Problem | Max. | Grade |
1. | 10 | |
2. | 10 | |
3. | 10 | |
4. | 15 | |
5. | 15 | |
6. | 10 | |
7. | 10 | |
8. | 10 | |
9. | 10 | |
Total: | 100 |
1. (10 pts) Find the value of A in the function y=Ax, that ,minimizes the sqare error with the following data:
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2. (10 pts) a) Derive the normal equations to find the least sqares curve f(x)=Acos(x)+Bsin(x). b) Use this result to find A and B for the following data:
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3. (10 pts.) Determine if the following functions are cubic splines:
(b) f(x) =
(c) f(x) =
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4. (15 pts) Verify that Simpson's rule (M=1, h=1) is exact for polynomials of degree <= 3 of the form f(x)=c3x3+c2x2+c1x+c0
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5. (15 pts) Solve the differential equation y'=e-2t-2y with y(0)=1/10. Let h=0.1 and do four steps.
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6. (10 pts.) Use both the Gauss-Seidel and the Jacobi iteration methods to find (xk,yk) for k = 1, 2, 3. Start at the point (x0,y0)=(0,0). Choose iteration equations that will make the method converge.
-x +3y = 1 |
7. (10 pts.) Let f(x)= 2sin(3.14x/6), where x is in radians. Use quadratic Lagrange interpolation based on the nodes x0=0, x 1=1, and x2=3 to approximate f(4)
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8. (10 pts.) Given the following table, compute the divided-difference table for the function x1/2. Also, write down the Newthon polynomials P1(x), P2(x), and P3(x).
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9. (10 pts.) Using Pade' approximations, find R2,2(x) for f(x)=arctan(x1/2)/x1/2. Start with the Maclaurin expansion:f(x)=1 - x/3 + x2/5 - x3/7 + x4/9
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