SOME INTERPOLATION FORMULAS IN TWO VARIABLES Thesis for the Degree of M. A. MICHIGAN STATE COLLEGE Frank Saidel 1941 ”:34: W3“? um“ Michigan State ' Unfiversity, , M‘ “— _L- ' MSU LIBRARIES n. RETURNING MATERIALS: PIace in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped beIow. SOME INTERPOLATION FORMULAS IN TWO VARIABLES by FRANK SAIDEL A TFESIS Submitted to the Graduate School of Michigan State College of Agriculture and Applied Science in partial fulfilment of the requirements for the degree of MASTER OF ARTS Department of Mathematics 1941 ACKNOWLEDGMENT To Doctor William.Dowell.Baten whose encouragement and many suggestions have made this thesis possible. 13412:} O l. 2. 5o 4. CONTENTS Introduction Euler‘s Polynomials In Two Variables Extension of Tschebyscheff's Formula of Mechanical Quadrature To a Cubature Formula Cubature Formulas Involving Differences Bibliography 17 25 31 SOME INTERPOLATION FORMULAS IN TWO VARIABLES 1. INTRODUCTION PrOperties of two classes of polynomials in one variable, which play an important part in the finite calculus, namely the polynomials of Bernoulli and the poly- nomials of Euler, have been develOped.1 One of the objects of this thesis is to ex- tend the polynomials of Euler to two variables and develop some important properties of these polynomials. An example is given showing the use of these polynomials in evaluating a double sum. A cubature formula for approximating the val- ue of double integrals is obtained by extending Tscheby- 2 in one variable scheff's formula of mechanical quadrature to two variables. A remainder term is found and an example illustrating the use of the formula is given. Finally by using Newton's interpolation form- ula with divided differences of functions of two variabless, results are obtained from which a variety of cubature form- 1 L. M. Milne-Thomson, The Calculus of Finite Differences, London, Macmillan and Company, Limited, 1955, pp. 124 ~ 150. Hereafter referred to as Milne-Thomson. 2 Ibid, p. 1770 5 J. F. Steffensen, Interpolation, Baltimore, The Williams and Wilkins Company, 1927, p. 205. Hereafter referred to as Steffensen. 2. ulas may be deduced. Remainder terms are given as well as an illustrative example. Notation will play an important part in simplifying and condensing the work of developing the afore- mentioned formulas. In one variable the operators A and V are defined as follows; (1) Aim) = 19““) ‘ I“) (2) V4“) : lz[-:£LX+\) + {I’d-I . For two variables the operator AK A] is defined as follows; (5) Axnfiuq) 2 A, [fcx,‘a+I)-— Imp] : AIY‘IUH’A) - {09%)1 = 4’th #09:“) - WM) 400:), from which it may be concluded that the symbols Afix'and [1%_ are commutative. Also, the operator Vac v.1, is defined so that (4-) VXV11€Lng <7x [11%er ““20“ ‘73 \iIiWW) " IWA’II ; 21"[tflx‘I’b‘a’I'O‘t-fo‘q“)+£LXH)3)+£(XJ'£—L which shows that Vx and V} are likewise commutative. )l 5. Divided differences are used in Newton's interpolation formula. For one variable these are defined as follows; — Ia.) Imps.) = rm.) f ao-av/ Ia”) '- ffituafl.) a, - a1, {(‘cmauafl :: Ia. and in general ;fi(¢ a" ....’an) : £(a'oa"";a.1~-,) .. 