63
Internat. J. Math. & Math. Sci.
Vol. 10 No..1
(1987)63-67
LOCAL BEHAVIOUR OF THE DERIVATIVE OF A
MID POINT CUBIC SPLINE INTERPOLATOR
H.P. DIKSHIT and S.S. RANA
Department of Mathematics and Computer Sciences
R.D. University
Jabalpur 482001 India
(Received January 16, 1985 and in revised form June 5, 1985)
In the present paper, we obtain an asymptotically precise estimate for the
ABSTRACT.
between the cubic spllne interpolating at the mid points
derivative of the difference
of a uniform partition and the function interpolated.
KEY WORDS AND PHRASES.
polation.
Local behaviour, derivative, mid point, cubic spline inter-
Primary 41 A05, 65 DO7
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
1.
INTRODUCTION.
P
such that x
functions
0=x
<x
<
< x
p for all
xi_
i
s
o
of [0,I] defined by
P
Consider a partition
P
defined over
n
Let
i
such that the restriction
is a polynomial of degree 3 or less for i
P
spllnes over
S(3,P)
be the class of all plecewlse polynomial
P3
1,2
n.
[xi_
i]
s
of s over
x
i
The class of periodic cubic
is defined by
{s: s e
s e C
P3
2
[0,i],
s
(j)
(0)
s
(j)
(,I), j
0,1,2}.
Under certain restrictions on the choice of y, Mehr and Sharma [I] have studied
convergence properties of the interpolant from S(3,P) matching a given function at the
points
xi-I +
Yi
(0=<y
yp
I), i
1,2,
n.
However, the interpolation at the
1/2 is not covered in [i]. Assuming
to a nonnegatlve measure dg, where g(x+p)
K (constant), one of the authors
g(x)
(Dikshit [2], Theorem 2) has proved the following which covers the case y
I/2.
THEOREM I. Suppose n is odd and
mid points, which corresponds to the choice
I p0
(6px
2
4x
3
p3)dg
Then there exists a unique
ixf
xi_
(f(x)
s
s(x))dg
I p0 dg
0
y
> 0
(I I)
S(3,P), which satisfies the interpolatory condition:
1,2
0, i
We observe that the case in which
g
(1.2)
n.
has a single jump of
at
p/2, (I.I) holds
and the interpolatory condition (1.2) reduces to the condition:
s(t i)
f(t i)
t
i
(x i +
Xi_l)/2,
i
n.
(1.3)
H.P. DIKSHIT and S. S. RANA
64
It may be mentioned that the derivative of a cubic spline interpolator has been
used for smoothing of histograms (see Boneva, Kendall and Stefanov [3] and Schoenberg
4
C and its unique spline interpolant s
[4]). Considering a function f
S(3,P)
n
<
matching at the knots
Rosenblatt [5] has obtained asymptotically
xi >i=0
In the present paper, we obtain a similar precise
f
precise estimate for s
estimate for the cubic spline interpolating at the mid points between the successive
knots.
2.
ERROR BOUNDS.
4
Let f
C and be periodic with period I.
In this section, we shall estimate
Let the number of mesh points of
(s
f)
satisfying the interpolatory condition (1.3).
Let
even.
of [M I, M
2
and
F
where
12p-2
l
j
f(tj+l)
aj
Mn_l]
and [F I, F 2,
I
j
n
+
2f(tj)
(-i)
=+i
folows
i+j
+ 22
6i_lj
cij
denote the transpose
respectively, with M i
(2.1)
j
s e
S(3,P) for which
+
C
(cij)
F"
i+lj
F"n_l]
respectively, with
F
where 2 8
F
f"(xi_ I)
+ 22
mi
E
En_ I]
2
and
e"(x i) and
f"(xi)
(2 4)
f"(Xi+l).
[2], p. 108) and we first obtain the following
preliminary results for determining the elements of
LEMMA 2.1.
For this we
B.1
i
+
i
is of course invertible (see
C
e"(xi).
(2.3)
are the transpose of the matrices [E
F
[Fy,
Thus it
F"
CE
and
O.
is given by
e’, we first determine an upper bound for
notice that the equation (2.2) yields
E
s" (0)
(2.2)
In order to estimate
where
s"(x i)
F
ij
be
that
where n-I x n-i coefficient matrix
2
F
and
P
is the cubic spline
s
For convenience, we consider in the rest of
f(tj_l).
C M
M
Fn_l]
this section, the class S*(3,P) of splines
from the proof of Theorem
where
For given real numbers a and b with b
C
-.
->a,
let
Dn(a,b)
(dij)
be a n x n
matrix with
(l-a)
dij
and
(b2
8
a
2 8
6i-lj
+ a2
b
6ij
+
a
6i+lj
(2.5)
Then
(b+g) n+l
Dn(a,b)
Proof of Lemma 2.1.
