User:Eml4507.s13.team3.berthoumieux

Given
A L2-ODE-CC system with the following conjugate roots:

$$ \lambda{_{1}}=-0.5+i2$$

$$ \lambda{_{2}}=-0.5-i2$$

Find 2.1
Find the homogeneous L2-ODE-CC in standard form in terms of $$y{_{h}}(t) $$  with the above roots.

Solution 2.1
$$y{_{h}}''+ay{_{h}}'+by{_{h}}=0$$

$$\lambda^{2}+a\lambda+b=0$$

So this system is underdamped.

$$\Delta=a^{2}-4b< 0$$

'''$$\left ( \lambda-\lambda_{1}\right )\left ( \lambda-\lambda_{2} \right )=0 $$'''

'''$$\left ( \lambda-(-0.5+i2) \right )\left ( \lambda-(-0.5-i2) \right )=0 $$'''

'''$$\left ( \lambda+0.5-i2\right )\left ( \lambda+0.5+i2 \right )=0 $$'''

'''$$\lambda^{2}+0.5\lambda+i2\lambda+0.5\lambda+0.25+0.5i2-i2\lambda-0.5i2-(i2)^{2}=0 $$'''

$$\lambda^{2}+\lambda+0.25+2=0$$

$$\lambda^{2}+\lambda+2.25=0$$

$$ {y_{h}}''+{y_{h}}'+2.25y{_h}=0 $$

Find 2.2
Find the damping ratio $$ \zeta $$

Solution 2.2
$$ 2w\zeta=a$$

$$ a=1$$

$$ w^{2}=b$$

$$ b=2.25$$

$$ w=1.5$$

$$ \zeta=\frac{1}{2w}$$

$$ \zeta=\frac{1}{3}$$

Given
The following are the initial conditions

$$ y(0)=1, y{}'(0)=0$$

Consider the case with no excitation

Find 2.3
Find and plot the solution satisfying the initial condition.

Solution 2.3
$$ y=c_{1}e^{(\alpha x)}cos(\beta x)+c_{2}e^{(\alpha x)}sin(\beta x)$$

From the conjugate roots, we know that:

$$ \alpha=-0.5, \beta= 2$$

$$ y=c_{1}e^{(-0.5 x)}cos(2 x)+c_{2}e^{(-0.5 x)}sin(2 x)$$

$$y{}'=-0.5c_{1}e^{(-0.5 x)}cos(2 x)-2c_{1}e^{(-0.5x)}sin(2x)-0.5c_{2}e^{(-0.5 x)}sin(2 x)+2c_{2}e^{(-0.5x)}cos(2x)$$

The initial conditions are plugged in the two above equations to solve the equations.

$$ y(0)=c_{1}e^{(0)}cos(0)+c_{2}e^{(0)}sin(0)$$

This gives:

$$c_{1}=1$$

$$y{}'(0)=-0.5c_{1}e^{(0)}cos(0)-2c_{1}e^{(0)}sin(0)-0.5c_{2}e^{(0)}sin(0)+2c_{2}e^{(0)}cos$$

This reduces to:

$$ -0.5+2c_{2}=0$$

$$ c_{2}=0.25 $$

Plugging in the coefficients, the solution is:

$$y=e^{-0.5x}cos(2x)+0.25e^{-0.5x}sin(2x)$$

Find 2.4
Using the plot of the solution, find the (approximate) amplitudes of oscillation in a cycle (i.e. two successive maximum values of $$y_{h}(t)$$ , denoted by  $$y_{h}^{(n)}$$  and  $$y_{h}^{(n+1)}$$  with  $$ n $$  being the cycle number.)

Find the log decrements for $$ n=1,2,3 $$  using:

$$\delta^{(n)}=log\frac{y_{h}^{(n)}}{y_{h}^{(n+1)}}$$

Solution 2.4
$$y_{h}^{(1)}=1$$

$$y_{h}^{(2)}=0.2$$

$$y_{h}^{(3)}=0.09$$

$$y_{h}^{(4)}=0.02$$

$$\delta^{(1)}=log\frac{1}{0.2}=0.699 $$

$$\delta^{(2)}=log\frac{0.2}{0.09}=0.347$$

$$\delta^{(3)}=log\frac{0.09}{0.02}=0.653$$

Find 2.5
Find the average log decrement:

$$ \delta\approx\frac13[\delta^((1))+\delta^((2))+\delta^((3))]$$

Solution 2.5
$$ \delta\approx\frac13[\delta^{(1)}+\delta^{(2)}+\delta^{(3)}]$$

$$ \delta\approx\frac13[0.699+0.347+0.653]$$

$$ \delta\approx 1.699 $$

Find 2.6
Verify the value of $$\delta$$  using:

$$\delta=\frac{2\pi \zeta}{\sqrt{1-\zeta^{2}}}$$

Solution 2.6
$$\delta=\frac{2\pi (1/3)}{\sqrt{1-(1/3)^{2}}}$$

$$\delta= 2.221$$

Find 2.7
Find the quality factor $$ Q $$  and the loss factor  $$ \eta $$  for the system.

Solution 2.7
$$Q=\frac{\sqrt{1-\zeta ^{2}}}{2\zeta}$$

$$Q=\frac{\sqrt{1-(1/3 ^{2}}}{2(1/3)}$$

$$Q=1.414$$

$$\eta=\frac{1}{Q}$$

$$\eta=\frac{1}{1.414}$$

$$\eta=0.707$$

Problem 3.5: 2 Finding Displacement and Forces in 2 Element Truss System
On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given
A truss has two elements, as shown in the figure. The members are made of aluminum hollow square cross-section. The outer dimension of the square is 12 mm, and the inner dimension is 9 mm. (The wall thickness is 1.5 mm in all four sides.) Assume Young's Modulus E= 70 GPa and yield strenght $$\sigma _{Y}$$  = 70 MPa. The magnitude of the force at Global Node 3 (F) is equal to 1,000 N.