1FLG'IJ""Iau) as “an. where this last expression is an n-th order divided diff- erence of f(x) with respect to the argmnents ao,a'l;""la"‘ ' Similarly, for two variables, (40:49) ‘" £La'u "3) ao-‘a, if «Rudd/en) 7‘ 4F . _ f(&.fl,}&.) " :fimwali 4!) 7-?(a’ua'l)’&e)’&'|) " r ’4. — ,6, :fmwgo) " £(aué) "' fang!) +£(QU4) (do-4'!) (’60 ”’63) and in general ' _ ”an 4‘. "{(ao/LU-vganj/d'.) :: flaw [0,...”40) ¥(4., J ) a.-aw and {(4M~-,au}4°,m,gn) : fl“01”‘2a’1~1;£’w"7£h-3) -.f@,-~a,,_,;4,.@ (a. we») (ea—4“) + - {(al...,a,.,;4,,,-..&h_,) +_f@“..,a,‘;4,l..,4h) (“a " an) (Lo-’4u) 4. 1 An important formula for placing limits on the remainder in Newton's fonnula is H-H MM {ago/“an;3,4)...Igh‘)___ 1); DV, {($31) (KM)! Us“)! whereDE andD,I denote partial differentiation, and where 4.9% I éan ) 4°; 9‘4““ . Important also will be the following Theorem of Mean Value2 for integrals; Let f(x) and g(x) be integrable functions of which f(x) is continuous in the closed interval OvéXé‘é’, where g(x) does not change sign in the interval. There exists, then, at least one point E inside the interval such that [Ahmflcxmx = :fmggwdx where ¢<§<4. l Steffensen, p. 205. 2 Ibid, p. 3. bl r\'i, - Ilka- ”I 5. 2. EULER'S POLYNOMIALS IN Th0 VARIABLES Defining E polynomials Efierx,y) of degree v in x and u in y and order n in x and m in y by the rel- ations (5) D: DIE E“ w (x?) ___ my.) LI E; “1.03) and (6) {I71 {17" 9:32)“)? 2 Xv II“, where 1;“): v(v—I)-~-(1I—i+l) 1 it may be shown by the method suggested by Batenl that these E polynomials for two independent variables x and y are the products of Euler polynomials for the single variables. By Taylor's Theorem E‘”:’(x+ h ,)-—~«3+k E‘L‘f’ (x «3) + W E11, “fix 3) +I= (x - 2>=(x - 2-)(x1- x - %—>9- =LZ— ‘° <2“. Is:- 0 b 7-0 twp“) COMM) where _ ‘9.“ "' ”,W K250) is called a ‘0 number of order n,m. In (8) let x:x+r and y=y+s giving :9 0' t1, my“h (MM) 7: 22;; T?0L(X1’L‘\*5)=¥ (flew) «50 ° ( 10) eolflxM-J +2~>L3tsfl+3wlt~9 f u femur,“ M) yams»? ) ~ Now let G(Xt ryw):xt +yw so that (10) becomes (11) if,“ f” i“- ’D u’ MOW”. 3+sze exflaWZZ-g: 23:16” :iQfi) ‘0' LL! «:0 ”=0 ((59 ”:0 and (8) becomes (12) t (e H) (mm) fmmL tw)e ex +aw-ra _ “Z 2%212 u, v Wong) a.» V’O 1. Substituting ‘ .‘. th.‘. ‘3”) 4. W42. ... for ext-r3»: in the right member of (11) and equating 2.. . V w . coef11c1ents of t 90' gives q) um:)(x+ml—318)_, (5:: :thL 3) +LVJKLQ1L.“::1L'115) “2231322 222-202 2: $2122 22) 9. +6) €sz {‘32: 2).” 