HW7 Solution

HW7 Solution
(7.1-7.2 on paper, 7.3 online)
7.1
Lagrange Polynomials

(t)
(H)
4 pts
You just watched some poor kid let go of her balloon in a 8-m high mall food
0
0

court. The four data points at right represent the height of a balloon (H, in







meters ) as a funcDon of Dme (t, in seconds).

1
1

(a) Create the 3rd-order Lagrange polynomial that exactly passes through all H(t)
2
7

data, and use it to esDmate (extrapolate) the balloon height at 4me = 4 seconds.




3
8

Show your work!!





(b) Create the 3rd-order Lagrange polynomial to interpolate the height of the balloon at 4me = 2.5 seconds (before hiHng the ceiling).

(c) Create a 1st-order Lagrange polynomial to interpolate the height of the balloon at 4me = 2.5 seconds. This is basically doing a LINEAR SPLINE to interpolate h at t = 2.5 sec.

(d) Make a plot by-hand of the data (on the paper you’re handing in), with the three interpolated points above super-imposed, and then comment on concerns using the Lagrange polynomials above for answering (a) – (c). Do your results above make “common-sense”? Do you “trust” them, or are these just “bogus mathemaDcal outputs”?

7.2
Making a Cubic Spline (by-hand)


8 pts
Take just the last three points from the H(t) data above (re-printed at right),
(t)
(H)

and develop a cubic-spline BY-HAND that matches that data, and use it to



esDmate the height at t = 1.5 and t = 2.5 seconds. For the two end points
1
1

(t = 1 and t = 3) apply “natural






spline” (y’’ = 0) condiDons.
2
7

When I say “by-hand” what I mean is (and show your work!!!)




3
8

•  write out the form of the cubic interpolaDng polynomials,







Fi(x) = ai + bi x + ci x2 + di x3 , in both intervals (i = 1 and 2),



•  apply all matching condiDons to create a system of 8 equaDons with 8 unknowns, •  re-write these equaDons in matrix form (Ax=b),

•  use MATLAB to solve for the 8 unknown coefficients, [OK to use MATLAB for just this step!]

•  Finally, by-hand again, use the resulDng funcDons to interpolate the force required at both t = 1.5 and t = 2.5.

Then feel free to compare the interpolated value at t = 2.5 from the spline with the value you got with the cubic and linear polynomials in 7.1. What do you think is “best”?

7.3 Using MATLAB to Explore Two Interpola4ng Methods

The data below represents the percent of cars (Ri) needing repairs as a funcDon of Dme in years (ti) a_er the model is released:
9 pts
ti (yr)
0
2
4
6
8
10
15
20

Ri (%)
3
4
7
9
12
13
19
21

Develop a short MATLAB script that calculates and plots the interpolaDon of this repair data over a finer range of every year from t = 0 to 20. Make sure ALL your steps and calculaDons below are included in your script so we can evaluate your methodology and answers (i.e. don’t do things off-line on paper that we

can’t see).
Con)nued next page


HW7 due (7.1-7.2 on paper, 7.3 online)

7.3 (con4nued …)

Start by creaDng a new vector of Dme: tnew = [0:20]. Now create three new vectors that interpolate the data over tnew using each of three different methods:

a) Exact Polynomial: Use polyfit to calculate the single polynomial that exactly fits all eight data points, and polyval to evaluate it over the tnew vector.

b) Linear Spline: Use interp1 with the “linear” opDon.

c) Cubic Spline: Use interp1 with the ”spline” opDon (which defaults to a cubic spline with “not-a-knot” end condiDons).

Make a single plot with the raw data (as circles) plus all three interpolated lines, where exact polynomial line (a) is dashed red, linear spline (b) is solid black, cubic spline (c) is dash-dot blue, maybe using something similar to the command below:

plot(ti,Ri,'ok', tnew,Ra,’--r', tnew,Rb,’-k', tnew,Rc,’-.b’)

Constrain the plot over 0 ≤ Repair % ≤ 25 and 0 ≤ Fme ≤ 20 with the command axis([0 20 0 25]).

Finally, interrogate the plot and the interpolated vectors over the range in tnew to answer the following quesDons:

i.  What does each model say the interpolated repair rate is at t = 3 years?

ii.  What does each model say the interpolated repair rate is at t = 18 years?

iii.  If you trust the data 100%, use the plot and your engineering judgment to determine which model you think did the best job at interpolaFng the data over the enDre range.


Please submit the following online in Carmen for problem 7.3 only

•  Your commented final script (HW7_3.m) that does all the above

•  The .pdf of your final (labeled!) plot.

•  In the comment box, enter your answers for quesDons (i) through (iii) above. Be sure to write values accurate to at least one decimal, and clearly say what answer goes with what quesDon. For example, you could write:

(i) Rate at t=3: (a) 4.5%, (b) 7.6%, (c) 12.9%

(ii) Rate at t=18: (a) 20.0%, (b) -14.3%, (c) 1.1%
(iii) I have no opinion – just tell me what I should like best.

Please don’t write anything from 7.3 on paper. Your code should be complete and show all your steps,