Piecewise rational interpolation

by Jan GeIfgren

UNIVERSITY OF UMEÅ

DEPARTMENT OF MATHEMATICS

S-90187 UMEÅ

SWEDEN

by Jan Gslfgren

No 9 1975

0. Introduction.

Wien f is a formal power series at z = 0, the [n,m] Padé approximant can be defined to be the unique rational function P (z)/G) (z), which has n m contact of highest order at the origin, where Pⁿ and are polynomials of degree n and m respectively (see El], [23).

If f is analytic in a small region containing the origin, we can use Hermite's interpolation formula there to get an estimate of the error e(z) = |(f - P /Q )(z)|.

' n m '

If, however, the function f is to be approximated on a real interval, the picture changes. If we use just one Padé approximant, we cannot in general use Hermite's interpolation formula to get a good error estimate in the whole region. To avoid this we partition the interval and determine one rational approximating function on each subinterval, and then we try to tie them together. The approximation result is then compared with the outcome of piece wise polynomial approximation.

The same technique can be used for error estimation in the complex plane.

In this case, however, we cannot tie the approximating rational functions together.

Another approach is to use rational splines (see [11], [15], [12], [1]).

With this kind of approximation a key problem is to solve the nonlinear equations that arise.

Acknowledgement ; I want to express my gratitude to professor Hans Wallin for his guidance and encouragement during my work with this paper.

1. Definition of the Padé approximant.

If f is a formal power series (1.1) f(z) « I a. z^k,

k=0 ^K

then (see I13H the tn,m] Padê approximant to f is a rational function P /Q . where n m

n u m ,

(1.2) P (z) ®

I

I

i 0,

n k=0 ^m k=0 ^{k m}

and the coefficients and 8^ are determined so that

(1.3) f(z) • Q^m(z) - Ppfz) ® A • ^2m+n+y' + higher degree terms, where A is a constant. From (1.1), (1.2) and (1.3) it follows that the numbers and 8^ must satisfy the system of equations

m f t » i U "

(1.4) \ a. , ß, « A

k=0 ^ to n+1 <_ j < n+m, a. = 0 if j < 0.

* J

Since the last m equations have m+1 unknowns 8^, it is possible to choose the 8^ss to satisfy these equations. After that the numbers are determined from the n+1 first equations.

Although the numbers and 8^ are not uniquely determined by (1.4), the (n,m) Padê approximant Pⁿ/Q^m is* which is proved in the following way [5, p. 2].

* *

Let Pn/0m be another (n,m) Padé approximant to f and consider the following expression

(1.5) (f

0

V

On the left side all the coefficients of terms of degree at most n+m are zero by the relation (1.3). The right side is a polynomial of degree

* afe

at most n+m and hence identically zero, which means that P /Q n m ^a P /Q . n m

2. The approximation procedure.

If a function f is analytic at the points Zg and z^, then we can determine the following special Newton series at these points

(see [14, p. 52-54]).

00

(2.1) f(z) ~ Ï a, i,(z), where k=0 ^{k k}

(2.2) w^k(z) -

Def. 2.1. The symmetric [n,m] Padé approximant to f at Zg and z^

is a rational function P /Q , where ₁ n m

n m .

(2.3) Ptz) ⁼ I ^ak ^wi<^^z^ ⁼ Ï K ^z > O^m('^z) i 0,

n k=0 ^m k=0 ^m

n+m+1 = 2p, and the coefficients and are determined so that

(2.4) f(z)G)^m(z) - P^p(z) ⁼ ACz-ZgiPfz-z^)¹³ + higher degree terms, where A is a constant.

In [8, p. 6 def. 1.1] Karlsson has defined the best interpolating function Pⁿ/Q^m of type [n,m] to f at ß²» •••» ^ßn+m+<|' to be the one that is determined by the relation (f Q -P Hz) » 0 m n for z ^a 0^, $2» •••' 6^n+m+i> ^anc* ^{we see} that our Padé approximant satisfies this definition if we let Zg and 2^ be multiple zeros of order p.