5) +6233 9‘,“ “(0%) 21,8) + .. .. +(V)(u)>43 5 1‘43 (4. V u. and equating: coefficients oft w this results in ”(X-5M-Q ( 21) m \M) V‘BV N “1’1”“? “m M But V vx Nurm) was); Z‘IERM: (x .9) ‘31") “Li-“\h M)“ 4"! MVP)» . W1)! ,3“) +M:W2X|3) ‘3“ (h-l ’u- " Then,l/HM'::(X,'1)-:'N Lx+n,‘+n)+Hu“(x-elqhfivuiwauybr my!) Letting x: O, y=O gives the recurrence relation for N nwnbers, Ll ”(ELM U “A” :15” )+Nb‘ u ::UI 0)*Nh.: +\\L‘:, “W: The simplest N polynomials are obtained by putting: g(t,w):O, n20, m:O in (20). This gives xt+aw_ 991 39‘ iv AICONIO" “so Uzo and equating coefficients off :3“ after expanding the left hand side shows that 0) (Q, g 17‘k These simplest N polynomials shall be regard- ed as Euler's polynomials for two variables of order zero, 00;) denoted byE (X,J)o E polynomials of order n,m are now defined by “"VM a... PM. 2. m“ w U3,“ (2‘1;e :. "1"“TE— ) W(€t|)(€—3 242415.14. :1 Following Norlund's notation for Euler's polynomialsl, let " UL M3 '(‘H'UQ “MW" (20’ E ‘ (0,0) 2 c 139» v,u. || The generating function for the 0 numbers is therefore “*M fl 2. (24) f u, M, 7- ""‘ getfl)“ (Q- ‘H) use {7:5 1’! (F‘W) The values of 1"”“EU’ ‘ ‘03:.) are called V t“ 0") ’JB ‘U’u. K‘ bd’ ‘TITITCC E numbers, E‘: :jof order n ,m; (25) (“I“) \f-rkk (“0“) E 3 1— E \‘1‘1. '5.) b.“ ‘U,\~ A“ l Milne-Thomson, p. 145. 13. Some fundamental properties of E polynomials may be listed by noting that E polynomials are N polynomials and therefore (6 polynomials. Hence they have the proper- ties of polynomials as well as of N polynomials. E\:::M POLE.“ )(0) +x]\!’ [EM (0) +13)“ from (14) <26> sway—— [f in [19% 1 (28) urn.) "CHM-1)L;1 <7 3‘K7xrz' ’ w::“3): t; ’ ) By mrepe)ated applicationL of 28) at 0‘0“” EC“ ) °)(x E u .23) ‘3) It follows that CK‘W (29) 337‘ E» QC¥I‘§) :X‘, “A“. Slrlce E‘o’u)kx‘\s) :ZX‘ Notice that properties (27) and (29) are I rom (21) exactly the same as the definitions (5) and (6) upon which the first method depended. An interesting theorem will now be proved showing that E numbers in which either subscript is odd, are all zero. This is the complementary argument theorem for two variables. The arguments X and n - x, y and m - y, are called complementary. Replacing x by n - x, y by m - y in (22) iv g 83 1 1M». chtam-AM -£V “ u~nj _______,__ "' ”Li Ev 0‘" “'33 “(ea PPM)“ 1"! “a! 3“ 4d: - v infihaea :5#‘ - (e't... .)V\(eoI-JT ‘ M 2 M8 q $2.0 :0 p0 -.-. 2i LL10? L’w) EMWL "U! LU u u. x13) Kzo uzo Equating coefficients of (’W“ shows that (30) 0' m; (mm) £1,“ 01- x,-m -;;) xflU) Ev.“ (Ly), Equation (50) is the complementary argument theorem for two variables and it is true for any N poly- nomial in whose generating function g(t,w) is an even function. Letting sz, y:O and v22r, u =25 in (30) “"1"" (h. M) £147.15 (mm) — E1415 (0’ 0) ~(1 +15) (h : 2- n C; 'b" Inns “Mm, _ "(10+15/ (hfl‘fl) ‘ Thus £112.15 .1 1m“. has zeros at x -o, y- 0 and xzn, yzm. Putting); X15113: ’7‘: and v~_2r+ l, 11:23 in (30) .