gauss quadrature rules involving some non classical weight functions

8/4/2019 Gauss Quadrature Rules Involving Some Non Classical Weight Functions

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-T E C H N IC A L N O T E

N A S A T N D - 5 7 8 5_ _ _ L - a, 1

Paul F. Byrd and David C . Galant94035

R O N A U T IC S A N D SP A C E A D M I N I S T R A T I O N 0 W A S H I N G T O N , D. C. M A Y 1970


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111TECH LIBRARY KAFB, NMIllllllIIIII III 111Ill11l~0332503 3. Recip ien t s Cata log No.

5. Repor t DateMay 1970

6. Per fo rm ing o rgan iza t ion Code

1. Report No. 2. Government Acc ess io n No.NASA TN ~ - 5 1 8 5 I

4. T i t l e a n d Su b t i t l eGAUSS QUADRATURE RULES INVOLVING SOME NONCLASSICAL WEIGHTFUNCTIONS

7. Author(s)Paul F. Byrd and David C. Galant9. Performing Organizat ion Name and Address

NASA Ames Resea rch CenterMoffett Field, Calif. 94035

2. Sponsoring Agency Nome and AddressNational Aeronautics and Space Administrati onWashington, D. C. 20546

15. Supplementary Note s

16. Abst rac t

8.I O .

11.

13.

14.

Per fo rm ing o rgan iza t ion Repor t No.A-3553Work Uni t No.129-04-04-02-00-21Cont rac t o r Gran t No . Type o f Repor t and Per io d Covered

Technical Note

Sponsoring Agency Code

Gauss quadra ture rules ar e given for numerica l ly evaluat ing integra ls on the interval [ 0, 11 for som e weight functionsof the form w(t) = t ? ( l - L a ) 0,where (2 is a positive integer 2 2 an d 0,7 > -1. The nodes, weight coefficients, and theparameters in the three- term recurrence re la t ion fo r the sequence of orthogonal polynomials associated with w(t \ ar etabulated to 25 significant fibure s fo r N-point ru les with N = 2, 4, 6, 8, 12, 16, and 24 . Numerica l tables to 5s ar e a l sogiven fo r a coeff ic ient in the exp ress ion of the er ro r te r m.

17. Key Words Suggested by A uthor ( s ) 18. Dis t r i b u t ion Sta tementGaussian quadra tu re Orthogonal polynomials Unclassified - Unlimited Elliptic intebTals

19. Secur i ty C la ss i f . ( o f t h i s repor t ) 20. Secur i ty C la ss i f . (o f t h i s page) 21. No. o f Pa g e s 22 . Pr ice Unclassified Unrlassified I 40 1 $3.00


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TABLE OF CONTENTS Page

SYMBOLS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv SUMMARY.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 GENERAL DISCUSSION OF CONSTRUCTION . . . . . . . . . . . . . . . . . 4 Algebraic Approach . . . . . . . . . . . . . . . . . . . . . . . . 5 Approach via Orthogonal Polynomials . . . . . . . . . . . . . . . 5 NUMERICAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Determination of the Moments . . . . . . . . . . . . . . . . . . . 6 Calculation of Parameters in the Three-Term Recurrence Relation ... 7 Calculation of the Nodes and Weight Coefficients . . . . . . . . . 8

Calculation of the Error Coefficient . . . . . . . . . . . . . . . 10 Details of the Calculations . . . . . . . . . . . . . . . . . . . . 11 RESULTS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 APPENDIX A.- QR-ALGORITHM PROGRAM OF GOLUB AND WELSCH . . . . . . . . 12 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

iii


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SYMBOLS

bn

E NgnkNMnN

S

p a ra m et er i n r e c u r r e nc e r e l a t i o n ( s e e e q . (15))

m o di fi ed d i f f e r e n c e s i n Q-D schemee r r o r term o f a quad ra tur e formula ( see eq . ( 2 ) )p a ra m et er i n r e c u rr e n c e r e l a t i o n ( s e e e q . (15))c o e f f i c i e n t i n t h e e r r o r te rm ( s ee eq . ( 3 ) )nth moment of weight fun ct io n w(t ) (see eq. (11))number of nodes i n quad ra tur e r u l e ( see eq . (1 ) )m od i fi ed quo t i e n t s i n Q - D schemes i g n i f i c a n t f i g u r e s j = 1(1)N , noda l po i n t s i n quad ra t u re fo rmu l a ( s ee eq . (1 ) ) we ig ht funct o n j = 1 (1 )N , w ei gh t c o e f f i c i e n t s i n q ua d r a t u r e r u l e ( s ee e q . ( 1 ) ) gamma f u n c t io n of argument z orthogonal polynomial of degree k ( see eq . (15) )

i v


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GAUSS QUADRATURE RULES INVOLVING SOME NONCLASSICALWEIGHT FUNCTIONS

Paul F . Byrd and David C . Galant

Ames Research CenterSUMMARY

Gauss qua dra tu re ru le s are g iv e n f o r n u m er i c al l y e v a l u a t i n g i n t e g r a l s ont h e i n t e r v a l [0 , 11 f o r some weigh t fun c t io ns o f t he form w( t ) = tY(1 - t u ) @ ,a i s a p o s i t i v e i n t e g e r h 2 and B y y > -1. The nodes , weight coeff i c i e n t s , and t h e p ar a me te r s i n t h e t h r e e -t e rm r e c u r re n c e r e l a t i o n f o r t h e .sequence of or thogonal polynomials ass oc ia te d with w ( t ) a re t a b u l at e d t o 25s i g n i f i c a n t f i g u r e s f o r N -po in t r u l e s w i t h N = 2 , 4 , 6 , 8 , 1 2 , 16, and 24.

merical t ab le s t o 5s a r e a l s o g iv e n f o r a c o e f f i c i e n t i n t h e e x p r e ss i o n o fterm.

INTRODUCTION

A wide v ar ie ty o f p rob lems req u i r ing numer ica l approx imat ions t os p e c i f i c d e f i n i t e i n t e g r a l s a r i s e v e ry o f te n i n p r a c t i c e , u s u a l l y as anin te rm edia te s te p i n a more ex ten s iv e computa tion . There are many methodsfo r approx imat ing such in te g ra l s , bu t among a l l quadra tu re fo rmulas o f theformb N

I = J b g ( t ) d t E w ( t ) f ( t ) d t >:WjNf(tjN) (1)a a j = ie we igh ted gauss ian typ es have a l l t h e i r pa ramete rs chosen t o maximizeformance. For th es e formulas , t he nodes t i N and th e co ef f i c i en t s WjNar e un ique ly de termined s o t h a t a l l a p pr ox i ma tl o ns t o t h e g i ve n i n t e g r a l a reo r a r b i t r a r y p o ly no mi al s f of t h e h i g h e s t p o s s i b l e d e g r e e, 2 N - 1.I f t h e f u n c t i o n f i s o t h e r t ha n a p5lynomial of degree 4 2 N - 1, t h e i n t e a t io n fo rmulas (1 ) have an e r r o r term g iv e n ( e .g . , s e e r e f . 1) by

EN =-N f ( 2 N ) ( T ) , a < T < b(2N) !($ N be ing th e assoc ia te d Nth-degree o r thogona l po lynomial .


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The l i t e r a t u r e c o n c er n ing we ig ht ed g a u s s i a n f o rm u la s i s v e r y l a r g e , b o thc l a s s i c a l and r e c e n t ( r e f s . 2 , 3 , and 4 ) . Widespread use of high-speed compu te rs , wi th emphasis on economizing th e number o f o pe ra t io ns re qui red i n comp u t a t i o n s , h a s r e s u l t e d i n c o n s id e r a b l e renewed i n t e r e s t i n t h e power of t h egauss ian techn ique; gauss quadr a tu re rules a re be ing tabu la ted and used wi thinc rea s in g f requency . Very acc ura te ru les , for example , were r e c e n t l y t a b u l a t e d i n r e f er e n c e s 1 and 5 f o r cases a s s o c i a t e d w it h t h e c l a s s i c a l o r th o go n apolynomials o f Legendre, Laguerre, Chebyshev, Jacobi, and Hermite. Recentn u me ri c al t a b l e s t o many s i g n i f i c a n t f i g u r e s h av e a l s o b ee n g iv e n f o r somen o n c l a s s i c a l cases such as a weight funct ion on [ O , 11 h a v in g l o g a r i t h m ic s i n g u l a r i t i e s ( r e f . 6) and th e weigh t func t i on e - X 2 o n t h e i n t e r v a l [ O , m]( r e f . 7 ) .

