Finite Differences, Interpolation, and Numerical Differentiation

Transcription

1 4 Fnte Dfferences, Interpolaton, and Numercal Dfferentaton 4. INTRODUCTION Lnear nterpolaton s dscussed n the precedng chapter as a method for fndng a partcular root of a polynomal, or, transcendental equaton when the upper and lower bounds of the nterval for search are provded. To contnue the dscusson of the general topc of nterpolatons whch not necessarly lnear could be quadratc (parabolc, cubc, quartc, and so on, we n ths chapter present methods for ths general need of nterpolaton n engneerng analyses by treatng not only equatons but also a set of N tabulated data, (x,y ) for = N. Fnte dfference table wll be ntroduced and constructed for the equally-spaced data, that s x 2 x = x 3 x 2 = = x N -x N. Ths table can be utlzed as a forward-dfference, backward- dfference, or, central-dfference table dependng on how ts s appled for the nterpolaton use. Taylor s seres and a shftng operator are to be used n dervaton of the nterpolaton formulas n terms of the forward-dfference, backward-dfference, and central-dfference operators. A program DffTabl has been developed for prntng out a dfference table of a set of equally-spaced data. Dfferentaton operator wll also be ntroduced for the dervaton of the numercal dfferentaton needs. When a set of equally-spaced data, (x,y ) for = N, are gven, formulas n terms of the forward-dfference, backward-dfference, and centraldfference operators are derved for the need of calculatng the value of dy/dx at a lsted x value or unlsted. If x s not equal to one of the x, nterpolaton and dfferentaton have to be done combnedly through a modfcaton of the Taylor s seres expanson. For curve-ft by polynomals and for nterpolaton, applcatons of the versatle Lagrangan nterpolaton formula are also dscussed. A program called LagrangI s made avalable for ths need. QuckBASIC, FORTRAN, and MATLAB versons of the above-mentoned programs are to be provded. Applcaton of the Mathematca s functon Interpolatng Polynomal n place of LagrangI s demonstrated. In soluton of the problems governed by a system of ordnary dfferental equatons wth ether some ntal and/or boundary condtons specfed, the fnte dfferences wll be appled. In Chapter 6, such method for fndng the approxmate answer to the problem s dscussed. Accuracy of such approxmate soluton wll depend on the ncrement of the ndependent varable, stepsze, adopted and on whch approxmate method s employed.

2 Because numercal dfferentaton s hghly naccurate, whenever possble numercal ntegraton should be preferred over numercal dfferentaton. In case that one needs to fnd the velocty of a certan moton study and has the opton of collectng the dsplacement or acceleraton data, then the acceleraton data should be taken not the dsplacement data. The reason s that one has the choce of applyng numercal dfferentaton to the dsplacement data or numercal ntegraton to the acceleraton data to obtan the velocty results. The numercal ntegraton whch s the topc of Chapter 5 has the smoothng effect and hence s more accurate! Graphcally, dfferentaton s of a local evaluaton of determnng the slope at a selected pont on a curve whch could be the result of fttng a number of data ponts dscussed n Chapter 3 whle ntegraton s of a global evaluaton of fndng the area under the curve between two specfed lmts of the ndependent varable. For a set of three gven ponts ftted lnearly by two lnear segments and quadratcally by a parabola, the slope at the md-pont could have very dfferent slope values whle the areas under the lnear segments and under the parabola would not dffer too sgnfcantly. Hence, t s worthy of emphaszng that learnng the computatonal methods s easer when compared to makng decson of whch method s best to solve the problem at hand. 4.2 PROGRAM DIFFTABL APPLICATIONS OF FINITE-DIFFERENCE TABLE Program DffTabl has been developed for the need of constructng a table of fnte dfferences of a gven set of N two-dmensonal ponts, (x,y ) for = N. The x values are assumed to be equally spaced,.e.,, x 2 x = x 3 x 2 = = x N -x N = h, h beng called the ncrement, or, stepsze. Ths so-called dfference table can be appled for nterpolaton of the y value for a specfed, unlsted x value nsde the range of x = x and x = x N (extrapolaton f outsde the range), and dfferentaton. Table shows a typcal dfference table. The symbol used n Table s called Forward Dfference Operator. If we refer the numbers lsted n the x and y columns as x to x 6 and y to y 6, respectvely, the frst number lsted under y,.9495, s obtaned from the calculaton of y 2 y and s dentfed as y. The last number lsted n the y column, 5.305, s equal to y 6 y 5 and referred to as y 5. Or, we may wrte the general formula as, for = to 5, y = y y + () y s called the frst forward dfference of y at x. The hgher order forward dfferences lsted n Table are obtaned by extended applcaton of Equaton. That s, 2 y = ( y y )= y y = y 2y + y (2)