— (n m) 14141-15 (31 m) t. ’ l) ' El I’D-“’15 (1) 1—):E 5‘" "441-5 (‘le 1-) (mm; (2;, $.13) O. . -H ‘m -r _ . Putting x-1’ 3,- J. and v Mar, 11 \231-1 in (50) M. 2.5+: 1- ' :1. Thus, E numbers in which either of the subscripts is odd, are all zero. E polynomials of the first order may be obtained by placing n: m a l in (22). Then, 11 _= Kt+3w_. tVW0“ (51) (Q‘HMCWH) € ‘ Z: Z 7 <7:- E‘Cua'g) and letting Xzyzo gives 15. _ —- c (etflne‘vu) V '4' W4. Q20 ‘VEO ' Letting x =y-. :3— shows that JL—€£t+iw ii tv’w“ ‘ 1 l (FHHQ‘W) ‘ 17? 7! firm (1’ 1—) uzo V=0 c» 00 1 33,51qu E V! u! 1"“ ’Vm (4:0 1/20 From (31) t ‘2‘ °° q "*3W-C€H’fl)(€t__l)2 E 57w : C ‘ Y ; LVNA[X';/v “:0 ~10 ' Expanding the exponentials on both sides and equating coefficients office“ , p:O,l,2,"° , q:O,l,2,-H gives the same polynomials as those found by the first method. To find the E numbers, let x::y:-% in the first order E polynomials which shows that EON : 1 Elm 2-1, EO’L: -l EAL=L l and it could further be shown that E30 : 5’ EON 2 5 EV,L:-5’ El’c.‘ ‘2 "'5 E.“ 1 25 E‘.O: ’61, EOI‘ : -61 16. E63. : 6]., E2,“ :: 61 Eh” = -505, Ex; : -505 An example will now be taken up to show how E polynomials could be used in evaluating a double sum. To evalu te a :thW :I make use of (29) for n»=m:=l, which gives Va Vx Eva“ 0"?) =XV3R. It follows that E: : WWW = 3' ZliéU/M VsVrEmIM) e, ,i, J—[qua I)+(- -EI)"“ (In-I, I) +61)”1 Emu hm)+( U ")5er Mn]. In particular to evaluate :{iQ—ommv [emu/2L 5”,(%//+E.,.,(/,é)—éjfl(7,6j]. It may be shown that 5‘”! (X13): X (K‘U (AV-lKLlA‘f 3X (3)3(‘7‘0 (az_y_” so that E5”, CH) 2-55“, (7,!) 7. E9” 0,5) 2.0 and Em (cm): 3/1, 383, 770. Then a)” #8": ~ 73 57.53%0. By actual caldulation, it is seen that this is the true value of the double sum. l7. 5. EXTENSION OF TSCHEBYSCHEBF'S FORMULA OF MECHANICAL QUADRATUHE TO A CUBATURE FOHhULA Let F(x,y) be a given function, and E(x,y) an arbitrary function which is assumed to have continuous derivatives in x and y up to and including the (n+—l)th. Points X“ X," - . -, X" and a“ 31V"! 7" are sought such that 'l (52) fl Fag) E(X.3)Jxflg 1‘ 44:2: 2: £01.75) ”in. , where hk and the points Ann, "3X” and 3”,“ ..., y" are independent of the particular function E(x,y) and where the remainder term Hz”, depends upon the above points and (X0,+gD,)M-‘E(X,g)’ where a v _ Juwv D,‘ 03 5mg) : two”): mil—(X050. By Maclaurin's Theorem I 150,3) -. E(o,o) + (xox+3o,)5(o,o) + W Ema) (x0 + D " (x0 +30 "" +... + 47p! Ema) + 0:”);— EM) where ogq‘éXJOéfiég. Consider from (52) l l h -. Rh... {[1, F(x.;)EIx.:)&xIIa ~41. : 2: Hindi). .I is! Substituting for E(x,y) from Maclaurin's expansion gives Rm : ['1' F093)[Eco,m+(xox+3v,)ecqo)+““ZD’LEmo) x D a C D n+1 +... + W E(0,0) + %ff,{’ EI4.6I]M&' 1L3. ( 0+ 1 ‘44; : LE{o,o) +(xgo,‘+yjo,)£(o,o) + K " :3? 