NAG Fortran Library Routine Document D01BBF.1
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1 D01 Qudrture NAG Fortrn Librry Routine Document Note: before using this routine, plese red the Users Note for your implementtion to check the interprettion of bold itlicised terms nd other implementtion-dependent detils. 1 Purpose returns the weights nd bscisse pproprite to Gussin qudrture formul with specified number of bscisse. The formule provided re Guss Legendre, Guss Rtionl, Guss Lguerre nd Guss Hermite. 2 Specifiction SUBROUTINE (D01XXX, A, B, ITYPE, N, WEIGHT, ABSCIS, IFAIL) INTEGER ITYPE, N, IFAIL rel A, B, WEIGHT(N), ABSCIS(N) EXTERNAL D01XXX 3 Description This routine returns the weights nd bscisse for use in the Gussin qudrture of function fðxþ. The qudrture tkes the form S ¼ Xn w i fðx i Þ i¼1 where w i re the weights nd x i re the bscisse (see Dvis nd Rbinowitz (1975), Fröberg (1970), Rlston (1965) or Stroud nd Secrest (1966)). Weights nd bscisse re vilble for Guss Legendre, Guss Rtionl, Guss Lguerre nd Guss Hermite qudrture, nd for selection of vlues of n (see Section 5). () Guss Legendre Qudrture: Z b where nd b re finite nd it will be exct for ny function of the form (b) (c) Guss Rtionl qudrture: fðxþ ¼ X2n ð þ b>0þ or Z P fðxþ ¼ X2nþ1 c 2n i i¼2 ðx þ bþ i ¼ c 2nþ1 i ðx þ bþ i ðx þ bþ 2nþ1 : Guss Lguerre qudrture, djusted weights option: ðb > 0Þ or Z ð þ b<0þ ðb <0Þ.1
2 NAG Fortrn Librry Mnul (d) (e) (f) fðxþ ¼e Guss Hermite qudrture, djusted weights option: bx X2n Z þ1 2n X fðxþ ¼e bðx Þ2 c i x i ðb>0þ: Guss Lguerre qudrture, norml weights option: e bx ðb >0Þ or fðxþ ¼ X2n Guss Hermite qudrture, norml weights option: Z þ1 Z e bðx Þ2 e bx ðb <0Þ fðxþ ¼ X2n Note: tht the Guss Legendre bscisse, with ¼, b ¼þ1, re the zeros of the Legendre polynomils; the Guss Lguerre bscisse, with ¼ 0, b ¼ 1, re the zeros of the Lguerre polynomils; nd the Guss Hermite bscisse, with ¼ 0, b ¼ 1, re the zeros of the Hermite polynomils. 4 References Dvis P J nd Rbinowitz P (1975) Methods of Numericl Integrtion Acdemic Press Fröberg C E (1970) Introduction to Numericl Anlysis Addison-Wesley Rlston A (1965) A First Course in Numericl Anlysis pp McGrw-Hill Stroud A H nd Secrest D (1966) Gussin Qudrture Formuls Prentice-Hll 5 Prmeters 1: D01XXX SUBROUTINE, supplied by the NAG Fortrn Librry. Externl Procedure The nme of the routine indictes the qudrture formul: D01BAZ, for Guss Legendre weights nd bscisse; D01BAY, for Guss Rtionl weights nd bscisse; D01BAX, for Guss Lguerre weights nd bscisse; D01BAW, for Guss Hermite weights nd bscisse. The nme used must be declred s EXTERNAL in the (sub)progrm from which is clled..2
3 D01 Qudrture In certin implementtions, to void nme clshes between single nd double precision versions, nmes of uxiliry routines hve been chnged, e.g., D01BAX to BAXD01. Plese refer to the Users Note for your implementtion. 2: A rel Input 3: B rel Input On entry: the quntities nd b s described in the pproprite sub-section of Section 3. 4: ITYPE INTEGER Input On entry: indictes the type of weights for Guss Lguerre or Guss Hermite qudrture (see Section 3): if ITYPE ¼ 1, djusted weights will be returned; if ITYPE ¼ 0, norml weights will be returned. Constrint: ITYPE ¼ 0or1. For Guss Legendre or Guss Rtionl qudrture, this prmeter is not used. 5: N INTEGER Input On entry: the number of weights nd bscisse to be returned, n. Constrint: N¼ 1,2,3,4,5,6,8,10,12,14,16,20,24,32,48 or 64. 