+ 2
(b-
)n+l
It is easily seen that
(2.6)
D (a,b)
n
satisfies the difference
equation:
D (a,b)
2 b
n
with
D_l(a,b)
O,
Dn_ l(a
Do(a,b)
b)
I,
+ a(l-a)
D l(a,b)
Dn_2(a,b)
2 b.
0
The lemma follows from the
DERIVATIVE OF A MID POINT CUBIC SPLINE INTERPOLATOR
65
above difference equation by using the induction hypothesis.
LEMMA 2.2.
q
where r
-n
(2b+r)
Qn(a, b)
(e b)
r2n
i/4)
and
IDn_l(I/2
IQl(e,b)
n
(I/2 b) by
r2n-2
+ ar (I
(2.7)
1/2).
It follows from the definition of
4b
IQo(e,b)
with
2
is the matrix obtained from D
b)
Then
2 b (I
-2(b-(b
(-i/2q)
IQn
Qn(
in its first row.
Proof of Lemma 2.2.
2
I/2 and
Suppose b
replacing I/2 by
-IDn-2 (i/2
b)
Qn(a,b)
that
b)
(2 8)
The result of Lemma 2.2 follows from (2.8)
2b.
by an application of Lemma 2.1.
LEMMA 2.3.
The coefficient matrix of (2.2) is invertible and if C
8.. can just be approximated asymptotically as n
-I
(Sij),
by
2 r lj-il (22 + r)
where 0 < e < i/n
REMARK 2.1.
then
(2.9)
2 3/- II.
j/n < l-e and r
It is interesting to note that the estimate (2.9) is sharper than that
obtained in terms of the infimum of the excess of the positive value of the leading
diagonal elements over the sum of the positive values of other elements in each row.
For, adopting the latter approach we see from (2.2) that
II C-I II
0.i whereas (2.9)
together with the fact that
Z
i
shows that the
2
22+r
r
lj-il_
II C-i II
Proof of Lemma 2.3.
2 (l+r)
(l-r) (22+r)
does not exceed 0.084.
I/2 in
ii and a
Taking 2b
Qn(a,b)
observe that the
coefficient matrix C satisifes the following difference equation
4
ICI
44
IQn_ 2
(1/2,11/2)
I- IQn_ 3
(I/2,11/2)
I.
(2.10)
Thus, it follows from Lemma 2.2 that
(ll+r) q
-n+2
ICI
In order to estimate
of the transpose matrix.
(ll+r/2)
C
-i
2
r
(0ij)
2n-6
(llr + i/2)
2
(2.11)
we obtain the elements
Thus, for 0 < i
j
n-2
or i
0ij
from the cofactors
0 (confer Ahlberg, et
j
al. [6], pp. 35-38)
(qr) j-i
ICI O ij
ICI Ooj
Qi(I/2,11/2) Qn_2_j(I/2,11/2)
(qr)J Qn-2-j
and for 0 < j < n-2,
(I/2,11/2).
Thus using the result of Lemma 2.2 and (2.10), we observe that for 0 < i j n-2,
(ll+r)
(l-r2n)sij
(l-r2n)Sin_2
(ll+r/2)
(ll+r/2)
j-i
(l-r2n-2j-2),
(l-r 2i+2)
r
n-2-i
(l-r
2 i+2
2n-2-2j
(ll+r/2)
and
r
2
(l-r2n)8oj rj (l-r
(l-r2n)8on_2 rn-2 (ll+r)
for 0
i
-<
for 0 < j < n-2
n-2
H. P. DIKSHIT and S. S. RANA
66
From the above expressions for 8..
C
Since
the result of Lemma 2.3 follows directly.
or more precisely
is invertible, it follows from the proof of Theorem
S*(3,P) satisfying the interpolatory
from (2.3), that there exists a unique spline s
condition (1.3).
f
S* (3,P) be the spline interpolant of a
Let s
THEOREM 2.1.
satisfying (1.3).
f(4)
Let
+
f(4)
f’(x)
10.08(xi_x
2
such that 0 < x <
x
tion, then for any fixed point
s’(x)
(x)
x)
[((ti+
12.92(x_xi_l)2
+
periodic function
exist and be a nonnegative monotonic continuous func-
4
(t
13.92p
i
-x)4)/p
2) /4]
p((xi+l-X)
2
/24 + o(p 3)
(2.12)
as n +=.
Proof of Theorem 2.
I.
We first proceed to obtain the derivative s" of the mid
S*(3,P) of f.
point spline interpolant s e
2
2
=-Mi_l(Xi-X) + Mix-xi_ I) +
2p s’(x)
’s
C
where the constants
[Xi_l,Xi],
Considering the interval
observe that, since s" is quadratic in the interval
we
[Xi_l,X i]
2pC
the
are to be determined by
(2.13)
i
requirement that s e C
2
[0,I].