Weighted g auss i an in te gr a t io n fo rmulas are e x c e l l e n t f o r l a r g e c l a s s e s of un c ti on s a r i s i n g i n p r a c t i c e . They are p a r t i c u l a r l y u s e f u l w h e n a completei n t e g r a n d g ( t ) i s f a c t o r a b l e i n t o two f u n c t i o n s f ( t ) and w( t ) , where f i swel l approximated by a polynomial , w i th th e nonnega t ive weight fun c t i on whaving a s i n g u l a r i t y o r a s i n g u l a r d e r i v a t i v e a t one o r both ends of thei n t e r v a l . S in c e n e i t h e r t h e e r r o r te rm ( 2 ) n o r t h e sum i n th e approximations( 1) d i r e c t l y i n v o lv e s t h e w e ig h t f u n c t i o n w , o nc e t h e q u a d r a t u r e r u l e s a r eco n st ru ct ed and programmed on a compute r, they can be repea te d ly ap p l ied t oin t e g r a l s of t h e form (1) w i t h d i f f e r e n t f u n c t i o n s f ( t ) and th e same w(t ) .There are s e v e r a l o th e r a d va n ta g es o f g a u s s - ty p e f or m u la s : ( a ) s i n c e a l l t h ec o e f f i c i e n t s W j N t u r n o u t t o b e p o s i t i v e , t h e r u l e s h a ve good r o u nd o f f p r op e r t i e s ; (b ) th e sequence o f formulas (1 ) f o r N = 1, 2 , . . . w i l l alwaysconverge to th e va lue o f th e Riemann i n t eg ra l I w henever t h i s i n t e g r a le x i s t s ; ( c ) t h e r u l e s h av e more f a v or a b l e e r r o r terms and genera l ly ach ievehi gh er a ccura cy wit h t h e use of fewer nodes th an Newton-Cotes methods.

T h i s r e p o r t p r e s e n t s t a b u l a t e d g a u s s i a n q u a d r a tu r e f o rm u la s f o r

where the weight fu nc t i on w i s of the formw(a, 6, y; t ) E w ( t ) = tY(1 - ta B

with a as a p o s i t i v e i n t e g e r 2 ; B y y > -1and w o r i t s d e r i v a t i v es i n g u l a r a t one o r b o th e nd s o f t h e i n t e r v a l . For t h e s p e c i a l case a = 1 ,t h e a p p r o x im a te r e l a t i o n ( 4 ) becomes th e c l a ss i c al Gauss-Jacobi formula

U j = i

2


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lov, Lugin, and Ja no vi c (r e f . 8) r e c e n t l y g av e t h e s e q u a d r a t u re r u l e s t os i g n i f i c a n t f i g u r e s f o r B , y = -0 .9 (0 .1 )3 .0 , B < y and N = 1 ( 1 ) 8 .W s h a l l c o n s id e r v a r i o u s c a s e s where t h e p a r a m e te r s (a,6 , y ) are

a = 2 , 3, 4 , 6 , 8B = -3/4, - 2 / 3 , - 1 / 2 , 1 / 2y = - 1 / 2 , 0 , 1 / 2

ne o f the se , f o r example, i s t h e i n t e g r a l

(7 )

, when f ( t ) i s simply a ra t i on a l fun c t i on , may be reduced by a s u b s t i o n i n vo lv i ng J a co b ia n e l l i p t i c f u n ct i on s t o a combina t ion o f th ree s t a n r d t a bu l at e d e l l i p t i c i n t e g r a l s ( r e f . 9 ) . A s i mi l a r s t at e m en t a p p l i e s t oB y y; f ) w i t h r e ga rd t o t h e o t h e r v al u e s a , B y y g iv e n i n( 7 ) .I f f i s no t a r a t i o n a l f u n c t io n ( e. g . , e - c t ) , none o f t h e i n t e g r a l sB y y) a r e e x p r e s s i b l e i n f i n i t e te rm s o f known t a b u l a t e d f u n c t i o n s f o r

(a, B y y ) , and the on ly recourse i s numerical approximai o n v i a some qua dra tur e formula . Even when such in te gr al s can be w ri t t en asm bi na ti on s o f s t a n da r d e l l i p t i c i n t e g r a l s e t c . , it i s probab ly more e f f i t o u se q u a d r a t u r e r u l e s h av in g t h e h i g h e s t d e gr e e of a l g e b r a i c p r e c i s i o no o b t a in a ns we rs . I n t e g r a l s w i th o u r w eigh t f u n c t i o n s w ( t ) o c cu r s u f f i f r e q u en t l y i n p r a c t i c e t o wa rr an t t h e c o n s t ru c t i o n o f s uc h r u l e s .Numerical ta bl es f o r N-point gauss ia n r u le s IWjN, t jN) wi th

= 2 , 4 , 6 , 8 , 1 2 , 16, and 24 a re g iv en t o 25 s i g n i f i c a n t f i g u r e s f o r 1 2I ( a , B y y ; f ) . These a re f o r

he c o e f f i c i e n t s b n, gn i n t h e t h re e - te r m r e c u r s i o n r e l a t i o n ( 15) f o r t h eence of monic or thogo nal polynomials corresponding t o w(t ) are a l s oe d t o 25s f o r N = 1 ( 1 )2 6 . S i n ce t h e e r r o r c o e f f i c i e n t s kN i n t h eterm (2) may be useful when a bound on t h e 2Nth de r i va t i ve can bewe g iv e t a b u l a t i o n s o f them t o 5s.3


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L i t t l e d i f f i c u l t y i s e n t a i l e d i n t h e a c t u a l c o n s t r u c ti o n of g au ss -t yp eq u a d r a t u r e f o rm u la s r e l a t i v e t o c l a s s i ca l weight fu nc t io ns where the th ree -term r e c u r r e n c e r e l a t i o n f o r t h e a s s o c i a t e d o r t h o g on a l p ol yn om ia ls are knowne x p l i c i t l y . T h i s i s n o t g e n e r a l ly t r u e f o r n o n c l a s s i c a l w ( t ) s uc h as o u r swhere t he sequenc e o f r e l a t e d unknown orthogo nal po lynomials must begenerated using the moments of w .GENERAL DISCUSSION OF CONSTRUCTION

For a given nonnegat ive weight fun c t i on w( t ) on [0 , 11, th e problem o fcons t ruc t i ng t he N -po in t gauss i an quad ra t u reN

j = l 0c o n s i s t s i n d e t er m i ni n g t h e n odes t j N and th e weight co ef f i c i en t s WjN ( 2 Np ar am et er s i n a l l ) s o t h a t t h e e r r o r EN(f) = 0 whenever f i s a polynomialof degree n & 2N - 1. There are es se n t i a l l y on l y two app roaches t o t hede te rmina t ion of th es e unknown parameters , and they s h a l l be d i sc ussedb r i e f l y .

Before proceed ing, however, we wish t o emphasize i n th e s t ro ng es tp o s s i b l e t er ms t h e n u me ri ca l d i f f i c u l t i e s i n h e r e n t i n t h e c o n s t r u c t io n o f t h equad ra t u re r u l e (10) from th e moments

Mk = Ik w ( t ) d t , k = O(1)2N - 10

G au t sch i ( r e f . 4 ) has shown that an approxi m at e uppe r bound t o t he r e l a t i ve ,asympto t ic con di t ion number of t h i s problem i s

C N = (17 + 1 2 f i ) N64N2

This means th a t t he computat ion i s ex t rem e ly i l l - con d i t i on ed num er i ca l l y , andt h a t t h e number o f s i g n i f i c a n t f i g u r e s l o s t i n d e t er m in i ng t h e r u l e (10) us i ngt h e moments (11) i s approximate ly

SN = l og l o C N = 1.53N - 2 loglo(8N)4


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Thus, i f one wants t o have t he nodes and weights of G N a c c u ra t e t o ks i gn if ic an t f ig ur es , then he must work with more than

k + 1.53N - 2 lOglo(8N)

s ig n i f i c a n t f i g u r e s . F or e xam ple, t o o b t a in a 1 0 - p o in t q u a d r a tu r e r u l e w i thnodes and w ei gh ts a c c u r a t e t o 10 s i g n i f i c a n t d i g i t s ( i . e . , N = k = l o ) , o n emust work with more than 2 2 s i g ni f i ca n t d i g i t s a t a c e r t a i n s t a g e i n t h eca l cu l a t i on o f t he nodes and weigh ts .

Algebraic ApproachI f th e nodes t jN and th e weight co ef f i c i en t s W j N i n equa tion (10) a r echosen so t h a t t h e f o r m u l a i s e x a c t f o r a l l polynomials f ( t ) o f degreen 4 2 N - 1, then it i s s u f f i c i e n t t o c hoose t h e f o rm ul a t o be e x a c t f o r t h emonomials t k , k = 0(1)2N - 1 , so th a t one has th e system

N

of 2 N n o n l in e a r a l g e b r a i c e q u a t i o n s i n 2 N unknowns t o so lv e, where Mkar e given by equat ion (11) . The drawback t o th i s formulat i on of determiningth e quadrat ure r u l e from th e moments i s tha t any a t t empt a t a d i r e c t s o l u t i o nr e q u i r e s t h e h ig h ly p r e c i s e a r i t h m e t i c i n d i c a t e d by e q u a t io n (13) a t everys t e p o f t h e c a l c u l a t i o n s . Fo r t h i s r e a s on , t h e a lg e b r a i c a pp ro ac h i s t o ber e j e c t e d e xc ep t f o r ve r y small N .Approach V i a Orthogonal Polynomials

Or thogona l po lynomia l s p lay an impor tant ro le i n gauss i an .q uadr a tu r es .I n f a c t , i t can e as i l y be shown th a t i f th e noda l po i n t s t jN i n fo rmula (10)a re se l ec t ed t o be the roo ts o f th e Nth-degree polynomial + , ( t) o f th esequence o f o r thogona l po lynomial s {+ n( t ) ) a sso c ia t ed wi th t he g iven weigh tf u n c t i o n w ( t ) , t h e n t h e f or mu la i s e x a c t f o r any p olyn om ia l f ( t ) o f d e g r e e< 2N - 1; t h e c o n v e r s e o f t h i s i s a l s o t r u e ( e .g . , s e e r e f . 1 0 ) .The or thogonal system of polynomials ass oci a te d with w(t) i n (a , b) i su niqu e and a ny t h r e e c o n s ec u t i ve p olyn om ia ls s a t i s f y t h e r e c u r r e n c e r e l a t i o n

w i t h $ - l ( t ) = 0 , $ O ( t ) = 1, an d

5


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The u s u a l a pp ro ac h i n t h e c o n s t r u c t i o n o f g a u s s i a n r u l e s i s based on th e useo f r e l a t i o n (15) and th e moments t o gen era te th e sequence of polynomials( $ k ( t ) ) . A f t e r t h e s e are ca lc u la t ed , one so lve s , i n some way, f o r t h e r o o t st jN of $N(t ) , and then f i nd s th e weight co ef f i c i en ts WjN by ot he r means .However, any procedure that r e l i e s on the moments i s e s s e n t i a l l y e q u i v a l e n t ts o lv in g t h e b a s i c a l g e b r a i c s ys te m (14) and i s h ig h ly u n s t a b l e n u m e r i c a l l y .

