Numerical Algorithms 5 (1993) 51-62
51
An efficient algorithm for periodic Hermite
spline interpolation with shifted nodes
Gerlind P l o n k a
Fachbereich Mathematik, UniversitdtRostock, D-18051 Rostoek, Germany
Generalized Hermite spline interpolation with periodic splines of defect 2 on an equidistant
lattice is considered. Then the classic periodic Hermite spline interpolation with shifted interpolation nodes is obtained as a special case.
By means of a new generalization of Euler-Frobenius polynomials the symbol of the considered interpolation problem is defined. Using this symbol, a simple representation of the fundamental splines can be given. Furthermore, an efficient algorithm for the computation of the
Hermite spline interpolant is obtained, which is mainly based on the fast Fourier transform.
1. I n t r o d u c t i o n
A periodic spline interpolation p r o b l e m based on spline functions of defect 2
with two shifted interpolation nodes TO + j and T 1 -[-j ( 0 < T 0 ~ T 1 ~ 1) in each k n o t
interval [j,j + 1] is considered. T h e n we obtain the periodic H e r m i t e spline interpolation p r o b l e m in the special case TO = ~'1.
In [3] a simple criterion for the existence and uniqueness of solutions of our
extended spline interpolation p r o b l e m was found.
T h e p u r p o s e of this paper is to present a new efficient algorithm for the c o m p u t a tion of the H e r m i t e spline interpolant by means of the fast Fourier transform. In
contrast to other papers dealing with H e r m i t e spline interpolation (cf. [2,5]) we
shall introduce a generalization of E u l e r - F r o b e n i u s polynomials based on Bsplines with double knots. In section 2, a new representation of the symbol of the
considered p r o b l e m will be given and its i m p o r t a n t properties will be investigated.
W i t h the help of this symbol, new explicit formulas for the periodic H e r m i t e fundam e n t a l splines will be f o u n d in section 3. Finally, in section 4 an efficient algorithm
for solving the H e r m i t e interpolation p r o b l e m will be presented.
2. S y m b o l o f p e r i o d i c Hermite spline interpolation
Let N, m e N (m >t 2) be fixed. By S~,2(Z ) we denote the linear space of all N-periodic real functions s s Cm-2(~) with
9 J.C. Baltzer AG, Science Publishers
G. Plonka/ PeriodicHermitesplineinterpolation
52
s ( j + t) = pj(t),
pj = Pj+N 99~m,
for all t 9 [0, 1] and for allj 9Z, where Tm signifies the set of all real polynomials of
degree ~<m definedon [0, 1].
Furthermore, let y~ 9I1~ (j e Z, v = 0, 1) with y~V)__-Yj+N"
(v) be given N-periodic
data, which can be completely described by the vectors
Y(~)=~Y0"(~),yl(~),''',y~)-I)T 9
(v=0,1).
We consider the following generalized Hermite spline interpolation problem.
For given shift parameters r0, rl e R with 0 < r0 ~<7"1 ~< 1, w e t r y to find an N-periodic
spline function s 9StuN,2(Z) such that
s ( j + 7"0) = yS0) ( j 9
[ s ( j + . ) : 7"0,rl] =Y~l)
(jeZ),
(2.1)
where [s(j + -) : 7-0,7-1] := [s(j + rl) - s ( j + 7"o)]/(7"I - 7"1) denotes the divided difference.
In order to solve this problem we shall introduce the symbol of the generalized
periodic Hermite spline interpolation problem (2.1) with the help of B-splines. Consider equidistant knots with multiplicity 2
x j : = [j/2]
(j 9
(~ = 0, 1) denote
where [j/2J denotes the integer part of j/2. Let NZveCm-2(~)
2
the normalized B-splines of degree m (m~>2) and defect 2 with the knots
xv, x~,+l, 9 9 X~+m+l. With the help of the B-splines with double knots the generalized Euler-Frobenius polynomials of multiplicity 2 with shift parameter x 9I~ can
be defined by
0(3
2
HmAX, Z)
N~,~,(S+X)Z
j
2
:=
(v-- O, 1 ; z e C ) .
