Consider the following population data for the state of California. Year Population

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1 Assigmets for Bradie Fall 2016 for Chapter 5 Assigmet sheet for Sectios 5.1, 5.3, 5.5, 5.6, 5.7, 5.8 Read Pages Exercises for Sectio 5.1 Lagrage Iterpolatio #1, #4, #7, #13, #14 For #1 use MATLAB to draw the graphs. For #4 do part (d) carefully. For #13 use MATLAB routie piterp; capture the formula ad graph to iclude with your aswer to the questio. For #14 part (b) use MATLAB routie piterp capture the formula ad graph to iclude with your aswer to part (c) Read Pages Exercises for Sectio 5.3 Divided Differece Form of Iterpolatio #3, #6, #7, #10, #11, #15, Califoria Problem, DD-Table Problem For #3 ad #6 do the calculatios by had; show the work. For #7 costruct the divided differece table by had; do ot compute the values of ay logarithms. For #15 use MATLAB routies divdiff ad divpoly; iclude prit outs from MATALB. Califoria Problem Cosider the followig populatio data for the state of Califoria. Year Populatio i Millios Table 1. We are to build a iterpolatio polyomial model to this data. To simplify the values ivolved ad also possibly to aid i the prevetio of roudoff error we 'rescale' the data as follows. Year Populatio i Millios Table 2.

2 Eter the data i Table 2, ito MATLAB. Call the years x ad the populatios y. Use routie divpoly i MATLAB do the followig. a) The populatio i 1930 was about 5.7 millio. Evaluate the iterpolatio polyomial to the data i Table 2 at x = 30. Compute the absolute ad relative error i the value obtaied from the evaluatio of the iterpolat. b) Use divpoly as i part a, but predict the populatios i 1985, 2000, ad Do these values seem reasoable? Explai. c) Use routie piterp with the data i Table 2 ad sketch the iterpolat over 1900 to 2010 by settig the miimum value of x to 0 ad the maximum to 110. Prit out the graph ad aotate it to explai your aswers to parts a ad b. (Geerate the sketch with y-coordiates 0.) DD-Table Problem Give the followig divided differece table. 0th DD 1st DD 2d DD 3rd DD 4th DD 4 a 7 b m f g 9 c t 10 d p 12 e h k r s (a) Write a expressio for the iterpolat that goes through the set of ordered pairs {(4,a), (7,b), (9, c)}. (b) Write a expressio for the iterpolat that goes through the set of ordered pairs { (7,b), (9, c), (10, d), (12, e)}. (c) The poit (15, w) is to be added to the table. Write ad expressio for the iterpolat through the ordered pairs {(12, e), (15, w)}

3 Sectio 5.5 Piecewise Iterpolatio Read my Notes for Piecewise Iterpolatio Do the followig Exercises. #1. Give a curve, collect a sample of poits, geerate the correspodig polyomial iterpolat, ad the compare the graph of the iterpolat with the origial curve. Sice a limited sample of poits will be permitted it is importat to take the sample from portios of the curve that i some sese cotrol or sigificatly affect its shape. Such a strategy ca aid the model costructed by the iterpolat, but is ot a guaratee that the model produced will match the shape of the etire curve. Eter the followig MATLAB commads: help iprob To see a picture of the curve you are asked to sample ad build a iterpolatio model for, type iprob,figure(gcf) The x-coordiates are i the vector x ad the y-coordiates i the vector y. To retai this image so we ca superimpose models you create, type axis(axis),hold o PRACTICE SAMPLING THE CURVE MATLAB has a commad that lets us sample a curve usig your mouse. For a descriptio type help giput To try this commad we will collect a sample of 3 poits from the curve. Type commad [px py] =giput(3) Positio your mouse over the curve ad click the left mouse butto whe you are at the poit you wat to iclude i the sample. Press ENTER after you have selected the 3 poits. The coordiates will be displayed. To see your poits o the graph use the followig commads.

