Gauss Radau formulae for Jacobi and Laguerre weight functions

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1 Mathematics ad Computers i Simulatio 54 () Gauss Radau formulae for Jacobi ad Laguerre weight fuctios Walter Gautschi Departmet of Computer Scieces, Purdue Uiversity, West Lafayette, IN , USA Dedicated to Joh R. Rice, o the occasio of his 65th birthday Abstract Explicit expressios are obtaied for the weights of the Gauss Radau quadrature formula for itegratio over the iterval [, 1] relative to the Jacobi weight fuctio (1 t) α (1+t) β, α>, β>. The odes are ow to be the eigevalues of a symmetric tridiagoal matrix, which is also obtaied explicitly. Similar results hold for Gauss Radau quadrature over the iterval [, ) relative to the Laguerre weight t α e t, α>. IMACS. Published by Elsevier Sciece B.V. All rights reserved. Keywords: Gauss Radau formula; Jacobi weight fuctio; Laguerre weight fuctio 1. Itroductio For ay positive measure dλ supported o the iterval [a, b], with a a fiite real umber, ad dλ havig momets of all orders, there exists a quadrature rule of the form b a f(t)dλ(t) λ f(a)+ λ f(t ) + R (f ) (1.1) 1 which is exact for polyomials of degree, R (f ), f P. (1.) It is called the ((+1)-poit) Gauss Radau rule for the measure dλ. Its iterior odes t are ow to be the zeros of π ( ; dλ a ), the polyomial of degree orthogoal with respect to the modified measure dλ a (t)(t a)dλ(t). The weights are obtaiable by iterpolatio at the odes a, t 1, t,..., t. Golub [5] i 1973 observed that the formula ca be obtaied, more elegatly, via eigevalues ad eigevectors of a modified Jacobi matrix of order +1, //$. IMACS. Published by Elsevier Sciece B.V. All rights reserved. PII: S ()179-8

2 44 W. Gautschi / Mathematics ad Computers i Simulatio 54 () α β1 β1 α 1 β J+1 (dλ). β (1.3)... α β β α Here, α ad β are the coefficiets i the recurrece relatio π +1 (t) (t α )π (t) β π (t),, 1,,..., π (t), π (t) 1 (1.4) satisfied by the (moic) orthogoal polyomials π ( )π ( ; dλ), ad α is give by α a β π (a) π (a). (1.5) The odes of Eq. (1.1) (icludig a) are the precisely the eigevalues of J+1 (dλ), whereas the weights λ j are expressible i terms of the first compoets v j,1 of the associated ormalized eigevectors v j, λ j β vj,1, j, 1,,...,, (1.6) where β b a dλ(t) (cf. also [,3]). We show here that i the case of the Jacobi measure o [, 1], dλ (α,β) (t) (1 t) α (1 + t) β dt, α >, β>, (1.7) the quatity α i Eq. (1.5) as well as all the weights λ j λ (α,β) j i Eq. (1.1) ca be computed explicitly i terms of, α, ad β, thus obviatig the eed of computig eigevectors. Our results geeralize well-ow formulae for the Legedre measure dλ (,) (cf., e.g. [1], p. 13), λ (,) ( + 1), λ(,) 1 (1 t )[P (t )] 1 1 t, 1,,...,, (1.8) ( + 1) [P (t )] where P is the Legedre polyomial of degree. We obtai, i fact, a additioal formula, which i the case of the Legedre measure becomes [ ] λ (,) t ( + 1)( + ) [P (,1) +1 (t )], (1.9) with P (,1) +1 the Jacobi polyomial of degree +1 relative to the Jacobi parameters α ad β1. Similar techiques ca be used to derive explicit Gauss Lobatto formulae for geeral Jacobi weight fuctios (cf. [4]). Aalogous results hold for the Laguerre measure o [, ), dλ (α) (t) t α e t dt, α >. (1.1) Sectios 4 are dealig with the Jacobi measure (1.7). After the explicit formula for the modified elemet α i Eq. (1.3) is obtaied i Sectio, the boudary weight λ(α,β) will be developed i Sectio 3, ad the iterior weights λ (α,β) i Sectio 4. Sectio 5 deals aalogously with the Laguerre measure (1.1).