3 TABLE Dfference Table (y = to 2x + 3x 2 to 4x 3 + 5x 4 ). x y y 2 y 3 y 4 y 5 y y = ( y y )= y y + + = ( y 2y + y ) y 2y + y = y 3y + 3y y (3) and so on. We shall show later how the thrd through seven columns of Table can be nterpreted dfferently when the backward and central dfference operators are ntroduced. Frst, we wll demonstrate how Table can be appled for nterpolaton of the y value at an unlsted x value, say y(x =.24). To do that, the shftng operator, E, needs to be ntroduced. The defnton of E s such that: Ey = y + (4) That s, f E s operatng on y, the y value s shfted down to the next provded y value. Interpolaton s a problem of not shftng a full step but a fractonal step. For the need of fndng y at x =.24, the x value falls between x 2 =.2 and x 3 =.3. Snce the stepsze, h, s equal to 0., a full shft from y 2 = would lead to y 3 whch s equal to We expect the value of y(x =.24) to be between y 2 and y 3. Instead of E y 2, the value of E 0.4 y 2 s to be calculated by shftng only 40%. To fnd the meanng of E 0.24, or, more generally E r for 0<r<, we substtute Equaton 4 nto Equaton to obtan: y = y y = Ey y = E y + = E, or, E = + (5,6)

4 By applcaton of bnomal expanson, we can then have: E r r r = + k = k= 0 k (7) where the bnomal coeffcents are defned as: r ( 0)= and ( r k)= [ ] rr ( ) r k 2 Lk (8) We can now use Equaton 7 to obtan: yx ( = 24. )= E yx ( = 2. )= E y = + y = y = y 2 (9) Equaton 9 can be appled for lnear nterpolaton f up to the y 2 terms are adopted; for parabolc nterpolaton f up to the 2 y 2 terms are adopted; and so on. Snce Table has up to the ffth order forward dfferences avalable but the last column contans a zero value, Equaton 9 can therefore be effectvely up to the fourth-order forward-dfference nterpolaton. The numercal results of y(x =.24) usng lnear, parabolc, cubc, and fourth-order are 7.406, 7.390, , and , respectvely. Snce we know y = 2x + 3x 2 4x 3 + 5x 4, the exact value of y(x =.24) s equal to An explanaton for dscrepances n all of these four attempts of nterpolatons, relatve to the exact value, s provded n a homework exercse gven n the Problems set. BACKWARD-DIFFERENCE OPERATOR Notce that the frst numbers lsted n columns three through seven n Table are the fve forward dfferences of y, and that only four forward dfferences (the second numbers n columns three through sx) of y 2 are avalable. Lesser and lesser forward dfferences are avalable for later y s untl there s only y 5 for y 5. That s to say, to nterpolate y(x) for an x value between x 5 =.5 and x 6 =.6, Equaton 9 can only be used up to the y 5 term. To remedy ths stuaton and to make most use of the provded set of 6 (x,y) data, t s approprate at ths tme to ntroduce the backward-dfference operator,, whch s defned as: y = y y (0)

