Lakireddy Bali Reddy College of Engineering, Mylavaram (Autonomous)

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1 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4 T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Iterpolatio: Itroductio Errors i polyomial Iterpolatio Fiite differeces Forward Differeces Backward Differeces Cetral Differeces Symbolic relatios ad separatio of symbols Differeces of a polyomial Newto s formulae for iterpolatio Lagrage s Iterpolatio formula.

2 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. ECE - B Sectio T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram 3. Itroductio Plaed Topics 3. Errors i polyomial Iterpolatio 3.3 Fiite Differeces i Itroductio ii Forward Differeces iii Forward Differece Table iv Backward Differeces v Backward Differece Table vi Cetral Differeces vii Cetral Differece Table 3.4 Symbolic relatios ad Separatio of symbols 3.5 Relatioship betwee ad E ; operators D ad E ad some more relatios 3.6 Differeces of a polyomial 3.7 Iterpolatio 3.8 Errors i polyomial Iterpolatio 3.9 Newto s Formula Iterpolatio Formula 3. Newto s Backward Iterpolatio Formula 3. Formula for Error i polyomial Iterpolatio 3. Cetral Differece Iterpolatio 3.3 Gauss s Forward Iterpolatio formula 3.4 Gauss s Backward Iterpolatio formula 3.5 Iterpolatio with uevely spaced poits 3.6 Lagrage s Iterpolatio Formula Lectures

3 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Lecture- Itroductio: If we cosider the statemet y = f, we uderstad that we ca fid the value of y, correspodig to every value of i the rage. Defiitio: If the fuctio f is sigle valued ad cotiuous ad is kow as eplicitly the the values of f for certai values of like,,,..., ca be calculated. The problem ow is if we are give the set of tabular values X Y y y y... y- y- y Satisfyig the relatio y = f ad the eplicit defiitio of f is ot kow, is it possible to fid a simple fuctio say such that f ad agree at the set of tabulated poits. This process of fidig is called Iterpolatio. Defiitio: If is a polyomial the the process is called polyomial iterpolatio ad is called iterpolatig polyomial. Note: Throughout this chapter we study polyomial iterpolatio. 3. Errors i polyomial Iterpolatio Types of Errors: Let be the value ad * be a approimatio to the value. The the differece * is called a Error i *. eact = Approimatio + error Iheret Roudig Trucatio Abs. Relative % Errors

4 Defiitio: Iheret Errors: Errors which are already ivolved i the statemet of a problem before its solutio are called Iheret Errors. These types of errors arise either due to the give data beig approimate or due to the limitatios of math tables calculatios or the digital computer. Iheret errors ca be miimized by takig better date or by usig precisio computig skills or aids. Defiitio: Roudig errors: Errors which arise from the process of roudig off the umber durig the calculatios are kow as roudig error. Let two umbers each havig digits be multiplied ad the resultig umber which will cotai digits be rouded to digits. Idoig so, a roud off error is itroduced. Similarly, i a decimal system, there may be ifiity of digits to the right of the decimal poit, ad it may ot be possible for us to use ifiity of digits i a computatioal problems. So, we use oly fiite umber of digits oly i our calculatios. This is the source of the so called Roudig of Errors. Roudig errors ca be reduced by i chagig the calculatio procedure or divisio of a small umber. ii retaiig at least oe or more sigificat figure at each step tha that give i the data ad roudig off at the last step. Defiitio: Trucatio Errors: These errors arise due to approimatio of results or o replacig a ifiite process by a fiite oe Eample: if si = i this is replaced 3! 5! 7! 3! 5! 7! by the si = -. Trucatio error type is algorithm error. The magitude of the error i the value of the fuctio due to cuttig trucatio of its series is equal to the sum of all the discarded terms. It may be large ad may eve eceed the sum of the terms retaied, thus makig calculated result meaigful.

5 Defiitio: Absolute error or relative or % errors: AE / RE / PE: If is the value of a quatity ad * is the its approimate value ad the * is called Absolute error E. Relative error: Er = - *. % Error: Ep = Er = - *. If such that * = the is a upper limit o the magitude of absolute error ad measures the absolute accuracy. Note: The relative error, % errors are idepedet ad absolute error is the epressio i terms of Er, Ep of these uits. Note: If a umber is correct to decimal places, the the error is.. Note: if the first sigificat figure of a umber is k ad the umber is correct to sigificat figures, the the relative error Er < k.. Defiitio: Errors i the approimatio of a fuctio: Let y = f, for all,. If, be the errors i, the the error y i y is give by y + y = +, + = f, + f f terms ivolvig high powers of, are ifiitesimal their squares ad higher powers ca be eglected. The, y + y = f, + f f. + f. y = f. +. Igeeral, y i y = f,, 3, 4,.... correspodig to errors i i i for i =,,3,..., is give by y = y y y. Relative error is Er = y y y =. y + y. y y. y

6 Eample: Rd = sigificat figure Eample: Rd865 = Ea = * = = 5 Er = - * = X Ep=ErX= - * = X 6.7 X 5 = 6.7 X 3 Eample: Rd = sigificat figure = Ea = * = =.35 Er = - * = X Ep=ErX= - * = X 6.7 X 5 = 6.7 X 3

