1. A Note on Polynomial Interpolation
W. A. G. Cecilio, C. J. Cordeiro, I. S. Mill´eo
C. D. Santiago, R. A. D. Zanardini, J. Y. Yuan∗
Departamento de Matem´atica
Universidade Federal do Paran´a
Centro Polit´ecnico, CP: 19.081
CEP: 81.531-990, Curitiba, Paran´a
Abstract
The Neville’s algorithm and the Aitken’s algorithm are successively lin-
ear interpolation approach to high degree Lagrangian interpolation. This
note proposes a new approach with iteratively quadratic interpolation to
high degree Lagrangian interpolation. The new algorithm here is cheaper
(about 20% cheaper) than the Neville’s algorithm. Several functions were
tested. Numerical experiments coincide with the theoretical analysis. The
combination of linear approach and quadratic approach is considered too.
AMS subject classification:
Key words: Aitken’s Algorithm, Neville’s Algorithm, Lagrangian Interpolation,
Successively Quadratic Interpolation, Successively Linear Interpolation, Polynomial
Interpolation, Parallel Computation
1 Introduction
The classic interpolation method is the Lagrangian polynomial interpolation,
which is a linear combination of basic functions, given by
p(x) ≡
n∑
i=0
yiLi(x) ≡
n∑
i=0
yi
n∏
ylek=0
k̸=i
(x − xk)
(xi − xk)
(1)
∗The work of this author was supported by CNPq and FUNPAR, Brazil
1

2. where (xi, yi = f(xi)) are given, and
Li(x) =
(x − x0) · · · (x − xi−1)(x − xi+1) · · · (x − xn)
(xi − x0)ots(xi − xi−1)(xi − xi+1) · · · (xi − xn)
, (2)
basic functions.
This interpolation formula needs all ordinates yi to find an estimate to the
solution of the interpolation problem. For large problems this method is not
efficient because it cannot make use of the previous interpolating result when we
add more interpolating points to get better result. To achieve better precision
of the solution by adding more points, Aitken developed a method to make use
of the previous interpolating results to save multiplications. The method can
generally be defined as follows[1, pp.40]:
Ii(x) = yi (3)
Ii0,i1,...,ik
(x) =
(x − xi0 )Ii1,i2,...,ik
(x) − (x − xik
)Ii0,i1,...,ik−1
(xik
− xi0 )
. (4)
This algorithm generates a sequence Ii0,i1,...,in which converges to an ap-
proximation to y = f(x) for a given x.
For the sake of the computational consideration, Neville established a similar
process to obtain the desired interpolation. The Neville’s algorithm is theoret-
ically equivalent to the Aitken’s algorithm but more efficient in computational
point of view[?]. A variation of the Neville algorithm proposed in [?] consists of
the following recursion.
Setting Ii+k,k = Ii,i+1,...,i+k, we have
Ii0
= yi (5)
Iik = Ii,k−1 +
Ii,k−1 − Ii−1,k−1
x−xi−k
x−xi
− 1
.belstoer (6)
We can obtain another alternative to this recursive process as follows.
Iik = Ii−1,k−1 +
(x − xi−k)(Ii−1,k−1 − Ii,k−1)
xi−k − xi
belline (7)
2

3. which has an equivalent computational gain compared with (??).
In fact, the Aitken’s algorithm and the Neville’s algorithm are an iterative
process of linear inetrpolation to approach the high degree Lagrangian interpo-
lation polynomial. At each step, just two points are used. This idea motivates us
to consider a new iterative process with quadratic interpolation since quadratic
interpolation has better approximation property than linear interpolation, and
just requies three points which is not expensive.
To obtain better results in large interpolation problems we propose here a
method based on quadratic approximations successively. The new approach re-
duces the computational costs and the CPU time greatly compared with the
Aitken’s algorithm and the Neville’s algorithm. we shall present our new ap-
proach in next section and some numerical results in the last section.
2 Successively Quadratic Interpolation
The quadratic interpolation formula is given by
Ii+2,j+2 =
(x − xi+2)(x − xi−j+1)
(xi−j − xi+2)(xi−j − xi−j+1)
Ii,j+
(x − xi−j)(x − xi+2)
(xi−j+1 − xi−j)(xi−j+1 − xi+2)
Ii+1,j +
(x − xi−j)(x − xi−j+1)
(xi+2 − xi−j+1)(xi+2 − xi−j)
Ii+2,j
(8)
with interpolation conditions
Ii+2,j+2(xi−j) = Ii,j,
Ii+2,j+2(xi−j+1) = Ii+1,j,
e
Ii+2,j+2(xi+2) = Ii+2,j.
This formula is not efficient under the computational point of view because it
requires 12 multiplications at each iteration while the Neville’s formula obtians
the same quadratic polynomial with just 6 multiplications.
3