The advantage of th e approach v i a or tho gonal polynomials , as w e s h a l ls e e , i s t h a t t h e i l l - c o n d i t i o n e d n a t u r e o f t h e pr ob le m i s encounte red i n th ede te rmina t ion o f th e th ree - t e rm recur renc e re la t i on f rom the moments, bu t a tn o p o in t i n t h e d e t e r m in a t i o n o f t h e n od es and w eig ht s o f t h e q u a d ra tu r e r u l eS i n c e t h e c a l c u l a t i o n o f bn, gn, n = 1(1)N, needs l e s s than 4N2 ari thmetico;\3erations, the amount o f m u l t ip l y p r e c i s e a r i t h m e t i c i s small and manageableth u s making t h i s a pp ro ac h ve r y t r a c t a b l e i n c o mp ar is on t o t h e a lg e b r a i capproach.

N U M E R I C A L METHODS

The numerical construct ion of o u r g au s si a n r u l e s p r oc ee ds i n t h r e e s t e p s(1) Determine, t o su ff ic ie nt accuracy, th e moments M k f o r k = 0(1)2N - 1 ;(2) us ing th e moments, ca lc ul a t e th e parame ters bk, gk, k = 1 (1 )N , i n t h eth r e e - t e r m r e c u r r e n c e r e l a t i o n (15) f o r t h e f i r s t N orthogonal polynomialsa s s o c i a t e d w i th t h e w eigh t f u n c t i o n w ( t ) on (0 , 1 ) ; (3) determine the quadr a t u r e n od es t j N and the weigh t co ef f ic i en t s W ~ N f o r j = l(1)N. Thet h r e e s t e p s w i l l b e d is c u s se d i n t h a t o r d e r .

Determ ination of t h e MomentsThe moments f o r t h e we ig ht f u n c t i o n o f i n t e r e s t are

f o r n = 0(1)2N - 1. I f we l e t z = t a , eq uat ion (17) becomes6


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o r

where B and I ar e th e be ta and gamma fun ct i ons , res pe ct iv el y . Using t hewell-known re cu rre nc e r e l a t i o n f o r th e gamma fu nc ti on , we have th e re cu rs io nr e1a t i o nMk-a

~~k = 1 + [ a ( l + Bj/(k + y - a + 1 > 1 , k = a(1)2N - 1 (19)s o t h a t a l l the moments can be generated from M o l M 1 , . . . , Ma-l.

C a lc u l a t i o n of Parameters i n th e Three-Term Recurrence Relati onThere a re seve ra l methods o f ca l cu l a t i ng the th ree - te rm r ecur rencere la t i on from th e moments i n th e genera l case ( r e f s . 1 and 11 ). However, f o rsev era l reaso ns , mainly because of t he volume of a r i th me t i c involve d, we havechosen a more sp ec ia l i ze d approach. W u s e t h e q u o t i e n t - d i f f e r e n c e a lg o ri t h m( r e f . 1 2) t o o b t a in t h e t h r e e -t e r m r e c u r r e n c e r e l a t i o n p a ra m e ter s from th emoments. Th is method w i l l always work when th e i n te rv al of in te gr at io n doesn o t h a v e t h e o r i g in as an i n t e r i o r p o i n t. The procedure i s b r i e f l y asfol lows: Def ine

G en er at e t h e r e s t o f t h e t a b l e

7


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us ing t he rhombus ru le s

with k = l (1 )N - 1. The paramete rs f o r re la t i on (15) are then expressed by

wi th g l = 0 .Ca lcu la tio n of t he Nodes and Weight C oe ff ic ie nt s

The th ree - te rm recur rence re l a t io n (15) f o r th e f i r s t N orthogonalpo lynomia ls can be wr i t t en ( re f . 13) as t h e m a x t r i x e q u a t i o n

where -000

@ ( t ) = B = ( 2 4 1

01

I -

and

8


14/42

001 0

=

From equation (23) it i s immedia te ly seen t ha t th e roo ts o f $N( t ) , which a r eth e n odes o f t h e q u a d r a tu r e r u l e , a re a l s o t h e e ig e n v a lu e s o f t h e m a tr i x A .Applying a d ia go na l s i m i l a r i t y t r a n s f o r m at i o n t o t h e m a t r i x (25) y i e ld s t h esymmetric Jacobi matrix

0 6 3 b3

b N - 1Gwhich has the same eigenvalues as A . In re fe renc e 13 , i t i s fu r t he r shownt h a t i f Q1,j i s t h e - f i r s t component o f th e o r thonormal e igenvec tor o f JNa s s o c i a t e d w ith t h e e ig e n v a lu e t jN , t h e n

2w j N = Q 1 a j Mo ( 2 7 )

Hence, bo th the nodes and th e weigh t co ef f i c i en t s o f t he quadra tu re ru le (10)can be ca lcu la t ed f rom th e e igensys tem of th e symmetric t r i d i a g o n a l m a t r i x(26) . The vers ion o f the ve ry f a s t n u m e r i c a l l y s t a b l e QR a lg o r i t h m ( r e f .14) given i n r e f e r e n c e 1 2 t o accompli sh t h i s computa t ion was used.

9


15/42

Calculation of the Error CoefPicient The coefficient of the error term (2) can be obtained from the gn. Fromequations (16) and equation ( 3 ) it immediately follows that

These constants are useful in obtaining error bounds if one has an estimate ofmax I f(2N) (.) I .O


16/42

were used. I t i s o u r e x pe r ie n c e t h a t , f o r q ua d r at u re ru les on a f i n i t ei n t e r v a l , t h e s e a re su f f ic ie n t t o de te rmine th e computed accuracy o f th e nodesand w e ig h t c o e f f i c i e n t s . I n no i n s t a n c e was th e disagreem ent more than ar e l a t i v e e r r o r of 10-28 .

RESULTS

I n t a b l e s 1 through 1 2 , th e nodes and weigh t co ef f i c i en t s o f th e N-pointq u a d r a t u r e r u l e s for N = 2(2)8(4)16,24 are g iv en f o r t h e 1 2 p r e v i o u s l yenumerated weight fun ct i ons . Each of th e nodes and weight co ef f i c i en ts i srounded t o 25s . In t a b l e 13 t h e e r r o r c o n s t a n t s kN, f o r t h e same v a lu e s o fN , a re g iv e n t o 5 s i g n i f i c a n t f i g u r e s .

Tables 14 through 25 are t a b u l a t i o n s o f M o and of bny gn, n = 1(1)26o f t h e re c u r r e nc e r e l a t i o n f o r t he monic o r thogona l po lynomia l s as soc ia te dwith t h e enumerated weight fun ct i on w( t) . T h e s e t a b u l a t i o n s are also roundedt o 25 s i g n i f i c a n t f i g u r e s .Appendix A i s a l i s t i n g of a d o u b le p r e c i s i o n v e r s io n of t h e a lg o r i t h mu se d t o o b t a i n t h e quad ra tu re ru le s . Th is p rogram can be used t o de te rminep r a c t i c a l a cc u ra c y q u a d r a tu r e r u i e s from t h e t h re e -t e r m r e c u r re n c e r e l a t i o nparameters . I n p a r t i c u l a r , it can be employed t o f in d qu adr atu re formulas no tt a bu l at e d i n t h i s r e p o r t .The parameters a ss oc ia te d with each ta b le a re summarized by:

Table c1 B Y1, 14 3 - 1 / 2 02 , 153, 16 34 1 / 2- 1 / 2 004, 175, 18 46 1 / 2- i / 2 006, 197 , 20 68 1 / 2- i / 2 008, 2 1 8 1 / 2 09 , 2 2 3 - 1 / 2 - 1 / 210, 23 3 1 / 2 1/211, 24 2 -3/4 01 2 , 25 2 -2/3 0

I n t h e t a b l e s , a 5 b denotes a - l0'b; f o r in s t an ce , 0 .17 - 6 means0.17Ames Research CenterNat ional Aeronaut ics and Space Adminis t ra t ionM o f f e t t F i e ld , C a l i f . 94035, Jan ua ry 19, 1970

11


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PN

CCCCCCCCCCCCCCCCCCCCCCCCCC

APPENDIX A

QR ALGORITHM PROGRAM OF GOLUB AND WELSCH

T H E R O U T I N E I S AC C ES S ED BY T H E F O R T R AN I V STATEMENT

WHEREM I S T H E N U MB ER O F N O D ES I N T H E Q U A DR A TU R E R U L E T O B E C A L C U L A T E DB wG A RE D O U B L E P R E C I S I O N AR R A YS O F L E N G T H A T L E A S T M W H I C HCO NT A IN T H E P A RA ME T E RS OF T HE T HRE E T ERM RE CURRENCER E L A T I O N FO R TH E F I R S T M MONIC ORT HOGONA L P OL Y NOMIA L S

Z M I S TH E Z ER O MO MENT OF T H E WE IG H T F U N C T I O N ( I N DO U BL E P R E C I S I O N I eALL OF T HE A B OVE P A RA ME T E RS ARE IN P U T S a T H E OUT P UT S ARE T HE DOUB L EP R E C I S I O N A R R AY S OF L EN G H T M

T a W W H I C H C O N T A I N T HE NODES A ND WE IGHT S OF T HE DE S IR E DM-P OIN T GA US S QUA DRA TURE RUL E 6T HE CONT E NT S OF B AN D G ARE DESTROYED BY THE ROUTINE.