5 By combnng Equatons, 7, and 0, we notce that: and So, y = y y = y + + y = y+ y = E y + = E, or, E = () (2) (3,4) Equaton 2 s an mportant result because t ndcates that the last numbers lsted n columns three through seven of Table are the frst through fve backward dfferences of y 6. If we could derve an nterpolaton formula n terms of, there are up to ffth-order backward dfference of y 6 avalable. Toward that end, let us consder the need of nterpolatng the value of y(x =.56). Ths y value can be reached by shftng backward by 0.4 step from x =.6 snce the stepsze for Table s h = 0.. By usng Equaton 4 and notcng Equatons 7 and 8, we can have: yx ( = 56. )= E yx ( = 6. )= E y = y = ( ) ( ) + 6 y 6 = y 6 (5) One can then apply Equaton 5 to obtan the nterpolated y(x =.56) values usng up to the ffth order backward dfferences. Ths s left as a homework exercse gven n the Problems set. CENTRAL-DIFFERENCE OPERATOR For the nterest of completeness and later applcaton n numercal soluton of ordnary dfferental equatons, we also ntroduce the central dfference operator,. When t s operatng on y, the defnton s: δy h h = y x + y x 2 2 (6) The frst-order central dfference s seldom used and the second-order central dfference s frequently appled, whch s: 2 h h δ y = δ y x + δ y x yx h 2yx yx h 2 2 = ( + ) + = y 2y + y + (7)

6 DIFFERENTIATION OPERATOR Another mportant operator that needs to be ntroduced n connecton wth the applcaton of dfference table s the dfferentaton operator, D, whch s defned as: dy x Dy( x = x ) dx Dy x= x (8) As t s our ntenton to apply an avalable dfference table for numercal dfferentaton at one of the lsted x values, or, at an unlsted x value, by usng ether the forward or backward dfferences of y values. For example, we may want to fnd Dy at x =.2, or, at x =.24. To derve an expresson for D n terms of, we recall the Taylor s seres of a functon y(x = a + h) near the neghborhood of x = a for a small ncrement of h: = ()+ () ya+ h ya () j y a y a y a h h 2 () j h +! 2! j! (9) where y (j) s the jth dervatve wth respect to x. Usng the notaton of dfferentaton operator D and the shftng operator E, the above expresson can be wrtten as: or, () + () + + () j j hdy a h D y a h D y a Ey()= a y()+ a! 2! j! [ ] ()= () 2 2 j j hd = + hd + h D + + h D + y a e y a hd E = e and D = lne h (20) (2) In order to use the dfference table for numercal dfferentaton, we substtute Equaton 6 nto Equaton 2 to obtan: D = ln + h (22) By substtutng the logarthmc functon n Equaton 2 wth an nfnte seres and applyng the D operator for y, the result s: Dy = y h 2 3 (23) Hence, to fnd Dy(x =.2) by usng the fnte dfferences n Table, Equaton 23 can be appled to obtan:

7 Dy 2 = x =. Notce that the above result s when up to the fourth-order forward dfferences of y 2 are all utlzed. Lnear, parabolc, and cubc numercal dfferentatons at x =.2 could also be calculated by takng only one, two, and three terms nsde the parentheses of the above expresson. The respectve results are , 22.05, and Snce y(x) = 2x + 3x 2 4x 3 + 5x 4 and y (x) = 2 + 6x 2x x 3, the exact value of y(x =.2) = = ndcates that the fourth-order calculaton s the best. When Dy s needed for x near the end of x lst, t s better to express D n terms of the backward-dfference operaton, whch based on Equatons 4 and 2 s: Dy x [ ] h n E y h n l l ( ) y = = = 2 3 y h 2 3 = y h (24) The shftng operator E and dfferentaton operator D can be combned to derve formulas for numercal dfferentaton of y(x) at x values unlsted n the dfference table ether n terms of forward-dfference operator or backward-dfference. Frst, let recall Equatons 7 and 23 and apply them to fnd y (x = x + rh) n terms of the forward-dfference operator as follows: y ( x + rh)= = ( + )( + ) h DE r y h n 4 y l 2 3 rr 2 = h r y (25) r 2 3r 6r r 9r + r 3 4 = y h Smlarly, y (x -rh) can be expressed n terms of backward-dfferental operator, as: y ( x rh)= = ( ) h DE r y ln h [ ] r y = rr + + h r y (26) r r r+ r r + r = y h 2 6 2

