Traub On Lagrange-Hermite Interpolation
Traub On Lagrange-Hermite Interpolation
Traub On Lagrange-Hermite Interpolation
INDUST.
APPL.
MATH.
ON LAGRANGE-HERMITE INTERPOLATION*
J. F. TRAUBt
(1-1)
= yi m)
O <
0 :5 m < p
i < n,
-1
Pn,
p( m=) yB
iP=O
p-1
z
m*(t)H
(1.2)
6j,mar,i%
< r < n,
= B,(co; ga , **
qi _g9
gqn),
B.(w; gX
g.)
= E
kayO
U.,k(
g.k+l)c
886
ON LAGRANGE-HERMITE
887
INTERPOLATION
Then
Unk
!kc!DW-Bn(0).
0
The Bn( 1; 91 *
9n) were studied by Bell [1]. (See also Schlomilch
[11, p. 4].) An explicitformulaforBn is
co
xII1 (9-b
Bn = n!
wherej = EiZ- bi and wherethe sum is taken over all nonnegativeintegersbi forwhich nj=i ibi = n.
Let F(t) = f[g(t)]. Then
'n
...
Zf(k)
g(n-k+1)
q
(2.2 n)=
(g
fUnk
k-O
or
= Bn(f; ',
g(n)),
f(k)
fk.
Bo =
12
B1 = wgq,
B2 =
W2g2+ ?42,
B33= W3+
B4 = C4O9
32
g2 +
W/3,
+ 6Cw
3q212 + co2(4g,93+
3922) + Wg4
polynomial.Let P(t)/Q(t) be a
3. The formulafor the interpolatory
properrationalfunctionand let Q(t) have a zero of multiplicityp at xi.
Let
1
PI," .+
Q(t)
j=i (t - xi)
P(t)
E (t-x__
77(t)
+ x(t).
i
13P'P-j=
Zap
k=O
p-k
p (*-k)(Xi
(-)
888
J. F. TRAUB
r = 11
= (t
i=O
(3.2)
Q(t)
xi),
RiR(t)tt= t - ( )txi
L()
LiMt)-R(()
Ri(xi)
k3.0p-
j5
Z
-Q)
(t (X)
( j-k)
-
(j
P(t)
)+
kRi
A
~RPx) (xi)
1aptip-k
k)!
Using
p(Fk)(Xi)
yi=
we obtain,aftersome manipulation,
E
Pnp,(t)= LiP(t) m=0
(t
(in)
(33i
V=O
Mn!
Let
(3.4)
Sv(xi)
(-1)v(v-1)!
r=O;r i
7
(xi
Xr)
RiP(xi)DtvRi P(xi)
(3.6)
Pnp(t)
__
(
Li(t)
i=O
m=0
) Yi
V=O
_*
_1
*B,(p; SI,
S,S).
formulais contained
Thus theessenceofthe pthorderLagrange-Hermite
in the B,(p; SI, ** , ), O < v p - 1. Let
Gpim =
v=O
(t-x)
v.
B(p; Si I
S,).
889
INTERPOLATION
ON LAGRANGE-HERMITE
G2=
1,
1 + (t
x)2S1,
+ (t - xi)3S1 +
1 + (t - xi)4S, +
G3 = 1
2(t
Xi)2[32S12
+ 3S2],
G4 =
2(t
xi)2[42Si2
+ 4S2]
+ I(t
G=
1 + (t
+
1(t
-_xi)3[43S13
-_xi)5S1
-
4S3j,
3.42S1S2
Xi)3[53S13
352S1S2
5S2]
2(t -_xi)2[52SI2
5S3]
+ 3S2
+ 52(4S1S3
+ 653S12S2
2) +
(t -_xi)[54S14
Equation (3.6) may be writtenin a numberof otherways. Let
T, = Tv(x*)= (v -1)!
x_-
r=OfrpziXi
5S4].
Xr
Then
n
p-i
t-X
i()
P-1-m
r
v=k
V.I
*..*
(Ti),
P-i-rn
E,
B,(p; T,
i!v
Tv-k,
T,).