SUBROUTINE GAUSSG! M b B s G s ZM, T 9 W )CC


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C I N T E R N A L V A R I A B L E SC D O U B L E P R E C I S I O N AArs A JI , B R 9 B Z E R 0 9 82 , C J , C T 9 DEPS, D E T 9 E I G M A X ,* F, L AM , L A M 1 9 L A M 2 9 NORM, Q, R H O 9 R R P S T , WJC C C C CCC

100

2 00

C

D E P S I S THE L A R GE S T P O S I T I V E N U MB ER S U CH T H A T 1 + D E PS = 1TO M A C H I N E DO U BL E P R E C I S I O NDEPS = 0,500 +* 54M N = M - 1

I N I T I A L I Z E W c S Y M M E T R I Z E M A T R I X , A N D D E T E R M I N E M A XI M U M ROWNORM

DO 10 0 J = 1, M NG ( J ) = D S Q R T ( G ( J + l ) )W(J+1) = 0.000C O N T I N U E W ( 1 ) = l a O D 0 NORM = D A B S ( B ( 1 ) ) -I- G ( 1 ) DO 200 J = 2 -9 MN NORM = D M A X l ( G I J ) + D A B S ( B ( J ) ) + G ( J - 1 1 , N O R M 1CONT I NUENORM = D M A X 1 ( G ( M N ) + D A B S ( B ( M ) ) , N ORM )DEPS = NORM*DEPSL A M = NORML A M 1 = NORML A M 2 = NORMRHO = NORMMN = MB Z E RO = O a O D O


19/42

F P CC.1 0 1

110

CCC 1 2 0

C C C

L O O K F O R C O N V E R G E N C E O F L O W E R D I A G O N A L E L E M E N TI F ( MN oEQo 0 1 R E T U R NM 1 = MN -c 1K = M 1I = KI F ( MN /*EQo 1 1 GO TO 110I F f D AB S ( G ( M 1 ) ) O C T O DEPS 1 GO TO 1 2 0T ( M N ) = B ( M N ) W ( M N I f Z M + W ( M N ) * * 2 RHO = D M I N l ( L A M 1 , L A M 2 1 MN = M 1 GO TO 1 0 1

S M A L L O F F - D I A G O N A L E L E ME N T ME AN S M A T R I X C AN B E S P L I TK = I I = 1-1 I F ( I E Q o 0 ) GO TO 130 I F ( D A B S ( G ( 1 ) ) o G To D EP S 1 GO TO 12 0

F I N D E I G E N V A L U E S OF LOWER 2 X 2 M A T R I X A ND S E L E C T A C C E L E R A T I N GS H I F T

130 82 = G ( M 1 ) * * 2D E T = D SQ RT ( ( B ( M N ) - B ( M 1 ) ) * * 2 4 4 0 0 D 0 * 8 2 A A = B ( M N ) + B ( M 1 ) L A M 2 = A A + D S I G N ( D E T 9 A A ) L A M 1 = ( B ( M N ) * B ( M l ) - 82 / L A M 2 E I G M A X = D M A X l ( L A M l p L A M 2 1 I F ( D A B S ( E I G M A X - R H 0 ) o G To O o 1 2 5 D O * Q A B S ( E I G M A X ) 1 GO TO 135

L A M = E I G M A X1 3 5 R HO = E I G M A XC


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C TRANSFORM BLOCK F RO M K T O MC CJ = G ( K )I F ( K oEQm Y BZERO = B ( K ) - LAMI F ( K O G T O 1 1 G ( K - 1 ) = B ( K ) L A MC DO 140 J = K, M 1 J M = J - 1I F ( J M o G T o 0 1 BR = G ( J M 1I F ( JM o E Q o 0 j dR = BZEROR R = DSQ RT ( BR* * 2 + C J * * 2 1S T = C J / R R

C T = BR / RRA J = B ( J )BR = R RC J = G ( J + l ) * S TG ( J + 1 ) = -C T ++ G ( J + l )F = C T * A J + S T * G ( J )Q = G ( J ) * C T -t S T * B ( J + l )B ( J ) = F * C T + Q * S TG ( J ) = F * S T - Q * CTWJ = W(J) B ( J + l ) = AJ + B ( J + l ) * B ( J 1 W(J) = WJ * C T + W(J+11 * ST W(J+1) W J * S T - W(J+1) * C T C I F ( J M o G T a 0 1 G ( J M ) = BRI F ( J M o E Q e 0 ) BZERO = BR140 C O N T I N U EC

I F ( K oEQo 1 1 BZERO = OoODOI F ( K o G T o 1 1 G ( K - 1 ) = 0 . 0 0 0C GO TO 10 1E N D


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REFERENCES 1. 2 .

3.

4. 5.

6.

7.

8.

9. 10. 11. 12. 13.

Stroud, A,; and Secrest, D.: Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, N. J., 1966. Gauss, C. F . : Methodus Nova Integralium Valores per ApproximationenInveniendi. Comment. S OC. Regiae Sci. Gottingensis Recentiores, v01.3, p. 1816; Werke, vol. 3, pp. 163-196.Stroud, A. H. : A Bibliography on Approximate Integration. MathematicalTables and Other Aids to Computations. Math. Comput., vol. 15, 1961,pp. 52-80.Gautschi, W.: Construction of Gauss-Christoffel Quadrature Formulas. Math. Comput., vol. 22, no. 102, April 1968, pp. 251-270. Shao, T. S.; Chen, T. C .; and Frank, R. M.: Tables of Zeros andGaussian Weights of Certain Associated Laguerre Polynomials and theRelated Generalized Hermite Polynomials. Math. Comput., vol. 18, no.88, Oct. 1964, pp. 598-616.Krylov, V. I.; and Pal'cev, A. A.: Numerical Integration of Functions Having Logarithmic and Power Singularities. Vesci Akad. Navuk BSSR Ser. Fiz.-Tghn. Navuk, no. 1, 1963, pp. 14-23 (Russian). Galant, D.: Gauss Quadrature Rules for the Evaluation of (2/&)I" exp(-x2)f(x)dx. Math. Comput., vol. 23, no. 107, July 1969, p..674.

0 Krylov, V . I.; Lugin, V. V.; and Janovic, L. A.: Tables for Numerical

Integration of Functions With Power Singularities I1 xB(l - x)clf (x)dx.Izdat. Akad. Nauk. Belorussk. SSR, Minsk, 1963 (Russian).Byrd, P. F.; and Friedman, M. D.: Handbook of Elliptic Integrals forEngineers and Physicists. Springer-Verlag, Berlin, Germany, 1954.Davis, P. J.; and Rabinowitz, P.: Numerical Integration. Blaisdell Pub. Co., Waltham, Mass., 1967. Golub, G. H.; and Welsch, J. H.: Calculation of Gauss Quadrature Rules. Math. Comput., vol. 23, no. 106, April 1969, pp. 221-230. Henrici, P.: The Quotient-Difference Algorithm. Appl. Math. Ser. 49, Natl. Bur. Standards, 1958, pp. 23-46. Wilf, H.: Mathematics for the Physical Sciences. John Wiley and Sons, Inc., N. Y., 1962.

16


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14. Franc i s , J . G . F . : The Q- R Transformat ion: A Unitar y Analogue t o t heL- R Transformation. I , 11. Comput. J . , v o l . 4, 1961, pp. 265-271;332-345.15. Wilkinson, J . H . : The Al ge br ai c Eige nvalu e Problem. Oxford Univ. Press(Clarendon), London and N . Y . , 1965.16. Galant , D . C . ; and Byrd, P . F . : High Accuracy Gamma Funct ion Values f o rSome Ra ti ona l Arguments. Math. Comput., vo l . 2 2 , no. 104, Oct. 1968,pp. 885-888.

