User:Egm6341.s11.team4.fields/HW2.9

=Problem 2.9: Roots of P5(x) and P10(x)=

Refer to lecture slides [[media:nm1.s11.mtg10.djvu|10-1]], [[media:nm1.s11.mtg8.djvu|8-1]] and [[media:nm1.s11.mtg8.djvu|7-5]] for the problem statement.

Given

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(9.1)
 * $$\displaystyle P_n (x) = \sum_{i=0}^{[n/2]} (-1)^i \frac{(2n-2i)! x^{n-2i}}{2^n i! (n-i)! (n-2i)!}$$
 * 
 * }
 * }

Objectives
a.) Use 9.1 to generate P5(x) and compute the roots of P5(x) to check values in the table on lecture slide [[media:nm1.s11.mtg8.djvu|7-5]]. Plot the roots on [-1,1].

b.) Repear steps in a. for P10(x). Observe the location of the roots near endpoints -1 and +1.

Solution
a.)Using Wolfram Alpha to compute P5(x) we get:


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(9.2)
 * $$\displaystyle P_5 (x) = \frac{63 x^5}{8} - \frac{35 x^3}{4} + \frac{15 x}{8}$$
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 * }
 * }

Matlab code to produce the roots of P5(x) and the plot of the roots:

>> C = [63/8 0 -35/4 0 15/8 0]; >> x = roots(C)

x =

0 -0.906179845938664  -0.538469310105683   0.906179845938664   0.538469310105683 >> y = [0;0;0;0;0]; >> plot(x,y,'.','markersize',15)

These roots compare well with the values in the table in lecture [[media:nm1.s11.mtg8.djvu|7-5]].


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(9.3)
 * $$\displaystyle 0 = 0$$
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 * }
 * }


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(9.4)
 * $$\displaystyle \pm0.538469310105683 \approx \pm \frac{1}{3} \sqrt{5 - 2\sqrt{10/7}}$$
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 * }
 * }


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(9.5)
 * $$\displaystyle \pm0.906179845938664 \approx \pm \frac{1}{3} \sqrt{5 - 2\sqrt{10/7}}$$
 * 
 * }
 * }

Below is a plot of the roots of P5(x):



b.)Using Wolfram Alpha to compute P10(x) we get:


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(9.6)
 * $$\displaystyle P_{10} (x) = \frac{46189 x^{10}}{256} - \frac{109395 x^8}{256} + \frac{45045 x^6}{128} - \frac{15015 x^4}{128} + \frac{3465 x^2}{256} - \frac{63}{256}$$
 * 
 * }
 * }

Matlab code to produce the roots of P10(x) and the plot of the roots:

>> C = [46189/256 0 -109395/256 0 45045/128 0 -15015/128 0 3465/256 0 -63/256]; >> x = roots(C)

x =

-0.973906528517167 -0.865063366688989  -0.679409568299026   0.973906528517167   0.865063366688989   0.679409568299024  -0.433395394129247   0.433395394129247  -0.148874338981631   0.148874338981631

>> y = [0;0;0;0;0;0;0;0;0;0]; >> plot(x,y,'.','markersize',15)

Below is a plot of the roots of P10(x):