By using a lerrma of Karlsson (see [0, p. 10]) to express f(z)Q^m(z) as a Newton series we can solve (2.4), and in the same way as before we can prove the uniqueness of Pⁿ/Q^m-

Let I = [a,b] be a real interval, and let r be a real number that is greater than zero. Let ß = (z|d(z,I) <_ r), where d is the usual

distance function. Let I have the partition x: a = Xg < x^ < ... < x^ = b, where every subinterval A. J = [x. ,, x.] has equal length J J A » |A«J .| » x.-x. „ J ^ for j = 1, 2, ..., N and A < r.

When approximating f on I we determine P ./Q ., that is the n,j m,j syrrmetric [n,m] Padê approximant to f at the endpoints x._. and x. _{J «J} of each subinterval A.. In this way we get a piecewise rational

J

function S (IHz). This function, however, does not necessarily •••• n,m have p-1 continuous derivatives at the nodes, but from (2.4) we see that if P . and Q . have no corrmon factors, it does, n,j m,j

Remark. When m = 0, then by S_ n(r)(z) we denote the piecewise

1 n

,u

polynomial function which on A. is defined to be j

m ci n ^t i ^2pr¹ . rir .k-[k/23 ^r Jk/2]

(2.5) P_ .(z) Y a, w, (z), where w,(z) = (z-x. .) (z-x.)

^p-I k_Q K k K J^-1 J

We see, that it is the partial sum of the Newton series (2.1) of f at x._. and x.. J J

3. Error estimation.

The capacity (transfinite diameter) of the compact set F c $ can be defined by

(3.1) cap. F = lim.inf. max. | h ( z ) | ^ , k-*=° b-tp. z€F

where p^ is the class of all polynomials h(z) = z + ... of degree k k (see [B, ch. 7], [7, ch. 16]). It satisfies

(3.2) raeaa F _< n(cap F)² (see [10]).

Lemma 3.1 Let r>0 be given and let g(z) be a polynomial of degree < m that satisfies

max |g(z) I 1, where y = {z|z = g|t), 0 <_ t <_ 1, g^O) • g|1) and g^ contj and has diameter 2r.

Let 0 < e < 1/(2/3+1) and let B = {z|z € the interior of y , |g(z)| <_ em).

Then cap B ± (2^+1)re.

This lemnna is a modification of a lerrma by Ponrrerenke [10] and is proved in the same way. Porrmerenke considers y ⁼ tz| |z| = r} and instead of 2v^i+1 his constant is 3.

Now let I, T, ft and S (x)(z) be as in section 2. Let G(A) = n,rn

= {z = x+iy| X 6 I, Iy| _< (3/4)A}. Then we have

Theorem 3.2 Let f be analytic in ft, and let I have the partition T.

Let n > 0 be arbitrarily chosen. Then it is possible to find a real number k so that, when n >_ k*m and z € G(A), we have

(3.3) |(f - S^{n m}(t)(z.)| ± K(A) max |f|

except on a set F of capacity <_ fj.

Remark 1. K(A) is a constant which is less than 4+4/it, and k is the smallest real number that satisfies

(3.4) fx 4§ A)" • ^N -ÜE-» < 1.

r ^ ^ ' /(b-a)² + (9/4) A²

Remark 2. When m = 0 we get the polynomial case, where we have no exceptional set.

Remark 3. If we approximate just on I, that is if z € I in (3.3), then the constant y-yg- can be replaced by 1/2. We see that the

fis

polynomial case is to prefer mainly for two reasons. In the first place the Newton series is easier to determine than the Padé approximant, and

in the second place we do not get any exceptional set in the polynomial case.

If f is real valued on [a,b], we get the following stronger result

for piece wise polynomial approximation. We use the same notation as before.

Theorem 3.3. Let f be analytic in ft and real valued on [a,b].

Let [a,b] have the partition T. Then, for x € [a,b], we have

l^fl D D-i

(3.5) |(f^(j) - ⁿ C T

) ) (x)|

P

P

P~ r P J (2p-j)

P"j

where j = O, 1, 2, ..., p-1 and ^2p)j ⁼ (2p-j)' "

Remark 4. If we put j = O we get

(3.6) | (f - S2p_<| Q(T))(X)| ^<_ max |f| (^p)²'³» * ^€ ta,b].

From Remark 3 and (3.3) we see, that now the constant K(A) has been reduced to unity. All the time we have had n+m+1 = 2p.