17


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TABLE 1 . - G A U S S I A N Q U A D R A TU R E R U L E FOR. W(T) = l/mVI11

tF2T r I )

w 2 4

e 1 2


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~T A B L E 2 . - GAUSSIAN Q U A D R A T U R E R U L E F O R Ji- T 3

w 1 2. 2 3 1 6 9 9 3 8 4 5 0 8 1 3 7 7 4 - 1.529353833W39478?33977-1. 7 W 5 4 6 3 1 5 4 S 3 1 9 1 2 7 7 1 8 - 1.S91Q%Q3759061R3038438132-1. 1 1 2 6 6 T Z 4 1 ~ 5 8 8 8 2 6 1 D w ).117149M17914983ocE416213+0.1115679434129505966942145+0. 962675295773558210541-1.~ K l 7 4 8 5 1 ~ 7 6 6 7 8 9 5 7 5 2 O - 1.47406714962986311E75478-1.23224667110954660428136R-1. 6 1 5 4 7 9 4 4 8 2 0 4 3 8 0 8 8 7 2


25/42

N .0 T A B L E 3 , - GAUSSIAN Q U A D R A T U R E R U L E F O R W(T) = l/m


26/42

TABLE 4 . - G A U S S I A N Q U A D R A T U R E RULE F O R W(T) = 41 - T 4T < I l d i T I I u l I

112 IF16.1342377!33241)142694644-1.3(176863205&49485166371~1-1.47(P5S53l3654051e 3 77642- 1. G l 5 7 ? 3 ? 1 4 0 7 ~ 6 6 4 6 ? 2 1 l Y ~ - l.73JXbSS53037661W624417644-1.&32$83512~5142775tt2YH36-I. 8337256235627H6255e\5759-1.9 15 49 12 5% 51 ~13977e\533M-l.W37oQL*)7449e!i391315139-1.@26561046?J63399741437-1. 7 1 6 9 5 5 6 7 1 ~ 7 0 8 9 % ? 2 6 1 - 1. 574J iCC3136197056551~52~. 4 1 3 2 1 ~ R ! i 3 4 1 ~ - 1. 2 5 4 & 7 9 3 5 1 E 7 3 ~ 4 5 S l. 1 2 l ~ ~ l k 5 9 4 - 1.314888e~84341675161S69-2

W 2

Nc


27/42

NN T A B L E 5 .- G A U S S I A N Q U A D R A T U R E R U L E FO R = 1/K-T6

.457184752904U9737521aODS*0. 5 ( a 3 3 8 9 1 4 1 1 9 + 0

.647~771240!B821437712+0

.73707310573147197764i452986*0

.81773op976215449eoTo67267ow3.3fE&e7466593P362Q79P95*0.94LBl2476679108873515042~.97642729681946M241.997586P131015394275640+0

w 1 2.p89393891292663344950964-1 .5416747?2855957439045E)385-1. 8 1 0 9 5 6 7 7 9 9 5 7 9 3 1.10296051SJ751151oBB7753J94w3. 1 1 & % 4 S S B 9 5 6 5 1 6 4 3 1 7 + 0.l26970187789e3789@3511544+0. 1 2 B B 9 4 7 ~ 1 8 1 7 3 4 Z 4 4 R 8 4 + 0.l258824054659614489R8578+0. 1 2 5 3 1 5 5 2 4 2 4 4 9 7 9 6 1 8 R 5 + 0.11444EC2159786554418766757+0.1090515170235062445182994w3.1073801W553m6552307126+0


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T A B L E 6 . - G A U S S I M Q UA DR AT UR E R U L E FOR W(T) = f i6T I 1 1 W I I I

K1 6

FFB

NW


29/42

NP T A B L E 7 . - G A U S S I A N Q U A D R A T U R ET I 1 1 U ( I 1

K 2

.5146468835T33D52233122436-1

.113074954577E752988U705+0

.15957~126853763015016761+0

. 1 8 4 8 m 7 C B E E 9 5 1 3 3 9 5 1 4 + 0

. 1 8 6 9 0 7 6 6 1 7 5 4 2 0 7 9 8 1 7 + 0.lR15540R368919204384287+0

. 1~539 ! X612 lCC .%97?UJ~

. 1 = 4 4 Z J 5 7 1 + 0

N=12

R U L E F O R W(T) = l/m. 1 ~ 5 3 7 1 3 1 0 3 4 0 - ~.313881LL)4(*718~o6uoL114-1.4?%33?53447%se5397590m3-1. 6 2 ~ 7 4 3 o L \ ~ 4 7 1 3 5 1 1 ~ 1 - 1.7 5439361OD44509e-69272.8r)18187274237739050F3333B-l.921394t37&3l3937RW334-1.9572101~715SiB71631311901-1.9612713!3497132576002~59-1.9377Z23R48l2?S%15322139-1.89471419%53167!SOD~724-.&IpeR67516a096322S3471-1.74510489oRO6236539766873-1. 7 4 7 1 5 l ~ P d ~ r P S - l. 7 1 6 2 9 7 4 2 9 3 1 5 1 0 1 - 1.T o W 3 l B D 5 ~ 1 6 B 9 9 n 9 1 - 1

. 9322753745~75675824757-2

. 4 8 4 7 8 2 3 ~ 1 4 3 8 0 4 4 5 9 4 4 3 3 - 1

.11634139736098625241W+0

. 2 C 9 6 7 3 1 5 8 5 9 5 7 8 8 3 1 W 7 6 + 0

.31%931852592804476105785+0.44247(B9Y47191266180961+0.%9371691257872159898345O+Ll. 692473518040437424056~+0.8o38M363303317194531057+0. 8 9 5 7 6 0 4 5 9 7 5 3 7 8 4 9 7 5 4 4 9 ~.%142)2RFxp526e36~13C619+u.99%542896153438813700887+0

.2385146651163786116117-1

.540697601333674S636371?05-1.f G W 4 2 4 8 7 4 7 5 7 6 4 9 5 7 7 5 - 1.KP7496385618,1181341055436329574150682+0.1~S93197428ooQl109162+0.12703438590613129+0.121808577411610164331~+0. 1132185%ElZ675543411~. 1 0 4 3 2 6 ? 6 1 2 2 4 ~ 7 5 8 2 9 W.9761MQ42540497916343516-1.9377543271589204243&21865-1

1


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TABLE 8.- GAUSSIAN Q U A D R A T U R EW I I )

N=2

.2334679306384943762@3?9t35-1

. ~ 1 8 9 3 5 6 7 1 5 o E a 8 5 ~ 4 4 8 - 1

. 7 % 3 3 5 9 2 4 3 8 2 6 4 2 7 6 3 9 1

. 1 ~ 1 7 3 4 5 5 6 6 7 1 ~ 1 C e 2 8 7 + 0.11548647696-7CBH

. l P O T 1 9 5 8 3 3 8 7 ~ 5 R + 0

. l 2 2 4 3 9 3 7 9 5 3 6 5 2 8 l s J 4 + 0

.1l2617%75Oe5951=17+0

.9e9518766358T3ooq18367146-1

.648885-8%1680844-1

. 3 3 9 9 1 8 6 3 5 3 J 3 m 3 9 9 3 - 1

.941598792R)61955675749333-2

R U L E FO R W(T) = 1T I I ) U l l l

K 1 6

.6137E3~7%ZO4!DS29J539-2

. 1 4 1 8 9 6 3 0 B Q 1 ~ 5 4 - 1. ~ m D 9 9 e 0 2 8 7 - 1.-731685-1. 3 W 7 6 i ? 9 B 7 4 " 5 8 5 1 3 3 5 1. ~ 3 6 6 7 3 8 4 8 2 6 U o R 8 1 9 7 - 1.48544P8m42185549e32512-1.539274x)4p56896711%66613-1.574JOR9QP41-4%!560-1. 8 D 0 8 4 6 " 5 6 8 1 0 1. 6 2 ~ 5 4 3 ? 9 7 1 m 3 8 1 ~ 1 5 1 - 1.6350917747744361379165615-1. 6 3 3 6 2 8 3 5 2 5 1 s 3 3 6 i n l m l - l.62MR5D56VY1871338LD173-1.-391-71 5867-1.554m77482982#789m78R-l. 5 0 1 B 8 7 8 7 5 o Q o T 0 5 2 8 4 - 1.43378564316918y145559328-1. 3 5 6 6 8 9 oon 6 8 9 4 p 3 8 8 6 7 5 1 ~ 1.2734596744591389112474242-1.Em 39 44 94 1!E2469IeP381-1. 1 1 4 3 5 5 4 1 8 L p 2 3 4 4 1 7 1.5 3 3 3 5 2 5 3 7 6 5 9 8 7 8 8 2 4 1 2 - 2. 1 3 R 9 4 6 9 o e 9 6 1 1 0 5 g 7 5 - 2


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N cn T A B L E 9 . - G A U S S I A N Q U A D R A T U R E R U L E FOR W(T) = 1 / m

N=24. 1 3 x S 6 ? 4 0 9 l 4 4 2 7 1 7 + 0. ~ a 3 E 3 ? 4 4 2 1 ~ 1 1 ~ + 0~ ~ 7 1 4 4 0 7 6 2 ~ 5 S 3 ~.Ssa8397424. 2 4 7 4 3 6 s ~ 1 9 l 6 8 1 1 ~ 5 1 9 1 + 0. ~ 1 ~ 7 ~ 5 9 5 3 ~. 1 1 9 l s r E ? 1 9 5 [ 3 4 e 8 2 3 7 ~. l S B 1 9 8 1 1 ~ ? 5 9 3 6 1 6 - + 0. l i P 3 3 ~ 1 9 1 9 6 1 ~. ( 3 8 4 9 6 ~ 7 1 3 1 ~ 1 7 U J.lW7oEEO14S&K5951579562+0.a3960T65e1316SZ6467lE278UJ~ 9 7 3 3 5 7 7 9 6 3 7 ? ? l B R l B 7 -1.909rm28833731421P6719-1. Q l 1 1 9 7 - 1. 8 7 7 B 9 8 3 4 ? 9 ~ 1 7 1 - 1. 8 5 1 7 8 m 3 4 5 8 7 1 6 1 4 - 1.E QaS 56%3m5s351525-1.809e(EPmR56BE26393568-1 . P s ' l s l - l. 7 7 9 6 3 2 1 ~ 7 0 6 4 7 3 4 5 - 1.7-7774873-1.7Q8[pm69513M3542119557-1. 7 ~ 9 1 3 4 5 9 4 7 7 9 1 1 1 7 - 1


