Computation for Jacobi-Gauss Lobatto Quadrature Based on Derivative Relation

Transcription

1 Computatio for acobi-auss obatto Quadrature Based o Derivative Relatio Z.S. Zheg uaghui Huag Abstract. The three-term recurrece relatio for derivatives of acobi-type polyomial is derived ad the auss-obatto poits are also the eigevalues of some special acobi matrix istead of.h. olub s modified acobi matrix. I additio, explicit weights are also preseted much more simply. Numerical experimets are carried out to illustrate the effectiveess of the method, ad the compariso is also give to demostrate that it s more robust ad accuracy based o three-term recurrece relatio for derivatives. Key words. Three-term recurrece relatio for derivatives; acobi-auss obatto quadrature; Modified acobi matrix AMS subject classificatios. 4A55, 65D3. Itroductio. I may sciece ad egieerig applicatios, PDEs will be ivolved with high precisio quadrature o the referece elemet after some umerical discretizatio lie FEM. auss-type quadrature is appreciated by practical egieers with its high accuracy. May years ago, it had bee proved that the auss poits were exactly the zeros of some orthogoal polyomials with respect to its oegative weight fuctio (cf.[5]). But util 969,.H olub[] successfully chaged the problem of locatig zeros of orthogoal polyomials ito computig the eigevalues of a tridiagoal acobi matrix. The method for computig (+)-poits auss-obatto quadrature rule for ay measure of itegratio is to geerate the acobi matrix of order + for the measure firstly, the modify the three elemets at the right lower corer of the matrix i a maer proposed by olub, ad fially compute the eigevalues ad the first compoet of the respective eigevectors to produce the odes ad weights of the quadrature rule. This wor has bee doe by.h. olub[] i 973. eerally, it wors quite well, but whe becomes large, uderflow problem causes the method to fail. This crucial pheomeo was first reported by Walter Cautschi[4]. To avoid the uderflow problem, Cautschi computed the modified elemets directly for arbitrary acobi measures istead of solvig the system of liear equatios that was used to compute the modified acobi matrix elemets ad gave explicit expressio of all the weights of the quadrature rules. The wor of auss-obatto formulae seems to be a ed, however, It is of course a good idea mathematically to compute the modified acobi matrix to get the auss-obatto odes, but ot directly, ad the explicit formulae of weights is ot so elegatly but somewhat complicated. The aim of the paper is to derive the three-term recurrece relatio of the derivatives of acobi polyomials which is ew to our best owledge, ad compute eigevalues of the special acobi matrix of order which is differet from the modified acobi matrix proposed by olub. The we develop aother more simple

2 explicit expressio of weights of the quadrature rule which is quite practical i the actual computatio. The rest of the paper is orgaized as follows. I Sect., we simply recall some crucial results proposed by.h. olub ad improved later by Walter Cautschi. I Sect3, the three-term recurrece relatio for the derivatives of acobi polyomial is derived. I the followig, explicit formula for weights of the quadrature rule is preseted for arbitrary acobi measures. Numerical comparisos ad cocludig remars are give i the fial sectio.. Some basis results for auss-obatto quadrature formulae. ive a positive measure d supported o the iterval [-,], ad assume that all its momets exist, We restrict our attetio to auss-obatto quadrature of the form f ( xd ) ( x) 0f( ) + f( x) + + f() + R( f), (.) which is exact wheever f is a polyomial of degree +, R ( f ) 0 f Ρ + It is called auss-obatto rule with (+) poits relative to the measure d. As is well ow, the iterior odes x are the zeros of π (; d ± ) or the polyomials of degree orthogoal with respect to the modified measure d ( x) ( x ) d( x) ±, whereπ ( ; d) are (moic) orthogoal polyomial of degree relative to measure d ad satisfy the followig three-term recurrece relatio π ( x) ( x α ) π ( x) β π ( x), 0,,,, (.) + withπ ( x) 0, π 0( x), α α( d), β β( d) > 0 ad β0 d( x). The acobi matrix of order is defied by α0 β 0 β α β ( d) (.3) β β α 0 emma. (olub, Welsch[]) The auss odes x are the eigevalues of ad the auss weights are give by β ( u ),,,3,, (.4) 0, where is the ormalized eigevector of correspodig to the eigevalues u T x (i.e. ( u ) u ) ad is its first compoet. u, For auss-obatto formula (.) the acobi-obatto matrix of order + is modified as ( d) βe 0 ( ) T + + d βe α β + e (.5) + T 0 β+ e+ α +