4 plot(px,py,'*r'), figure(gcf) Experimet with this a few times to get the feel for the mouse ad thik about where you will collect the sample data #2. Next type hold off,close(gcf) followed by iprob,figure(gcf),axis(axis),hold o to get a fresh picture. Use the followig commads to sample the curve at 6 poits ad geerate a polyomial iterpolat of degree 5. [sx sy]=giput(6) %collect the 6 poit sample usig your mouse plot(sx,sy,'*r'),figure(gcf) %plottig the sample A=vader(sx);c5=A\sy; % set up Vadermode matrix & %solve for coefficiets of quitic iterpolat z5=polyval(c5,x); % evaluate quitic model at x-coords for the curve plot(x,z5,':k'),figure(gcf) % plot the cubic model If your model is t very good to start aew type hold off,close(gcf) followed by iprob,figure(gcf),axis(axis),hold o the repeat the code for samplig. Oce you get a good approximatio to the curve, put your ame i the title of the graph ad prit it out. Tur i your graph. #3. Repeat Exercise 2, but this time sample the curve at 20 poit istead of 6. If your model is t very good to start aew type hold off,close(gcf) followed by iprob,figure(gcf),axis(axis),hold o the repeat the code for samplig. Were there ay warigs from MATLAB? If so copy them as part of your solutio. Try to get a good approximatio to the curve, put your

5 ame i the title of the graph ad prit it out. Were you able to get a good approximatio? Discuss you attempts. Tur i your graph #4. A alterative to polyomial iterpolatio through a etire data set is to costruct polyomial iterpolats to subsets of the data. Thus differet polyomials ca be used over subitervals of the data poits ivolved. The simplest such procedure is piecewise liear iterpolatio which amouts to coectig successive data poits with straight lies ( coect-the-dots ). While the iterpolatig fuctio costructed i this way is cotiuous, it is usually ot differetiable because sharp poits occur at the data poits. Such procedures geeralize to piecewise quadratic ad piecewise cubics by appropriately subdividig the data set ito successive subsets of three ad four poits respectively with oe poit overlap to esure cotiuity. To experimet with piecewise polyomial iterpolatio of this type use routie pwiterp. A brief descriptio follows. PWINTERP Piecewise polyomial iterpolatio usig: LINEAR QUADRATIC or CUBIC pieces. INPUT: Vectors x ad y must cotai the x ad y coordiates of the data poits respectively. OUTPUT: The coefficiets the pieces are i arrays lf qf ad cf respectively. Various iterpolats ca be superimposed graphically. => pwiterp(x,y) or [lf,qf,cf]=pwiterp(x,y) <= x Cosider the serpetie curve' f(x) over [-2, 2] x I MATLAB type figure (a) Sketch the curve f(x) usig routie ezplot. Prit out the curve or draw a facsimile. (b) Let x = [-2, -1, -0.5, -0.25, 0, 0.25, 0.5, 1, 2] be the x-coordiates of sample poits. Eter this vector of data i to MATLAB. Compute the correspodig y-coordiates i MATLAB usig commad y = x./(0.25+x.*x); Record the data set. (c) Use piterp to costruct the polyomial iterpolat to this data ad graph it. (Set the viewig widow so that -2 x 2 ad -11 y 11. Superimpose the graph of f(x) eterig the formula as x/( x^2). Briefly discuss the approximatio qualities of the polyomial model. Prit out the superimposed graph or draw a facsimile.

6 (d) Use pwiterp o the data from part (b). Prit out each of the three possible models. Briefly discuss the approximatio qualities of the piecewise polyomial models. Which model seems better tha the others ad why? Read Pages ; Read the documet spliedefs. See the website. Exercises for Sectio 5.6 Cubic Splie Iterpolatio #1, 2, 3, 10, 15, ad H1 Use MATLAB files for splie computatios. Display the coefficiets of the cubic pieces i all the problems. The mfiles are splie, csplie, ad aksplie. It is assumed that you have read the documet spliedefs. See also Sectio 5_6 Example with details. H1. Sectio 5.6 The 1995 Ketucky Derby was wo by a horse amed Thuder Gulch i a time of 2:01 1/5 (2 mi 1 ad 1/5 sec) for the 1.25 mile race course. Times at the quarter mile, halfmile, ad mile poles were 22.4 secs, 45.8 secs, ad 1 mi 35.6 secs. a) Use the previous data together with the iitial data, t = 0, dist = 0, to costruct a atural cubic splie for Thuder Gulch's race. b) Use the splie to predict the time at the three-quarter-mile pole, ad compare this to the actual time of 1 mi 10.2 secs. c) Use the splie to predict the speed at which Thuder Gulch left the startig gate ad the speed at which he crossed the fiish lie. Give aswers i miles per hour. Show all your work. For #1, discuss the questio also discuss the type of relatioship betwee z a T implied by the splie. For #2, discuss the questio also discuss the type of relatioship betwee z a p implied by the splie. For #10 use the MATLAB routies to geerate the requested graphs. Iclude a copy of the graphs as part of your solutio. For #15, iclude a graph of the error for the atural cubic splie