8 It should be partcularly ponted out that n usng Equaton 26 for fndng y (x) where x <x<x, r s to be calculated as (x -x)/h and not as (x-x )/h. For example, n usng Table, to calculate y (x =.56) based on Equaton 26 r should be equal to (.6.56)/0. = 0.4 and not equal to (.56.5)/0. = 0.6, and equal to 6 not 5 because n Table x 6 =.6 and x 5 =.5. Program DffTabl has been prepared for nteractve nterpolaton and dfferentaton usng a dfference table such as Table. User can nteractvely specfy the data ponts and where the nterpolaton or dfferentaton s to be calculated and also up to what order of fnte dfferences should the computaton be performed. Both QuckBASIC and FORTRAN versons of the program are made avalable. Lstngs are gven below along wth some sample applcatons. At present, the hghest order of fnte dfference allowed s the fourth. QUICKBASIC VERSION

9 Sample Applcaton FORTRAN VERSION

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12 Sample Applcaton Usng the nput data and dfference table as for the QuckBASIC verson, the nteractve applcaton of the FORTRAN verson gves a sample run as follows: MATLAB APPLICATION A fle DffTabl.m can be created and added to MATLAB m fles for prntng out the dfference table. Ths fle may be wrtten as:

13 Ths m fle then can be appled as llustrated by the followng examples: The statement format compact requests the results to be dsplayed wthout unnecessary lne spaces on screen. It s approprate at ths tme to demonstrate how some graphc capablty of MATLAB can be effectvely utlzed here n connecton wth the dfference table. Frst, the calculaton of the frst dervatves can be graphcally nterpreted as the slope of the lnear segments connectng the gven ponts as shown n Fgure whch s obtaned wth the followng nteractvely entered statements:

14 FIGURE. The calculaton of the frst dervatves can be graphcally nterpreted as the slope of the lnear segments connectng the gven ponts. The frst, X =, statement creates an array havng 6 elements whose values start at. and ends at.6 and have a unform ncrement of 0.. In the plot statement, the character nsde the frst set of sngle quotaton sgns requests that the gven set of ponts specfed by the coordnates arrays X and Y are to be connected by sold lnes whle the character * nsde the second set of sngle quotaton sgns s for markng those ponts. It also s approprate at ths tme to ntroduce the bar graph feature of MATLAB when we consder data set and dfference table. Fgure 2 s presented to show the use of bar and num2str commands of MATLAB. The bar command plots a seres of vertcal bars based on a set of coordnates arrays X and Y where X values must be equally spaced. The num2str command converts a numercal value nto a strng, t often facltates the dsplay of numercal values n conjuncton wth the text command. The followng nteractvely entered statements have enabled Fgure 2 to be dsplayed:

15 FIGURE 2. Notce that the frst two arguments for text are where the text strng should be placed whereas the thrd argument converts the value of Y(I) to be prnted as a strng. The for-end loop allows all Y values to be placed at proper heghts. MATHEMATICA APPLICATIONS To produce a plot smlar to Fgure 3 n the program DffTabl by applcaton of Mathematca, we may enter statements and obtan the followng: Input[]: = X = Table[,{,.,.6,0.}; Y = Exp[X]; Input[2]: = g = Show[Graphcs[Lne[Table[{X[[]],Y[[]]},{,,6}]]]] Input[3]: = g2 = Show[g, Frame->True, AspectRato->, FrameLabel->{ X-axs, Y-axs }] Input[4]: = g3 = Show[g2,Graphcs[Table[Text[ X,{X[[]],Y[[]]}, {,,6}]] Input[5]: = Show[g3,Graphcs[Text[ Lnearly Connected,{.2,4.8}, {,0}],Text[ Data Ponts,{.2,4.6},{,0}]]]