7 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Lecture- Defiitio: Fiite Differeces: The calculus of fiite differeces deals with the chages that take place i the value of a fuctio due to fiite chages i the idepedet variable. Let y, y, y, , y be a set of values of ay fuctio y = f. Defiitio: Forward differeces: The differece y - y, y - y, y3 - y, , y - y- deoted by y, y, y, y3,..., y- respectively are called the First Forward Differeces ad operator is kow as Forward differece operator. y = y+ - y The differeces of the First Forward Differeces are called Secod Forward Differeces ad are deoted by y, y, y, y3,..., y etc., y = y - y For ay r, r y = r- y+ - r- y y = y - y y = y+ - y r th forward differece. X Y y y 3 y 4 y 5 y y y y y y 3 y y y 4 y y 3 y 5 y 3 y3 y 4 y y3 3 y 4 y4 y3 y4 5 y5

8 This table is called a Diagoal Differece Table. X is called the argumet ad y the fuctio or the etry. y the first etry is called leadig term ad y, y, 3 y,... are called the leadig differeces. Note: if is operator the y = y - y y = y + y = y + y = y + = + y y = y - y y = y + y = y = y, y = y y = y + y = y + y = y + = + y y = y +.y + y = y + = + y y = + y y3 = + 3 y Similarly, yk = y + k C.y + k C. y k Ck. k y Therefore for k, yk = + k y Note: y = y - y = y - y y - y = y - y - y 3 y = y - y = y3 - y + y y - y + y = y3-3 y + 3y - y 4 y = 3 y - 3 y = y4-3 y3 + 3 y - y y3-3 y + 3 y - y = y4-4 y3 + 6y 4 y + y y = y C y- + C y y Note: is liear operator if it satisfies below i, ii aioms: i f + g = f + g ii c.f = c.f for ay c > ad iii m f = m+ f for m, are positive itegers.

9 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Problems o errors, Diagoal Differece tables Tutorial- Problem # : Costruct a diagoal differece table for the followig set of values: i yi [As:384 Problem # : Costruct a diagoal differece table for the followig set of values: i yi [As: Problem # 3: Fid i e a ii e a ad evaluate iii ta - [As: i e a e ah - ii e h - e iii ta - h + h+ Problem # 4: Prove i. = - ii - = + iii E = E= Problem #5: Evaluate i si a + b ii [3 e iii [ e a log b [As:i cosa + b + ah/ si ah/ ii 3e + h e + h + e iii e a + h.logb + h e a.logb Problem # 6: Evaluate beig oe uit? 5 + takig the iterval of differecig [As:

10 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Solutios o errors, Diagoal Differece tables Tutorial- Problem # : Costruct a diagoal differece table for the followig set of values: i yi [As: 384 Solutio: Forward, Backward ad cetral Differeces i yi yi yi 3 yi 4 yi 5 yi Forward Cetral Backward

11 Problem # : Costruct a diagoal differece table for the followig set of values: i yi [As: Solutio: Diagoal Differece Table is as below Y y y 3 y 4 y Hece the solutio is. 63

12 Problem # 3: Fid i e a [As: i e a e ah - ii e h - e iii ta - ii e a ad evaluate iii ta - h + h+ Solutio: i e a By Defiitio, e a = e a+h - e a = e ah [e a - [e a = e a [e ah - e a = e a [e ah - To fid ii Cosider e a = e a = e a+h - e a = e [e h = [e h - e = e h e +h - e = e h e To evaluate iii e a = e h e Cosider, ta - = ta - + h - ta - = ta - Hece the solutio. ta - = ta - h + h+ + h + + h = ta - h + h+ Problem # 4: Prove i. = - ii Solutio: to prove i. = - - = + iii E = E= Cosider,. = E - E = E + E - = E E - = To prove ii: + Cosider, - = + To prove iii: Cosider, E = E - E = E = E E - = Thus, E = E = Hece the solutio.

13 Problem #5: Evaluate i si a + b ii [3 e iii [ e a log b [As: i cosa + b + ah/ si ah/ ii 3e + h e + h + e iii e a + h.logb + h e a.logb Solutio: To evaluate i si a + b Cosider, si a + b = si a a + h + b si a + b = si a + b + ah si a + b = cosa b + ah/ si ah/ To evaluate ii Cosider, [3 e = 3 [e + h - e = 3 [e + h - e = 3 [ e + h e + h + e To evaluate iii Cosider, [ e a log b = e a+h log b + h e a logb Hece the solutio. Problem # 6: Evaluate beig oe uit? 5 + takig the iterval of differecig [As: Solutio: let f = ad give h = f = f = [ f + f = [ f + [f = f + f + + f = = Hece the solutio.

14 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Backward Differeces Defiitio: Differeces of a polyomial: Lecture-3 The th differeces of a polyomial of the th degree are costat whe the values of idepedet variable are at equal itervals ad all higher order differeces are zero. Coverse is true. Let the polyomial be y = f = a + a - + a a- + a... y+y = f + = a + h + a + h - + a + h a- + h + a... Here let = h, y = f + - f = a h [ a h + a -h +[ a h + a + a - h + - h Similarly, y = a! h th differece is costat, all higher differece =. Note: if h =, y = a! ad =!. Defiitio: Backward Differeces: Let y, y,y, ,y-, y deote the values of ay fuctio y = f. The The differece y - y, y - y, y3 - y, , y - y- deoted by y, y, y3, y4,..., y respectively are called the First Backward Differeces ad operator is kow as Backward differece operator. y = y - y- The differeces of the First Backward Differeces are called Secod Backward Differeces ad are deoted by y, y3,..., y etc., y = y - y = y y y y = y y + y 3 y3 = y3 - y = y 3 y + y y y + y = y 3 3y + 3 y y