2;) 5030) (x 0 +31) (no +30)” +u. + Afmw+ git-‘33; EMQQZI’ . 3 K "‘ ~— . Letting; gm, : L[' i 5.; “WNW? above gives Rn” Z 7:10 E0,o(o’0) + H0 EI”, (0:0) +73. Ea,‘(010) 1’ 71.0 57-10(010) +7.” Em (0(0) 1’ Ta. 50.21“") 4"” + 7:1.0 Ema {0, 0) + T51-“ EH,|(0’0) + “ ‘ "‘ 77.“ Elm-I (0:0)1'Em 50m (0’0) +4f' W E(q {)F(x.3)&xfly -—n {£59,020} “445.0(00):X«. ”£454.00: 3a 01*: _ H‘ - V145: 1:0.__IL_____(O,0) : 2; _£&::IWW E@£.4a') d-‘ c"-l where 0§ a; E x; , 05. e'a' : ’1' 50“ t.i=’o1:"‘a’l. The terms containing E 000 (0’0) 1 £50 (0’ 0) D . . .) 50,3.(010) can be made to disappear by taking hl‘gfi : T010 *l g‘é : KC 2 7:10 h44 : 33' 2 7’0" n 2. - 1! T '1 (4 if. X; “ ' "° 2: l‘giixt'w :l:TI,' =l C:. I 19. .124 Since ,1: 2—4 X534. ‘ 1:. ‘ i=I 3’ there are only 2n+l independent equations above. These are ”1‘4 2 Tom MIX" x; = 7;, Wig w: 7;, i: “a: x221! 72,, we}: a; :1! 73,, tut The 2n 1 numbers .“ + (in x‘)*‘- Ik‘llg'13l;"'13h may now be determined since they are expressed above in terms of the moments, Tq,-V'. Then (52) constitutes Tschebyschefr's formula for two variables, the remainder term being 20. I I k+l (53) RH" : f Lfl‘éfflfl E(q,c) F(X:y)&;(!g " h+a o '0 14% g (Xu9x+30 a) E(45,€j) (HI)! which vanishes when G(x,y) is a polynomial of degree n at most since then (XDX +303)"“G(x,3) :0. In this case formula (32) is exact, that is, there is no remainder term. In general, (53) will not vanish. In this case it is not a practical form for the remainder since each of the binomial expansions in the formula contains n+-2 terms and the total number of terms in the remainder is (nl+-l)(n+-2). The points X“ x,” “RX!“ 3,, 31. ”'7‘! may be obtained by making some assumptions as to the nature of F(x,y), E(x,y) and introducing a function of z and w, f(7,w), so that the problem reverts to the method of Tscheby- scheff for one variablel. The procedure is as follows; Let F(x,y) : G(x) H(y), a product of two given functions. Let f(z,w0 :,r(z) s(w) where r(z) : (z - x1)(z - xl)(z - x3)°" (z - x”) and s(w) : (w - yl)(w - y1)(w - y3)°°' (w - ya) so that xux,.-u, xh are the roots of r(z):(1 and '3‘,31'~.’ 3‘1 are the roots of s(w) :0. l Milne-Thomson, pp. 178-180. Take E(X,.V) : (AX) My) where g(X)-: (Z - XYJ and My) = (W - y)" It follows then, that [I .' F03) Exams: [1' 6(5)H(3}§(K)4(3)£X10I1 _ ' GM) HQ!) _ [4} xflxlw -3 I0, It has been shown1 that /’ GCX/&K _ it “(3)1, ~& ‘1- _ W + ‘ I o -‘ 3~x ,2(}) 1.5,??1 J'If3 I d o? ‘ H‘J) ‘ 1—”) I + ——i‘ + and Le, a”; [J ‘ 4 5g) + LV'H'L Wm: where gucl,nu and &01%I~- are independent of z and w respectively, so that ' FCK. ) (9).?) Cl C). [.