3 . The modified Jacobi matrix W. Gautschi / Mathematics ad Computers i Simulatio 54 () The moic polyomials orthogoal with respect to the Jacobi measure (1.7) will be deoted by π π (α,β), ad the Jacobi polyomials, as covetioally defied (cf., e.g., [6]), by P P (α,β). They are related by P (t) π (t), 1 ( ) + α + β. (.1) It is well ow, furthermore, that ( + β P () () By Eq. (1.5), we have α β π () π () ). (.) β P () P (), which, i view of Eqs. (.1) ad (.), gives α + 1 β ( + α + β)( + α + β 1), ( + β)( + α + β) 1. (.3) O the other had, β π π h, h where h P (α,β) [P (α,β) (t)] dλ (α,β) α+β+1 Γ(+ α + 1)Γ ( + β + 1) (t) + α + β + 1 Γ(+ 1)Γ ( + α + β + 1). O substitutio i Eq. (.3), this yields α + ( + α), ( + α + β)( + α + β + 1) 1. (.4) The modified Jacobi matrix J+1 (dλ(α,β) ) i Eq. (1.3) is ow readily computable, the recursio coefficiets α ad β for the Jacobi measure dλ (α,β) beig explicitly ow. This yields the Gauss Radau formula i terms of the eigevalues ad (first compoets of) the eigevectors of the matrix (1.3). I the ext two sectios we develop explicit formulae for the weights λ j λ (α,β) j, which allow us to bypass the formula (1.6). 3. The boudary weight Our cocer, i this ad the ext two sectios, is with the Gauss Jacobi Radau formula f(t)dλ (α,β) (t) λ f() + λ f(t ) + R (f ), (3.1) 1 where dλ (α,β) is the Jacobi measure (1.7). We first deal with the boudary weight λ λ (α,β).

4 46 W. Gautschi / Mathematics ad Computers i Simulatio 54 () For the Jacobi measure we have dλ (t)(1+t)dλ (α,β) (t)dλ (α,β+1) (t), so that the iterior odes t t (α,β) of Eq. (3.1) are the zeros of P (α,β+1), P (α,β+1) (t ), 1,,...,. (3.) Puttig f(t) λ () (t) i Eq. (3.1) the yields (t) dλ (α,β) (t). (3.3) After replacig β by β+1 i Eq. (.), the coefficiet o the left is give by ( ) + β + 1 P (α,β+1) () (). (3.4) I order to compute the itegral i Eq. (3.3), we begi writig, usig the secod relatio i [6] (Eq. (4.5.4)), (t) Observig from (.) that + α + β ( + 1) () P (α,β) () ( ), + β ( + 1)P (α,β) +1 (α,β) (t) + ( + β + 1)P (t). (3.5) 1 + t ad similarly + β + 1 ( + 1) ( )() +1 P (α,β) + β (), we ca write Eq. (3.5) as (t) () ( + 1) ( + β ( + α + β + ) ) P (α,β) +1 (α,β) (t)p () P (α,β) 1 + t O the other had, by the Christoffel Darboux formula ([6], Eq. (4.5.)), where P (α,β) +1 (α,β) (t)p () P (α,β) 1 + t +1 () c (α, β) (t)p (α,β) ν 1 h (α,β) ν α+β ( + α + β + )Γ ( + α + 1)Γ ( + β + 1) c (α, β) Γ(+ )Γ ( + α + β + ) (t)p (α,β) +1 (). (3.6) P ν (α,β) (t)p ν (α,β) (), (3.7) (3.8)

5 W. Gautschi / Mathematics ad Computers i Simulatio 54 () ad h (α,β) ν (α,β) [P ν (t)] dλ (α,β) (t). Now itegratig Eq. (3.6) with the measure dλ (α,β) gives, by Eq. (3.7) ad the orthogoality of the Jacobi polyomials, (t) dλ (α,β) (t) () ( + 1)c (α, β) ( ), + β ( + α + β + ) or by a elemetary computatio, usig Eq. (3.8), (t) dλ (α,β) (t) () α+β+1 Γ(+ α + 1) Γ(β+ 1) Γ(+ α + β + ). (3.9) Combiig Eqs. (3.3), (3.4), ad (3.9) fially gives the desired result α+β+1 Γ(β+ 1) Γ(+ α + 1) ( ) + β + 1 Γ(+ α + β + ). (3.1) λ (α,β) I the case αβ, we recover the first equatio of (1.8). I almost all cases computed, the boudary weight (3.1) tured out to be more accurate tha the boudary weight computed by Eq. (1.6) (for j), ofte sigificatly so. 4. The iterior weights We ow put f(t) (1 + t)p (α,β+1) (t)/(t t ) i Eq. (3.1) ad obtai, by virtue of Eq. (3.), (1 + t )λ P (α,β+1) P (α,β+1) (t) (t ) dλ (α,β+1) (t). t t Applyig oce more the Christoffel Darboux formula ([6], Eq. (4.5.) with replaced by, ad β replaced by β+1), we get similarly as i Sectio 3, usig Eq. (3.), where P (α,β+1) (t) dλ (α,β+1) (t) d (α, β) t t (t ), d (α, β) α+β+1 ( + α + β + 1)Γ ( + α)γ ( + β + 1). (4.1) Γ(+ 1)Γ ( + α + β + ) Thus, λ (α,β) d (α, β) (1 + t )P (α,β+1) (t ) (t ). (4.)