32/42

T A B L E 1 0 . - G A U S S I A N Q U A D R AT U R E R U L E F O R W (T ) = J T ( 1 - T 3 )T - ( l l WfII 1 1 1 1 w I )

K2 N=16

K 4.laW6312711$205741395353+0 .61224m7719452882~1744499&1.35S?J6748S4lCB91641577T320r0 . 1 7 7 ~ 1 4 9 7 8 1 9 3 8 5 7 4 3 5 4 T 4 + 0.~ 1 2 ? 4 9 0 6 8 0 B 5 3 2 1 3 + 0 .2oO14~%425177368489641+0-909011 6 S S ~ 8 9 7 U 9 2 e O 7 7 5 + 0 .84049111263368!3331014518-1

N=24

w 12


33/42

T A B L E 11.- G A U S S I A N Q U A D R A T U R E R U L E F O R W(T) = 1 / ( 1 - T 2 ) 3 / 4

K24

Kl2


34/42

T A B L E 1 2 . - GAUSSIAN Q U A D R A T U R E R U L E F O R W(T) = 1/(1 - T2 ) 2/3

K8


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w 0 T A B L E 13.- E R R O R C O N S T A N T S kN

I IWeight N ~ ~ ~~~~~~function 2 4 6 8

12

3456789

10111 2

0.88323-2 0.35333-4 0.13925-6 0.54651-9 0.29917-28~ .38568-2 .15381-4 .60574-7 .23766-9

.83723-2 .33428-4 .13195-6 ,51784-9 -79396-14 .12145-18.38488-14 .58808-19.74402-14 .11381-18.41029-14 .62755-19.71543-14 .10943-18,42669-14 .65265-19.16925-13 ,25831-18.18397-14 .28071-19-11935-13 .18198-18.10918-13 .16668-18

,13006-28.28347-28.13726-28.26564-28.14647-28.25543-28.15233-28.60142-28.65359-29.42341-28.38828-28


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T A B L E 1 4 . - T H R E E - TE R M R E C UR R E NC E R E L A T I O N P A R A M E T E R S F O R W E I G H T F U N C T I O N 1[Zero moment = 0.1402182105325454261175019+1]

J B I J ) G I J )123 4S67a9

10111213141 s16171819a, 21P 23 24252s

T A B L E 15.- T H R E E - TE R M R E C U R R E N C E R E L A T I O N P A R A M E T E R S F O R W E IG H T F U N C T I O N 2[Zero moment = 0.8413092631952725567050114+0]J

12 3 4S67a9

10111213141s16171819a, 21P 23242 526

31


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TABLE 16.- THREE-TERM RECURRENCE RELATION PARAMETERS FOR WEIGHT FUNCTION 3[Zero moment = 0.1311028777146059905232420+1]

J G ( J 11234S670S

1 01 1I21 31 41 5I61 7181 9202l P P 242 526

0.0.98061 5755213986646437un-10.651230668R96883783940316-10.6345674644453p46mo76581-10.8(p166p0436324487745683-10.62Ees46714342633412948ab10.6?RS9574S3263l642853379-10.62648934740R7Z8321655&10 . 6 2 6 ~ 6 1 4 1 4 1 3 ~ 1 9 1 3 1 - 10 .e 9 9 8 9 0 7 0 3 8 8 8 ) 5 ~ ~ 7 1mi 53-10 . 6 2 5 T 9 B 8 ~ 7 4 2 0 4 - 10 . 6 2 5 6 6 5 8 7 3 5 S S J ~ 7 5 5 5 10.255P49QOB(P44789Ri383-10.6254470043TIJOLH34820Tl0 . 6 2 5 4 ) 4 7 6 2 9 9 P 9 3 ~ R 4 9 - 10 . 6 ~ 5 P n r ? 6 1 2 8 3 0 8 w S S 7 ~ 10.653092524613064958977411-10 . 6 5 2 7 3 ~ 7 0 4 6 7 3 9 7 1 - 10.6S?439834Cj417O7708777642-1 0.6S?1881180874624tP414351-10. 251973613204479J11178897-10.6517891T134056964261P33-I0 . 651~ 12M CEI % ?74169945-10 . 6 5 1 4 9 M 8 2 9 7 7 7 ~ 7 0 4 3 9 7 l - 10 .651368M 561EXl3435750-10 . 6 2 5 1 2 K K 5 3 1 ~ 5 5 1 5 7 - 1


J1 234S6 780

1 011I21 31 41 sI61 71 81 sa, 2l P 23 242 526

G ( J10 .o . m 9 s i 3 3 o i s s s 4 i w w x s s - i0.6439953981l?J63914711742CB-10.633384424251~1%3630555-10 . 6 3 7 4 0 4 1 4 3 5 5 1 9 7 8 9 9 3 6 5 10. ~ 1 5 3 5 7 3 8 4 9 1 7 p 2 6 7 1 3 4 6 - 10. 271 l B P 4 5 5 6 9 9 6 8 3 9 ~ ~ 10 . 6 6 V P m 7 - 10. 6261968906429339953aqa9%10.6994733p475693J43116?2-10.65768467495498796535T991-i0 . 6 2 m S B 9 5 R O l 0 10.655J4Sx1876194717321l291-10.654~108885519e4405T162-10.6539369153752757941453-1O . B a 4 3 2 a 3 4 1 2 8 6 7 8 7 8 8 5 3 ~ 10 . 6 5 3 0 1 8 3 7 6 8 4 4 8 W 3 T 7 - 10.62sz67~7s9OmpBTT356511-10 . 6 5 2 3 8 7 5 3 6 9 3 9 7 8 5 3 7 7 10 . 6 2 P 1 4 3 8 5 o o 9 6 4 3 o 1 3 10 . 6519356116453~ 1% 196-10.65175639499765R75n1424-10.65183093a3e31142!Z19419-1 0 . 6 5 1 4 6 5 2 ~ 4 9 9 4 0 8 9 3 9 8 - 10.651348386276909425429012-10 . 6 5 1 2 4 0 9 0 5 4 5 4 1 9 9 3 9 4 1

32


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123456789101112131415161718192021

123456789

101112131415161 718192021


J B I J ) C L J I0 . 57 7 3%E691 8962 5764MiW 0 .m-a .I.I. 0 . SO73244852932461 776984732443 0.9 785 1 59320496503 91w91 917- 10.49487 951056 762227 333886 12+a 0.5 607 1 51331L135z1159(1372786-10.49R6R07813098141DBT3aW3 a.~ e s 9 5 6 6 ~ 3 9 5 2 7 3501IeI0.49H38250394211850R184619+0 O . Q O 1 7 7 3 5 9 7 7 5 M t 1 6 9 H 9 M 3 5 3 8 - 10.4989e32903804297O251218sna 0 . 6 2 H 2 6 0 6 0 6 6 9 9 7 6 4 4 H 9 4 3 ~ - 10~4992378482546192918SZ0371443 0. 62724Xa394244149U14tW337-10.49943521881043530463695%5+0 0.626639BLu96645CU33WOSi71-10.49955792382103361275567+0 0. 6 2 6 2 s L p 7 4 9 1 0 4 7 9 4 8 3 5 10.4996420238734425491- 0 . 6 2 5 9 8 4 8 1 1 7 7 8 0 4 3 4 6 1 2 - 10.4997116540121911954372261443 0.6?5795783@Z288n5l31'375-10.49976oon780114012ooe73S1443 0. 625656415915994613se22356-10.4997971QlP76292@5?81298+0 0.62559071323757373057282-10.499828m827589831848744Ew3 0.62546864077176?559707157H-l0 .499849~7965B83e7641945MI 0.6254m6433351425331~712-10.4931\68474757580021R%73J3Ul 0.253S12914322nm834HS744-10.4 998 840 191 953 451 561375c.46+0 0 . 6 2 5 3 0 8 ~ 5 o l m 6 4 - 10.4-61 7132075425616461443 0.~527308eJ3o4a1~78019&10.49990785231701570996161oMo 0 . 6 2 s 2 4 3 4 3 8 M 6 5 6 8 2 ~ 3 5 10 . 4 9 9 9 1 7 1 ~ 1 9 8 0 4 7 Q 9 m 4 4 4 3 0.625218368993R937162528-10.49992fa2722625797a309e354443 0 . 6 2 5 1 9 6 9 7 9 6 1 0 9 2 1 3 6 5 9 - 1

zz 0 . 4 9 9 3 3 1 8 6 7 0 7 2 6 3 3 U 2 5 7 4 ~ 0.625178538217894788657XEl-123 a.49 93 37 ~174722044830iu474+0 0.6 251 626 576 968 1 113@6451-12 4 0 . 4 9 9 9 4 3 0 1 ~ 1 5 0 9 0 5 5 3 8 9 5 4 4 4 3 0.6251487677616924519194694-12 5 0 . 4 ~ 7 5 8 4 2 6 5 1 0 1 1 2 1 c e O 7 O 2 5 c o 0. 6 2 5 1 3 6 5 8 4 1 6 2 6 8 4 3 9 7 7 ~ 1 - 126 0 . 4 ~ 5 i e ~ 3 2 9 5 5 a a ~ 5 1 m . 625125838401639O71016247-l