16 FIGURE 3. Only the fnal plot s presented here. The ntermedate plots desgnated as g, g2, and g3 can be recalled and dsplayed f necessary. The Lne command n Input[3] drects the specfed pars of coordnates to be lnearly connected. A bar graph can be drawn by applcaton of Mathematca command Rectangle and ther respectve values by the command Text. The followng statements recreate Fgure 4 n the program DffTabl.: Input[]: = X = {,2,3,4,5}; Y = {2,4,7,,24}; Input[2]: = g = Show[Graphcs[Table[Rectangle[{X[[]] 0.4,0}, X[[]] + 0.4,Y[[]]}],{,,5}]]] Input[3]: = g2 = Show[g,Graphcs[Table[Text[Y[[]], {X[[]] 0.,Y[[]] + }],{,,5}]]] Input[4]: = g3 = Show[g2, Frame->True, AspectRato->] Input[5]: = g4 = Show[g3,Graphcs[Text[ Bar Graph of X Y Data, {0.5,8},{,0}]]] Input[6]: = Show[%,FrameLabel->{ X-axs, Y-axs }]

17 FIGURE 4. Notce that when no expresson nsde a par of doubt quotes s provded for the command Text, the value of the specfed varable wll be prnted at the desred locaton. Ths s demonstrated n Input[3]. Mathematca also has a functon called BarChart n ts Graphcs package whch can be appled to plot Fgure 5 as follows (agan, some ntermedate Output responses are omtted): FIGURE 5.

18 Input[]: = Y = {2,4,7,,24}; Input[2]: = <<Graphcs`Graphcs` Input[3]: = g = BarChart[Y] Input[4]: = g2 = Show[g,Graphcs[Table[Text[Y[[]], {,Y[[]] + }],{,,5}]] To prnt out a dfference table of a gven set of n y values, we can arrange the y values and up to the n-st order of ther dfferences n a matrx form. The y values are to be lsted n the frst column and ther th-order dferences are to be lsted n the + st column for =,2,,n. The followng Mathematca nput and ouput statements demonstrate the prnt out of a set of 6 y values: Input[]: = y = {,3,7,2,44,78}; Input[2]: = n = Length[y]; yanddys = Table[x,{,n},{j,n}]; MatrxForm[yanddys] Output[2] = x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Input[3]: = Do[yanddys[[,]] = y[[]]; MatrxForm[yanddys] Output[3] = x x x x x 3 x x x x x 7 x x x x x 2 x x x x x 44 x x x x x 78 x x x x x Input[4]: = Do[Do[yanddys[[,j]] = yanddys[[ +,j ]]-yanddys[[,j ], {,n-j + }],{j,2,n}]; MatrxForm[yanddys] Output[4] = x x x x x x x x x x 78 x x x x x