6 48 W. Gautschi / Mathematics ad Computers i Simulatio 54 () The iterior weights i terms of +1 (t ) Here, we use the secod relatio i ([6], Eq. (4.5.7) with β replaced by β+1) i combiatio with Eq. (3.) to obtai (1 + t )P (α,β+1) (t ) 1 t ( + 1)( + α + β + ) +1 (t ) (t ). (4.3) 1 t + α + β t The recurrece relatio for the Jacobi polyomials (cf. ([6], Eq. (4.5.1) with replaced by +1)), o the other had, yields, agai usig Eq. (3.), ( + 1)( + α + β + )( + α + β + 1) +1 (t ) ( + α)( + β + 1)( + α + β + 3) (t ), that is ( + 1)( + α + β + )( + α + β + 1) (t ) +1 (t ). (4.4) ( + α)( + β + 1)( + α + β + 3) It suffices ow to isert Eqs. (4.1), (4.3), ad (4.4) ito Eq. (4.) to obtai, after some computatio, λ (α,β) α+β ( + α + β + 3) (+1)(+α+β +) where t t (α,β) Γ(+α+1)Γ (+β +) Γ(+)Γ ( + α + β + 3) are the zeros of P (α,β+1). This is Eq. (1.9) whe αβ. 4.. The iterior weights i terms of P (α,β) (t ) 1 t [ +1 (t )], 1, (4.5) We rewrite Eq. (4.) by first otig from the first relatio i ([6], Eq. (4.5.4) with replaced by, ad β replaced by β+1), ad also recallig Eq. (3.), that P (α+1,β+1) (t ) Furthermore, by ([6], Eq. (4.1.7)), P (α+1,β+1) (t ) Therefore, ( + α) (t ). + α + β t + α + β + 1 P (α,β) (t ). + α + β + 1 (t ) (1 t ) ( + α)( + α + β + 1) P (α,β) (t ). (4.6) O the other had, usig Eq. (4.3) ad the relatio +1 (t ) ( + α)( + β + 1)( + α + β + 3) ( + 1)( + α + β + )( + α + β + 1) (t ),

7 W. Gautschi / Mathematics ad Computers i Simulatio 54 () which follows from the recurrece relatio for (cf. ([6], Eq. (4.5.1) with replaced by +1)) ad from Eq. (3.), we obtai (1 + t ) (t ) ( + α)( + β + 1) (t ). + α + β t This, together with Eq. (4.6), allows us to write Eq. (4.) i the form λ (α,β) ( + α)( + α + β + 1) d (α, β) 1 ( + β + 1)( + α + β + 1) (1 t )[P (α,β) (t )]. Substitutig the expressio (4.1) for d (α, β) the gives α+β + α + β + 1 Γ(+ α + 1)Γ ( + β + 1) + β + 1 Γ(+ 1)Γ ( + α + β + 1) λ (α,β) 1 (1 t )[P (α,β) (t )]. (4.7) I the case of αβ, we recover the secod relatio i Eq. (1.8). For computatioal purposes, oe will probably wat to replace P (α,β) (t ) by (1/)( + α + β + 1)P (α+1,β+1) (t ) ad compute P (α+1,β+1) (t ) from its recurrece relatio The iterior weights i terms of P (α,β) (t ) We ow rewrite Eq. (4.7) by usig the secod relatio i ([6], Eq. (4.5.7)), (1 t (α,β) )P (t ) 1 + α + β + {( + α + β + 1)[( + α + β + )t +α β]p (α,β) (t ) ( + 1)( + α + β + 1)P (α,β) +1 (t )}, ad combiig it with the relatio (t ) + α + β + ( + β + 1)P (α,β) (t ) + ( + 1)P (α,β) +1 (t ), 1 + t which follows from Eq. (3.) ad the secod relatio i ([6], Eq. (4.5.4)), that is, with P (α,β) +1 (t ) + β + 1 P (α,β) (t ). + 1 The result is λ (α,β) α+β + β + 1 Γ(+ α + 1)Γ ( + β + 1) Γ(+ 1)Γ ( + α + β + ) 1 t [P (α,β) (t )]. (4.8) I the Legedre case αβ, this reduces to the last relatio i Eq. (1.8). Numerically, the three formulae (4.5), (4.7), ad (4.8) behave similarly. They produce less accurate results tha the formula (1.6) i about two-thirds of the cases computed, ad more accurate results otherwise.