TABLE 19.- THREE-TERM RECURRENCE RELATION PARAMETERS FOR WEIGHT FUNCTION 6[Zero moment = 0 .9107439929578431044324781+0 ]

J

P 232 42 526

G ( J 10.461ee~a?I5551701 36 11 Q7 31 M o._-.. .._..0.484292247438311105313734bbO 0 . 7 4 1 2 3 2 0 4 3 ~ 5 6 1 4 9 9 4 8 3 9 0 L F I0.495 1303 4 %86909e3590814542+0 0.64289624956P37301P5733K+l0 . 4 9 7 ~ 7 5 6 9 9 6 1 0 @ 2 1 9 4 3 + 0 0 . 8 3 6 2 R 3 0 ~ 4 3 4 1 7 8 - 10 . 4 9 8 4 9 0 1 4 5 3 7 5 T 5 U X J + O 0.629774432410BQB85oBce4I38-10 . 4 9 8 9 0 7 1 2 1 3 7 8 2 9 7 9 1 0 1 5 ~ 0.62806W5143Ua54830~168-10 . 4 ~ 7 3 4 2 4 ~ 1 ~ 1 0 9 Q l O l ~. 62713158831129O5743978424-10.4994533 523708811746413716+00.4995738m83624814313487574430.4996583926532631487561469+00 . 4 ~ R m 7 0 9 8 4 ~ 4 7 8 1 6 2 1 9 2 1 + O0.49976612759174141m8861O3+00.4998(P1494164393515R911pl+O0 . 4 9 3 B 3 [ P ~ 6 1 6 0 5 3 7 9 1 6 1 7 4 4 30.49385277441677142386944414430.4~710883e42144546337165+00 . 4 9 9 B S 6 1 8 6 0 8 7 6 6 1 5 1 5 9 ~ 1 ~0 . 4 9 9 8 9 8 7 7 7 8 2 4 0 7 5 4 6 5 s . 9 u 30.49990938945S4215274819310+00 . 49991841 5487790161~ 742+ 00 . 4 9 9 9 2 6 1 ~ 4 5 5 0 R o Q 5 2 4 5 1 6 + 00.499932848309334921 1903314430.49993866aDBM1678291345DtO0 . 4 9 9 9 4 3 7 8 0 7 1 2 7 0 5 8 T l M0.49994t245918112FZP380462a?+O0~4999S?2151399067780115B39+0

0 . 6 2 6 5 6 E W - 9 7 1 4 - 10 . 6 X 2 C E 9 9 3 7 0 5 5 R 7 0 5 8 ~ ~ 10 . 6 2 5 9 5 1 3 5 1 1 8 3 2 4 1 3 ~ 10 . 6 2 5 7 7 1 4 0 5 2 0 8 8 7 4 3 6 1 8 10 . 6 2 ~ 1 0 7 2 9 3 3 8 3 4 1 6 10 . 6 2 5 S 6 6 1 9 9 2 8 3 6 4 1 2 6 7 6 10.6254S7S536732714490780611-10.62539476977356885581679*10.62~44O7806957315121648213-10.6253[p561611P64421992leDi0.62~68131993Q444905585M-I0.6252392Q"8939265671-10 . 6 2 ~ 1 4 8 1 8 4 n 5 4 6 6 5 10.625193937!344014426972p956-10.625175983853507258935403510.62516037268617U?E87955144-10.62514676811LDi?4842797R876-10.6251348242644m065837OR-l a.62512e8140~a3~3~366385i-i

33


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J

1254Ie7e9

1011l 2131 415161 71019a,2lP23242s26

0 .0.968459893871660943011S549-10.66957340493676~68194364510. Q 4 6 7 1 U 9 3 3 s B S T r 9 - 10.63012138283?6785c2981076%-10 . 8 8 2 4 8 0 4 3 7 7 9 7 9 7 4 6 4 7 ~ 5 10. 6R3657U47411175p29Bsn-10 .62-1 rsrs1smienmTP-i0.628468099S7T5oej12091~10 . 6 9 9 s p e 7 1 1 1 1 ~ D 5 9 9 3 3 7 ~ 1o . ~ s 7 9 4 p l o l o 5 5 e l0 . 2 5 6 S ~ ? 8 O l 6 1 6 4 ( p B o I - i0.62554983~716889e1347TB70-10.62546?84291835Z31OQl2l87-10.62S4U3al536753?6OS1334XV-l0 . 6 5 3 9 0 ~ 1 6 ~ 10.6253080~89686oo24454F10 . 6 P R 7 3 0 ~ 1 2 R 1 9 B 1 3 9 4 5 1 - 10 . 6 2 P 4 3 1 3 6 9 8 3 0 9 8 6 1 0 8 1 0 10 . 6 P 1 8 l l 2 m l s 1 m ~ 10 . 6 5 1 9 6 7 6 1 6 9 ~ 1 1 T B I D l0.6S178484339ms24641996-10.62S18942911162!53P39B17Sl0 . 6 2 S 1 4 e 6 ? 4 1 0 - 10 . 6 S 1 3 6 4 5 8 4 3 8 9 5 3 1 9 1 4 5 10.62512SR718353371P141732tKbl

TABLE 21.- THREE-TERM RECURRENCE RELATION PARAMETERS FOR WEIGHT FUNCTION 8 [Zero moment = 0.9308740569746155002420145+0]

J B ( J ) G I J )_ _ _ _ _1 0 . U I O ~m s e w c m m x ii 7 4 0 ~ 0 0 .

ssi846-1 0 . 4 8 4 9 ~ 6 9 o x ) 1 3 5 0 1 2 1 3 0 ~ ~ 0 . r w ~ ~ m 7 1 2 u a i5 0 . ~ s 4 a r 7 i m m ~ s i ~ ~ i 2 5 s o m a0.6423043428644120996395976-14 0.497Sl?S74267123663351QW+O 0.63368847a618849e308061~1

0.~ 7 6 8 3 4 3 6 7 4 S 1 O P 4 7 S l 0 . 4 ~ ~ I I E ~ 1 ~ 8 0 4 1 + 06 9 . 4 9 8 9 0 3 3 1 2 1 8 7 ' 9 S 2 6 3 4 ~ ~ 3 C o 0 . 8 8 0 7 7 0 6 S 7 1 3 7 4 ~ 7 ~ 7 6 1 9 7 - 17 0.4992'1D98312~270786105513OMl 0 . 6 7 1 3 9 4 R ~ T 9 2 9 9 2 S 5 8 3 - 1e 0 . 4 9 9 4 S 1 ? ~ 7 3 8 1 l 4 S 9 2 9 6 6 1 3 ~ 0.8 65 73 41 7694356875621689019 0 . 4 9 9 5 R X Y ) 7 1 ~ 1 4 4 ~ 0.6Bo9377B261778376SR3J-l

10 0 . 4 9 3 6 5 7 6 o Q o 8 6 1 0 + 0 0 . 6 2 5 9 9 5 5 6 6 p s 1 2 6 3 2 9 6 1 4 111 0 . 4 9 9 7 1 9 4 8 7 4 4 4 8 1 3 7 l ~ p 3 6 BBc o 0. 62S773OS34242126%%3397511 2 0.43976 598271 783761 7867405W3 0.6S63S37829381134601S9~7-115 0.499801BcpSl9491901ml4i.94H3 0.625~7SSSS?~S71SQ5%?461-11 4 0 . 4 9 9 8 2 9 9 8 2 7 S 1 2 8 2 8 T S 7 3 3 c 0 0.62~4583274OPl8343815943Sl1s 0 . 4 9 9 8 ~ 5 S 1 6 8 1 1 8 J 4 5 7 7 4 1 ? + 0 0.62539538831R79817453484?-116 0.4998'104)641283T88%678%Z+Cl 0.853445814328289?31'?8+11 7 0.499888)34668829~44s4040284+0 0.62aOe9767076[P5371695T91-Ile 0.49989869081 7 7 1 8 1 0 " 3 a 0 . 6 ? P 6 8 4 7 8 3 1 2 6 7 1 8 3 8 m 7 119 0.4993o9eB1Ba6pS?36?27411S+O 0.8P3955496571682LPMS693-120 0.49991~S7843414793&3249Co 0 . 6 2 P 1 5 0 6 6 7 6 8 9 m P 5 121 0.4935QQ77629(324517lP1437+0 0 . 6 2 5 1 9 4 1 5 1 F lP 0 . 4 ~ 7 7 7 S 9 1 0 6 S m 7 1Z?U3&W 0 . 6 2 5 1 7 6 1 4 4 % 3 8 2 1 8 M 1 4 125 0.4999588)61232S33-m 0.62516J5Mo55893731091556-124 0.4999437081054?9613777497+0 0.62514SQ338312C%42491298-12 5 0.49394819947654743=@2~613+0 0.6251349477~181258S631-126 0.49995217393~53R6476169+0 0 . 6 2 5 1 2 4 3 9 O 7 1 4 6 4 9 1 ~ 6 - 1