Find
Use FEM to determine the displacement at Global Node 3 and axial forces in Elements 1 and 2. Use von Mises' yield theory to determine if the elements will yield or not. Use Euler buckling load to determine if the elements under compressive loads will buckle.

Redesign the truss so that both the stress and buckling constraints are satisfied with a Safety Factor of N not less than 2 for stresses and N not less than 1.2 for buckling. The design should reduce the weight as much as possible.

Draw the truss designed and provide the nodal coordinates and element connectivity in the form of a table. Include the nodal displacements, forces in each member, and safety factors for stresses and buckling for each element in the form of a table.

Solution
The Matlab Code from Problem 1 is used and the specific inputs for this problem were used.

The first thing that is inputed in the Matlab code is the total number of global nodes. Next, the position of the nodes is input by putting the value of the x, y, and z coordinates for each node. G=3; x=[0 0 2]; y=[0 1 0]; z=[0 0 0];

Next the number of elements in the system is input. Next we put which global node corresponds to each local node is input for l1 and l2. E=2; l1=[1 2]; l2=[3 3];

The known forces are then input. There is only one force so we input the components of the force at the degrees of freedom 7 and 8, which is at global node 3. F_in=[7 0; 8 -1000];

Next we input the known displacements. These displacements include the forces at the reactions which are the only known forces. Q_in=[1 0; 2 0; 3 0; 4 0; 5 0; 6 0; 9 0];

Lastly, we input the Young's modulus and the cross-sectional area. The members have hollow square cross-sections. Modulus=[7e10 7e10 7e10]; Area=[6.3e-5 6.3e-5 6.3e-5];

The following image represents the initial system with a blue dashed line and shows the displaced system with a solid green line.

The displacement at Global Node 3 is: Q_out = 7.0000  -0.0009    8.0000   -0.0043

The axial forces in Elements 1 and 2 are: F = 0        0   -0.0009    0.0011

The von Mises theory states that a material will yield when the von Mises stress exceeds the yield stress for the material. $$\sigma _{VM}\geqslant \sigma _{Y}$$

The von Mises stress in the Elements of the truss is found to be 15 MPa and the yield stress is given to be 70 MPa. This means that $$\sigma _{VM}< \sigma _{Y}$$  so the elements will NOT yield. The Euler buckling load is equal to $$P_{cr}=\pi ^{2}EI/L^{^{2}}$$ where the moment of inertia of the cross section is given by $$I=\frac{a_{0}^{4}-a_{i}^{4}}{12}$$

For element 1, the Euler buckling load is 0.8161 N. For element 2, the Euler buckling is 0.1632 N.

Element 1: $$P_{cr}=\frac{\pi ^{2}(70 GPa)\frac{(12 mm)^{4}-(9 mm)^{4}}{12}}{1 m^{2}}=0.8161 N$$

Element 2: $$P_{cr}=\frac{\pi ^{2}(70 GPa)\frac{(12 mm)^{4}-(9 mm)^{4}}{12}}{\sqrt{5 m}^{2}}=0.1632 N$$

For Elements 1 and 2: $$F< P_{cr}$$  so the elements will NOT buckle under compressive force.

After conducting a brief analysis of different truss designs, the best one was found to be the same design as provided in the problem. The different designs that were looked at were a rectangular truss system with vertical, horizontal, and diagonal elements, an isoceles trianglular truss system with the elements being connected to the wall at the same angle, and the square triangular truss system from the problem. The right triangular system was chosen because it has the best mass to stress handling ratio. The figure shows the truss system.

Once the design was chosen, different dimensions were tested. Here is a list of the tests and the results. The original weight of the truss was 74.72 kg. The new weight of the truss is: $$W= (6.3*10^{-3}m^{2})(1.25 m+1.458 m)(2800\frac{kg}{m^{3}})=47.77 kg$$

The nodal coordinates, element connectivity and nodal displacements are included in the following table.

The displacements and forces were found using the same Matlab program used in the previous section. The displacement matrix is: Q_out = 7.0000  -0.0000    8.0000   -0.0020 The force matrix is: F = 1.0e-003 * 0        0 -0.4724   0.6425 = Report 5 =

= Problem 6.2 = On my honor, I have neither given nor received unauthorized aid in doing this assignment.

Given: Plane Truss
Consider the plane truss in the figure above. The horizontal and vertical members have length L, while inclined members have length 1.414*L. Assume the Young's modulus E = 100 GPa, cross-sectional area A = 1.0 cm2, L = 0.3 m, and the density if 5000 kg/m3.

Find: Transform GEP into SEP.
Transform the generalized eigenvalue problem into a standard eigenvalue problem.

Solution
The mass matrix was computed in the same way as it was in Report 5. The following code was used to find it.

