Lebesgue Constant Minimizing Bivariate Barycentric Rational Interpolation

Transcription

1 Appl. Math. If. Sci. 8, No. 1, (2014) 187 Applied Matheatics & Iforatio Scieces A Iteratioal Joural Lebesgue Costat Miiizig Bivariate Barycetric Ratioal Iterpolatio Qiaji Zhao ad Bigbig Wag School of Sciece, Ahui Uiversity of Sciece ad Techology, Huaia , P.R.Chia Received: 5 Ju. 2013, Revised: 4 Oct. 2013, Accepted: 5 Oct Published olie: 1 Ja Abstract: The barycetric for is the ost stable forula for a ratioal iterpolat o a fiite iterval. The choice of the barycetric weights ca esure the absece of poles o the real lie, so how to choose the optial weights becoes a key questio for bivariate barycetric ratioal iterpolatio. A ew optiizatio algorith is proposed for the best iterpolatio weights based o the Lebesgue costat iiizig. Several uerical exaples are give to show the effectiveess of the ew ethod. Keywords: Lebesgue costat, bivariate, barycetric ratioal iterpolatio, weight 1 Itroductio Iterpolatio is oe of the iportat tools of atheatics. It ca solve ay origial probles like differetiatio, ultistep ethods for ordiary differetial equatios, collocatio ethods for partial differetial equatios, etc. So iterpolatio is a key subject i uerical aalysis. I recet years, barycetric ratioal iterpolatio has bee oe of the focuses for researchers [1, 2]. As a iportat ethod for o-liear approxiatio, it possesses various advatages i copariso with other iterpolatio, such as sall calculatio quatity, good uerical stability, o poles ad o uattaiable poits [3]. I 1945, W. Taylor discovered the barycetric forula for evaluatig the iterpolatig polyoial [4]. I 1984, W. Werer preseted the barycetric ratioal iterpolatio [5]. The barycetric ratioal iterpolatio ca have o poles ad o uattaiable poits by cotrollig the barycetric weights [6]. The barycetric ratioal iterpolatio is deteried whe its weights are give. How to choose the optial weights becoes a key questio for barycetric ratioal iterpolatio. I the sigle-variable case this proble has bee solved by usig a quatity of differet approaches. Oe of the, the iiizig Lebesgue costat approach, has bee developed i a series of papers [6,7]. I 1996, Berrut obtaied the optial weights o a fiite iterval by iiizig the Lebesgue costat for the give odes [6]. I this way, the liearity of the iterpolatio process with respect to the iterpolated fuctios is preserved, ad it ca be writte as fuctio of barycetric weights. Lebesgue costat is based o Lagrage bases of polyoials. I this paper we study the proble of two-variable barycetric ratioal iterpolatio. The reaso for our iterest i this proble is that i a ore geeral cotext, barycetric ratioal iterpolatio of ultivariate fuctios play a cetral role i odel reductio of liear systes which deped o paraeters. A ew optiizatio algorith for bivariate barycetric ratioal iterpolatio is preseted. A crucial optiizatio algorith is obtaied for the best iterpolatio weights based o the iiizig Lebesgue costat. Miiizig Lebesgue costat is bee as the objective fuctio, the weights are bee as decisio variables ad satisfy soe costrait coditios to esure the bivariate barycetric ratioal fuctio has o poles ad o uattaied poits ad uiqueess. The solutio of the optiizatio odel is obtaied usig the software LINGO. The paper is structured as follows. Sectio 2 is dedicated to the barycetric ratioal iterpolatio. I Sectio 3 our ai tool, the Lebesgue costat, is itroduced by eas of Lagrage bases of polyoials. I particular, after a brief overview of the sigle-variable case, the two-variable extesio is developed. I Sectio 4, soe uerical exaples are preseted which illustrate the effectiveess of the ew ethod. Correspodig author e-ail: qiajizhao@yahoo.co.c; wagbigbig510811@163.co