Thu s,
M
p
i
C
Ci+
(2.14)
i
and we have
6 p s(x)
Mi_l(Xi-X)
3
+
Mi(x-xi_l)
3
+
3pCi(2x-xi-xi_l) +
6pb
(2.15)
i.
Again using the continuity requirement, we get
p (C
+
i
Ci+ I)
2
-b
(bi+
(2.16)
i)
(2.16) and the interpolatory condition (1.3), we have
Using (2.14)
Mi_l[p2-24(xi-x) 23 + 24Mi [(x-xi_l)2
48ps" (x)
p
]2
_p2Mi+l
+ 48
Thus
replacing
M
[f(ti+ I) f(ti)3
by e" (x i) in (2.17),
i
[p2-24(xi-x) 2 ]
48 p s’(x)
p
where
Ri(f)
2e,, (xi+ I)
[p2 24(xi_x) 2
+ 48
[f(ti+ I)
e" (xi_ I) + 24
+
Ri(f)
f,,
(xi_ I) +
uj
(x-xi_ I) 2_p2 ]e"(x i)
(2.18)
24
[(x-xi_l)2-p 2]
appropriate points of
necessarily the same at each occurence.
f’(x) +
+
we see that
f"(x i)
p
2 f,,
(xi+
I)
f(ti)]
Fbr convenience, we denote by
Ri(f)/48 p
(2.17)
24(xi-x)
f(4)(u i)
2
(xj_2,xj+ I)
which are not
Thus by Taylor’s Theorem, we have
[{(ti+l-X)4-(ti-x)4}
(Xi_l-X) 2}
/4] /24 + o(p
3)
/p- p
{(Xi+l-X)
(2.19)
DERIVATIVE OF A MID POINT CUBIC SPLINE INTERPOLATOR
Now writing 2B
i
i
n
J E--I
j=i+l
n
=
2F’.’1
Z
j=i+l
Ta1or’s Theorem,
so that by
i
(-i)
n
j -i ),
(24p-2j
(-I) i+j f
+ Y.
j=l
j=i+l
Z
(_
i+j
+
Bj
(4)
(u.) + o(p
Recalling the equation (2.3) and noticing that C
(e" (xi))
say, where
nx
as n
l
IR-il
m
(T I) +
(T2),
m
(eiR
F
-I
Suppose that
x
=-
xn and n-i
-
is a fixed given point in
(l-x)n.
2).
(2.20)
we have
R
denotes the largest integer not greater than
i
Bj_ I)
(Sij)
is a sufficiently large positive integer.
m
separately.
where
+
l
IR-i
we have
we have
i
p2
F,,
)(-1) i+j (
r.
We shall estimate T and T 2
[nx] /n
(0, I) and let x
nx.
Then it is clear that
Now using the monotonicity of
Abel’s Lemma to the inner sums, we have for some positive constant
(TI)I
67
m I(0-29) m
p
f(4)
and
applying
K
2
(2.21)
by virtue of Lemma 2.3.
Next we see that for the
x
R
x
vales
of R occuring in T
2
(2.22)
0(p)
Thus, using the hypothesis that
f(4)
is continuous and applying the result of Lemma
2.3,
we have
l(T2)-(2/(22+r))
Z
R-il
< m
Combining the estimates of
rIR-ilp 2 (-jlR
(TI)
and
(T 2)
+
n
l
)(-I) j+Rf
(4) (x)
o(p
2).
j=R+I
and noticing that
m
is arbitrary, we
complete the proof of Theorem 2.1 in view of (2.18).
REFERENCES
I.
2.
3.
MEIR, A. and SHARMA, A.. Convergence of a Class of Interpolatory Splines, Jm Approx.
Theory ! (1968), 243-250.
DIKSHIT, H.P. On Cubic Interpolatory Splines, J.. Approx. Theory 22 (1978), 105-110.
BONEVA, L.I., KENDALL, D.G., and STEFANOV, I. Spline Transformations: Three New
Diagnostic Aids for the Statistical Data Analyst, J..Royal Stato Soc. Ser. B
33 (1971), 1-70.
4.
SCHOENBERG, l.J.
5.
ROSENBLATT, M. The Local Behavior of the Derivative of Cubic Spline Interpolator,
J. Approx. Theory.
(1975), 382-387.
AHLBERG, J.H., NILSON, E.N., and WALSH, J.L. The Theory of. Splines and Their
Applications, Academic Press, New York, 1967.
6.
Splines and Histograms in Spline Function and Approximation
Theory Proceeding, Editors Meir, A. and Sharma, A. University of Alberta,
Edmonton, 1972, ISNM 21 (1973), Birkhuserverlag.