Upcoming SlideShare
×

# 2002 santiago et al

155 views

Published on

Published in: Technology
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
155
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
2
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 2002 santiago et al

1. 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. 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. 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. 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. 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. 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. 7. 00pt 0 Figure 1: f(x) = cos(x) + (x−3) (x2+1) 00pt 0 Figure 2: f(x) = 1 (25x2+1) 7
8. 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 8