19 Notce that n Input[2], the Mathematca functons Length has been appled to determne the number of components n the array y, Table s used to ntalze a matrx of n by n wth the character x, and MatrxForm allows the matrx, yanddys, to be prnted n a matrx form. Input[3] stores the y array nto the frst column of the matrx yanddys by applcaton of the Mathematca command Do. Such loopng s extended n Input[4] where the hgher order dfferences are generated by usng an nner ndex and an outer ndex j. The column number j of the matrx yanddys s ncreased from 2 to n but the length of each column s contnuously decreased to n-j +. Such DoDo arrangement s made possble by keepng the y values and ther dfferences n a column-by-column form. 4.3 PROGRAM LAGRANGI APPLICATIONS OF LAGRANGIAN INTERPOLATION FORMULA Program LagrangI s desgned to curve-ft a gven set of n ponts, (x,f ) for =, 2,,n, by a polynomal of n-st degree based on the Lagrangan Interpolaton Formula: n n fx = f ( x x) x x = k= k [ k ( k) ] () If only the value of the functon f(x) at a specfed value of x = x s s needed, then Equaton can be appled to compute n n fx ( s)= f ( x2 x) x x = k= k [ k ( k) ] (2) In Equatons and 2, the symbol s to represent a product of a specfed number of factors such as: n F = k FF 2 F k= n (3) Equaton can be proven f we wrte the equaton whch fts the n gven ponts (x,f ) for = to n by a combnaton of n functons L to n (x) as: = fx fl x fl x fl x 2 2 n n (4) Notce that the ordnates f to n are utlzed n Equaton 4. We expect the functons L to n (x) to behave n such a way that when x = x only the f L (x) term n Equaton 4 wll contrbute to f(x). That s to say when x = x, L (x ) should be equal to unty and the other L(x) should be equal to zero. Mathematcally, we wrte demand that:

20 = = L x and L x 0 for j j (5) The second condton of Equaton 5 suggests that x-x k are factors of L (x) for k =,2,,n but not x-x. Therefore, we may wrte: L ( x)= c ( x x )( x x ) ( x x )( x x ) x x 2 + n (6) The constant assocated wth L (x), c s to be determned by satsfyng the frst condton of Equaton 5. That s: n c = x x k= k k (7) Consequently, the complete expresson for L (x) s: [ ] n k xk k= k L ( x)= x x x (8) And, when Equaton 8 s substtuted nto Equaton 4, we arrve at Equaton. A numercal example wll clarfy the applcaton of Equaton 2. Consder the case of three gven ponts (x,f ) = (,2), (x 2,f 2 ) = (.5,2.5), (x 3,f 3 ) = (3,4), then n = 3. If we need to calculate f(x = 2), Equaton 2 can be used to fnd the equaton whch passes all three ponts. That s: fx = ( ) ( ) x x2 x x3 x x x x f x x x x3 + x x x x f 2 3 ( ) ( ) x x x x2 + x x x x f ( x 5. )( x 3) = ( ) ( ) ( x ) x ( ) ( ) ( x ) ( x 3) ( ) = 2( x 4. 5x+ 4. 5) ( x 4x+ 3)+ x 25. x = x + 2

21 When x = 2, f(x = 2) = 3. Actually, the value of f(x = 2) can be specfcally calculated as: fx ( = 2)= QUICKBASIC VERSION ( ) ( ) ( ) ( ) x x2 x x3 x x x x f x x x x3 + x x x x f 2 3 ( ) ( ) x x x x2 + x x x x f ( 2. 5) ( 2 3) = ( 3) ( ) ( 2 ) = = ( 2 ) ( 2 3) ( ) 2

22 Sample Applcaton FORTRAN VERSION

23 Sample Applcaton MATLAB APPLICATION A m fle called LagrangI.m can be created and added to MATLAB m fles for nterpolatng a Y value for a gvng X value based on a set of (X,Y) data pont usng the Lagrangan formula. Ths fle may be wrtten as: Ths m fle can then be appled by specfyng the data ponts, X vs. Y, as llustrated by the followng examples: The graphc capablty of MATLABcan also be utlzed here to nterpret Lagrangan nterpolaton. In Fgure 6, fve gven ponts marked wth the character * have been exactly ftted wth a fourth-order polynomal whch s plotted for X 5 wth a sold lne. The nterpolaton at X = 4.56 usng Lagrangan formula s llustrated by the broken lne and dotted lne. The nteractvely entered MATLAB statements, n addton to those already dsplayed above, are:

24 FIGURE 6. Fve gven ponts marked wth the character * have been exactly ftted wth a fourth-order polynomal whch s plotted for X 5 wth a sold lne. The nterpolaton at X = 4.56 usng Lagrangan formula s llustrated by the broken lne and dotted lne.