J'~ - oo
Because of the finite support ofN~,v (v = O, 1) the functions H2,~ are well defined.
We are especially interested in the behaviour of/-/~,~, (v = O, 1) on the unit circle.
Therefore we put, for u 9Ii~,
h2v(X,U) :=
2
e-iU)
(x 9
The following properties hold.
T H E O R E M 2.1
Let m e N (m i> 2) and u, x 9 ~ be fixed. Then
(i)
h2,~,(1, u) = eiUh2,u(0 , u)
(v = 0, 1);
53
G. Plonka / Periodic Hermite spline interpolation
(ii)
OJ-2
t
oJ
~xjnm,,ix+ 1,u) = e:Liu-ff~hE,(x,u)
(7X
(v = 0, 1;j = 0 , . . . m-- 2);
0 2
(1
2
--~xhm,o(X,U) : - m - ~ - ~ h m _ l , o ( X , U )
1
2
)
[ ( m + 1)/2J hm-l'l(X'u) '
(iii)
(
0 h2
-~X
m'l(x'u)=m
e-'U hi-l,O(X' u) ) .
1
2
[ ( m + 1)/2] hm-l'l(x'u)
[~/2]
Proof
(1) Since N ~ ~ Cm-2(N) we see for u = 0, 1, x e N andj = 0 , . . . , m - 2:
oJ hE,.(x + 1, u) = ~
OxJ
l=--oo
dj 2
Oj .2 ," u) (uEN).
-~N~,~,(x
+ l)e-iU(l~l) -_ e+iU~xjnm,~,tx,
The assertion (i) is a special case of(ii) forj = 0 and x = 0.
(2) The relation (iii) follows immediately from the recursion formulas for Bsplines with double knots:
N~,o(X ) = m
d
2
-~N~,,l(x ) = m
(
)'
1
)
Lm/2 j N2_i,o(X- 1) .
[ ( m + 1)/2] N2m-l'l(x)
Nm-l'~
1
2
[ ( m + 1)/2J N~'-I'i(x)
[]
Now for x, x0, xl, u ~ N, we define the following two determinants:
h2(xo,xl,u) := det(h2'~176
\ hE,o(X,,U)
h~,,(xo,u))
h2,1(Xl,U)
( hE,o(X,U)
h2 (x, [xo,xl],u) := det\ h2,o[Xo,Xl](U)
'
hE,l(x,u)
hL,l[xo,xll(U)]'
where
hEAxo,xl](u) :=
2
[hm,~(.,u):
xo,x,]
(v = 0, 1)
denote the divided differences ofh2m,~(., u).
THEOREM2.2
Let N, m ~ N (m >~2) and 0 < To ~ rl ~ 1 be given. Then the interpolation problem
(2.1) is uniquely solvable if and only if
G. Plonka / Periodic Hermite spline interpolation
54
h 2(TO, [To, "q], 27rj/N) • 0
(j = 0,..., N - 1)
(2.2)
is satisfied.
The relation (2.2) is equivalent to
IBm(') : 7"o,7"1] # 0,
where Bm denotes the ruth Bernoulli polynomial.
F o r a p r o o f of t h e o r e m 2.2 we refer to [3]. W e call h2m('ro, [7-o,7-1],-) the symbol of
the generalized Hermite spline interpolation problem (2.1).
EXAMPLES
F o r x, Xo, Xl ~ (0, 1], u ~/I~ we find
h20(x,u) = 2x(1 - x),
h2,l(X,U) = x 2 + (1 - x)2e -i",
h2(xo, [xo, xl],u) = 2 ( x o x l - (1 - xo)(1 - xl)e-iU) ,
h2,0(x,u) = 8 9
5x) + (1 - x)3e-iu),
X)2(Sx+
h],l(X,U) - - l~( x 3 + (1 -
1)e -i~) ,
h](xo, [xo, x,l,u) =3 {X~oX~ -[Xo(1 - Xo)
+ Xl(1 - xl) + 2x0xi(1 - x0)(1 - Xl )]e -iu
+ (1 - x0)2(1
-
Xl)2e-2iu}
.