15 r y = r- y - r- y-, r th Backward Differece. r, r y = r- y - r- y- rth Backward/Horizotal differece Table: Y y y 3 y 4 y 5 y y y y y y 3 y3 y y3 4 y4 y3 3 y4 5 y5 3 y3 y4 4 y5 y4 3 y5 4 y4 y5 y5 5 y5 Note: y = y - y- y- = y - y = - y y = y - y- ; y- = y - y y- = y- - y- = y - y y - y = y -.y + y y- = y -.y + y = - y So, y-3 = - 3 y y-k = - y= k C+ k C -...y= k C y + k C y -...

16 Defiitio: Error propagatio is a fiite differece Table :Error E i y3 values: y y y 3 y 4 y y y y y y3+ y4 y5 y y 3 y+ y+ y3+ 3 y-3 4 y - 4 y- y3-3 y+3 4 y + 6 y3+ y4 3 y3-4 y - 4 y4 y5 y6 Problem: Form the backward Differece Table for the give set of values: y Solutio: Backward Differece Table y y y 3 y 4 y

17 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Cetral Differeces Defiitio: Cetral Differeces: Aother system of differeces is cetral differeces. I the system, the Cetral Differeces Operator C.D.O. is defied by the relatios. Lecture-4 Let y, y,y, ,y-, y deote the values of ay fuctio y = f. The The differece y - y, y - y, y3 - y, , y - y- deoted by y, y, 3..., y- respectively are called the First Cetral Differeces ad operator is kow as Cetral differece operator. This is liear operator. y = y = y ; y For =,,,... 3 = y = y ; y 5 = y = y3 ;... y+/= y = y+ The first cetral differeces of the first cetral differeces are called the secod cetral differeces ad are deoted by y, y, y3,...thus, y = y 3 - y, y = y 5 - y 3,..., y = y+/ - y-/. Similarly, Higher Order Cetral Differeces are defied as y - y = y3/ ad so o. Cetral Differeces Table as below: X y st diff d diff 3 rd diff 4 th diff 5 th diff y/ y y3/ 3 y3/ y 4 y y5/ 3 y5/ 5 y5/ 3 3 y3 4 y3 y7/ 3 y7/ 4 4 y4 y9/ 5 5

18 Differece Operator E ad :. Shift Operator E: The operator E icreases the argumet by h so that E f = f + h, E f = f + h, E 3 f = f + 3h,..., E f = f + h The operator E - is a iverse operator is defied by E - f = f h if f = y, the E y = y+h E - y = y-h E y = y+h, for ay real umber y.. Averagig Operator : is defied by, y = y / h y / h I the differece calculus, E is regarded as the fudametal operator ad - Delta operator - Del operator - small delta operator - mue epressed i terms of E.

19 Some Importat Formulae: Relatios betwee Operatios & Idetities. = E or E = + Proof: y = y +h - y = Ey - y = E - y Where ad E coected as = E or E = +. = E Proof: y = y - y-h = y E - y = - E - y Where ad E coected as = - E - or E - = - 3. = E / E / Proof: y = y +/.h - y-/.h = E / y E -/ y = E / E- / y Where ad E coected as = E / E- / / -/ 4. = E - E y +/.h y /. h Proof: y = / -/ = E - E Where ad E / / -/ coected as = E - E 5. = E = E = E / Proof: by makig use of, ad 3 formulae = E or E = + ; = E ad = E / E- / Cosider, E =.E / = E = Where, E, ad coected as = E = E = E / = I Proof: f = f + h f = E f = E or E = + f = f f - h = - E f = - E or E = = EE - = I y

20 7. = + 4 By 3, we have = E / E / Squarig both sides, = E + E - / -/ = E - E = - E - E + 4 = = + 4 = E = e hd Proof: Ef = f + h = f + h. f + h! f +... = f + h. D f + D +...! = f [ + hd + h d +... Ef = f.e hd E = e hd h

21 Factorial Powers: Fuctios: = h h [ - [ = [-.h [ = [ = [ =.h. [- [ = [- h m [ = -...[-m- - m. h m m [ = for all m >. Formula for fiite Summatio: Sm = m i m i =.h. [+... [ = h, S m = = m i + i [ + [ + m + Similar to Newto Leibitz formula for a positive itegral.

22 Iverse Operator - for fiite Itegratio: If y = u the y = - u Here - is called fiite Itegratio operator or iverse of operator. His operator - or ordiary itegratio may be deoted by D - or. D Formulae:. - e a+ b = e e a+ b ab if a =, b = the - e = e e h. - a = a a h a 3. - u + v = - u + - v 4. - cu = c. - u 5. - a + b = a + b +.hb +, = + -, h = a + b = b a+ b, 8. - = - - Summatio series: Let S = v + v + v3 + v v = vi for i=,,..., Let vi = ui ui = - vi vi = ui = ui+ - ui = v = u = u u v = u = u3 u v = u = u+ u S = v + v + v3 + v v = vi = u+ u = u+ u = [ - v +

23 Momorts theorem result: u u + u + u u u +... = u + u + u +... = u + E u + E u +... = [ + E + E +... u = [ - E - u = u E = u + sice, E = + = u = u - = u = u i i u u u u i =