25 Notce that plot.m automatcally uses sold, broken, and dotted lnes to plot the four-order polynomal curve based on arrays XC and YC, and the vertcal lne based on arrays XV, YV, and the horzontal lne based on arrays XH, YH, respectvely. The detals nvolved n exact curve-ft of the fve gven pont by applyng Least- SqG.m already has been dscussed n the program Gauss. The coeffcents, {C}, of the fourth-order polynomal determned by LeastSqG.m are arranged n descendng order. In order to apply polyval.m of MATLAB, the order of {C} has to be reversed and stored n {Creverse} whch s mplemented above by the for and end loop. MATHEMATICA APPLICATIONS Dervaton of the polynomal whch passng through a set of gven (x,y) ponts based on the Lagrangan formula can be acheved by applcaton of the Interpolatng Polynomal functon of Mathematca. For example, a fourth-order polynomal can be derved for a gven set of 5 (x,y) data ponts as follows: In[]: = pofx = InterpolatngPolynomal [{{,2},{2,4},{3,6},{4,8},{5,}},x] Out[]: = x x x ( + x) 24 To nterpolate the y value of usng the derved polynomal at x equal to 4.56, we replace all x s appearng n the above expresson (saved n pofx) wth a value of 4.56 by nteractvely enterng In[2]: = pofx/. x -> 4.56 Out[2]: = Lnear and parabolc nterpolatons can also be mplemented by selectng approprate data ponts from the gven set. For example, to nterpolate the y value at x =.25 by lnear nterpolaton, we enter: In[3]: = p = InterpolatngPolynomal[{{,2},{2,4}},x] Out[3]: = ( + x) In[4]: = p/. x ->.25 Out[4]: = 2.5 ( + )( + ) To parabolcally nterpolate the y value at x = 3.75 usng the ponts (3,6), (4,8), and (5,), the nteractve applcaton of Mathematca goes as: In[5]: = p2 = InterpolatngPolynomal[{{3,6},{4,8},{5,}},x]

26 4 Out[5]: = x 3 2 ( + x ) In[6]: = p2/. x -> 3.75 Out[6]: = PROBLEMS DIFFTABL. Construct the dfference table based on the followng lsted data and then fnd the y value at x = 4.5 by usng the backward-dfference formula up to the thrd-order dfference. x y Explan why nterpolatons usng Equaton 9 by the frst through fourth orders all fal to match the exact value of y(x =.24) = by makng 4 plots for x values rangng from.2 to.3 wth an ncrement of x = These 4 plots are to be generated wth the 4 equatons obtaned when the frst 2, 3, 4, and 5 ponts are ftted by a frst-, second-, thrd-, and fourth-degree polynomals, respectvely. Also, draw a x =.24, vertcal lne crossng all 4 curves. 3. Fnd the frst-, second-, thrd-, and fourth-order results of y(x =.56) by use of Equaton Wrte E r n terms of bnomal coeffcent and the backward-dfference operator, smlar to Equaton Fnd the frst-, second-, thrd-, and fourth-order results of y (x =.24) by use of Equaton Fnd the frst-, second-, thrd-, and fourth-order results of y(x =.56) by use of Equaton Gven 6 (x,y) ponts (,0.2), (2,0.4), (3,0.7), (4,.5), (5,2.9), and (6,4.7), parabolcally nterpolate y(x = 3.4) frst by use of forward dfferences and then by use of backward dfferences. 8. Modfy ether the QuckBASIC or FORTRAN verson of the program DffTabl to nclude the ffth dfference for the need of forward or backward nterpolaton and numercal dfferentaton. 9. Gven 5 (x,y) ponts (0,0), (,), (2,8), (3,27), and (4,64), construct a complete dfference table based on these data. Compute () y value at x =.25 usng a forward, parabolc (second-order) nterpolaton, (2) y value at x = 3.7 usng a backward, cubc (thrd-order) nterpolaton, and (3) dy/dx value at x = 0 usng a forward, thrd-order approxmaton. 0. Based on Equaton 2, derve the forward-dfference formulas for D 2 y and D 3 y.. Use the result of Problem 0 to compute D 2 y 2 and D 3 y by adoptng the forward-dfference terms n Table as hgh as avalable.