3 where ( d ) is as before, ad, α + β+ is the solutio of the liear system π+ ( ) π( ) α + π+ ( ) π+ () π() β π () (.6) + + so we have emma. (olub[]) The auss-obatto odes x0, x,, x, x + are the eigevalues of + ad the auss-obatto weights are give by β ( u ), 0,,,3,,, + (.7) 0, where u is the ormalized eigevector of + correspodig to the eigevalues x (i.e. ( ) T u u ) ad, is its first compoet. u auss-obatto formulae are therefore computable by the QR algorithm applied to +. But whe becomes large, the elemets of the matrix (.6) become very small ad the products is eve smaller, this will create a sigular system. For the egedre measure d () x ad sigle-precisio IEEE arithmetic, this happes begiig with 79, ad i double precisio, begiig with 543. This crucial pheomeo was first reported by Walter Cautschi[4]. I order to avoid uderflow problem, Cautschi directly computed the modified elemets for arbitrary acobi measures istead of solvig the system (.6), ad meawhile he also derived the explicit expressio of associated weights. Numerical results showed that his method is more superior to olub s. 3. The relatio for derivative of acobi polyomials. We cosider acobi polyomials deoted by ( x) with respect to weight fuctio As is well ow, ( x) ( x) α β ω ( + x) α, β >. (3.) ( x) are the solutios of sigular Sturm-iouville problem: α β d + α + β d ( x) ( + x) ( x) ( + x) ( x) + ( + + α+ β) ( x) 0, (3.) which ca be rewritte as d d ( x) ( ) x + ( β α ( β+ α+ ) x) ( x) + ( + + α+ β) ( x) 0. (3.3) It is easy to have from (3.) α β d α, β d α, β ( x) + ( + x) + ( ) ( ) 0, x m x m, d, which implies that αβ ( x ) are also mutually orthogoal with respect to the + weight fuctioω( x) ( x) α ( x) + + β d,, so αβ ( x ) must tae the form (.).

4 This motivates us to see for the correspodig α, β. Recall that acobi polyomials satisfy the followig three-term recurrece relatio (cf. Eq.(4.5.) i Szego s[3]) ( x) ( D x+ E ) ( x) + F ( x),,,... + With 0 ( x), ( x) ( α + β + ) x+ ( α β ), (+ α + β + )(+ α + β + ) D ( + )( + α + β + ) ( α β )( + α + β + ) where E ( + )( + α + β + )( + α + β) ( + α)( + β)(+ α + β + ) F,... ( + )( + α + β + )( + α + β) Aother importat formula used frequetly[3] is d α+, β+ ( x ) ( x) ( + α + β + )( x ) ( x) A + B + C + (3.4). (3.5) (3.6) where ( + α)( + β)( + α + β + ) A ( + α + β)(+ α + β + ) ( α β) ( + α + β + ) B. (3.7) ( + α + β)(+ α + β + ) ( + )( + α + β + ) C,... (+ α + β + )(+ α + β + ) Theorem 3. The derivatives of acobi polyomials satisfy the followig three-term recursive relatio d d d + ( x) ( ax+ b) ( x) c ( x), (3.8) d d with 0 ( x) 0, ( x) ( α + β + ), (+ α + β+ )(+ α+ β+ ) a ( + α+ β+ ) where b ( ) α β + ( α β) ( + α+ β+ ). (3.9) ( + α+ β+ )( + α+ β) ( + α)( + β)(+ α+ β+ ) c ( + α+ β)( + α+ β) Proof: Differetiate both side of the Eq. (3.4) to obtai d d d + ( x) D ( x) + ( Dx+ E) ( x) + F ( x). (3.0)