24 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Tutorial- Problem # : Prove that 3 y = 3 y5. Problem # : Fid the cubic polyomial i for the give data below: y Problem # 3: Show that. =. [As Problem # 4: Epress f = ad its differeces i L factorial otatio h =? [As. f = [3 + 3 [ + [ - Problem # 5: Fid the 7 th term of, 9, 8, 65, 6, 7 ad geeral term? [As. 7 th term = 344, y = + 3 +, y 6 = = 344. Problem # 6: Prove that y5 = y6 y5 + y4 Problem # 7: Evaluate? [As Problem # 8: Prove that log f = log f + f Problem # 9: Fid the differece table of f ad E f for give data: X f [As. =.6 is =.694 Problem # : Fid y6 whe y = 9, y = 8, y =, y3 = 4 third differeces beig costat? [As. y6 = E 6 y =38

25 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Solutios Problem # : Prove that 3 y = 3 y5. Tutorial- Solutio: L.H.S. = 3 y = E - 3 y = E 3 3 E 3E- y = y - 3 y4 + 3 y3 - y R.H.S. = 3 y5 = E - 3 y5 = E 3 3 E 3E- y5 = y8-3 y7 + 3 y6 y5 But, 3 y5 = - E - 3 y5 = 3E E - - E -3 y5 = y5-3 y4 + 3 y3 y Therefore, it implies that 3 y = 3 y5. Hece proved the result. Problem # : Fid the cubic polyomial i for the give data below: X Y Solutio:Fuctio be y, y be cubic polyomial so, 4 y is zero. [As y y y 3 y 4 y y= -3 6 y = y= y3= y4 = y5 =7 + 3! Now, y = E y = + 3 y = { + + y

26 3 = y + y + y + y 3! = ! = = Cubic polyomial of y = Problem # 3: Show that. =. Solutio: L.H.S..y = y+h y = y+h y = y+h y y y-h = y+h y + y-h = R.H.S. L.H.S. = R.H.S. i.e.,.y =. y For ay y fuctio. Hece Proved the result. Problem # 4: Epress f = ad its differeces i L factorial otatio h =? [As. f = [3 + 3 [ + [ - Solutio: let f = = A--+ B- + C + put =, D = - put =, C + D = -8 or C = put =, B + C + D = or B = 3 o equatig co-efficiets of 3 both sides, A = f = [3 + 3 [ + [ f =.3. [ [ +. [ f =.3.. [ [ 3 f =.3.. Hece the solutio.

27 Problem # 5: Fid the 7 th term of,9,8,65,6,7 ad geeral term? Solutio: [As. 7 th term = 344, y = + 3 +, y 6 = = 344. X y y y 3 y 4 y y= 7 y = y= y3= y4 = y5 =7 7 th term : y6 = y + 6C. y + 6C. y +6C3. 3 y + 6C4. 4 y + 6C5. 5 y + 6C6. 6 y = = = 344 So, y6 = 344. Geereal term: y = y + C. y + C. y + C3. 3 y + C4. 4 y + C5. 5 y + C6. 6 y+... = = = = y = implies y6 = = 344. Hece the solutio.

28 Problem # 6: Prove that y5 = y6 y5 + y4 Solutio: cosider, L.H.S. = y5 = y5 = y/ y9/ = y/ y9/ = y6 y5 - y5 y4 = y6 y5 + y4 y5 = y6 y5 + y4 Hece the solutio. Problem # 7: Evaluate? [As Solutio: Cosider, L.H.S. = = - Now, = - = ad Also, = + + O repeatig the process, we get = Hece the solutio. Problem # 8: Prove that log f = log f + f Solutio: let h be the iterval differecig f + h = E f E f = + f -h +h f + h = f + f implies that f + h = f f + f f + h f O takig logarithms both sides, we get log = log + f f log f + h log f = log + f f [log f + h log f = log f = log Hece Proved the result. + f f

29 Problem # 9: Fid the differece table of f ad E f for give data: X f [As. =.6 is =.694 Solutio: f f f 3 f O observatio 3 f = = Irregularity of is at.6 differece is =.694 crept i while copyig. Hece the solutio.

30 Problem # : Fid y6 whe y = 9, y = 8, y =, y3 = 4 third differeces beig costat? [As. y6 = E 6 y =38 Solutio: y6 = E 6 y = + 6 y = y Sice 3 rd differece of costat, sub sequet differeces are zero. Y 6 = y Y 6 = y + 6 y + 5 y + 3 y Table as : f f f Y 6 = y + 6 y + 5 y + 3 y = = 38 Y 6 = 38. Hece the solutio.

31 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Try Ur Self studet Tutorial-3 Problem #: Epress y = i factorial &show that y=. [As. y = [3 + 3 [ + 3 [ ad y = Problem # : Fid the missig data i the followig data table X 3 4 f [As. = 3, 3 3 = 7 Problem # 3: Prove that y- = y y- + y y. Problem #4:Prove That y y y + +!! y3 + 3! e y + y y +! 3 3 y + 3! +... Problem # 5: Show that u u u u + u3 u to = u + u u Problem # 6: Fid - [ +? [As. 3 Problem # 7: Fid ? [As Problem # 8: Fid where, u= + +. [As Dear studet Try Ur Self problem to problem 8...