27 2. Use the data n Table to compute the frst dervatve of y at x =.55 by ncludng terms up to the thrd-order forward dfference. 3. Apply MATLAB for the ponts gven n Problem to prnt out the rows of x, y, y, 2 y, 3 y, and 4 y. 4. Same as Problem 3 but the ponts n Problem Apply Mathematca and DO loops to prnt out a dfference table smlar to that shown n Mathematca Applcaton of Secton 4.2 for the ponts gven n Problem. 6. Apply Mathematca and DO loops to prnt out a dfference table smlar to that shown n Mathematca Applcaton of Secton 4.2 for the ponts gven n Problem Compute the bnomal coeffcent for r = 0.4 and k =,2,3,4,5 accordng to Equaton (8) n Secton 4.2 usng MATLAB. 8. Rework Problem 7 but usng Mathematca. LAGRANGI. Gven fve ponts (,), (2,3), (3,2), (4,5), and (5,4), use the last three ponts and Lagrangan nterpolaton formula to compute y value at x = A set of 5 (x,y) ponts s gven as (,2), (2,4), (4,5), (5,2), (6,0), apply the Lagrangan nterpolaton formulas to fnd the y for x = 3 by parabolc nterpolaton usng the mddle three ponts. Check the answer by (a) wthout fttng the three ponts by a parabolc equaton, and (b) by dervng the parabolc equaton and then substtutng x equal to 3 to fnd the y value. 3. Apply the Lagrangan formula to curve-ft the followng lsted data near x = 5 by a cubc equaton. Use the derved cubc equaton to fnd the y value at x = Use the data set gven n Problem 3 to exactly curve-ft them by a quartc equaton y(x) = a + a 2 x + a 3 x 2 + a 4 x 3 + a 5 x 4. Do ths manually based on the Lagrangan formula. 5. Wrte a program and call t ExactFt.Ln5 for computaton of the coeffcents a 5 n the y(x) expresson n Problem Generalze the need n Problem 4 by extendng the exact ft of N gven (x,y) ponts by a polynomal y(x) = a + a 2 x + + a x + + a N x N based on the Lagrangan formula. Call ths program ExactFt.LnN. 7. Based on the Lagrangan formula, use the frst four of the fve ponts gven n Problem to nterpolate the y value at x = 2.5 and then the last four of the fve ponts also at x = Wrte a program and call t Expand. whch wll expand the set of fve ponts gven n Problem 2 to a set of 2 ponts by usng an ncrement of x equal to 0.2 and lnear nterpolaton based on the Lagrangan formula. For any x value whch s not equal to any of the x values of the fve gven x y

28 ponts, ths x value s to be tested to determne between whch two ponts t s located. These two gven ponts are to be used n the nterpolaton process by settng N equal to 2 n the program LagrangI. Ths procedure s to be repeated for x values between and 6 n computaton of all new y values. 9. As for Problem 8 except parabolc nterpolaton s to be mplemented. Call the new program Expand Extend the concept dscussed n Problems 8 and 9 to develop a general program Expand.M for usng N gven ponts and Mth-order Lagrangan nterpolaton to obtan an expanded set.. Apply the functon InterpolatngPolynomal of Mathematca to solve Problems and Check the result of Problem 4 by Mathematca. 3. Apply LagrangI.m to solve Problem by MATLAB. 4. Apply LagrangI.m to solve Problem 2 by MATLAB. 5. Apply LagrangI.m to solve Problem 7 by MATLAB. 4.5 REFERENCE. R. C. Weast, Edtor-n-Chef, CRC Standard Mathematcal Tables, the Chemcal Rubber Co. (now CRC Press LLC), Cleveland, OH, 964, p. 38.