%Mass Matrix Construction m = zeros(3*G,3*G); for i=1:G for j=1:E if(l1(j)==i || l2(j)==i) m(3*i-2,3*i-2)=(Ro(j)*Area(j)*L(j))/2 + m(3*i-2,3*i-2); m(3*i-1,3*i-1)=(Ro(j)*Area(j)*L(j))/2 + m(3*i-1,3*i-1); m(3*i,3*i)=(Ro(j)*Area(j)*L(j))/2 + m(3*i,3*i); end end end

This is the resulting mass matrix

m = Columns 1 through 10 0.2561        0         0         0         0         0         0         0         0         0           0    0.2561         0         0         0         0         0         0         0         0           0         0    0.1500         0         0         0         0         0         0         0           0         0         0    0.1500         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 28 0        0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0           0         0         0    0.3311         0         0         0         0           0         0         0         0    0.1500         0         0         0           0         0         0         0         0    0.1500         0         0           0         0         0         0         0         0    0.2561         0           0         0         0         0         0         0         0    0.2561 The reduced mass matrix is as follows.

m_red = Columns 1 through 10 0.2561        0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      0.3311         0         0         0         0         0         0         0         0         0           0    0.3311         0         0         0         0         0         0         0         0           0         0    0.3311         0         0         0         0         0         0         0           0         0         0    0.3311         0         0         0         0         0         0           0         0         0         0    0.3311         0         0         0         0         0           0         0         0         0         0    0.3311         0         0         0         0           0         0         0         0         0         0    0.3311         0         0         0           0         0         0         0         0         0         0    0.3311         0         0           0         0         0         0         0         0         0         0    0.3311         0           0         0         0         0         0         0         0         0         0    0.3311           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0      0.3311         0         0         0         0           0    0.1500         0         0         0           0         0    0.1500         0         0           0         0         0    0.2561         0           0         0         0         0    0.2561

k_red was calculated using the same code as in Report 3. The resulting K matrix is as follows.

k_red = 1.0e+07 * Columns 1 through 10 4.5118        0         0   -1.1785   -1.1785         0         0         0         0         0           0    7.8452    1.1785         0         0   -3.3333         0   -1.1785   -1.1785         0           0    1.1785    4.5118         0   -3.3333         0         0   -1.1785   -1.1785         0     -1.1785         0         0    7.8452    1.1785         0         0   -3.3333         0         0     -1.1785         0   -3.3333    1.1785    4.5118         0         0         0         0         0           0   -3.3333         0         0         0    7.8452    1.1785         0         0   -3.3333           0         0         0         0         0    1.1785    4.5118         0   -3.3333         0           0   -1.1785   -1.1785   -3.3333         0         0         0    7.8452    1.1785         0           0   -1.1785   -1.1785         0         0         0   -3.3333    1.1785    4.5118         0           0         0         0         0         0   -3.3333         0         0         0    7.8452           0         0         0         0         0         0         0         0         0    1.1785           0         0         0         0         0   -1.1785   -1.1785   -3.3333         0         0           0         0         0         0         0   -1.1785   -1.1785         0         0         0           0         0         0         0         0         0         0         0         0   -3.3333           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0   -1.1785           0         0         0         0         0         0         0         0         0   -1.1785           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0   -1.1785   -1.1785         0         0         0         0         0         0         0           0   -1.1785   -1.1785         0         0         0         0         0         0         0           0   -3.3333         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      1.1785         0         0   -3.3333         0   -1.1785   -1.1785         0         0         0      4.5118         0   -3.3333         0         0   -1.1785   -1.1785         0         0         0           0    7.8452    1.1785         0         0   -3.3333         0         0         0         0     -3.3333    1.1785    4.5118         0         0         0         0         0         0         0           0         0         0    7.8452    1.1785         0         0   -3.3333         0   -1.1785           0         0         0    1.1785    4.5118         0   -3.3333         0         0   -1.1785     -1.1785   -3.3333         0         0         0    7.8452    1.1785         0         0   -3.3333     -1.1785         0         0         0   -3.3333    1.1785    4.5118         0         0         0           0         0         0   -3.3333         0         0         0    7.8452    1.1785         0           0         0         0         0         0         0         0    1.1785    4.5118         0           0         0         0   -1.1785   -1.1785   -3.3333         0         0         0    7.8452           0         0         0   -1.1785   -1.1785         0         0         0   -3.3333    1.1785           0         0         0         0         0         0         0   -3.3333         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0   -1.1785   -1.1785   -3.3333           0         0         0         0         0         0         0   -1.1785   -1.1785         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0     -1.1785         0         0         0         0     -1.1785         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0   -3.3333         0   -1.1785   -1.1785     -3.3333         0         0   -1.1785   -1.1785      1.1785         0         0   -3.3333         0      4.5118         0         0         0         0           0    3.3333         0         0         0           0         0    3.3333         0   -3.3333           0         0         0    4.5118    1.1785           0         0   -3.3333    1.1785    4.5118

Then we decompose the mass matrix:

$$M=M^{\frac{1}{2}}M^{\frac{1}{2}}$$

$$ M^{\frac{1}{2}}= $$

Columns 1 through 10 0.5061        0         0         0         0         0         0         0         0         0           0    0.5754         0         0         0         0         0         0         0         0           0         0    0.5754         0         0         0         0         0         0         0           0         0         0    0.5754         0         0         0         0         0         0           0         0         0         0    0.5754         0         0         0         0         0           0         0         0         0         0    0.5754         0         0         0         0           0         0         0         0         0         0    0.5754         0         0         0           0         0         0         0         0         0         0    0.5754         0         0           0         0         0         0         0         0         0         0    0.5754         0           0         0         0         0         0         0         0         0         0    0.5754            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      0.5754         0         0         0         0         0         0         0         0         0           0    0.5754         0         0         0         0         0         0         0         0           0         0    0.5754         0         0         0         0         0         0         0           0         0         0    0.5754         0         0         0         0         0         0           0         0         0         0    0.5754         0         0         0         0         0           0         0         0         0         0    0.5754         0         0         0         0           0         0         0         0         0         0    0.5754         0         0         0           0         0         0         0         0         0         0    0.5754         0         0           0         0         0         0         0         0         0         0    0.5754         0           0         0         0         0         0         0         0         0         0    0.5754            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0      0.5754         0         0         0         0           0    0.3873         0         0         0           0         0    0.3873         0         0           0         0         0    0.5061         0           0         0         0         0    0.5061