W e can prove the following properties for h 2.
THEOREM 2.3
Let m ~ N (m ~> 2), Xo, xl ~ 1t~and u ~ [-Tr, 7r] be fixed. Then
(i)
h2m(XO, xl, u) = - h m(xl,
2
X0,/,/),
h~(xo, [xo, xl], u) = h 2 (Xl, Ix0, Xl], u) ;
(ii)
hZ(xo, x,,u + 2kvr) = h2m(XO,Xl,U) ( k z Z ) ,
h2(xo • 1, xl , u) = e +iuh ,2. ( x o , x l , u ) ,
h2(xo, xl 4- 1 , u) = ~_• nm(XO, X l , U ) ,
h 2(xo, Xo, u) = 0 ;
G. Plonka / Periodic Hermite spline interpolation
55
(iii)
h2(xo -I- 1, [xo +
h2m(XO.
1,[xo
. .
1,xo + 1],u) = ~_2iu/2
,
nmtXo,
{Xo,X0], u ) ,
1,xo
1],u) ~_-2iu,_2
.
,,,.(xo, [xo, xo], u),
h2(xo,[Xo,xo+k],u)=O
(k~Z).
Proof
(1) The identities in (i) and (ii) are immediate consequences of the definition
and theorem 2.1.
(2) For the symbol we find by theorem 2.1 (ii)
h2m(xo+ 1,[Xo + l,xo + 1],u)
-det(
h2m,o(Xo+1,u)
-
It, h~,o[Xo + 1,Xo + 1](u)
eiUh2,l(xO'u) ~ =~2iu,.2/=
(eiUh2m, O(xO'u)
--
h2,1(x0 + l,u)
h2m,1[xo + 1,xo + l](u) ,]
detk,eiUh2m0[Xo,x0l(u)
eiUh2m,l[xO,xo](u),]
c
nmtX0 ,[xO,xOl,u).
Analogously it follows
h2m(XO-
1, [x0 - 1,xo - l],u) = e -2iu-2
, {x0, xo], u).
n,.cxo,
Finally we get by (ii)
h2m(XO,[Xo,Xo+k],u)
eiku
=--~-h2(xo,xo, u ) = 0
(k+Z).
[]
For the computation of the N-periodic fundamental splines and the N-periodic
Hermite spline interpolant by discrete Fourier transform the vectors
NJJj=o
,,:(,<o,x,>.:
(u = 0, 1;x~IR),
+=>
~
and the corresponding divided differences with respect to x,
h2,v[Xo, Xl]
:=
h~(xo,[xo,xl]) :=
hm,u{xo, x1]~-~-j jj= 0
((
+=o,,/,
20)1,=o
h~ xo,[Xo,x,],- 9 -
,
with xo, xl ~ ]R are needed. We shall make available a fast algorithm for the computation of these vectors.
56
G. Plonka / Periodic Hermite spline interpolation
Let the N-periodic B-splines P~,~ of degree m (m >~2) and defect 2 be explained
by the N-periodization of the B-splines N~2,~(v = 0, 1), i.e.,
oo
Nm,~,(x-lN )
(x~,u
= 0, 1).
1=--oo
We put
nm,u(x)
2
:--- (Pm,~,(x
+ k'~'~N-I
/Jk=0
( u = O , 1).
Then by
N-1
Pm,~(x + k)~ k
=
hZ,(x, 27rj/N)
(u = 0, 1,j = 0 , . . . , N -
1),
k=0
with w := exp(-27ri/N), it follows that
h 2 v ( x ) = Punrn,v(X )
Here
FN
(/2 = 0, 1).
(,,,],k~N-1
:= ~,Jj,k=0denotes the Fourier matrix of order N. We obtain
ALGORITHM I
Computation of the vectors h2m,~(x)(u -- 0, 1)
Input:
m
N
splinedegree (meN, m/>2),
period (NEN),
x
Compute P2,(x+j) (u = 0, 1,j = 0 , . . . , N - 1) by the B-spline recursion formula.