4. Proofs of Theorem 3.2 and Theorem 3.3.

Proof of Theorem 3.2. If z belongs to the region

G(A,j) = {z = x+iy| X € A., |yj £ (3/4)A}, then Hermite's interpolation formula (see [14, p. 50]) gives

(f Q^m . - Pⁿ .)(z) . (fQ.Ht)

(4.1) —^ / üUJ dt,

(z - X, -) (z - x,)^{p 11} Y- (t - X. - X.)P(t - z)

J-1 J J J-1 J

where 0 . is normed so that max |Q .(t)| =1 and

m'^J

tey.

Yj = {z|z = Xj_^/| + r • e^is, s € [ir/2, } u

U {z|z = Xj + r • e*^s, s € [- U

U {z = x+iy| X € A., Iy J = r), and J

P ./(D . is the symmetric Padê approximant to f at the endpoints of A.. n,j m,j ^J . J From (4.1) we get

max (z-x. J^p(z-x,)^p

(4"²' l^(f °m,j -^Pn.j^,(z,l Ì5T _y. ' ' F - lo/4B _{mm (t-x.} . ^r. * r l t - x ìPa nP .r

J ^V U J

Yj

Since max |z - x. * \\ z - x.| = (15/16)A , *?

z£fl(A,j) ^{J J}

n y

and min jt - x. ^||t - x.| = r (1 + (A/2r)~), we get

«ïj ^J

(4.3) |(f (D . - P' m,j n,j ⁿ .)(z)| < K(A) max |f K\/^| ¹ — ^{1 1} v ib r ^)^m+n+1. Now j

(4.4) |(f -^Ì)(z)| < K(A) max |f | -(\/|f A)^m+n+1 .

Yj ^ej

except on a set F of capacity <_ 5(r + A/2)e. (see Lemma 4.1).

»J

We put e. = e > 0 for j = 1, 2, .... N, and then use the definition J

of S (n,m T)(Z). For z € G(A) this gives

(4.5) j Cf - Sⁿ (T)(Z)| < K(A) max |f| (v/jl A)^m+n+1 (l)^m,

1 n,m — ⁰

' ' v

except on a set F = U F.. N 1 J

The "subadditivity" of the capacity (see ^[9, p. ^259]) gives

(4.6) cap F. <_ 5(r + A/2)e for all j ==> cap F (5(r + A/^îe^^diam F)'' vJ

r/i -?•> i „4. . ! n _ l . v/(b-a)² + (9/4)A²

(4,7) Let e " t ^7l = J 5TFTÂ75T— '

\V(b-ar + (9/4)Ay then cap F <_ r).

If k is the smallest real number that satisfies (3.4), then (4.5)

gives (3.3) except on a set F of capacity rj. From (4.2) we see, that K(A) < 4 + 4/ir since A < r.

Q. E. D.

Proof of Theorem 3.3. The usual real interpolation formula [4, p. 67]

gives

-j(2p)^m

(4.8) (f - ^p2p-i,k^{)(x) =} (2p): ^(x " ^xk-1* ^(x " ^xk⁵ ' x and Ç 6 Ak*

The function f - ^P2p-1 k ^^{as a Z0ro ort}^^er P ^XK-1 ^anc' x^, which means that (f - ^P2p-1 k^ ^^{as a zero orc}'^er P"^

those points, and according to Rolle's theorem it has a zero at z = c inside x^]. This means that

(4.9) f'(x) - ^p2p-i,k^(x^ * C2p-1)I^ ^(x_xk-1^{)P fx}"^xk^{)P (x-c)}' where x, Ç<j, and c G A^.

The first factor on the right side is the derivative of order 2p-1 of f' divided by (2p-1)!. By using the definition of S2^p_>| Q(T)(Z) and by iterating j times we get

f^(2p'u.)

(4.10)

S

-c

)...

P B . ( X""C • ) , J

where x, Çj, c^, C2> • ••» and k£ {1,2,..., N}.

Let

(4.11) g(x) = I tx-x^^ ^(X-X|^<.)ⁱ³ ^(x-c^Hx-^) ••• (x-Cj)|, where x, c^, C2» •••> Cj € A^.

We see that g(x) has its largest maximum when the points c^ are either equal to x^ or to x^.

If we let £ c^:s equal x^ and j-£ c^:s equal x^, and if we then determine the maximum of g, we get

f4-¹²⁾ W ⁼ (P-J*¹¹'""¹ t^Xk2p-j^k"¹)^2P"^J-

Now, if we let £ >_ j-£, we see that £ = j gives the largest maximum of g

(4.13, . pVjìP-J

r

a

(4.14) f (z) - -gr / ^[t.^z)2p«1 ^dt- where y = {t||t-z| = r>.