5 α β Multiply both side of Eq.(3.6) by ( x) ( + x), ad differetiate it to get d d d d d ( ω ) ( ω ) ( + ) + α + β ( x) ( + x) () x A () x () x + B () x () x + C () x () x It s trivial to show that ω ( x) ( l ω( x) ) ( αl( x) + β l( + x) ) α + β ω( x) x + x For the left part of Eq. (3.), it yields usig the relatio of formula (3.) d d + α + β ( x) ( + x) ( x) ( + + α+ β) ( x) ω( ) ad For the right part of Eq. (3.), we get by computig directly ω. (3.) (3.) x, (3.3) d d d A ( () xω() x ) + B ( ω() x ) + C ( + ω() x ) d d d ω () x ω() x A () x + B () x + C + () x + ( A () x + B () x + C + () x ) ω() x Usig the above formulae i combiatio with (3.6),(3.), (3.) ad (3.3), the followig formula ca be derived d d d α β d ( + + α+ β) ( x) A ( x) + B ( x) + C + ( x) + + ( x ) ( x) x + x which ca be simplified ito d d d ( + + α+ β) ( x) A ( x) + ( ( α+ β) x+ B α+ β) ( x) + C + ( x). (3.4) Thas to (3.0), we ca have d d d ( x) ( x) + ( D x+ E ) ( x) + F ( x) +, D ad substitute it ito formula (3.4), the fial three-term recursive relatio for the derivative of acobi polyomials ca be gotte after complicated computatio, which completes the proof. d, To mae αβ d ( x ) be moic, we set q () x which is of degree -. It is ca gotte from (3.8) ( xq q b q + c q. (3.5) + a + ) Therefore, the followig should be satisfied (for cosistecy with (.)) b c, α, β. a + a a.

6 Iferrig from the above equatio, we have ( ) 4( )( )( )( ) α α β + α β, β + α + β + α+ β+ (3.6) ( + a+ b)(+ a+ b+ ) ( + α+ β) (+ α+ β+ )(+ α+ β ) Remar: Ifα + β, the deomiator of β is vaished, but the omiator - is also zeros, this time we ca set ( + α)( + β ) β ( ) directly from (3.6). Though it happes, it maes sese to our acobi matrix. I fact, our acobi matrix eed t β, which ca be igored aturally. From (3.5) ad (3.6), we ow that (replacig by ) with q 0, q. 0 xq ),,, q+ + αq + βq (3.7) So a special acobi matrix of oly order - for the zeros of ewly-established relatio (3.7) is listed as followig ˆ α β β α β3 0 β 0, β α d ( x ) with the ad the details of costructig geeral acobi matrix relative to the orthogoal polyomials ca be foud i [5]. Our acobi matrix is somewhat aalogous to ulob s acobi matrix for auss quadrature odes istead of modified acobi matrix for auss- obatto quadrature odes which we have metioed before. We recall that the iterior odes x of auss-obatto quadrature (.) are the zeros of π (; d ± ) or the polyomials of degree orthogoal with respect to the modified measure d () x ( x ) d () x ±. For acobi weight fuctio ( x) ( x) α β ω ( + x), α, β d () x ( x) ω() x d + + ± α+, β+ () x, the iterior odes x are therefore the zeros of () x. Note that the first equality of Eq.(3.6), so we have d α+, β+ + ( x) ( + α + β + ) ( x )