[I (01 MM}; 3) A“2:51 [4/27 4'3““ +JTIT+WJ $1M)+ IQ! 4£5_ [fis 5(W)+ wax—L + wh+3flnj Integrating both sides with respect to z and w gives [lF(Kg)[v](3'KI/¢J(W7)JIMJ=[A£H1...”) .E—_.’ _‘_;.‘ ..] we?“ my“ ‘ [(1778 5(W) qH)wn'—'L—?|- W-h] where C and D are constants. It follows that l Milne-Thomson, p.178. 22. ['G(x)floa(a‘}()&x [1' “(Im’aW-Wg {.4147 mast __c_,_ ...j (hfi)9htl [Aggie 50d) [hf-()Ld.” Q'fiH‘L] . Since this is now a function of 2 times a function of w equal to a function of 2 times a function of w, the functions of the like variables may be equated giving I: we m We“ and I. 09, W I' Henge/w; M75" m This throws the problem of determining the points directly upon the case of one variable where it is shown that for1 n: 2, -xI : x; : .5775502'7 n==5, -x'; x3; .707l0678, x1: 0 n = 4, -x, .-. x1 : .7946544'7 -x2_:-. x3 : .1875924'7 n::5, ~x’; xy: .83249749 -x,_:x.‘ ; .57454141, x3: 0, where G(x) has been taken equal to one. The same values, of course, apply to the y's for corresponding values of n and again, H(y) has been taken equal to one. To determine hk, proceed as follows; ‘7‘“. =51} Irma/mo; but F(x,y)=: G(x) H(y) : 1 so l Milne-Thomson, p. 180. (0 (fl 0 h). . I I {Q - I, [I did; and An example will show how formula (52) is used. Suppose it is desired to evaluate 3 3 ' [/1 X\— 3?! (Ma He ,. J. - t P A ‘ re E(x'3)‘X‘+y‘- ) F(X:&)v-l. A ransformation of variables must first be made that will change the limits of integration so that they will be from -1 to 1. Let x : p1-2, y : q+-2. Then 3 34.. ML [[\L X"+g 4x13 2/”! (ow-z) +(8+1)11¢Jg J’1-2. where ( , : ll ‘ E M) warren» Taking n::3, the result is, exclusive of the remainder, pa-l [.l, (I+1)‘+(5+z) do”; = %L[.33473r.llm Jr. l‘f365’t35167 +, 150001. (7655'r. 30079+. 1330+, K6770] 21.00533. Taking n::4, the result is, exclusive of the remainder, ['I' “1 M5-=L’1183+lw [Quit-I301 -: c (ft-J.) 141311.)" ‘I V 7/111 1‘, 381.56 1". 1.756% t 11757 +16 335 +, 35067 1: 17/06 1-. 12M. +. 036% 1-. 30170 +. 15%? 4-. 111$7+J78QU 2 1.00431. However, the integration may be performed directly in this case, giving [73 swag = a,” wwwws ~.2 2:3 : I. 003 8.2. This shows that by increasing n, more accurate results may be obtained. 25. 4. CUBATUHE FORMULAS INVOLVING DIFFERENCES From Newton's interpolation formula with divided differences for a function of two variablesl (56’ 1’09?!) r: i X 3.1 m +R “=0 «4.:0 9 f(X:Y) where the following notation has been used; X, = (K- a.) (X- 4m) ‘a, = (21-5.) (3-4.-.) XO 2 3. = I iv“ 1' 7: (“am “V; 40'" (‘1) X mu run m“ R = (ff; Dg f(é,1)+(:~—j;',0:f( u -,"~, (335; D. D, How) Assume f(x,y) to have continuous derivatives in x and y up to and including the (n1-l)th in x and the (nr+l)th in y in the region being considered, and a, §X,£,a( gal... 6. .5- may... Let a and a4vw be numbers such that in the interval a< x 1». ”'iliifilTHIIIIflifliLifliflflifiifliwflmfs