$$ M^{\frac{-1}{2}}= $$

Columns 1 through 10 1.9760        0         0         0         0         0         0         0         0         0           0    1.7379         0         0         0         0         0         0         0         0           0         0    1.7379         0         0         0         0         0         0         0           0         0         0    1.7379         0         0         0         0         0         0           0         0         0         0    1.7379         0         0         0         0         0           0         0         0         0         0    1.7379         0         0         0         0           0         0         0         0         0         0    1.7379         0         0         0           0         0         0         0         0         0         0    1.7379         0         0           0         0         0         0         0         0         0         0    1.7379         0           0         0         0         0         0         0         0         0         0    1.7379            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 11 through 20 0        0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0      1.7379         0         0         0         0         0         0         0         0         0           0    1.7379         0         0         0         0         0         0         0         0           0         0    1.7379         0         0         0         0         0         0         0           0         0         0    1.7379         0         0         0         0         0         0           0         0         0         0    1.7379         0         0         0         0         0           0         0         0         0         0    1.7379         0         0         0         0           0         0         0         0         0         0    1.7379         0         0         0           0         0         0         0         0         0         0    1.7379         0         0           0         0         0         0         0         0         0         0    1.7379         0           0         0         0         0         0         0         0         0         0    1.7379            0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0           0         0         0         0         0         0         0         0         0         0    Columns 21 through 25 0        0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0           0         0         0         0         0      1.7379         0         0         0         0           0    2.5819         0         0         0           0         0    2.5819         0         0           0         0         0    1.9760         0           0         0         0         0    1.9760 The following steps are taken to transform the generalized eigenvalue problem,  $$ Kx=\lambda Mx $$ into the standard eigenvalue problem, $$ K^{*}x^{*}=\lambda x^{*} $$  :

$$ M^{\frac{-1}{2}}Kx=\lambda M^{\frac{-1}{2}} Mx $$

$$ M^{\frac{-1}{2}}Kx=\lambda M^{\frac{-1}{2}} Mx=\lambda M^{\frac{1}{2}} $$

$$ x^{*}:=M^{\frac{1}{2}}x\Leftrightarrow x=M^{\frac{-1}{2}}x^{*} $$

$$ \left (M ^{\frac{-1}{2}}K M ^{\frac{1}{2}} \right )x^{*}=Kx^{*} $$ $$ K^{*}x^{*}=\lambda x^{*} $$

The matrix $$\mathbf K^\star$$ is as follows: kstar= 1.0e+008 * Columns 1 through 7 1.7618        0         0   -0.4047   -0.4047         0         0         0    2.3694    0.3559         0         0   -1.0067         0         0    0.3559    1.3627         0   -1.0067         0         0   -0.4047         0         0    2.3694    0.3559         0         0   -0.4047         0   -1.0067    0.3559    1.3627         0         0         0   -1.0067         0         0         0    2.3694    0.3559         0         0         0         0         0    0.3559    1.3627         0   -0.3559   -0.3559   -1.0067         0         0         0         0   -0.3559   -0.3559         0         0         0   -1.0067         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 14 0        0         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -1.0067         0   -0.3559   -0.3559         0         0   -1.0067         0         0   -0.3559   -0.3559         0    2.3694    0.3559         0         0   -1.0067         0         0    0.3559    1.3627         0         0         0         0         0         0         0    2.3694    0.3559         0         0   -1.0067         0         0    0.3559    1.3627         0   -1.0067         0   -1.0067         0         0         0    2.3694    0.3559         0         0         0         0   -1.0067    0.3559    1.3627         0         0         0   -1.0067         0         0         0    2.3694         0         0         0         0         0         0    0.3559         0         0   -0.3559   -0.3559   -1.0067         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.3559         0         0         0         0         0         0   -0.3559         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 15 through 21 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -0.3559   -0.3559         0         0         0         0         0   -1.0067         0         0         0         0         0         0         0         0         0         0         0         0    0.3559         0         0   -1.0067         0   -0.3559   -0.3559    1.3627         0   -1.0067         0         0   -0.3559   -0.3559         0    2.3694    0.3559         0         0   -1.0067         0   -1.0067    0.3559    1.3627         0         0         0         0         0         0         0    2.3694    0.3559         0         0         0         0         0    0.3559    1.3627         0   -1.0067   -0.3559   -1.0067         0         0         0    2.3694    0.3559   -0.3559         0         0         0   -1.0067    0.3559    1.3627         0         0         0   -1.4957         0         0         0         0         0         0         0         0         0         0         0         0         0   -0.4047   -0.4047   -1.1447         0         0         0         0   -0.4047   -0.4047         0         0  Columns 22 through 25 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0   -1.4957         0   -0.4047   -0.4047         0         0   -0.4047   -0.4047         0         0   -1.1447         0         0         0         0         0    2.2222         0         0         0         0    2.2222         0   -1.7007         0         0    1.7618    0.4602         0   -1.7007    0.4602    1.7618 The eig command in MATLAB is used to calculate the eigenvectors for $$\mathbf K^\star$$.The eigenvectors are represented by the matrix V, and D is the vector of eigenvalues.