34


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J

125 4S 67e9IO

B ( J )0.431 184926538298422492?P%O0 . 5344112677185342651850937+00.49970816653571 532451144akO0.499997992XZ3652642CE16647+0 0. R1111137535S946480494s4+0O . ~ 1 K 1 3 3 1 2 1 ~ 7 9 4 6 9 5 + 00 . 4 ~ % 3 8 6 8 0 1 8 5 4 7 8 6 5 7 7 + 00 . 4 ~ 6 2 4 2 6 0 S 1 5 4 5 3 + 00. -776644347e4+00.~ 1 0 9 3 X l 3 1 4 9 S + O

11 0 5*0 12 0 mzm-

C ( J )0. ..0.121 %M9W607895?1822357+O0.63Lp33 5 181643522a3[IJ32703- 10 .625079775ml3a3E34203533610.2499961(P298;?3557297S7-10 . 6 2 4 %9 9 6 6 ? 2 1 6 6 1 3 7 6 9 -10.625o1111xI1iu4277852856404-10. Brrm3oOP32057766949852-10. 6 2 p m O I z I D D 4 9 5 Q 4 9 3 4 ~ 10 . 6 2 4 ~ 1 0 8 3 6 8 3 0 9 5 9 - 10.624--67165-10.62- 16723660-10 . 6 Z ~ ~ l ~ T X 7 - 10624 59.93-10 . 2 4 P E 8 510.e- 144-10.amL t u r p I -1

13 1 41 s1617101920 21 22 23 2 4 2 s 26

0. lQcpSS+o0.P llwo+O0 13co00. - "Lo o.:0 . c0 . m r r r . 0.0 . =40 .pm.Lu- 0 . 0 . 0.r

00.62. 0 . 6 2 2 1 5 i - 10.62. 1oe3*=100. 2Wl.1. - _.-.It 10.6221111 10.629 1


J B I J ) C I J )12 3 4 56 7 e 9 1011 1213 1 4 1 s 16 17 18 1920 21 2223 2 4 2s 2 6

0. ....__~0.6 1 78132e2217U?IX3232457- 10 . 6 2 5 2 9 0 1 9 9 3 7 R 1 0 3 9 5 S 3 - 10 . 6 2 5 m6 8 8 7 4 6 9 X " l Z 4 Z 3 5 5 - 10.6249397553471393iU7690941-10 . 6 2 4 ~ 1 8 9 8 3 0 4 0 T J 8 4 7 8 7 2 2 -10.623XUXLE2341827357316S3-10.62- 1886554093610 . 6 2 4 - 7 9 8 1 85- 10 . 6 2 4 ~ 7 2 1 1 4 0 7 8 7 1 - 10.62-. 75Ec90B590-10. 2- ~ .L. 1 5 5 6 3 9 5 1 0 1024 67-10 . 6 2 4 P 1 0 5 5 2 - 10 . 6 2 r T r l 3 . < . ~ l r f i ~-10. 31-1o . 2 r 10 . 6 2 P l0. 253 1 I1 . I 1 1I m . l l0.62Y .. 10.62 9m 10eT ..0. 29 1 n0.625 r ~ . . .0 . a - II 1

1.24I r 1

3 5


41/42

123456789

101112131415161 718192021

I1 I 1 I

T A B L E 2 4 . - T H R E E - T E R M R E C U R R E NC E R E L A T I O N P A RA M E T E R S F O R W E I G H T F U N C T I O N 11[Zero moment = 0.2622057554292119810464840+1]J B ' J ) G I J )

0.76 2 759 763 501813188062326ocg o.- - - ~0.43564062201 772%40697Z94+0 O . & 2 8 6 4 2 C B R 4 3 2 4 6 1 4 6 1 ~ 10.48366222 5381225795751342O4+0 0.601 5784166190404885715378-10 . 4 9 2 6 3 5 9 6 3 7 7 5 7 8 3 4 9 1 ~ 1 + 0 0.6193678264971~E57191Ul-10.49%16082403241212&39954+0 0.6P5ZP585m762349e8lt274-10.4973cE57536834t219R594964H3 0~836205886612335061715978-10.49Rll64296636529946910406+0 0.841260383439648456735817-10.49R61c P3 55Q)l2693H51H7147+0 0 . 6 2 4 3 9 8 6 n 4 3 3 6 2 4 0 e 1 1 - 10 .49R9523~ 19128576~ 120~ 67+ 0 ~~~~~~~~~~~~~~~~10 .4 99 1 5 4 1 0 5 Z 6 3 8 6 0 7 5 3 9 6 1 2 4 ~ 0.846668234WD447JX3665392-10.49331326702#2797%37930142+0 0.62473CU 7678932a23O5718368- 10.49343136816?9S15576863%5+0 0.62478~545166552066Z235-10.4995214144758561080R43496+0 O . 2 4 8 2 6 6 4 # 5 2 1 2 4 4 ~ 1 e 3 # 9 4 6 10.49959164IJ109~59207520R7+0 D.24#54955224m585277H449+10.49964 7464 73H7156655116376+0 0.624#76H7955oH7R5713376H1610.49%9257411~501241471&l+0 0.24#942om3416#9749557342-10.499R95457223993m6H53533085eO 0.249oH11#4S3394243165 534-10.4997Ce2599-135R554M 0 . 2491 -1 7#5359645619747- 10.4997859657#7912aa31233R+D 0.62492#34X?1769769A3229232-10.499eo7771B4333151074733RJ+o 0.624936675534m574H0597428-10 . 4 ! 3 s e ~ 1 6 2 7 8 3 ~ 1 7 + 0 O.249432H529524016541109!Fl

P 0.49904245737869#55- 0.62494#913#5259961Q5674#1-123 0 . 4 9 9 ~ 5 ~ ~ 1 " 5 i857375893eo 0.24953745#977#3#7784U3676-12 4 0 . 4 ~ 5 3 5 3 0 3 0 4 5 2 2 ~ 0.249579246HiU4935279#7D4612 5 0 .4 9 9 0 E 2 0 9 9 M # E 5 9 9 2 1 6 5 6 1 + 0 0 .24961 5271P92163Ke32851-126 0 . 4 9 9 B B B 6 3 5 Z E 2 3 Z 4 4 9 3 1 f f l 0 . 6 2 4 9 6 4 7 4 9 2 5 0 4 5 m 7 4 4 7 - 1

T A B L E 2 5 . - T H R E E -T E R M R E C U R R E N CE R E L A T I O N P A R A M E T E R S FO R W E IG H T F U N C T I O N 1 2[Zero moment = 0.2103273157988181391762529+1]J

1234S6789

1011121314151617181920 2122 23 2 42 52 6

E I J )0.71 31 7412781265985Xa76291*00.4574649756UUOC1297#073342+00.488000316#W32957R92(P167+00.4944457136921354Rt)6c174163+00.4968059789R226520923H6CUJ

G I J )

0.0.913826341H65190505355904-10.6214157546585639457435741-10.62562WJ#9751P2097592RW-l0 . 8 5 6 i ? 9 9 8 o i U 3 5 4 6 5 9 1 6 9 ~ - 10.6?54RO3%91R9#19131332-1. 4 9 ~ ~ o i c " ~ 3 ~ 9 7 0 9 3 ~ 4 * 0

0 . 4 9 8 5 4 5 ~ 7 6 6 9 ~ 6 4 ~ 5 6 7 0 1 9 ~ 7 * 0 0.655641066555R03461296587-10.49R~31367944321599964#18+00.4991 70704605515 6 317 2 5353+00.4993417D3Ce3752420121273~0 . 4 9 9 4 6 4 7 5 5 5 9 5 R 2 8 7 5 9 4 f f l0 . 4 9 9 5 5 6 2 W 6 2 4 4 7 3 4 3 5 Rm4 l W0.499661382321380163244n7+00.499680718205169263541238yo0.49 972 41 5RR9RRm57#4313R9R+o0.499759e9R49m97omcu65944+00.4997R012!532432201121933#+00 . 4 9 9 R l 2 D 6 6 1 R m ~ 7 2 ~ e - 4 4 9 + 00.493H3216613O5429612525275+00.499#4W051120634935611432+(30 . 4 9 9 R 6 3 7 7 4 6 3 4 # 9 0 7 3 9 R R 7 + 00.4990763299112KJ4176981253+00.499RRE25RR3042R9P3#E17DR6*00.4990967428374243971934465+00.4999051E44#329RR407#4CD+Ll0.4399124RR312553739655583DeO

0 . 6 2 5 2 R 2 1 0 1 ~ 3 3 4 4 3 8 5 5 C + l0.2~3876#26361766?811P-l0.85181~3RO~!332RoR26464-10.6251499Z33!37178aR4R937R-l0.625125R7603250055l55-1 0.62510709875W38o83a99149-10. 8 3 3 9 2 1 9 9 1 R R 4 M 9 9 0 4 4 5 7 - 10. 6255301H71H990127953261+10. 2R7iU366RTXP6534SZZ-tS34-10 . 6 2 X l e 2 3 e 4 " 4 9 5 7 6 ~ 10 . 2 5 0 5 5 4 3 6 3 2 1 9 3 1 2 R 6 3 2 1 ~ 10.625049687859374437556P-1 0.6250447#6#911064312231361-10.62504057511934#93333#9131O.25m6329443144Ee873961 0 . 6 2 5 0 3 3 7 5 2 9 9 R 3 4 0 1 6 1O.25U3U36#7354327R31~-1 0.29ceR514767H126cp6373!371-10.625ZP634092230R1R42llH-l

36 NASA-Langley, 1970- 19 A-3553


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gauss quadrature rules involving some non classical weight functions

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