32 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Problems Tutorial-4 Problem #. The followig data give the meltig poit of a alloy of lead ad zic, is the temperature i degrees cetigrade is percet of load. : : Fid whe = 43 ad whe = 84? [As , Problem #. Fid the value of y from the followig data at =.47? : y: [As..799 Problem #3. The followig data give, the values of y are cosecutive terms of which 3.6 is the 6 th term. Fid the first ad teth terms of the series. : y: [As. 3., Problem #4. Usi the Newto s forward iterpolatio formula ad fid the value of si 5 from the followig data. Estimate the error : y=si : [As..39 Problem #5. The followig data gives the distace i autical miles of the visible horizo for the give heights i feet above the earths surface. Height : Distace:y Fid y whe = 8 ft. ad whe = 4 ft.? [As. 5.7,.53 aut. miles Problem #6. The followig data of the fuctio f = / fid /.7 usig quadratic iterpolatio. Fid a Estimate Error. X f [As..5 X -5

33 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Problems Tutorial-5 Problem #. The followig data give, fid the umber of studets whose weight is betwee 6 ad 7. Wt Kg : No of Studets As. y7- y6 = 54 Problem #. The followig data give the populatio of a tow. Year : Pop. y: Estimate the populatio icreases durig the period ? [As. 9.4 lakhs Problem #3. Fid the cubic polyomial which takes the followig values. X 3 f Hece or otherwise evaluate f 4? [As. 4 Problem #4. The followig data give, estimate the umber of studets who obtaied marks betwee 4 ad 45. Marks: N of stu: [As. 7 Problem #5. Fid a polyomial of degree two which takes the values : y: [As. ½ + +

34 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Summary. Newto s Forward Iterpolatio Formula: p p = + ph = y + py + y! p p p...[ p! y p p p 3! 3 + y Error i Newto s Forward Differece Iterpolatio Formula:... E = y - = y where < <! SUM- 3. Newto s Backward Iterpolatio Formula: p p + = + ph = y + py + y +! p p + p + 3 y p p + p +...[ p + h y 3! 4. Error i Newto s Backward Differece Iterpolatio Formula:... E = y - = y where < < +!

35 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Summary o Formulae SUM- NEWTON S FORWARD INTERPOLATION FORMULA: Let y = f be a polyomial of degree ad take i the followig form y = f = b + b + b + b b A This polyomial passes through all the poits [i, yi for i =,,,...,. therefore, we ca obtai the yi s by substitutig the correspodig i s as: At =, y = b At =, y = b + b At =, y = b + b + b... Let h be the legth of iterval such that i s represet as below:, +h, +h, +3h,..., + h. =h, = h, 3 = 3h,..., = h... From ad we get, y = b y = b + bh y = b + bh + bhh y3 = b + b3h + b3hhh + b3hhh y3 = b + bh + bh-h bh-h-h... B

36 O solvig the above equatios for b, b, b,..., b we get b = y y b b = = h b = y y b b h = h y b =! h y y h = y h y y y h h h 3 y Similarly, we ca see that b3 = 3 3! h = y y y h 4 y, b4 = 4 4! h y = y y h 5 y, b5 = 5 5! h y = h y y,..., b =! h, y = f = y + y h y +! h y ! h 3 y + 3 3! h If we us the relatioship = + ph - = ph where p =,,,..., the = + h = h = ph h = p - h = + h = h = p - h h = p - h i = p i h = [p -h Equatio 3 becomes, y = f = f + ph p p =y+p y + y! p p p 3! 3 + y p p p...[ p y +!... 4 This formula is kow as Newto s forward iterpolatio formula or Newto s Gregory forward iterpolatio formula. Uses: This formula for iterpolatio ear the begiig of a set of tabular values.

37 NEWTON S BACKWARD INTERPOLATION FORMULA: Let y = f be a polyomial of degree ad take i the followig form y = f = a + a + a - + a a ad impose the coditio that y ad y should agree at the tabulated poits, -, -, -3,...,,,. we obtai y = p p + p p + p + 3 y+p y + y + y ! 3! Where p = h p p + p +...[ p + +! y... 6 Uses: This formula is useful for tabular values to the left of y ad also useful for iterpolatio ear the ed of the tabular values. Formulae for error i polyomial Iterpolatio: If y = f is the eact curve ad y = is the iterpolatig polyomial curve, the the error i polyomial iterpolatio is give by... Error = f - = f +! For ay, where < < ad < <. The error i Newto s Forward Iterpolatio formula is give p p p... p + f - = f +! where p = h... 8 The error i Newto s Backward Iterpolatio formula is give by p p + p +... p f - = h f +! where p = h... 9

38 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Summary o Formulae SUM-3 CENTRAL DIFFERENCE INTERPOLATION As metioed earlier, Newto s forward iterpolatio formula is useful to fid the value of y = f at a poit which is ear the begiig value of ad the Newto s Backward Iterpolatio Formula is useful to fid the value of ad the Newto s Backward Iterpolatio formula is useful to fid the value of y at a poit which is ear the termial value of. We ow derive the Iterpolatio Formulas that ca be employed to fid the value of which is aroud the middle to the specified values. For this purpose, we take as oe of the specified values of that lies aroud the middle of the differece table ad deote rh by -r ad the correspodig value of y by y-r. The the middle part of the forward differece table will appear as show below: y y y 3 y 4 y 5 y y-4 y-4-3 y-3 y-4 y-3 3 y-4 - y- y-3 4 y-4 y- 3 y-3 5 y-4 - y- y- 4 y-3 y- 3 y- 5 y-3 y y- 4 y- y 3 y- 5 y- y y 4 y- y 3 y 5 y- y y 4 y y 3 y 3 y3 y 3 4 y