EDU>> [V,D]=eig(kstar)

V = Columns 1 through 7 -0.0110  -0.0315   -0.0474    0.0455   -0.0792   -0.0889   -0.1931    0.1557    0.2277   -0.1435   -0.3004   -0.1617    0.2041   -0.4005    0.0395    0.0784    0.0091   -0.1050    0.0065    0.1288   -0.2570    0.0821    0.1971    0.2085   -0.2278    0.3017    0.2841    0.3472   -0.0018    0.0014    0.0341    0.0026    0.0593    0.0043    0.2715   -0.2742   -0.2610    0.3190    0.2376    0.3316    0.0538    0.1634   -0.0563   -0.0673    0.0268    0.0048    0.0601    0.2775    0.0064   -0.1877   -0.3668   -0.2722    0.2983   -0.3174   -0.1619   -0.0642   -0.0237   -0.0583   -0.0308    0.1022   -0.0527   -0.2660    0.1207    0.3496    0.0886   -0.3889    0.0026   -0.2603    0.0434    0.3646    0.0647    0.0170   -0.0334    0.0985   -0.0458   -0.0028    0.2341    0.2782    0.3812    0.2241   -0.0062    0.1242   -0.3824   -0.0281    0.0452    0.0799   -0.0039   -0.0781   -0.0219   -0.1467   -0.1805   -0.3809    0.1944    0.2865   -0.0505   -0.0603   -0.2520   -0.2453   -0.0628    0.0417    0.0091   -0.0879   -0.0446   -0.2807    0.1110   -0.3281   -0.2238   -0.1774   -0.4031    0.0188    0.2610   -0.0404   -0.0599   -0.0540    0.0316   -0.0379    0.0716    0.2108   -0.2000    0.3820   -0.4494   -0.0719   -0.1934    0.3121   -0.1435   -0.0836    0.0661   -0.1100    0.1125   -0.0884   -0.0389   -0.1323   -0.0628    0.3192   -0.0062    0.1944    0.5157   -0.0137    0.2596   -0.2266    0.0611    0.0083   -0.0604    0.1340    0.0319    0.2425    0.0179   -0.2290    0.3219    0.0668    0.1875   -0.3376    0.2516    0.1494    0.1054   -0.2025    0.4186   -0.2325   -0.4546    0.0415    0.1507   -0.2091    0.1365   -0.2034   -0.1894    0.0408   -0.1486    0.2193   -0.1545    0.2486   -0.3967    0.2109    0.3697   -0.0208   -0.0742  Columns 8 through 14 -0.1082   0.2590    0.1593    0.0995    0.0920   -0.5045    0.4512   -0.1368   -0.1865   -0.2806   -0.0426    0.0795   -0.1556   -0.4220   -0.2536    0.2527    0.1979    0.2892    0.0125    0.3888   -0.1230    0.1028   -0.2806    0.0152    0.2436    0.0029    0.2378    0.1403    0.2126   -0.3367   -0.2557   -0.3286   -0.0702   -0.2821   -0.0295   -0.1487    0.2270    0.1129   -0.0085   -0.0606   -0.0487   -0.2568   -0.3531    0.0392   -0.0611   -0.3971   -0.2713    0.0113    0.2019    0.0602   -0.1237   -0.1545   -0.0928   -0.0609    0.2550   -0.0933    0.3267   -0.0786    0.0356    0.3746    0.3146   -0.0098   -0.1418   -0.0781    0.0143    0.1930   -0.0162   -0.0455    0.1400    0.3221   -0.3730   -0.0322   -0.1677    0.0755    0.4672   -0.0899   -0.0142    0.1165    0.2732    0.0272   -0.2226    0.0569   -0.1608   -0.0633    0.3772    0.0256    0.1585   -0.0146   -0.4776    0.1282    0.0376   -0.1190   -0.2665   -0.2509    0.0779    0.0728    0.1526    0.3975   -0.2858   -0.1814    0.0304    0.2665   -0.3923   -0.1843   -0.1236    0.1844   -0.0581    0.1921    0.1731    0.0270   -0.2992    0.0815    0.3254    0.1235    0.0285   -0.3059    0.3661    0.1684    0.1462   -0.0360   -0.0120   -0.0337   -0.0057   -0.0012   -0.0598   -0.1177   -0.1124   -0.2951    0.4610   -0.2556    0.1775    0.0683   -0.0818    0.0533   -0.2026   -0.0266    0.1385   -0.0682    0.0114    0.0633    0.1749    0.2819   -0.3911    0.2471   -0.1284   -0.1470    0.0246    0.0748    0.0357    0.3347   -0.0738   -0.0106   -0.1803   -0.3144    0.0467    0.1988   -0.1670    0.0211   -0.0367    0.0616    0.0180   -0.0085    0.3412   -0.2219   -0.1508    0.0171    0.2337   -0.1101   -0.0198   -0.0589    0.0148    0.0014   -0.0036    0.0180    0.0059  Columns 15 through 21 -0.5574   0.1148    0.0041    0.0377    0.0178   -0.0802   -0.1034   -0.2613   -0.1480    0.0161   -0.0131    0.1253    0.0463    0.0756    0.1005    0.0286    0.0436    0.2403    0.0239   -0.4054   -0.3811   -0.3591    0.3670   -0.0209   -0.0421    0.0693    0.0377   -0.0098   -0.0288   -0.1465    0.0389    0.2037    0.0051   -0.3640   -0.3777   -0.1200   -0.2985    0.0276   -0.0560    0.2248    0.0823    0.0434    0.0332    0.1448    0.1141    0.3747    0.0589   -0.2894    0.1347   -0.1033    0.4069   -0.0371   -0.0411    0.1528   -0.0121   -0.1127   -0.1098   -0.1100    0.1097    0.3538    0.0449   -0.3326    0.0600    0.1036   -0.2702    0.0348   -0.1084    0.2987    0.0324   -0.0414    0.1442    0.1126    0.2060    0.3598    0.0738    0.1333    0.3411    0.2859    0.1959   -0.0487   -0.0088    0.2429   -0.0554   -0.0987   -0.1063   -0.0294    0.2022    0.3583    0.0688    0.0643    0.3955    0.1364   -0.0095    0.0385   -0.1498    0.3518   -0.0749    0.0214    0.0349    0.0324    0.3098    0.1805    0.0432    0.3491   -0.1906    0.3490   -0.0934   -0.0560    0.0347    0.3259   -0.0159   -0.0187    0.0075    0.0259    0.3068    0.1938    0.0465    0.3336   -0.0703   -0.0074    0.2857    0.0397   -0.1692    0.3867   -0.1490    0.1529   -0.1021   -0.0206    0.4168   -0.1050   -0.0146    0.0868   -0.3058   -0.0625   -0.2595   -0.0597    0.0691    0.3861    0.0790   -0.0716    0.1662    0.2111    0.4150   -0.0894   -0.0089    0.1258   -0.3066   -0.0149    0.3453    0.0267   -0.1153    0.2688   -0.1058    0.1157   -0.0761   -0.1647    0.3505   -0.2686   -0.0376   -0.2421    0.1823   -0.3571   -0.1733   -0.0536    0.0736    0.3633    0.1235   -0.1592   -0.0332   -0.1199    0.4577   -0.3468   -0.0476   -0.2999    0.2119  Columns 22 through 25 -0.0882   0.0483    0.0971   -0.0110    0.0980    0.3005    0.0433   -0.1621   -0.2775    0.0483    0.1312    0.0617    0.0451    0.0476    0.1733   -0.1872   -0.3426    0.0782    0.1281    0.1545   -0.0169    0.4537    0.1277   -0.1073    0.3627   -0.0396   -0.2270   -0.2080    0.0028    0.0868    0.3271   -0.2778    0.3107    0.0815   -0.3302   -0.1818   -0.0344    0.3814   -0.0693   -0.0822   -0.1177   -0.0489    0.3473    0.2131    0.1336    0.0013    0.2536   -0.3414   -0.0444    0.1491    0.2615    0.2781   -0.0114    0.2047   -0.1428    0.1717   -0.2388   -0.1493   -0.2096   -0.2993    0.1211   -0.1785    0.2699   -0.1266   -0.3494    0.0481   -0.2430   -0.2606   -0.1885   -0.0232   -0.2832    0.2031    0.3830   -0.0360    0.1279    0.2418    0.0340   -0.3964    0.0618   -0.0248    0.2943    0.1901   -0.0218    0.2570   -0.1544   -0.0229   -0.2468    0.1826   -0.0974   -0.1963    0.1089   -0.1756    0.1215   -0.3768   -0.0340    0.1917   -0.1046   -0.1748    0.1099   -0.1718