6: WEIGHT(N) rel rry Output On exit: the N weights. For Guss Lguerre nd Guss Hermite qudrture, these will be the djusted weights if ITYPE ¼ 1, nd the norml weights if ITYPE ¼ 0. 7: ABSCIS(N) rel rry Output On exit: the N bscisse. 8: IFAIL INTEGER Input/Output On entry: IFAIL must be set to 0, or 1. Users who re unfmilir with this prmeter should refer to Chpter P01 for detils. On exit: IFAIL ¼ 0 unless the routine detects n error (see Section 6). For environments where it might be inpproprite to hlt progrm execution when n error is detected, the vlue or 1 is recommended. If the output of error messges is undesirble, then the vlue 1 is recommended. Otherwise, for users not fmilir with this prmeter the recommended vlue is 0. When the vlue or 1 is used it is essentil to test the vlue of IFAIL on exit. 6 Error Indictors nd Wrnings If on entry IFAIL ¼ 0or, explntory error messges re output on the current error messge unit (s defined by X04AAF). Errors or wrnings detected by the routine: IFAIL ¼ 1 The N-point rule is not mong those stored. If the soft fil option is used, the weights nd bscisse returned will be those for the lrgest vlid vlue of N less thn the requested vlue, nd the excess elements of WEIGHT nd ABSCIS (i.e., up to the requested N) will be filled with zeros. IFAIL ¼ 2 The vlue of A nd/or B is invlid..3
4 NAG Fortrn Librry Mnul Guss Rtionl: A þ B ¼ 0 Guss Lguerre: B ¼ 0 Guss Hermite: B 0 If the soft fil option is used the weights nd bscisse re returned s zero. IFAIL ¼ 3 Lguerre nd Hermite norml weights only: underflow is occurring in evluting one or more of the norml weights. If the soft fil option is used, the underflowing weights re returned s zero. A smller vlue of N must be used; or djusted weights should be used (ITYPE ¼ 1). In the ltter cse, tke cre tht underflow does not occur when evluting the integrnd pproprite for djusted weights. 7 Accurcy The weights nd bscisse re stored for stndrd vlues of A nd B to full mchine ccurcy. 8 Further Comments Timing is negligible. 9 Exmple This exmple progrm returns the bscisse nd (djusted) weights for the six-point Guss Lguerre formul. 9.1 Progrm Text Note: the listing of the exmple progrm presented below uses bold itlicised terms to denote precision-dependent detils. Plese red the Users Note for your implementtion to check the interprettion of these terms. As explined in the Essentil Introduction to this mnul, the results produced my not be identicl for ll implementtions. Exmple Progrm Text Mrk 14 Revised. NAG Copyright Prmeters.. INTEGER N PARAMETER (N=6) INTEGER NOUT PARAMETER (NOUT=6).. Locl Sclrs.. rel A, B INTEGER IFAIL, ITYPE, J.. Locl Arrys.. rel ABSCIS(N), WEIGHT(N).. Externl Subroutines.. EXTERNAL D01BAX,.. Executble Sttements.. Exmple Progrm Results A = 0.0e0 B = 1.0e0 ITYPE = 1 IFAIL = 0 CALL (D01BAX,A,B,ITYPE,N,WEIGHT,ABSCIS,IFAIL) WRITE (NOUT,99998) Lguerre formul,, N, points Abscisse Weights WRITE (NOUT,99999) (ABSCIS(J),WEIGHT(J),J=1,N) STOP FORMAT (1X,2e15.6) FORMAT (1X,A,I3,A).4
5 D01 Qudrture END 9.2 Progrm Dt None. 9.3 Progrm Results Exmple Progrm Results Lguerre formul, 6 points Abscisse E E E E E E+02 Weights E E E E E E+01.5 (lst)
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