39 From the table, we ote the followig: y = y- + y-, y = y- + 3 y-, 3 y = 3 y- + 4 y-, 4 y = 4 y- + 5 y-... ad so o. Ad y- = y- + y-, y- = y- + 3 y-, 3 y- = 3 y- + 4 y-, 4 y- = 4 y- + 5 y- 5 y- = 5 y- + 6 y-... ad so o. By usig the epressios ad, we ow obtai two versios of the followig Newto s Forward iterpolatio Formula: yp = p p p p p 3 p p p p 3 4 y + p y + y + y + y +...! 3! 4!... 3 here yp is the value of y at = p = + ph.

40 GAUSS s FORWARD INTERPOLATION FORMULA: Substitutig for y, 3 y... from i the formula 3 we get, yp = p p y + p y + y! p p p p 3 4 y 4! 3 + y 5 + y p p p 3 + y 3! y + yp = y p p p p p 3 p p p p p y + y + y ! 3! 4! y Substitutig for 4 y- from, this becomes yp = p p p p p 3 p p p p 3 4 y + p y + y + y + y +...! 3! 4!... 4 This versio of Newto s Forward Iterpolatio Formula is kow as the Gauss s Forward Iterpolatio Formula. We observe that the formula 4 cotais y ad the eve differeces y-, 4 y-,... which lie o the lie cotaiig called the cetral lie ad the odd differeces y, 3 y-... which lie o the lie just below this lie, i the differece table. Note: we observe the followig differece table that y = y/ y- = y 3 y- = 3 y/ 4 y- = 4 y ad so o. Accordigly the formula 4 ca be re-writte i the otatio of cetral differeces as give below: yp = y p p p p p 3 p p p p p y/ + y + y/ ! 3! 4!... 5 y

41 GAUSS s BACKWARD INTERPOLATION FORMULA: Net, let us substitute for y, y, 3 y... from i the formula 3. Thus we obtai yp = p p 3 y + p y + y + y + y +! p p p p y + y ! p p p 3 y 3! 4 + y + yp = y p p + p p p + 3 p p p + p 4 + p y + y + y ! 3! 4! y 3 4 Substitutig for y, y from, this becomes p p + p p p + 3 y + p y + y + y! 3! yp = p p p + p 4 5 y + y ! 4 + y + yp = y p p + p p p + 3 p p p + p 4 + p y + y + y ! 3! 4! y... 6 The versio of the Newto s Forward Iterpolatio Formula is kow as the Gauss s Backward Iterpolatio Formula. Observe the formula 6 cotais y ad the eve differeces y-, 4 y-... which lie o the cetral lie ad the odd differeces y-, 3 y-... which lie o the lie just above this lie. Note: I the otatio of cetral differeces the formula reads yp = y p p + p p p + 3 y 3! p p p + p ! p y / + y + / y!

42 INTERPOLATION WITH UNEVENLY SPACED POINTS: I the previous classes of lectures we have derived iterpolatio formulae which are of great importace. But i those formulae the disadvatage is that the values of the idepedet variables are to be equally spaced. We desire to have iterpolatio formulae with uequally spaced values of the idepedet variables. We discuss Lagrage s Iterpolatio Formula which uses oly fuctio values. LAGRANGE S INTERPOLATION FORMULA: Let,,,..., be the + values of which are ot ecessarily equally spaced. Let y, y, y,..., y be the + values of y = f. Let the polyomial of degree for the fuctio y = f passig through the + poits, f,, f,, f,...,, f be i the followig form y = f = a... + a... + a a where a, a, a,..., a are costats. Sice the polyomial passes through, f,, f,, f,...,, f, the costats ca be determied by substitutig oe of the values of,,,..., for i the above equatio. Puttig = i we get, f = a... f a =... Puttig = i we get, f = a... f a =... Similarly, Substitutig = i we get, f = a... f a =... Cotiuig i this maer ad Puttig = i we get, f = a... - a = f...

43 Substitutig these values of a, a, a,..., a we get, c f f f f f This is kow as Lagrage s Iterpolatio Formula. This ca be epressed as k j j j k j k k f f Aother form is : f f f f

44 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Solutios Tutorial-4 Problem #. The followig data give the meltig poit of a alloy of lead ad zic, is the temperature i degrees cetigrade is percet of load. : : Fid whe = 43 ad whe = 84? [As , Solutio: The values of ad differeces of are tabulated below: p p By Newto s Iterpolatio Formula, = + p! Here = 84, =, = 43 4 To fid 43 p =. 3 h p = X ! Sice = 84 is ear the ed of the table, we have to use by Newto s Backward Iterpolatio Formula = + p + p p Also p=.6, p = -.6, = -.6! h = X Hece the solutio.

45 Problem #. Fid the value of y from the followig data at =.47? : y: [As..799 Solutio: Sice we require the value of correspodig to a value of ear the ed of the table, Newto s Backward Formula is to be used. The backward table is give as below: Y y y 3 y 4 y Here h =., p = h..3. implies p = -.3 Newto s Backward Differece Formula is y = y + py + p p + p p + p + 3 y + y! 3! p p + p + p y 4! = ! 3! ! =.799 Hece the solutio.

