User:Eml5526.sp11.team1.adam/HW2

=Homework 2=

=Problem 7= Consider the family $$F=\begin{bmatrix}x,x^2,x^3,x^4 \end{bmatrix} $$

Is $$ F $$ an othogonal family evaluated at $$\Omega=\begin{bmatrix}0,1 \end{bmatrix} $$?

From Lecture Slide Mtg.10

Solution
Let's start by calculating $$\Gamma (F)$$

$$(F)=\int_{0}^{1} \begin{bmatrix} 1,x,x^2,x^3,x^4\\ x,x^2,x^3,x^4,x^5\\ x^2,x^3,x^4,x^5,x^6\\ x^3,x^4,x^5,x^6,x^7\\ x^4,x^5,x^6,x^7,x^8\\ \end{bmatrix}dx $$

Performing the integration we ogtain:

$$\Gamma(F)= \begin{bmatrix} x,\frac{x^2}{2},\frac{x^3}{3},\frac{x^4}{4},\frac{x^5}{5}\\ \frac{x^2}{2},\frac{x^3}{3},\frac{x^4}{4},\frac{x^5}{5},\frac{x^6}{6}\\ \frac{x^3}{3},\frac{x^4}{4},\frac{x^5}{5},\frac{x^6}{6},\frac{x^7}{7}\\ \frac{x^4}{4},\frac{x^5}{5},\frac{x^6}{6},\frac{x^7}{7}\frac{x^8}{8}\\ \frac{x^5}{5},\frac{x^6}{6},\frac{x^7}{7}\frac{x^8}{8},\frac{x^9}{9}\\ \end{bmatrix}$$

When the latter equation is evaluated at (0,1) we obtain:

$$\Gamma(F)= \begin{bmatrix} 1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}\\ \frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6}\\ \frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7}\\ \frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7}\frac{1}{8}\\ \frac{1}{5},\frac{1}{6},\frac{1}{7}\frac{1}{8},\frac{1}{9}\\ \end{bmatrix}$$

Where in can clearly be observed that $$\Gamma(F)$$ is not orthogonal

Contributing Members
=Problem 8= Show that equation (1) and equation (2) on page 4 Meeting 10 are equilavent.

From Lecture Slide Mtg.11

Given
Let's define our equations:

Equation (1) on page 4 Meeting 10


 * {| style="width:100%" border="0"

$$\int_{\Omega }{}w^{h}(x)P(U^{h}(x))dx=0;\forall w^{h}(x)$$
 * $$\displaystyle (Eq.8.1) $$
 * }
 * }

and Equation (2) on page 4 Meeting 10


 * {| style="width:100%" border="0"

$$\int_{\Omega}{} b_{i}(x)P(U^{h}(x))dx=0 $$
 * $$\displaystyle (Eq.8.2) $$
 * }
 * }

With
 * {| style="width:100%" border="0"

$$w^{h}(x)=\sum_{i=1}^{n}c_{i}b_{i}(x)$$
 * $$\displaystyle (Eq.8.3) $$
 * }
 * }

Solution
If we chose $$c_{i}$$ to be orthogonal, we have:

$$c_{1}=\begin{bmatrix}1,0,...,0 \end{bmatrix} $$

$$c_{2}=\begin{bmatrix}0,1,...,0 \end{bmatrix} $$

$$c_{3}=\begin{bmatrix}0,0,...,1 \end{bmatrix} $$

When we multiply $$c_{i}$$ times $$b_{i}$$ we observe that multiply by $$c$$ is equivalent to multiplying by the identity matrix, therefore replacing $$w^{f}(x)$$ in $$eq. 8.1$$ we have:

$$\int_{\Omega}^{ }w^{h}(x)P(U^{h}(x))dx=\int_{\Omega}^{ }\sum_{i=1}^{n}c_{i}b_{i}(x)P(U^{h}(x))dx=\int_{\Omega}^{ }I.b.P(U^{h}(x))dx=\int_{\Omega}^{ }b_{i}P(U^{h}(x))dx$$