User:Yongtao-Li/HW2.2

=Problem 2.2: "MOTIVATION OF WRF"=

Given
$$ \{ {{\mathbf{b}}_i},i = 1, \cdot \cdot  \cdot ,n\} $$ are basis vectors for $$ {{\mathbf{R}}^n} $$ which are not necessarily orthonormal.

In order to find $$ \{ {{\mathbf{v}}_i} \} $$ s.t. $$ {\mathbf{v}} = \sum\limits_{i = 1}^n $$ , we consider $$ \{ {{\mathbf{a}}_i},i = 1, \cdot \cdot  \cdot ,n\} $$ are orthonormal basis in Cartesian coordinates, and $$ {{\mathbf{b}}_j} = b{_{jk}}{{\mathbf{a}}_k} $$ , $$ {{\mathbf{v}}=5{{\mathbf{a}}_1}-7{{\mathbf{a}}_2}-4{{\mathbf{a}}_3}} $$

where $$ \left[ \right] = \left[ {\begin{array}{ccccccccccccccc} 1&1&1 \\ 2&{ - 1}&3 \\  3&2&6 \end{array}} \right] $$

Find
A) Find $$ \det \left[ \right] $$

B) Find $$ \det \Gamma $$ where $$ {\mathbf{\Gamma }} = \left\{ \right\} = {\mathbf{K}} $$

C) Find $$ {\mathbf{F}} = \left\{ \right\} = \left\{ {{{\mathbf{b}}_{\mathbf{i}}} \cdot {\mathbf{v}}} \right\}$$

D) Solve equation (5) P7-2 for $$ {\mathbf{d}} = \left\{ \right\} $$

E) Use equation (1) P7-4 to find $$ \overline {\mathbf{K}} {\mathbf{d} = \overline {\mathbf{F}} }$$ and what is $$ \overline {\mathbf{K}}, \overline {\mathbf{F}} $$

F) Solve for $$ {\mathbf{d}}$$ and compare of $$ {\mathbf{K}}$$ and $$ \overline {\mathbf{K}} $$

G) Observe symmetric properties of $$ {\mathbf{K}}$$ and $$ \overline {\mathbf{K}} $$ and discuss pros and cons of these two methods

Solution
A) The determinant of $$ \left[ \right] $$ is as following:

B) Write $$ \left[ \Gamma \right] = \left[ K \right] $$ in matrix form:

Then the determinant is as following

C) Write $$ \left[ K \right] $$ in matrix form:

D) Use equation (5) P7-2 and write it in matrix form:

$$ \left[ {\begin{array}{ccccccccccccccc} 3&4&{11} \\ 4&{14}&{22} \\  {11}&{22}&{49} \end{array}} \right] \times \left[ d \right] = \left[ {\begin{array}{ccccccccccccccc} { - 6} \\ 5 \\  { - 23} \end{array}} \right] $$

Multiply the above equation with the inverse of matrix K, we can get d in matrix form:

E) Consider $$ {w_i} = {a_i},i = 1, \cdot \cdot  \cdot ,n $$ and follow the steps above, it is easy to get K and F in matrix form:

F) Similar with step D, we can get d in matrix form:

G)Both $$ {\mathbf{K}}$$ and $$ \overline {\mathbf{K}} $$ are symmetric matrix. However $$ \overline {\mathbf{K}} $$ will be symmetric only if the basis are orthonormal otherwise not and $$ {\mathbf{K}}$$ will always be symmetric matrix.

Bubnov-Galerkin method is more difficult to compute, but can choose arbitrary vector to find the answer.

Petrov-Galerkin method is easy to use, but have no find orthonormal vectors which sometimes difficult in specific problems.