D = 1.0e+008 * Columns 1 through 7 4.7170        0         0         0         0         0         0         0    4.3103         0         0         0         0         0         0         0    3.8339         0         0         0         0         0         0         0    3.7647         0         0         0         0         0         0         0    3.6051         0         0         0         0         0         0         0    3.0752         0         0         0         0         0         0         0    3.0589         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 14 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    2.9417         0         0         0         0         0         0         0    2.7262         0         0         0         0         0         0         0    2.3728         0         0         0         0         0         0         0    2.1074         0         0         0         0         0         0         0    2.0577         0         0         0         0         0         0         0    1.7262         0         0         0         0         0         0         0    1.6623         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 15 through 21 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.4801         0         0         0         0         0         0         0    0.9847         0         0         0         0         0         0         0    0.0017         0         0         0         0         0         0         0    0.0267         0         0         0         0         0         0         0    0.0702         0         0         0         0         0         0         0    0.1154         0         0         0         0         0         0         0    0.2452         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 22 through 25 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    0.3962         0         0         0         0    0.7073         0         0         0         0    0.5060         0         0         0         0    0.5581 Next, the answers to the SEP need to be checked for accuracy.

The eigenvectors are recovered. EDU>> X=mm^(-1/2)*V

X = Columns 1 through 7 -0.0217  -0.0623   -0.0936    0.0899   -0.1566   -0.1757   -0.3815    0.2705    0.3957   -0.2493   -0.5221   -0.2809    0.3547   -0.6960    0.0686    0.1362    0.0158   -0.1824    0.0113    0.2238   -0.4467    0.1427    0.3425    0.3624   -0.3959    0.5242    0.4938    0.6035   -0.0032    0.0024    0.0592    0.0045    0.1030    0.0075    0.4718   -0.4766   -0.4536    0.5544    0.4130    0.5762    0.0934    0.2840   -0.0978   -0.1170    0.0465    0.0084    0.1044    0.4823    0.0112   -0.3262   -0.6375   -0.4731    0.5185   -0.5517   -0.2814   -0.1116   -0.0412   -0.1013   -0.0535    0.1777   -0.0916   -0.4622    0.2098    0.6076    0.1540   -0.6759    0.0046   -0.4523    0.0754    0.6337    0.1125    0.0295   -0.0581    0.1712   -0.0797   -0.0049    0.4068    0.4834    0.6625    0.3894   -0.0107    0.2158   -0.6645   -0.0489    0.0785    0.1388   -0.0068   -0.1358   -0.0380   -0.2549   -0.3137   -0.6620    0.3378    0.4979   -0.0878   -0.1048   -0.4379   -0.4264   -0.1091    0.0724    0.0158   -0.1527   -0.0775   -0.4877    0.1929   -0.5702   -0.3890   -0.3082   -0.7006    0.0327    0.4535   -0.0702   -0.1042   -0.0939    0.0549   -0.0658    0.1244    0.3663   -0.3476    0.6639   -0.7810   -0.1250   -0.3361    0.5424   -0.2494   -0.1453    0.1148   -0.1911    0.1956   -0.1537   -0.0676   -0.2300   -0.1092    0.5547   -0.0107    0.3379    0.8963   -0.0239    0.4512   -0.3938    0.1062    0.0145   -0.1050    0.2329    0.0555    0.4214    0.0311   -0.5913    0.8312    0.1724    0.4842   -0.8716    0.6497    0.3858    0.2720   -0.5228    1.0808   -0.6003   -1.1739    0.1072    0.3892   -0.4131    0.2697   -0.4020   -0.3742    0.0807   -0.2936    0.4334   -0.3054    0.4912   -0.7838    0.4167    0.7305   -0.0412   -0.1465  Columns 8 through 14 -0.2138   0.5118    0.3148    0.1967    0.1818   -0.9970    0.8917   -0.2377   -0.3242   -0.4876   -0.0740    0.1382   -0.2705   -0.7333   -0.4407    0.4392    0.3439    0.5027    0.0218    0.6756   -0.2138    0.1787   -0.4876    0.0264    0.4234    0.0051    0.4133    0.2439    0.3694   -0.5852   -0.4444   -0.5711   -0.1221   -0.4903   -0.0512   -0.2584    