2 188 Q. Zhao, B. Wag: Lebesgue Costat Miiizig Bivariate Barycetric... 2 Barycetric ratioal iterpolatio I 1984, Scheider ad Werer have bee the first to deterie barycetric represetatios of ratioal iterpolatios. 2.1 Uivariate barycetric ratioal iterpolatio Give +1 utually distict poits x 0,x 1,,x ad fuctio values f 0, f 1,, f, the ratioal fuctios r (x)= w i f i, w w i 0 (1) i iterpolate the values f i at the poits x i for ay ozero weights w i, i other words r (x i )= f i [4]. Fro the above, as log as the weights are ot all zero, it will ot appear uattaiable poits, ad it is easy to see that the degree i deoiator ad uerator of r (x) is at ost ad the barycetric weights are oly deteried up to a ultiplicative costat. A ecessary coditio for the barycetric weights to satisfy r (x) has o poles i[x 0,x ] is w i w i+1 < 0,,1,, 1. (2) 2.2 Bivariate barycetric ratioal iterpolatio I this sectio, the barycetric ratioal iterpolatio is geeralized to the two-variable case. Lea 2.1. [9]. Let x ={x i,1,,} (a,b), x 0 < x 1 < <x, y y 0 < y 1 < <y, ad, x,y ={y j j= 0,1,,} (c,d), ={(x i,y j ),1,,, j= 0,1,,} (a,b) (c,d), f(x i,y j )= f i, j (,1,,, j= 0,1,,). The here R(x,y)= r i (y)= w i w i r i (y) u j, w i 0 (3) y y j f i, j u j, u j 0. (4) y y j All the weights w i ad u j are both ozero so that the bivariate barycetric ratioal iterpolat has o uattaied poits. Ad the weights w i ad u j are chose for the ratioal fuctio r(x,y) has o poles i fiite iterval. Lea 2.2. [5]. Let x 0 < < x,w i 0,i = 0,,, y 0 < <y,u j 0, j= 0,,. If r(x,y) has o poles i D (x 0,x ) (y 0,y ), the sig(w i ) sig(w i+1 )= 1,,1,,, sig(u j ) sig(u j+1 )= 1, j= 0,1,,. Furtherore, for esurig the uiqueess of the optiizatio solutio, we take the orative costrait w i =1 (5) u j = 1. (6) 3 Lebesgue costat iiizig bivariate barycetric ratioal iterpolatio Theore 3.1. Let x 0,x 1,,x be +1 distict poits i [a,b]. The liear projectio P which i C[a,b] associates with ay fuctio f the polyoial P f P iterpolatig f betwee the x k s has the or P = Λ = ax a x b l k (x) (7) λ (x)= l k(x) is called the Lebesgue fuctio of the approxiatio operator ad Λ is Lebesgue costat [8]. Fro the above, l k (x) is Lagrage fudaetal polyoial, it ca be writte as l k (x)=w k (l(x)/(x x k )) here w k = 1/ (x k x i ),i k ad l(x)=(x x 0 )(x x 1 ) (x x ). So we ca get the Lebesgue costat of uivariate ratioal fuctio w k x x k Λ = ax a x b l k (x) = ax a x b w k. (8) x x k The Lebesgue costat of the uivariate ratioal fuctio be geeralized to the bivariate case as follows w i u j ( )(y y j ) Λ, = ax a x b,c y d w i u. (9) j ( )(y y j )

3 Appl. Math. If. Sci. 8, No. 1, (2014) / Bivariate barycetric ratioal iterpolat is deteried whe weights are give. The key questio is how to choose weights by iiizig the Lebesgue costat. The weighs are bee as decisio variables ad satisfy soe costrait coditios to esure the bivariate barycetric ratioal fuctio has o poles ad o uattaied poits ad uiqueess. The optiizatio odel for optial iterpolatio weights be costructed as follows i = ax a x b,c y d w i u j ( )(y y j ) w i u, j ( )(y y j ) sig(w i ) sig(w i+1 )= 1,,1,,, sig(u j ) sig(u j+1 )= 1,,1,,, w i 0,,1,,, u j 0, j= 0,1,,, w i =1, u j =1. Fig. 1: The iterpolated fuctio f 1 (x,y). The optial weights ca be obtaied by LINGO software. 4 Nuerical exaples Exaple 4.1. Let f 1 (x,y) = 1 1+x 2 + y 1+y 2 as the iterpolated fuctio ad x 0 = 0,x 1 = 0.25,x 2 = 0.5, y 0 = 0.5,y 1 = 0.75,y 2 = 1 as iterpolatio poits. We ca get the optial barycetric weights by the LINGO software as follows Fig. 2: The iterpolatio fuctio with the ew ethod. w 0 = ,w 1 = ,w 2 = u 0 = ,u 1 = ,u 2 = I order to show the effectiveess of the ew ethod, we give figures of the iterpolatio errors (see Fig. 4, Fig. 5), bivariate barycetric ratioal iterpolat (see Fig. 2) ad the iterpolatio fuctio with the ethod i paper [10] (see Fig. 3) by software MATLAB. Exaple 4.2. Let f 2 (x,y) = si(x + y 2 ) as the iterpolated fuctio ad let x 0 = 0, x 1 = 0.25, x 2 = 0.5 ad y 0 = 0.5,y 1 = 0.75,y 2 = 1 as the iterpolatio poits. We ca also get optial barycetric weights by the LINGO software as follows w 0 = ,w 1 = ,w 2 = u 0 = ,u 1 = ,u 2 = Fig. 3: The iterpolatio fuctio with the ethod i paper [10].

4 190 Q. Zhao, B. Wag: Lebesgue Costat Miiizig Bivariate Barycetric... Fig. 4: The iterpolatio error with the ew ethod. Fig. 7: The iterpolatio fuctio with the ew ethod. Fig. 5: The iterpolatio error with the ethod i paper [10]. Fig. 8: The iterpolatio fuctio with the ethod i paper [10]. Fig. 6: The iterpolated fuctio f 2 (x,y). Fig. 9: The iterpolatio error with the ew ethod.