Put n,,,~ (x) := (P2v(x q-j))No1 ~]~N (u = 0, 1).
Step 2. Compute h2m~(x) := FNnm~(X) (u = 0, 1) by the fast Fourier transform.
Output: h2m,~,(x)= (h~,~,(x, 27rj/N))j'_-o (u = 0, 1).
Step 1.
Using the fast Fourier transform, the algorithm requires only O(N log N) arithmetic operations. The divided differences h2,v[xo,xl] (u = 0, 1) may be computed
in the same manner. If x0 ~ xl (Ix0 - Xl[ >e), we apply algorithm 1 for x0 and xl
and compute the divided difference by definition. In the case x = x0 = xl
(IX 0 --Xl[ ~s we have to compute the derivatives dhZm,,(x,27rj/N)/dx (j = O,
2
. . . , N - 1, u = 0, 1) using theorem 2.1(iii). Finally, the vectors hm(xo,
xl) and
hE (x0, [x0, Xl]) may easily be computed by the definition and algorithm 1.
3. Periodic fundamental splines
Now we shall give a new explicit representation of periodic fundamental splines
for the generalized Hermite spline interpolation problem (2.1).
G. Plonka / Periodic Hermite spline interpolation
57
T H E O R E M 3.1
Let m, N e N (m~>2) and shift parameters 0<~'0~<'rl~l be given such that
N
ham(T0, ['r0,'rl], 2rrj/N) 7~ 0 (j = 0,... , N - 1) is satisfied. Then Lm,~eS~,2(z)N
(r, = 0, 1),
N-1
2
1 ~ hm(x,[To,~'t],27rj/N
LN,o(x) := -~ ~=o h2m(TO----~,
[7.o"--~'-~l]:21rj/N)
(xeR)
(3.1)
2
21r' N
LN
1 N-1
N-" hZ('r~
m,~(x) := N ~=o h2 (,ro,[,ro,,rl],2~rj/N)
(xeN)
(3.2)
are periodic fundamental splines for the Hermite spline interpolation problem
(2.1), i.e.,
N k + 7"0) =
Lm,o(
~N
0,k,
N k
= 0,
Lm,l(+'r0)
[L~,o(k+.):TO, T1]= 0
(keZ)
O,k (keN) ,
[LN,I( k + ' ) : To, ~-1] = 6N
with
(5N
(1, k=0(modN),
~,k:-- 0, k r
For the coefficients/jN~, 0 and lj,~,1 (u = 0, 1)in the representation
N-1
N
2
= F_,(G,oem,o(x-J)
+
N
j=O
the following hold forj = 0, . . . , N - 1:
lN
1 N-1 h2,1 [7.0, 7.1] (27rk/N)w-Jk
j,o,o = -~ k~.o h2(To, (To,.ri},2rck/N) ,
IN
1 ~-~ h2,o['-o, r,](2rck/N) w-jk
1 N-1
S-"
IL o -
IN
2 ('co, 2r
hm'x
-jk
2
N ) '
1~--~ h2,o('ro,2~rk/N)w -jk
Proof
By theorem 2.3 and the definition it follows for k e Z that
G. Plonka/ PeriodicHermitesplineinterpolation
58
12h2m(To + k,[ro,rl],Drj/N)
L,";'.,o(a:+ To) = ~ j:o h~(;;-o,Pg;i~i-~77~
1 N-lh2m(TO,[To,T1],27rj/N)w-kJ 1N-1
j~O
= -- Z w-kJ
h2m(To, [To, r,],27rj/N)
.:
~U
0,k"
N j~
Further, with h2,,([k + ro,k + r,], [TO,rl], u) := [h2m(k+ , [To, "q], u) "T0, T,] we have
1 ~-~h2m([k + To,k + T,],[To, r,],2rrj/U)
[LNo(k + "): To, 71] = ~ j=o
h2m(TO,[TO,T 1 ] I ~
1~
= N j=o
h~,([To,T,], [T0, T1]
27rj/N)w -jk
h2m(TO,[70, rl], 27rj/N)
= O.