7 That is to say, the iterior odes x are the zeros of d + x. ( ) Therefore, we ca compute the eigevalues of acobi matrix iterior odes x. ˆ to get the 4. Weights of the quadrature rule. Now, we restrict our attetio to the weights of auss-obatto formulae. For the boudary weights, a lot of the results have bee gotte, we refer the reader to Cautschi[4] ad Yag[], ad they both derived the same formula for the boudary weights i a differet way. Here, we give the result followig Cautschi. Sice the chage of variable t t covert α, β β, α that + 0, It suffices therefore to compute explicitly as ad α, β β, α + 0. α+ β+ 0 Γ ( α + ) Γ ( β + ) Γ ( α + β + 3), d α β ito α, β 0, d β α, oe easily sees ad it ca be expressible + α + ( ) + β + + α + β + ( )( ) autschi ad Yag give their formula for the iterior weights respectively, ad they are differet i the form. Cautschi s formula seems to be quite complicated, while Yag s is simpler comparig with the former, but ot practical i the actual computatio. Below are their formulas for the iterior weights. autschi s : α+ β+ Γ ( α + ) Γ ( β + ) Γ ( α + β + 3)( + ) + α + + β + ( )( ) + α + β + ( ) x 4( + α + )( + β + ) + ( α β) ( x ) ( + α + β + ) x + Yag s: α+ β+ 3 Γ ( α + + ) Γ ( β + + ) Γ + Γ ( α+, β+ ) ( ) ( α β 3) ( x) ( x) where ( α, β ) ( ) α+, β+ x deote as the value of derivative of ( x) at + + The secod aim of this paper is see more compact form ad more practical for computatio. Theorem 4. The iterior weights associated with the iterior odes listed as follows: x.

8 α+ β+ Γ ( α + + ) Γ ( β + + ) + Γ + Γ α + β ( ) ( ) ( 3) ( x ) (4.) Proof: Thas to (3.6), We rewrite it as (replacig by +) d α+, β+ + ( x) ( + α + β + ) ( x ). (4.) As metioed above, the iterior odes x are the zeros of, α+ β+ () x, amely d, ( x ) α+ β+ + ( x ) 0,,,,. (4.3) Differetiate (4.) to get d, αβ d x α+, β+ + + α + β + x ( ) ( ) ( ) (4.4) Recall that acobi polyomials ( x ) satisfy Sturm-iouville Eq.(3.3) (replacig as +) + d d ( x) + () x + ( β α ( β+ α+ ) x) + () x + ( + )( + + α+ β) + () x 0. (4.5) Substitute (4.4) ito (4.5), we ca get d ( )( ) α, β d (),, ( ( ) αβ ) () ( ) αβ + α+ β+ x + + x + β α β+ α+ x + x α+ β + () x 0. Usig the fact (4.3), the above formulae ca be simplified as d α, β α, β ( x) + + ( x) ( + ) + ( x ) (4.6) With Yag s formula ad Eq. (5.6), we develop a more simple ad practical formula for the iterior weights i the actual computatio α+ β+ This completes the proof. Γ ( α + + ) Γ ( β + + ) + Γ + Γ α + β ( ) ( ) ( 3) ( x ) This is true for egedre case with α β 0. I fact,. 0,0 Γ ( + ) Γ ( + ) ( ) ( ) ( 3) ( x ) [ ] 0,0 + Γ + Γ + ( + )( + ) ( x) Numerical Experimets ad Aalysis. I the umerical experimet described i this sectio, we compare our methods based o ewly-established recursive relatio for the derivatives of acobi polyomials with covetioal modified acobi matrix for auss-obtto odes, ad the associated weights are also cosidered.

9 We begi with the classical Chebyshev-auss-obatto formula, i.e. with the case α β, whose odes ad weights are ow explicitly π π x cos( ),, 0,,, + + c ( + ) where c 0 c, ad c,,,, Base o Derivatives Modified acobi Matrix Fig. 5. Nodes compariso of method based derivative ad modified acobi matrix for 00 to 000 per 00 i the case αβ 0.5 The absolute errors of our method ad modified acobi matrix with explicit formula[4] for odes of Chebyshev-auss-obatto compared with the exact formula are plotted i the Fig.5.,where the gree lie deote the our method, ad the blue lie autschi. Fig.5. demostrates that whe the ewly-established recursive relatio for derivative of acobi polyomials is used for computig the auss-obatto odes by QR algorithm, the results will be more robust ad efficiet compare to the method of based-o modified acobi matrix improved by autschi. We will be more iteded to use robust method i the computatio to some degree.

