas in Theorem (2.2.3.4) can now be used to calculate the values of rational expressions for prescribed x successively, alternately increasing

the degrees of numerators and denominators. This corresponds to a zigzag path in the (Jl, v)-diagram:

(2.2.3.6)

vJl 0 1 2 3

o

1 2

1···1

., ... ^,' .... ^~

Special recursive rules are still needed for initial straight portions-vertically and horizontally-of such paths.

As long as v

=

0 and only Jl is being increased, one has a case of pure polynomial interpolation. One uses Neville's formulas [see (2.1.2.1)]

<I>~' o(x)

:=1.,

ffill-l.0( ) ffill-l.0( )

<I>~.O(X):=lXs'VS+1 x -lXs+Il'Vs X ,

IXs - IXs+1l Jl = 1,2, ....

Actually these are specializations of Theorem (2.2.3.4a) for v = 0, provided the convention <I>~+

l" -

^{1 := 00} is adopted, which causes the quotient marked

*

(on page 69) to vanish.

If Jl = 0 and only v is being increased, then this case relates to polynomial interpolation with the support points (Xi' 1/};), and one can use the formulas

<I>~' O(x):=

1.,

(2.2.3.7) IXS - IXs+v

<I>~' V(x) := - - - ' - - - -

IXs IXs+v v = 1,2, ... ,

<I>~+"t I(X) <I>~' v I(X)

which arise from Theorem (2.2.3.4) if one defines <l>s-+1 i v-I (x) := O.

Experience has shown that the (Jl, v)-sequence

(0,0) -+ (0, 1) -+ (1, 1) -+ (1,2) -+ (2,2) -+ ...

-indicated by the dotted line in the diagram (2.2.3.6}-holds particular advantages, especially in the important application area of extrapolation methods (Sections 3.4 and 3.5), where interest focuses on the values <I>~' V(x) for x

=

O. If we refer to this particular sequence, then it suffices to indicate Jl

+

v, instead of both Jl and v, and this permits the shorter notation

1ik

:=<I>~'V(x) with i = ^S + Jl + v, k = Jl + v.

The formulas (2.2.3.4) combine with (2.2.3.7) to yield the algorithm

1i, -1 :=0,

(2.2.3.8) 'T' ._ 'T' 1i k-1 -1i-1 k-1

~ik'- ~i.k-1

+ [

X -_Xi - k 1 _ ^1i.k-1

=

1i-l,k-1 _ 1

X Xi 1i,k-1 1i-1,k-2

for 1 ~ k ~ i, i = 0, 1, .... Note that this recursion formula differs from the corresponding polynomial formula (2.1.2.5) only by the expression in brackets [ ... ], which assumes the value 1 in the polynomial case.

If we display the values ^1ik in the tableau below, letting i count the ascending diagonals and k the columns, then each instance of the recursion formula (2.2.3.8) interrelates the four corners of a rhombus:

(11, v) = (O,O) (0, 1) (1, 1) (1, 2) .. , fo = Too

0= TO, - l Tl1

f1 = T10 T22:>

0= Tl,-l T21 T33

f2 = T20 T32 : ".

0= T2,-1 T31

f3 = T30

If one is interested in the rational function itself, i.e. its coefficients, then the methods of Section 2.2.2, involving inverse or reciprocal differences, are suitable. However, if one desires the value of the interpolating function for just one single argument, then algorithms of the Neville type based on the formulas of Theorem (2.2.3.4) and (2.2.3.8) are to be preferred. The formula (2.2.3.8) is particularly useful in the context of extrapolation methods (see Sections 3.4, 3.5, 7.2.3, 7.2.14).

2.2.4 Comparing Rational and Polynomial Interpolations

Interpolation, as mentioned before, is frequently used for the purpose of approximating a given functionf(x}. In many such instances, interpolation by polynomials is entirely satisfactory. The situation is different if the loca- tion x for which one desires an approximate value off(x} lies in the prox- imity of a pole or some other singularity off(x )-like the value of tan x for x close to n12. In such cases, polynomial interpolation does not give satisfac- tory results, whereas rational interpolation does, because rational functions themselves may have poles.

EXAMPLE [taken from Bulirsch and Rutishauser (1968)]. For the function f(x) = cot x the values cot 1 0, cot 2°, ... have been tabulated. The problem is to determine an approximate value for cot 2°30'.

Polynomial interpolation of order 4, using the formulas (2.1.2.4), yields the tableau

Xi ^{/; = cot} (Xi) 1° 57.28996163

14.30939911

2° 28.63625328 21.47137102

23.85869499 22.36661762

3° 19.08113669 23.26186421 22.63519158

21.47137190 23.08281486

4° 14.30066626 22.18756808

18.60658719 5° 11.43005230

Rational interpolation with (j.l, v) = (2, 2) using the formulas (2.2.3.8) in contrast gives

1° 57.28996163

22.90760673

2° 28.63625328 22.90341624

22.90201805 22.90369573

3° 19.08113669 22.90411487 22.9037655J.

22.91041916 22.90384141

4° 14.30066626 22.90201975

22.94418151 5° 11.43005230

The exact value is cot 2°30' = 22.903 765 5484 ... ; incorrect digits are underlined.

A similar situation is encountered in extrapolation methods (see Sections 3.4,3.5, 7.2.3, 7.2.14). Here a function T{h) of the step length h is interpolated at small positive values of h.