Analogously we find
1 U-I h2m(rO,TO,2~j/U)w_Jk
LN,1(k + r0) = ~
h2m(T0' [TO,rl],ZTrj/U) = O,
/~o
and
=vl-~hm(TO_N-1
___2[,TOT,1]2,~2N
/j )w_-Jk____ _
.
_ _
[L.Nj(k + ") rO, rl] U~=o h2(TO,~T~,~IT,
~
I
1N-I
=-~ j~=OW-kj r0,k'
N
By
N-1
~ e2 ,(x +/),v,
=hm#(X, 2 n l / N )
(u = 0, 1)
j=0
we see that
1 U-1
ZTrl/N)w lj (u = 0,1).
P2, u(x -J) = N E hm#(x,
2
1=o
Thus,
N.(, N
(1N.)
j=0E -N k~=O-~m(-~O, ~O, ~
]
Z h2'~(x' 2rrl / N ) wi
+ ~ --N~=o~~o,7~I]:2~--~/N)JI-Nl~=oh2'l(x'2rrl/N)~`
1
hm, 1
k=0 1=0
j=0
/
G. Ptonka / PeriodicHermite spline interpolation
N,
)
E wJ(l-k)
hZ'~176
1 N-IN-1
59
j=0
1~
= N ~
h2(x,['ro,T1],27rk/N)
N
h2(T0-------~~o,-~l], 2~--~N) = Lm'~
[]
The assertion for L~, 1 is obtained analogously.
4. A n a l g o r i t h m for solving the generalized H e r m i t e spline i n t e r p o l a t i o n
problem
Let m, N e N (m >~2) and 0 < "r0 ~<Tl ~< 1 be given such that
hZ('ro,[To,'rl],ZTrj/N) # O ( j = 0 , . . . , U - 1 )
holds. Then a continuous solution s ~ sN2(z) of the generalized periodic Hermite
spline interplation problem (2.1) reads
N-1
s(xl =
+ yS'>LL(x-j)),
j=0
where y(0) :. . (y~0),
. (0) "~T and y(~) := (y~l),. .., y(Nl)_l)V are given vectors of
. .
,YN-I)
data.
We shall compute this solution efficiently by means of the fast Fourier transform.
Setting
S(t) := (s(t +/))IN; 1 ,
lm,o(t) := (LNo(t + I))N~I ,
lm,l(t) := (LN, l(t + l))N~ l
(te [0, 1]),
we obtain the discrete convolution equation
s(t) = y(O) . lm,o(t) + y(1) . /,.,~(t) ,
and using the convolution theorem of discrete Fourier transform
VNS(t) =
o (FNlm,o(t)) + j,(1) o (FNlm,l(t))
hZ(To, t,27rj/N) )N-1
\hZm(-~o,-~o[-T~,2--~U)J j=o '
=j,(0) o (h]"(t'[T0'Tl]'27rj/N) "~N-l+S0) ~ (
\hZ (To, [To,T1],ZTrj/X)J j=o
with .~(0) := FNy(O) and .~(1) := FNyO)" Thus, we find the following algorithm for
the computation of the Hermite spline interpolant s e Stun2(Z):
ALGORITHM 2
Computationof s(t + k) (k = O, . . . , N -
1; t~[0, 1])
G. Plonka / Periodic Hermite spline interpolation
60
Input:
Step 1 9
m
N
To, 71
y(O),yO)
t
Compute
spline degree (m E 5t, m >/2),
period (N e N),
shift parameters (0 < To ~<'q ~< 1),
vectorsofdata 0,(~ (1) e ~ g ) ,
(re [0, 1]).
2
.
s-1
2 t
9
N-1 and
t, 27rj/N)j_
o , ( h ,,(
, [TO,rl], 27rj/N))i_o
the vectors (h,,(ro,
9
N=I
9
9 9
(h,,2 (TO, ['tO,"q], 27rj/N))j=o
by algorithm
1 and the definition.
Step 2.
Compute
j,0) := FNyO),
j~(o) := FNy(0),
Step 3.
s-
with j,(") :--- (9~"),... ,.y~)_l) T (/: = 0, 1).