46 Problem #3. The followig data give, the values of y are cosecutive terms of which 3.6 is the 6 th term. Fid the first ad teth terms of the series. : y: [As. 3., Solutio: Sice we require the value of correspodig to a value of ear the ed of the table, Newto s Forward Formula is to be used. The Forward table is give as below: Y y y 3 y 4 y Here h =, =, = 3 ad 3 p = h implies p = - Newto s Forward Differece Formula is y = y - = = 3. 3 to fid the th term, use Newto s Backward Differece Formula with 9 = 9, =, h = ad p = h y = y = =! 3! Hece the solutio.

47 Problem #4. Usi the Newto s forward iterpolatio formula ad fid the value of si 5 from the followig data. Estimate the error : Solutio: y=si : [As..39 y = si y y 3 y p =. 4 h 5 From Newto s Forward Formula = + ph p p =y +py + y! p p p y ! 3 + p p p...[ p! y.4.4. = = =.7883 Si 5 =.7883 p p p... p + Error = y ; Takig = +! p p p Error = y =.7 3! 6 =.39 Hece the solutio.

48 Problem #5. The followig data gives the distace i autical miles of the visible horizo for the give heights i feet above the earths surface. Height : Distace:y Fid y whe = 8 ft. ad whe = 4 ft.? [As. 5.7,.53 aut. miles Solutio: X Y y y 3y 4y Takig =, = 8, h = 5, p =. 36 h 5 Newto s Forward Iterpolatio Formula is give by, Y8 = p p p p p 3 p p p...[ p y+py+ y + y +...+! 3!! y = = = = 5.7 autical miles To fid the value of y whe = 4 which is ear the ed of the table, we use Newto s Backward Iterpolatio Formula

49 4 4 = 4, p =. h 5 usig the lie of Backward Differeces y =.7, y =.37, y = -., 3 y =. etc., Newto s backward Formula is give by Y4 = p p + p p + p + 3 p p + p + p = y4 + py4 + y4 + y4 + y4! 3! 4! = =.53 autical miles. Hece the solutio. Problem #6. The followig data of the fuctio f = / fid /.7 usig quadratic iterpolatio. Fid a Estimate Error. X f [As..5 X -5 Solutio: The differece table is as below: X f f f =.7, p =. ; f.7 = = = = h Error = 3 f ;E.7 <. 4 3! 6.7 =.5-5. Hece the solutio.

50 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Solutios Tutorial-5 Problem #. The followig data give, fid the umber of studets whose weight is betwee 6 ad 7. Wt Kg : No of Studets As. y7- y6 = 54 Solutios: The cumulative values ad differeces table is as below: Wt: Stu: y y y 3 y 4 y < 4 5 < < < 54-5 < = 4, = 7, p =. 5 y7 = y + py + y = = = = 44 No of studets whose weight is betwee 6 ad 7 = y7 Y6 = = 54 Hece the solutio.

51 Problem #. The followig data give the populatio of a tow. Year : Pop. y: Estimate the populatio icreases durig the period ? [As. 9.4 lakhs Solutio: The differece table is y y y 3 y 4 y 4 y Whe = 946, p =. 5 p p p p p y946 = y + py + y + 3 y =+.54+ y = = Whe = 976, p =. 5 p p + p p + p + y976 = y + py + y + 3 y = = = Therefore icrease i populatio durig the period = 9.4 lakhs. Hece the solutio.

52 Problem #3. Fid the cubic polyomial which takes the followig values. X 3 f Hece or otherwise evaluate f 4? [As. 4 Solutio: The differece table is f f f 3 f p = h Usig Newto s Forward Iterpolatio formula, we get f = f + f + f + 3 f = = which is required polyomial To compute f 4, we take = 3, = 4 So that p = h Usig Newto s Backward Iterpolatio Formula, we get p p + p p + p + ` f 4 = f 3 + pf3 + f3 + 3 f = = 4 Which is the same value as that obtaied by substitutig = 4 i the cbic polyomial Hece the solutio.

53 Problem #4. The followig data give, estimate the umber of studets who obtaied marks betwee 4 ad 45. Marks: N of stu: [As. 7 Solutio: The differece table is as below: Y y y 3 y 4 y < < < < < = 45, = 4, p =. 5 h Usig Newto s Iterpolatio Formula we get, p p - p p - p - p p - p - p 3 y45 = y4 + py4+ y4 + 3 y4+ 3 y = = = The umber of studets with marks < 45 is i.e., 48. But the umber of studets with marks less tha 4 is 3. Hece the umber of studets gettig marks betwee 4&45 = 48 3 = 7. Hece the solutio. Problem #5. Fid a polyomial of degree two which takes the values : y: [As. ½ + + Try Urself...