8 41 W. Gautschi / Mathematics ad Computers i Simulatio 54 () The Gauss Radau formula for the Laguerre measure Techiques similar to those i Sectios 4, but much simpler, apply also to the Laguerre measure (1.1). The fial results (Eqs. (5.5) ad (5.9)) are i fact ow (cf. [1], pp. 3 4)), but for completeess we re-derive them from scratch. The moic ad covetioal Laguerre polyomials, π (α) ad L (α), are related by L (α) () (t)! ad we have ( + α L (α) () π (α) (t), ), β ( + α). (5.1) From Eq. (1.5), the modified elemet α i the matrix (1.3) is ow α β π (α) () π (α) () β L (α) () L (α) (), which, by Eq. (5.1), becomes, surprisigly simply, α. The Gauss Laguerre Radau formula to be cosidered is f(t)dλ (α) (t) λ f() + (5.) λ f(t ) + R (f ), (5.3) 1 where dλ (α) is the Laguerre measure (1.1), ad, as before, R (P ). Sice dλ (t)t dλ (α) (t)dλ (α+1) (t), the iterior odes t t (α) are the zeros of, (t ). (5.4) The boudary weight is obtaied by puttig f(t) (t) i Eq. (5.3), λ () (t) dλ (α) (t). The coefficiet o the left ca be computed by Eq. (5.1), ad the itegral o the right by otig from ([6], Eq. (5.1.13)) that (t) ν L (α) ν (t), ad hece, by orthogoality, (t) dλ (α) (t) ν L (α) ν (t) dλ (α) (t) dλ (α) (t) Γ(α+ 1).

9 W. Gautschi / Mathematics ad Computers i Simulatio 54 () There results λ (α) Γ(α+ 1) ( ). + α + 1 (5.5) For the iterior weights λ λ (α),weputf(t) t (t)/(t t ) i Eq. (5.3) to obtai λ t t (t) (t ) dλ (α) (t) (t) dλ (α+1) (t). (5.6) t t t t The Christoffel Darboux formula ([6], Eq. (5.1.11) with α replaced by α+1), similarly as i the begiig of Sectio 4, yields ( ) λ (α) Γ(α+ ) + α t (t )L (α) +1 (t ). (5.7) By ([6], Eq. (5.1.14) with α replaced by α+1), ad Eq. (5.4), t (t ) ( + α + 1) (t ), where, by ([6], Eq. (5.1.13)) ad Eq. (5.4), (t ) L (α) (t ). The recurrece relatio for the Laguerre polyomials, (cf. [6], Eq. (5.1.1)), i combiatio with Eqs. (5.4) ad (5.8), o the other had, gives +1 (t ) ( + α + 1) so that fially ( ) λ (α) Γ(α+ 1) + α 1 + α + 1 (t ) ( + α + 1)L (α) (t ), (5.8) [L (α) (t )]. (5.9) This is the aalogue of Eq. (4.8). The aalogue of Eq. (4.7) coicides with Eq. (5.9). This is because of the idetities ([6], Eqs. (5.1.13) ad (5.1.14)) L (α) (t) (t) L(α) (t) L(α+1) (t), which, for tt, i view of Eq. (5.4), gives L (α) (t ) L (α) (t ). The umerical experiece with these explicit formulae is much lie i the case of the Jacobi measure, i.e., the boudary weight (5.5) is almost always cosiderably more accurate tha the boudary weight obtaied via eigevectors, whereas for iterior weights, the formula (1.6) ivolvig eigevectors is geerally more accurate tha the explicit formula (5.9).

10 41 W. Gautschi / Mathematics ad Computers i Simulatio 54 () Refereces [1] P.J. Davis, P. Rabiowitz, Methods of Numerical Itegratio, d Editio, Academic Press, Orlado, [] W. Gautschi, Algorithm 76: ORTHPOL a pacage of routies for geeratig orthogoal polyomials ad Gauss-type quadrature rules, ACM Tras. Math. Software (1994) 1 6. [3] W. Gautschi, Orthogoal polyomials: applicatios ad computatio, i: A. Iserles (Ed.), Acta Numerica 1996, Cambridge Uiversity Press, Cambridge, 1996, pp [4] W. Gautschi, High-order Gauss Lobatto Formulae, Electr. Tras. Numer. Aalysis, accepted for publicatio. [5] G.H. Golub, Some modified matrix eigevalue problems, SIAM Rev. 15 (1973) [6] G. Szegö, Orthogoal Polyomials, 4th Editio, Vol. 3, America Mathematical Society Colloquium Publicatios, America Mathematical Society, Providece, RI, 1978.