Computeforj=0,...,N-1
y)o)L2
,,m (t,
T1],2.j/s)
Poj := hZ (ro, [ro, T1],27rj/N ) ,
^ (1).2
:
Y) nrn(To, t, 27rj/N)
P l j := h2m(7_o,[To, Tt],27rj/U) "
N-I
PutPo := (poj)7__oI andpl := (plj)j= 0 9
Step 4.
Compute
s(t) := FN1 (p0 "}-Pl) "
O u t p u t : s(t) = (s(t + k'?~N-1
::k=o"
2
9
N-1
2
9
N-1
The vectors (hm.(ro,27rj/N))j_
o , (hm,.[ro,
rl](27rj/N))j_
o (v = 0, 1) and (h2(ro,
[To, T1 ], 27rj/N)
)/N=01may be precomputed.
2'-
9
N-1
2
9
N-1
The vectors (hm(t, [To, T1],27rJ/N))j= 0 and (hm(TO, t, 27rj/N))j= 0 may be computed
directly with O(mN) arithmetic operations or by two discrete Fourier transforms
1.0
-4.0
-1,0
1.0
8.0
Fig. 1. Periodic fundamental spline L6S.ofor 7-0= 0.2, rl = 0.3.
G. Plonka / Periodic Hermite spline interpolation
61
1.0
9
i
--4.0
i
~
,
-1.
:
l
i
~"
~
~_-
0
Fig. 2. Periodic fundamental splineL~,1for ~'0= 0.2, ~'l = 0.3.
of length N. In steps 2 and 4 one has to perform three discrete Fourier transforms
of length N, which require O(N log N) arithmetic operations using the fast Fourier
transform.
Remarks
(1) The derivatives of the periodic Hermite spline interpolant may be computed
similarly using theorem 2.1 (iii).
(2) For the computation of the Hermite spline interpolant, not the fundamental
splines LN~ (v = 0, 1) themselves, but the discrete Fourier transforms of lm,~(t)
(v = 0, 1) have to be known.
Setting y(0) = (60,~, 0, 0 , . . . , 0), y(1) = (61,v, 0, 0 , . . . , 0), the fundamental splines
Lm~,~(~' = 0, 1) are obtained in algorithm 2.
5. Conclusions
It has been known for a long time that the periodic and cardinal Lagrange spline
interpolation problem with shifted nodes can be solved with the help of a symbol,
which is formed by a generalized Euler-Frobenius polynomial or by an exponential
Euler spline. We have shown that for solving Hermite spline interpolation problems, too, the introduction of a corresponding symbol is very helpful. The symbol
has been found to be the main tool for representing the fundamental splines and
the Hermite spline interpolant.
The considered periodic Hermite spline interpolation problem can be generalized to higher defects r > 2. Using the normalized B-splines with multiple knots, the
generalized Euler-Frobenius polynomials of multiplicity r can be introduced.
Then the periodic fundamental splines and the Hermite spline interpolant can be
represented in the same way as for r = 2.
62
G. Plonka / Periodic Hermite spline interpolation
References
[1] K. Jetter, S.D. Riemenschneider and N. Sivakumar, Schoenberg's exponential Euler spline
curves, Proc. Roy. Soc. Edinburgh 118A (1991) 21-33.
[2] G. Merz and W. Sippel, Zur Konstruktion periodischer Hermite-Interpolationssplines bei
~iquidistanter Knotenverteilung, J. Approx. Theory 54 (1988) 92-106.
[3] G. Plonka, Periodic spline interpolation with shifted nodes, J. Approx. Theory, to appear.
[4] G. Plonka and M. Tasche, Efficient algorithms for periodic Hermite-spline interpolation, Math.
Comp. 58 (1992) 693-703.
[5] M. Reimer, Zur reellen Darstellung periodischer Hermite-Spline-Interpolierender bei
fiquidistantem Gitter mit Knotenshift, Splines in Numerical Analysis, eds. J.W. Schmidt and H.
Sp~ith (Springer, 1989) pp. 125-134.
Gerlind Plonka-hoch