CH-02 (Pao - Engineering Analysis), страница 5

Given four points (1,0.5), (2.5,0.88), (3.2,1.35), and (4.5,2.76), they areto be least-squares fitted by a linear combination of two selected functions,f1(x) = e-x and f2(x) = e–2x, in the form of f(x) = c1f1(x) + c2f2(x). Find c1and c2.2. For a given set of 10 points, (xi,yi) for i = 1,2,…,10, the least-squaresmethod has been applied to fit these data by two students. Student Aselects 3 functions f1(x) = x, f2(x) = x3, and f3(x) = x5 to obtain the coefficients a1, a2, and a3 for the expressiony = a1x + a2x3 + a3x5. Student Bselects 3 other functions f1(x) = sinx, f2(x) = sin3x, and f3(x) = sin5x toobtain the coefficients b1, b2, and b3 for the expression y = b1sinx + b2sin3x+ b3sin5x.

Write a program to calculate the least-squares errors EA andEB for the curve-fit approaches taken by the students A and B, respectively.3. Apply the least-squares method to curve-fit the following three givenpoints by a linear combination of two selected functions f1(x) = x andf2(x) = x3, namely, y = c1x + c2x3:xy0–11022Find c1 and c2 by use of Cramer’s Rule.© 2001 by CRC Press LLC4. Given eight data points (Xi,Yi) for i = 1,2,…,8 as listed, fit them by theleast-squares method with the equation Y = a1 + a2X + a3X2.

Find a1–3 byapplying the computer program LeastSqG.XY11.1321.4531.7642.1952.4362.7973.5184.885. A set of three points are provided as listed: (1,0.2), (2,0.5), and (3,0.6).These points are to be fitted by application of the least-squares methodusing a linear combination of (a) two functions x and x2, or (b) two otherfunctions x0 and x3. Which fit will be better, a or b? Back up your answerwith detailed calculations.6. Given three points (1,2), (3,5), and (4,13), two selected functions f1(x) =x–1 and f2(x) = x3 are to be linearly combined to fit these points by theequation y = a1f1(x) + a2f2(x) = a1(x–1) + a2x3.

Derive two equationsneeded for solving a1 and a2 by use of the Least-Squares method.7. Given 7 points of which the coordinates are X(i) and Y(i) for i = 1 to 7,a least-squares fit of these points with a linear combination of 3 selectedfunctions f1(X) = X, f2(X) = sin2X, and f3(X) = e-X in the form of Y(X) =C(1)f1(X) + C(2)f2(X) + C(3)f3(X) has been conducted and the coefficientsC(1) to C(3) have been found.

Complete the following segment of FORTRAN program to calculate the total error E which is the sum of thesquares of the differences between Y(X(i)) and Y(i) for i = 1 to 7.DIMENSION X(7),Y(7),C(3)DATA X,Y,C/(17 real numbers separated by,)/insert statements for< - - - calculation of E involvingC, X, Y, and f1, f2, and f3.WRITE (*,*) ESTOPEND8. For a given set of data (1,–2), (2,0), (3,1), and (4,3), two equations havebeen suggested to fit these points. They are Y = X–2 and Y = (-X2 +7X–10)/2.

Based on the least-squares criterion, which equation should bechosen to provide a better fit? Explain why.9. Given 12 points of which the coordinates are X(i) and Y(i) for i = 1 to 7,a least-squares fit of these points with a linear combination of 4 selectedfunctions f1(X) = X, f2(X) = sin2X, f3(X) = cos3X, and f4(X) = e-X in theform of Y(X) = C(1)f1(X) + C(2)f2(X) + C(3)f3(X) + C(4)f4(X) has beenconducted and the coefficients C(1) to C(4) have been found. Completethe following segment of FORTRAN program to calculate the total errorE which is the sum of the squares of the differences between Y(X(i)) andY(i) for i = 1 to 12, using a FUNCTION subprogram F(I,X) which evaluatethe Ith selected function at a specified X value for i = 1 to 4.© 2001 by CRC Press LLCDIMENSION X(12),Y(12),C(4)DATA X,Y,C/(28 real numbers separated by,)/insert statements for< - - - calculation of E involvingC, X, Y, and f1, f2, and f3.WRITE (*,*) ESTOPEND10.

Any way one can solve the above-listed problem by application of MATLAB? Compare the computed results obtained by QuickBASIC, FORTRAN, and MATLAB approaches.11. Try Mathematica and compare results for the above problems.CUBIC SPLINE1. Presently, program CubeSpln is not interactive. Expand its capability toallow interactive input of the number of points, N, and coordinates (Xi,Yi)for i = 1 to N. Also, user should be able to specify the KK value so thatboth periodic or nonperiodic data points can be fitted. Call this programCubeSpln.X and rerun the case used in Sample Application.2.

Change the program CubeSpln slightly to allow a sixth point to be considered. Add a sixth point whose Y value is equal to that of the first pointthen run it as a periodic case by changing KK equal to 2. The resultingplot for X(6) = 5.5 should be as shown below.© 2001 by CRC Press LLC3. Use the program CubeSpln.X to run Problem 2.4. Apply spline.m of MATLAB to fit the points (1,2), (2,4), (3,7), and (4,13)and then plot the curve by using plot.m.

Mark the points by the character *.5. Apply spline.m of MATLAB to fit the points (0.5,3), (1.2,6), (2.5,5), and(3.7,11) and then plot the curve by using plot.m. Mark the points by thecharacter + .6. Apply spline.m of MATLAB to fit the points (3,3), (3.6,6), (4.2,8), and(5.1,11) and then plot the curve by using plot.m.7. Combine the curves obtained in Problems 4 to 6 into a composite graphby using solid, broken, and center lines which in use of plot.m require tospecify with ‘-’, ‘- - -’, and ‘-.’, respectively. The resulting compositegraph should look like the figure below.8. Use text command of MATLAB to add texts ‘Problem 4’, ‘Problem 5’,and ‘Problem 6’ near the respective curves already drawn in Problem 7.9. The temperature data in °F, collected during a period of seven days are (2,75),(3,80), (4,86), (5,92), (6,81), and (7,90).

Cubic-spline fit these data, plot thecurve, and label the horizontal axis with ‘Days’ and vertical axis with ‘Temperature, in Fahrenheit’ by use of xlabel and ylabel commands of MATLAB.© 2001 by CRC Press LLC10. Add to the graph obtained for Problem 9 by marking the data points withthe character * and also a text ‘Cubic Spline of Temperature Data’ at anappropriate location not touching the spline curve.2.7 REFERENCES1. Y.

C. Pao, “On Development of Engineering Animation Software,” in Computers inEngineering, K. Ishii, editor, ASME Publications, New York, 1994, pp. 851–855.2. A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.3. H. Flanders, R. R. Korfhage, and J. J. Price, A First Course in Calculus with AnalyticGeometry, Academic Press, New York, 1973.© 2001 by CRC Press LLC.