From this we see

f(2p) (4.15) max

z€I T2pTT

(z) £ max z€Q

f (z) ,2p

Now (4.15),(4.13) and (4.10) give (3.5).

Q. E. D.

5. Piecewise Padé approximation in the complex plane.

Let the function f be analytic in the domain

ft = {z = x+iy| IX j <_ r, I y ( _ < r}, where r is a real number which is greater than zero. This implies that for an arbitrarily chosen point

Z

00 u

(5.1) f(z) = I a^k(z-z^Q) . k=0

n k ^m k

If we put P (z) = £ a, (z-zn) and Q (z) = T |3, (z-zn) , we see

n k=0 ^m k=0

from (1.1), (1.2) and (1.3) hov; to define the [n,m] Padé approximant to f at this point.

We now divide Q into disjoint subsquares and determine the [n,m]

Padé approximant to f at the center of every subsquare. We shall make two different partitions of fì and compare the results.

The first partition.

Let p and j be positive integers such that p _< 4(2^J 1+1 - 3).

1+1 Let z. = x. + iy. , where 4

J'P J>P J'P *

fx. = (2p-1)r/2^J J»P

y^jip . r(l-3/2>¹)

when p <_ 2^ - 1 and

'x. « rd-3/2"³*¹) J'P

y. _ « r(2-(5+2p)/2^j+1), J 'H

when 2^ <_ p 2"^

rx . _{= r(1-3/2}j+1) J »P

Let z. = X. + iy. , where

J'P J>P J'P I io.* 4⁺T

Yj = (-r/2^ )(2p-2 + 5) for 2j+1 -3<p<3*2j-4,

rx. „ = (r/2^J'⁺¹)(2^j+3 - 11 - 2p )

i J»P i î+1

and < . ⁴ , for 3 • 2^J - 4 < p < 2(2^J -3), y. ^s -r(l-3/2^J )

J»P

For 2(2^ - 3) < p < 4(2^ - 3) we define z. in a similar way.

- J»P

Now we define

fi(j,p) = {z = x+iy||x-x, JP | < r/2^^— +1, |y-y. JIP | < r/2^+1} , and

Yj,^p - (z • xMyllx-x^l - ^ , ^yjp - < y < y^ » -^} U

U

(z '

Jfr

^fri.

(See fig. 1:) We also define

S U)(z) = {(P_ . VQ . „3(2)1 P_ i „AL . _ is the [n,m] Padé n,m n,j,p m,j,p n,j,p m,j,p approximant to f at the center of Q(j,p), and j < U,

Theorem 5.1. Let f be analytic in fi and let > 0 be arbitrarily chosen.

Then there exists a real number k so that, when n > k*m and z £ fi(j,p), we have

(5.2) I (f - S _n,m ^mUHz)| < i _{' — ir} max |f I • -i- (Ä^n+1_km

ß 2ri\j 3 except on a set F of capacity <_ t|.

Remark. Here k is the smallest real number that satisfies 15,2n£\N ^f^k+1 ^ , , .. ^Mf0, ^A r ^f —s+1 -4-I ) (y) < 1, ^where N = NU) = 4 \ {2

1 s=1

is the number of squares £Xj,p) in Q.

ß(j,p) ^s \

l ^(z =

The second partition.

Let s be a real number such that r/s = IM, where N > 1 is an integer.

Let p and j be positive integers that satisfy the relation p <_4(2j-1) and j < N. Now we define (see fig. 2)

f{z = x+iy| (p-1)s <_ X <_ ps, (j-1)s y <_ js}, if p < j

x+iy| (j-1)s < X £ js, (2j-p-1)s y £ (2j-p)s}, if j<p^2j-1, ( {z = ^x+iy| (j-1)s <_ ^{X <_} js, (p-2j-1)s ^{<_ -y <_} (p-2j)s}, ^if 2j<p<3j-1 nCj,p) = s

^{z = x+iy| (4j-p-2)s <_ x <_ (4j-p-1)s, (j-1 )s£-y<js}, if 3j<p<4j-2.