Then
n
P(t)
X mn
p-1 (t
L* (t) Yj
m=0
i=o
p-i1-m
y(m) k=O Hpimk
m.I
Li'j)(xi),
i (-1)kk!C(p
Rp(xi)DevR-P(xj) =
g(t) = Ri(t).
f(U) =u-P,
k=O
+ k - 1, k)Uv,k(L',
. L(v-k+1)
Hence
n
PBt,(t)
(3.8)
P-1
= j Lip(t) ,
p-1-m
izO
(t -xi)^
VI
(t
, (
k=O
yi(m)Epim
-Uvk(
L (P-k~rl)
890
J. F. TRAUB
Observe that EpXm may be obtainedfromthe polynomialEv Xo by truncating the highestmtterms.Hence foreach p, Pn, (t) may be easily obtainedfromEp - Epi ,o . The firstfiveEp are
El =
1,
E2=
1 + (t
xi)[-2L'],
E3 =1
+ (t
xi)[-3L'J + 2(t
xi)2[-3L"
?4 =1
+ (t
x)[-4L'J +
X)2[-44L"
+ 6( -xs)3[-5L"'
(t
+ 20(L')2]
6 (t -x)
E5 = 1 + (1 -x,)[-5L']
+
2(t
+ 12(L')2],
+ '(t
+ 90L'L"
+ 30(L')2]
210(L')3]
+ 120L'L"' + 90(L")2
xl)4[-5L(4)
x)2[-5L"
1260(L')2L"
+ 1680(L')4].
As faras calculationwiththeseformulasis concerned,observethat
Lij'(Xi) = Rij(Xi)-)
Ri (xi)-
ZxLit),
i=O
1,
*,
n.
(4.1)
E Li(t)
i=O
v=O
Y.
E
i=O
(4.2)
[r(i]
BP-,(p; Si, *
, SV-l)
= 0.
This generalizes
n
Ei= 7'(xi)
/()= O.
We can derive a formulaforthe confluentdivided difference
with the
same numberof repetitionsof all arguments,f[xo, p; x1, p;
; Xn
p].
ON LAGRANGE-HERMITE
891
INTERPOLATION
fAXO;x I ; S., ;n
(4.2)
Sp-p-m
Bp-i -m( p
f=0
Mm)
i)) n
(m)
(x
j-r (X.)
E
raz
REFERENCES
[1] E. T. BELL, Exponential polynomials, Ann. of Math., 35 (1934), pp. 258-277.
[2] T. FORT, Finite Differences,Clarendon Press, Oxford, 1948.
[3] T. N. E. GREVILLE, A generalization of Waring's formula, Ann. Math. Statist.,
15 (1944), pp. 218-219.
[4] C. HERMITE, Sur la formuled'interpolationde Lagrange, J. Reine Angew. Math.,
84 (1878), pp. 70-79.
[5] A. S. HOUSEHOLDER, Principles ofNumerical Analysis, McGraw-Hill, New York,
1953.
[6] V. I. KRYLOV, Approximate Calculation of Integrals, Macmillan, New York, 1962.
[7] J. KUNTZMANN, Mgthodes Numeriques Interpolation-D6riv6es, Dunod, Paris,
1959.
[8] J. RIORDAN, An Introductionto Combinatorial Analysis, Wiley, New York, 1958.
[9] H. E. SALZER, Formulae for hyperosculatoryinterpolation, direct and inverse,
Quart. J. Mech. Appl. Math., 12 (1959), pp. 100-110.
[10]
, Hermite's general osculatoryinterpolationformula, this Journal, 8 (1960),
pp. 18-27.
[11] 0. SCHL6MILCH,
Compendium der Hbheren Analysis, II, Friedrich Vieweg und
Sohn, Braunschweig, 1895.
[12] A. SPITZBART, A generalization of Hermite's interpolationformula, Amer. Math.
Monthly, 67 (1960), pp. 42-46.
[13] J. F. TRAUB, Iterative Methods for the Solution of Equations, Prentice-Hall,
Englewood Cliffs,New Jersey, 1964.