4. Now, rewritting (??) as
Ii+2,j+2 =
x − xi+2
xi+2 − xi−j+1
[(x − xi−j+1)(
Ii+2,j − Ii+1,j
xi+2 − xi−j+2
+
Ii+1,j − Ii,j
xi−j+1 − xi+1
)
+Ii+2,j − Ii,j] + Ii+2,j. (9)
we obtain an equivalent expression with only 5 multiplications at each iteration
to obtain the desired quadratic interpolation polynomial in (??) to reduce the
cost of the interpolation. This represents a computational gain about 20%.
Then with the iteratively quadratic interpolation the partial polynomials are
linked in the following table:
k = 0 1 2
x0 y0 = I0(x)
x1 y1 = I1(x) I012(x)
x2 y2 = I2(x) I123(x) I01234(x)
x3 y3 = I3(x) I234(x)
x4 y4 = I4(x)
(10)
In fact, the sequence {I01...k}n
k=0 converges to the value f(x) for given x.
ITherefore, we establish the following iteratively quadratic interpolation algo-
rithm.
ALGORITHM 2.1 (QII Algorithm)
Given (xi, yi), p e ε
Ii,1 = yi, i = 0, 1, . . . , n
While
∥Ii+2,j+2−Ii,j ∥
∥Ii,j ∥ ε
For j = 1, 3, 5, ..., i
Ii+2,j+2 = x−xi+2
xi+2−xi−j+1
[(x−xi−j+1)(
Ii+2,j −Ii+1,j
xi+2−xi−j+2
+
Ii+1,j −Ii,j
xi−j+1−xi+1
)+
Ii+2,j − Ii,j] + Ii+2,j
i = i + 1
end
4

5. end
Another algorithm of combining linear and quadratic interpolating formulas
iteratively is given as follows.
ALGORITHM 2.2 (QLII Algorithm)
Given (xi, yi), p e ε
Ii,1 = yi
Do
For j = 1, 3, 5, ..., i
Ii+1,j+1 = Ii,j +
(x−xi−j+1)(Ii,j −Ii+1,j )
xi−j+1−xi
end
Verify the convergence
i = i + 1
For j = 1, 3, 5, ..., i − 1
Ii+2,j+2 = x−xi+2
xi+2−xi−j+1
[(x−xi−j+1)(
Ii+2,j −Ii+1,j
xi+2−xi−j+2
+
Ii+1,j −Ii,j
xi−j+1−xi+1
)+
Ii+2,j − Ii,j] + Ii+2,j
end
Verify the convergence
i = i + 1
For j = 1, 3, 5, ..., i − 2
Ii+2,j+2 = x−xi+2
xi+2−xi−j+1
[(x−xi−j+1)(
Ii+2,j −Ii+1,j
xi+2−xi−j+2
+
Ii+1,j −Ii,j
xi−j+1−xi+1
)+
Ii+2,j − Ii,j] + Ii+2,j
end
end
3 Numerical Examples
There are some classical problems in the literature where bad results appear
when interpolating methods are used to approximate function values[?]. In this
5

6. section we shall give some numerical results for such functions with our new
algorithms, and also for comparison with performance of the Neville’s algorithm.
In Table 1 the following functions are considered
• f1(x) = cos(x) + (x−3)
(x2+1) ;
• f2(x) = 1
(25x2+1)
• f3(x) = −196
1125 x8
+ 144
125 x6
− 2777
1500 x4
− 569
4500 x2
+ 1
with the data vector x = −100 + 0.3i, i = 1, 2, . . . , 666. Here the interpolating
point is p = 27.93 and ε = 10−8
is a given tolerance.
Test functions Methods CPU time Solution
Neville 66.13 1.5694e × 006
f1(x) QII 42.65 1.5694e × 006
QLII 63.52 1.5694e × 006
Neville 66.06 -2.75533e × 008
f2(x) QII 42.55 -2.75533e × 008
QLII 63.47 -2.75533e × 008
Neville 62.5 -6.39708e × 010
f3(x) QII 40.53 -6.39708e × 010
QLII 60.16 -6.39708e × 010
Table 1: Numerical results.
All computations were done by MATLAB on Pentium III 450MHz Personal
Computer at CESEC Laboratory, the Federal University of Paran´a, Brazil.
Acknowledgments
The first five authors like to give their sincere thanks to Professor Jin Yun
Yuan for introducing us to the topic, to Nelson Haj Mussi J´unior for his pre-
liminaries discussions.
References
[1] J. Stoer and R. Bulirsch: Introduction to Numerical Analysis, Springer-
Verlag, 1980.
6

7. 00pt 0
Figure 1: f(x) = cos(x) + (x−3)
(x2+1)
00pt 0
Figure 2: f(x) = 1
(25x2+1)
7

8. 00pt 0
Figure 3: f(x) = −196
1125 x8
+ 144
125 x6
− 2777
1500 x4
− 569
4500 x2
+ 1
[2] M. C. K. Tweedie, A modification of the Aitken-Neville Linear Iterative
Procedures for Polynomial Interpolation, Math. tables and Other Aids to
Computation, 8(1954) 13-16.
[3] V. Pan, New Approach to Fast Polynomial Interpolation and Multipoint
Evaluation, Computers Math. Applic. Vol. 25, No. 9, pp. 25-30, 1993.
[4] R. P. Agarwal e B. S. Lalli, Discrete Polynomial Interpolation Green’s Func-
tions Maximum Principles Error Bounds and Boundary Value Problems,
Computers Math. Applic. Vol. 25, No. 8, pp. 3-39, 1993. 1999 Academic
Press, Inc., 1993
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