When 2(2j-1) < p £ 4 ( 2 j -1 ) we define S2(j,p) in a similar way.

If z. = x. + iy. is the center of Q(j,p) we define J.P J>P J>P

Tj.p * ^{z • "•iyllx-xjj - T • l^y-^yJ.pl -T> ^u

We also define

Y. " {z = x+iy||x| = (j+1)s, |y| <_ (j+1)s> U {z = x+iyjj |x| £ (j+1)s, |y|«(j*1)s>, J

and S (z) = {(P . /Q . _,Hz)|p . /Q . is the [n,m] Padê approximant to f at the center of ft(j,p)}<

See fig. 25

With these definitions we get

Theorem 5.2. Let f be analytic in fì and let n > 0 be arbitrarily chosen.

Then there exists a real number k so that, when n > k*m and z € fi(j,p), we have

(5.4) _' J (f - S )(z) I < - max |f| • (^)_n,m ¹ — TT ' ' j ^n_km+1 Y

j

except on a set F of capacity £ t|.

Fi ^urt

h

#> «tt «>•

! \n

%10

_ k . .... „ .

v

V

ih-

m

*t _WNB

%%t

ht u

v»

P*twt*ttotr. j,f> *n( I,p)

f t y u r e . i .

! V*

1 ^W

V

V

W

4,3 I*»

Tu] _{v« • » «}

\ï

!*

k»

L I

Denotation:

Remark. Here k is the srallest real number that satisfies

(5.5) (^)^k+1 [Idlf < 1, where M - 4(| - 1)² denotes the number of squares S2(j,p) in fi.

The proofs are omitted since the technique is the same as before.

We use Hermite's formula to get an error estimate and the "subadditivity"

of the capacity to measure the exceptional set.

Theorems 5.1 and 5.2 show that the degree n becomes very large, if the estimate is to be valid in the vicinity of the boundary, or if the exceptional set is to be small. The use of Hermite's formula in the estimation procedure is responsible for the fact that we cannot reach the boundary of fi, when we estimate the error.

Remark. If n + m = p, where p is a fixed integer, we see from (5.2) and (5.4), that if we put m = 0, we get at least as good error estimates as with piecewise Padê approximation, and we do not get any exceptional set.

When we use the first partition we see from (5.2) that the approximation procedure gives its best result near the boundary of fi, and from (5.4) we see that if we use the second partition the approximation is best in the middle of fi.

References

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"Advances in Theoretical Physics; 1" (Brueckner ed.), Academic Press, New York 1965, pp 1 - 58.

t3] : The Padê Approximant Method and Some Related Generalizations,

"The Pade Approximant in Theoretical Physics" (Baker and Gammel ed.).

Academic Press, New York 1970, pp 1 - 39.

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reihen, 3. für reine und angew. Math. 90 (1881), 1 - 17.

[6] Golusin, G.M.: Geometrische Funktionen theorie. Deutsch, Verlag Wiss., Berlin 1957.

[7] Hille, E.: Analytic function theory, vol. II Ginn and Company, Boston etc.

1962.

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Preprint University of Umeå, Department of Mathematics, no 6, 1972.

[9] Nevanlinna, R.: Eindeutige analytische Funktionen, 2:te Aufl. Springer-Verlag, Berlin etc. 1953.

[10] Ponrnerenke, Ch.: Padê Approximants and Convergence in Capacity, 3. Math.

Anal. Appi. 41_ (1973), 775 - 80.

[11] Schaback, R.: Spezielle rationale Splinefunktionen. 3. Approximation Theory 7 (1973), 281 - 92.

[12] Schömberg, H.: Tschebyscheff-Approximation durch rationale Spline-funktionen, Dissertation Münster 1973.

[13] Wallin, H.; On the Convergence Theory of Padê Approximants, "Linear Operators and Approximation", Proceedings of a conference in Oberwolfach 1971

(Butzer, Kahane and Sz-Nagy ed.) Birkhäuser, Stuttgart 1972, pp 461 - 69.

[14] Walsh, 3.L.: Interpolation and Approximation by Rational Functions in the Complex Domain. American Math. Soc. Colloquium Publications, Vol. XX 4:th ed. 1965.

[15] Werner, H.: Tschbyscheff-Approximation mit einer Klasse rationaler Spline-Funktionen, 3. Approximation Theory 10 (1974), 74 - 92,