0.3945    0.1962   -0.0147   -0.1054   -0.0846   -0.4462   -0.6136    0.0682   -0.1062   -0.6901   -0.4715    0.0196    0.3509    0.1046   -0.2151   -0.2685   -0.1613   -0.1059    0.4432   -0.1621    0.5677   -0.1365    0.0619    0.6510    0.5468   -0.0170   -0.2464   -0.1357    0.0249    0.3353   -0.0281   -0.0790    0.2433    0.5598   -0.6482   -0.0560   -0.2915    0.1312    0.8120   -0.1562   -0.0246    0.2024    0.4748    0.0472   -0.3868    0.0988   -0.2794   -0.1100    0.6555    0.0445    0.2754   -0.0254   -0.8301    0.2228    0.0653   -0.2069   -0.4632   -0.4360    0.1354    0.1265    0.2652    0.6907   -0.4966   -0.3153    0.0528    0.4631   -0.6818   -0.3203   -0.2148    0.3204   -0.1011    0.3339    0.3008    0.0470   -0.5199    0.1417    0.5656    0.2146    0.0496   -0.5316    0.6362    0.2927    0.2541   -0.0625   -0.0209   -0.0585   -0.0099   -0.0020   -0.1039   -0.2046   -0.1953   -0.5128    0.8011   -0.4442    0.3085    0.1188   -0.1421    0.0926   -0.3521   -0.0463    0.2406   -0.1185    0.0199    0.1101    0.3040    0.4899   -0.6797    0.4294   -0.2232   -0.2554    0.0427    0.1931    0.0921    0.8641   -0.1907   -0.0275   -0.4654   -0.8119    0.1206    0.5134   -0.4313    0.0544   -0.0948    0.1590    0.0465   -0.0168    0.6743   -0.4386   -0.2979    0.0338    0.4618   -0.2175   -0.0391   -0.1164    0.0292    0.0028   -0.0070    0.0355    0.0117  Columns 15 through 21 -1.1014   0.2269    0.0082    0.0745    0.0352   -0.1585   -0.2043   -0.4542   -0.2573    0.0279   -0.0227    0.2178    0.0805    0.1314    0.1746    0.0497    0.0757    0.4176    0.0416   -0.7045   -0.6624   -0.6241    0.6379   -0.0364   -0.0732    0.1204    0.0655   -0.0171   -0.0500   -0.2547    0.0677    0.3540    0.0089   -0.6326   -0.6564   -0.2085   -0.5187    0.0479   -0.0973    0.3907    0.1430    0.0754    0.0576    0.2516    0.1982    0.6511    0.1023   -0.5029    0.2342   -0.1796    0.7071   -0.0645   -0.0714    0.2656   -0.0210   -0.1958   -0.1908   -0.1912    0.1906    0.6149    0.0780   -0.5780    0.1042    0.1800   -0.4695    0.0605   -0.1884    0.5191    0.0564   -0.0719    0.2506    0.1956    0.3581    0.6252    0.1283    0.2317    0.5929    0.4969    0.3405   -0.0847   -0.0153    0.4221   -0.0962   -0.1715   -0.1847   -0.0511    0.3514    0.6227    0.1195    0.1118    0.6874    0.2370   -0.0165    0.0670   -0.2603    0.6115   -0.1301    0.0372    0.0607    0.0564    0.5384    0.3137    0.0751    0.6067   -0.3313    0.6066   -0.1624   -0.0974    0.0603    0.5663   -0.0277   -0.0325    0.0130    0.0451    0.5332    0.3367    0.0809    0.5798   -0.1222   -0.0128    0.4965    0.0690   -0.2941    0.6721   -0.2590    0.2657   -0.1774   -0.0358    0.7243   -0.1825   -0.0253    0.1508   -0.5315   -0.1087   -0.4511   -0.1038    0.1201    0.6709    0.1374   -0.1245    0.2889    0.3669    0.7213   -0.1553   -0.0154    0.2186   -0.5329   -0.0384    0.8916    0.0690   -0.2976    0.6941   -0.2732    0.2986   -0.1965   -0.4254    0.9051   -0.6936   -0.0971   -0.6251    0.4707   -0.7057   -0.3424   -0.1058    0.1455    0.7178    0.2440   -0.3145   -0.0656   -0.2369    0.9044   -0.6853   -0.0940   -0.5926    0.4187  Columns 22 through 25 -0.1742   0.0954    0.1920   -0.0217    0.1703    0.5222    0.0753   -0.2817   -0.4822    0.0840    0.2280    0.1072    0.0784    0.0828    0.3012   -0.3254   -0.5953    0.1359    0.2226    0.2685   -0.0293    0.7884    0.2220   -0.1865    0.6304   -0.0688   -0.3946   -0.3615    0.0048    0.1509    0.5684   -0.4828    0.5400    0.1416   -0.5738   -0.3160   -0.0597    0.6628   -0.1205   -0.1428   -0.2045   -0.0850    0.6036    0.3703    0.2322    0.0022    0.4407   -0.5933   -0.0772    0.2591    0.4545    0.4833   -0.0198    0.3558   -0.2481    0.2984   -0.4151   -0.2595   -0.3643   -0.5201    0.2105   -0.3102    0.4690   -0.2200   -0.6072    0.0837   -0.4223   -0.4529   -0.3275   -0.0403   -0.4921    0.3530    0.6657   -0.0626    0.2223    0.4202    0.0592   -0.6889    0.1074   -0.0430    0.5115    0.3304   -0.0379    0.4467   -0.3986   -0.0591   -0.6372    0.4714   -0.2516   -0.5067    0.2811   -0.4534    0.2400   -0.7445   -0.0672    0.3789   -0.2067   -0.3454    0.2171   -0.3395 Then, the GEP is solved and the resulting eigenvectors are found. If the two sets match up, then the results from the SEP are correct.