7 Read pages Sectio 5.7 Hermite Iterpolatio ad Cubic Hermite Iterpolatio #8, 10 (i the text) There is a mfile for the data set for #8 & 10; Bradie_data_Sec5_7_Exer8.m Dr. Hill s problems #1, 2, 3 For #8 use MATLAB routie pwhermitecubic.m. For #10 use MATLAB routie csplie.m. IMPORTANT: For Dr. Hill s problems use the geeralizatio of divided differeces to compute ay Hermite iterpolatio polyomial. Show the divided differece tables for each problem. 1. The followig sample of fuctio f is give. a) Approximate f(0.5) usig a polyomial iterpolat. b) Approximate f(0.5) usig a Hermite iterpolat. x f(x) f (x) c) The fuctio sample is from f(x) = 2xe x e 3x. Compute the absolute error i each of the results from parts a ad b. d) Graph the absolute error expressios for parts a ad b over [-1,1]. Commet o the accuracy of part a vs that i part b. 2. A switchig track is to be costructed betwee a pair of parallel trolley tracks as show i the figure. The trasitio betwee the two tracks is to be smooth. Assume that the tracks are horizotal ad the switch poits have coordiates (3,2) ad (0,0) o tracks #1 ad #2 respectively. Costruct a Hermite iterpolatig polyomial that goes through the poits ad matches the slope of the track at each of the two poits. Geerate a graph of your track over [0,3].

8 3. At aother costructio site a switchig track like that i Exercise 2 is to be costructed with a additioal requiremet. Namely, the track must also go through poit (1,1). Modify the result from Exercise 2 to iclude this requiremet. Geerate a graph of your track over [0,3]. Are there ay costructio difficulties i this case? Explai Read Pages Exercises for Sectio 5.8 Regressio #2, 5, 8, 13, ad Dr. Hill s problems Use MATLAB ad the matrix approach for computig least squares lies ad liearized fits. Use MATLAB to geerate appropriate graphs for #2 ad 13 Dr. Hill s problems. 1. The ormal system of equatios for the least squares lie has a uique solutio provided the matrix C L NM i1 x i1 x i 2 i i1 x i O QP is osigular. To verify that this is ideed true x1 1 Mx2 1 whe all the x i are distict we use the matrix formulatio, C = A T A, where A =. L Let k N M k1o Q P k 2 L M NM x 1. The we kow the followig: the homogeeous liear system Ck = 0 has O QP

9 oly the trivial solutio if ad oly if matrix C is osigular. Verify the followig to show that C is osigular. a) Explai why the colums of A are liearly idepedet. b) Explai why Ak ca oly be equal to the zero vector 0 if k = 0. c) Explai why k T A T = (Ak) T. d) Assume Ck = 0. Explai why (Ak) T (Ak) = 0. (Note: the right side is the zero scalar.) e) Explai why (Ak) T (Ak) = 0 is the same as the dot product (Ak)(Ak) = 0. f) Whe ca the dot product of a vector with itself be zero? g) Use parts b, e, ad f to explai why k = 0 ad hece C is osigular. 2. For a give data set S, let lie L be the least squares lie ad E L be the miimum value of the sum of the squares of the deviatios. For the same data set D let quadratic Q be the least squares quadratic ad E Q be the miimum value of the sum of the squares of its deviatios. Why must E Q E L? (Hit: We ca thik that the least squares process for determiig L looks over all polyomials of degree 1 or less to determie the oe that miimizes the sum of the squares of the deviatios. Similarly, the least squares process for determiig Q looks over all polyomials of degree 2 or less.)