54 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Problems Tutorial-6 Problem #: Fid f. if f = 76, f = 85, f = 94, f 3 = 3, f 4 =, f 5 =, f 6 = 9. [As Problem #:From the followig table, fid y whe =.85 ad =.4 by Newto s Iterpolatio Formulae y= e [As. 6.36,. Problem #3: Costruct a polyomial for the data give below. Fid also y = 5. : y: [As. y = , y Problem #4: give si 45 =.77, si 5 =.766, si 55 =.89, si 6 =.866 fid si 5 usig Newto s Forward Formulae? [A..788 Problem #5: Fid y 3 if y = 35.3, y 5 = 3.4, y = 9., y 5 = 6., y 3 = 3., y 35 =.5. [As..948 Problem #6: Estimate the values of f ad f 4 from the followig available data: : f : [As. 35, 9 Problem # 7: calculate 5. 5 give 5 =.36, 6 =.449, 7 =.646, 8 =.88. [As..344 Problem # 8: From the followig data, estimate the umber of persos havig icomes betwee ad 5. Icomes : < No of persos [As. 4,76

55 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Problems Tutorial-7 Problem # : From the followig data, estimate the umber of studets who secured marks betwee 4 ad 45. Marks : No of studets [As. 7 Problem # : From the followig data gives the populatio of a tow durig the last si cesus. Estimate the icrease i the populatio durig the period from 976 to 978. Year Populatio [As. 53 Problem # 3: Usig a polyomial of third degree, complete the record give below of the eport of a certai commodity durig five years. Year Eport T [As. 369 Problem # 4: The area A of a circle of a diameter d is give for the followig values. d: A: Calculate the area of a circle of diameter 5. [As. 369 Problem # 5: Costruct Newto s Forward Iterpolatio polyomial for the followig data. : y: Evaluate y for = 5. [As. 369

56 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Objective Type Questios OTQ-. If =, = 5, = 8, 3 = 3, 4 = 7, 5 = the 5 6 =... A -6 B 6 C -6 D 6 [ A. [f =... A f f h B f + h C f + h f D f h [ C 3. The relatio betwee ad is... A = + E - B. + E C = - E D - E - [ D 4. If the iterval of differecig is uity ad f = a fid which oe of the followig choices is wrog... A [f = a +B [f = a C 3 [f = D 4 [f = [ C 5. If 3 4 =, by bisectio method first two approimatios ad are ad the is... A,5 B. C.75 D.5 [ D 6. 3y5 = A y6+ 3y5 + 3y4 + y3b y5-3y4-3y3 yc y6-3y5 +3y4 -y3d y5+ 3y4 + 3y3 + y [ B 7. Gauss forward iterpolatio formula is used to iterpolate the values of y for A < p < - B - < p < C < p < D < p < [ D 8. If first two approimatios ad are roots of 3 + = are ad the by regula-falsi method is... A.5 B.5 C.5 D.35 [ B 9. If first two approimatios ad are roots of = is =.64 the by Newto Raphso Method N.R.M is... A.85 B.6565 C.7 D.674 [ B. Give that X 5 f The f =... A.58 B.96 C.5 D.54 [ D

57 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Objective Type Questios OTQ-.e =... takig h = A e e B e + e - C ec D e + e [ A. If f = + +, the the value of f is takig h = A + 3 B C + D + [ B 3. If y = which of the followig is ot true? A y = + 3 B 3 y = 6 C 4 y = D = 6 + y[ A 4. Give that X 3 f The f =... A B 3 C 4 D [ D 5. Newtos iterative formula to fid the value of N is... A + = 3 N B + = N C + = N D + = + N [ D 6. Which oe is wrog? A [f f = f. f B E + C [f + f = f + f D 5= [ A 7. Newto s iterative formula to fid the value of 3 N is... A+= 3 N B+= + 3 N C+ = 3 N D+= - N [ B 8. A liear versio of the lagrage s iterpolatio formula for f is... A f f B + f f C + f f D + f f [ D 9. If = the first approimatio ad are ad by bisectio method is... A.5 B.5 C.5 D [ A. The order of covergece i Newto-Raphso Method N.R.M is... A B 3 C D [ D

58 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Problems o cetral diff, gauss, ward Problem #: Fid f.5 usig the followig data table X 3 4 f Tutorial-8 [As Problem #: From the followig table values of ad y = e iterpolate values of y whe = e [As Problem # 3: From the followig table fid y whe = y [As.4.43 Problem # 4: From the followig table fid y whe = y [As.-.65 Problem #5: Use Gauss s Forward Iterpolatio Formula to fid f 3.3 from the followig table: y=f [As.4.9 Problem #6: Use Gauss s Forward Iterpolatio Formula to fid f 3 give that f = 8.478, f5 = 7.844, f9 = 7.7, f33 = 6.343, f37 = [As.6.9 Problem #7: Fid by Gauss s Backward Iterpolatio Formula the value of y at = 936, usig the followig table: y [As Problem #8: Use Gauss s Backward Iterpolatio Formula to fid f 3 give that f5 =.77, f3 =.37, f35 =.3386, f4 = [As..365

59 Lakireddy Bali Reddy College of Egieerig, Mylavaram Autoomous B.Tech I Year II-Semester May/ Jue 4. T 64- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagedram Problems o cetral diff, gauss, ward Tutorial-9 Problem #: Fid f.36 from the followig table: : y: [As..589 Problem #: Fid f from the Gauss s Forward formula: : f: [As Problem #3: Usig Gauss s Backward Formula,Fid f 8 from the table: : y: [As Problem #4: Fid y 5 give that y = 4, y4 = 3, y8 = 35, y3 = 4, Usig Gauss s Forward differece Formula. [As Problem #5: Give that 65 = 8.63, 65 = , 65 = , 653 = 8.884, 656 by usig Gauss s Backward Formula. [As