5 Appl. Math. If. Sci. 8, No. 1, (2014) / Fig. 10: The iterpolatio error with the ethod i paper [10]. Fig. 12: The iterpolatio fuctio with the ew ethod. Fig. 11: The iterpolated fuctio f 3 (x,y). Fig. 13: The iterpolatio fuctio with the ethod i paper [10]. So soe figures like above to show the effectiveess of the ew ethod are show i Figure 6, Figure 7, Figure 8, Figure 9 ad Figure 10. x2 Exaple 4.3. Let f 3 (x,y) = e +y as the iterpolated fuctio. Let x 0 = 0.5,x 1 = 0.25,x 2 = 0 ad y 0 = 0,y 1 = 0.25,y 2 = 0.5 as the iterpolatio poits. We ca get a set of optial weights by software LINGO fro the above optiizatio odel. w 0 = ,w 1 = ,w 2 = u 0 = ,u 1 = ,u 2 = We ca also get soe figures (see Fig. 11, Fig. 12, Fig. 13, Fig. 14 ad Fig. 15) like above to show the effectiveess of the ew ethod. Obviously, the ew ethod is better. Fig. 14: The iterpolatio error with the ew ethod.

6 192 Q. Zhao, B. Wag: Lebesgue Costat Miiizig Bivariate Barycetric... Refereces Fig. 15: The iterpolatio error with the ethod i paper [10]. 5 Coclusios I this paper, the optiizatio algorith odel for bivariate barycetric ratioal iterpolatio is studied. A ew optiizatio algorith is give for the best iterpolatio weights based o the Lebesgue costat iiizig. Miiizig Lebesgue costat is the objective fuctio; the weights are decisio variables ad satisfy soe costrait coditios to esure the bivariate barycetric ratioal fuctio has o poles ad o uattaied poits ad uiqueess. The solutio of the optiizatio odel is obtaied usig the software LINGO. I future work, the shape cotrol i bivariate barycetric ratioal iterpolatio based o this ew ethod will be studied. Ackowledgeets The authors wish to thak the helpful coets ad suggestios fro y teachers ad colleagues i Service coputig Lab. We would like to thak the support of the Natioal Natural Sciece Foudatio of Chia uder Grat No , No , No ad No , the Natioal High-Tech Research ad Developet Pla of Chia uder Grat No.2009AA01Z401, the Natural Sciece Foudatio of Educatioal Goveret of Ahui Provice of Chia (KJ2009A50, KJ2007B173, KJ2012A073, KJ2011A086), Ahui Provicial Natural Sciece Foudatio ( MF105), Ahui Provicial Soft Sciece Foudatio( ), Progra for Excellet Talets i Ahui ad Progra for New Cetury Excellet Talets i Uiversity(NCET ). [1] L. Kockaert, A siple accurate algorith for barycetric ratioal iterpolatio, IEEE Sigal processig letters, 15, (2008). [2] H. T. Nguye, A. Cuyt ad O. S. Celis, Coootoe ad cocovex ratioal iterpolatio ad approxiatio, Nuerical Algoriths, 58, 1-21 (2010). [3] S. M. Floater ad H. Kai, Barycetric ratioal iterpolatio with o poles ad high rates of approxiatio, Nuerische Matheatik, 107, (2007). [4] J. P. Berrut, R. Baltesperger ad H. D. Mittcla, Recet developets i barycetric ratioal iterpolatio, Iterpolatio Series of Nerical Matheatics, 151, (2005). [5] C. Scheider ad W. Werer, Soe ew aspects of ratioal iterpolatio, Matheatics of Coputatio, 47, (1986). [6] J. P. Berrut ad H. D. Mittela, Lebesgue costat iiizig liear ratioal iterpolatio of cotiuous fuctio over the iterval, Coputers & Matheatics with Applicatios, 33, (1997). [7] L. Bos, H. D. Marchi ad M. Viaello, O the Lebesgue costat for the Xu iterpolatio forula, 141, (2006). [8] S. J. Sith, Lebesgue costats i polyoial iterpolatio, Aales Matheatics et Iforaticae, 33, (2006). [9] H. T. Nguye, A. Cuyt ad O. S. Celis, Shape cotrol i ultivariate barycetric ratioal iterpolatio, Proc. ICNAAM 2010, 1281, (2010). [10] J. P. Berrut, Ratioal fuctios for guarateed ad experietally well-coditioed global iterpolatio, Coputers & Matheatics with Applicatios, 15, 1-16 (1988). Qiaji Zhao received the Ph.D. degree fro Hefei Uiversity of Techology, Chia. He is curretly a professor i School of Sciece at Ahui Uiversity of Sciece ad Techology. His research iterests are i the areas of ratioal iterpolatio, digital iage processig ad coputer-aided geoetric desig. Bigbig Wag is curretly a aster cadidate i Ahui Uiversity of Sciece ad Techology, Chia. Her research iterests are i the areas of ratioal iterpolatio ad geoetric desig.