10 0 - Base o Derivatives Modified acobi Matrix Fig.5. Weights compariso of method based derivative ad modified acobi matrix for 00 to 000 per 00 i the case αβ Base o Derivatives Modified acobi matrix Fig.5.3 Error of weights per compoet based derivative ad modified acobi matrix whe 00 i the case αβ 0.5 Compariso of weights is also plotted i the Fig.5.. The color of lie stads as before. From it, we ca see results based o our explicit formula (4.) are more accuracy tha the results give by autschi s explicit formula. The formula shows us a more stable property for computig weights. Fig.5.3 is used as the reiterpretatio of the iterior weights with great robust ad accuracy i the actual computatio. This paper maily derive the three-term recurrece relatio for the derivatives of acobi polyomials which is ew to our best owledge, ad compute eigevalues of the special acobi matrix of order which is differet from the modified acobi matrix by olub ad autschi. The we develop aother simpler explicit expressio ad practical for the computatio. Numerical experimets verify that our ewly-

11 established recursive relatio ad explicit weight formula is efficiet ad robust superior to other methods. Acowledgmet. This wor was supported by the Natioal Natural Sciece Foudatio if Chia uder Project , the Natioal Basic Research Program of Chia uder Project 006CB60507, the Hi-Tech Research ad Developmet Program of Chia uder Project 006AA06Z05. REFERENCES [].H. olub, Some modified matrix eigevalue problems, SIAM Review,5 (973), pp [].H. olub,.h.welsch, Calculatio of auss quadrature rule,math. Comp.,3 (969), pp.-30. [3].H. olub,. Meurat, Matrices, momets ad quadrature. Numerical Aalysis 993 (Dudee, 993), 05-56, Pitma Res. Notes Math. Ser., 303,ogma Sci. Tech., Harlow [4]Walter autschi, High-Order auss-obatto formulae, Electr. Tras. Numer. Algorithm, 5(000), pp. 3 [5]Walter autschi, Algorithm 76: ORTHPO a pacage of routies for geeratig orthogoal polyomials ad auss-type quadrature rules, ACM Trasactios o Mathematical Software, 0(994), pp.-6 [6]W. autschi, O geeratig orthogoal polyomials, SIAM. Sci. Statist. Comput. 3(98), pp [7]Walter autschi, Orthogoal polyomials ad Quadtrature. Electr. Tras. Numer. Aalysis, 9(999), pp [8]W. autschi, The iterplay betwee classical aalysis ad (umerical) liear algebra--a tribute to.h. olub, Electro. Tras. Numer. Aal. 3(00), pp.9-47 [9]W. autschi, Orthogoal Polyomials: Computatio ad Approximatio, Oxford Uiversity Press, Oxford, 004 [0]Walter autschi, Orthogoal polyomials (i Matlab),oural of Computatioal ad Applied Mathematics, 78(005), pp.5 34 [] a S. Hesheve, Sigal ottlieb, David ottlieb, Spectral Methods for Time-Depedet Problems, Cambridge U. Press, 007 [] Yag Shi-ju, auss-radau ad auss-obatto formulae for the acobi weight ad ori-micchelli weight fuctios, oural of Zhejiag Uiversity Sciece, 3(00), pp [3]. Szegö, Orthogoal Polyomials(4ed), America Mathematical Society Providece, 975 [4].H. olub,. Meurat, Matrices, momets ad quadrature II: How to compute the orm of the error iiterative methods, BIT,37 (997), pp [5]H. Wilf, Mathematics for the Physical Sciece, oh Wiley ad Sos Ic. New Yor,96