EDU>> [VV,DD]=eig(k_red,mm) VV = Columns 1 through 7 -0.0082  -0.0745   -0.0352   -0.1585   -0.2043    0.1742    0.1920   -0.0279    0.0227   -0.2178    0.0805    0.1314   -0.1703    0.0753   -0.0757   -0.4176   -0.0416   -0.7045   -0.6624    0.4822    0.2280    0.0364    0.0732   -0.1204    0.0655   -0.0171   -0.0784    0.3012   -0.0677   -0.3540   -0.0089   -0.6326   -0.6564    0.5953    0.2226   -0.0479    0.0973   -0.3907    0.1430    0.0754    0.0293    0.2220   -0.1982   -0.6511   -0.1023   -0.5029    0.2342   -0.6304   -0.3946    0.0645    0.0714   -0.2656   -0.0210   -0.1958   -0.0048    0.5684   -0.1906   -0.6149   -0.0780   -0.5780    0.1042   -0.5400   -0.5738   -0.0605    0.1884   -0.5191    0.0564   -0.0719    0.0597   -0.1205   -0.3581   -0.6252   -0.1283    0.2317    0.5929    0.2045    0.6036    0.0847    0.0153   -0.4221   -0.0962   -0.1715   -0.2322    0.4407   -0.3514   -0.6227   -0.1195    0.1118    0.6874    0.0772    0.4545   -0.0670    0.2603   -0.6115   -0.1301    0.0372    0.0198   -0.2481   -0.5384   -0.3137   -0.0751    0.6067   -0.3313    0.4151   -0.3643    0.0974   -0.0603   -0.5663   -0.0277   -0.0325   -0.2105    0.4690   -0.5332   -0.3367   -0.0809    0.5798   -0.1222    0.6072   -0.4223   -0.0690    0.2941   -0.6721   -0.2590    0.2657    0.3275   -0.4921   -0.7243    0.1825    0.0253    0.1508   -0.5315   -0.6657    0.2223    0.1038   -0.1201   -0.6709    0.1374   -0.1245   -0.0592    0.1074   -0.7213    0.1553    0.0154    0.2186   -0.5329   -0.5115   -0.0379   -0.0690    0.2976   -0.6941   -0.2732    0.2986    0.3986   -0.6372   -0.9051    0.6936    0.0971   -0.6251    0.4707    0.2516    0.2811    0.1058   -0.1455   -0.7178    0.2440   -0.3145   -0.2400   -0.0672   -0.9044    0.6853    0.0940   -0.5926    0.4187    0.2067    0.2171  Columns 8 through 14 0.0217  -0.0954   -0.2269    1.1014   -0.8917   -0.9970   -0.1818    0.2817   -0.5222    0.2573    0.4542    0.7333   -0.2705   -0.1382   -0.1072   -0.0840   -0.0497   -0.1746    0.2138    0.6756   -0.0218    0.3254   -0.0828   -0.6379    0.6241   -0.2439    0.4133   -0.0051   -0.2685   -0.1359    0.2547    0.0500    0.0512   -0.4903    0.1221    0.1865   -0.7884    0.5187    0.2085    0.4462   -0.0846    0.1054    0.3615    0.0688   -0.2516   -0.0576   -0.3509    0.0196    0.4715    0.4828   -0.1509   -0.7071    0.1796    0.1621    0.4432    0.1059    0.3160   -0.1416    0.1912    0.1908    0.2464   -0.0170   -0.5468    0.1428   -0.6628    0.4695   -0.1800   -0.5598    0.2433    0.0790   -0.3703    0.0850   -0.1956   -0.2506    0.0246   -0.1562   -0.8120    0.5933   -0.0022   -0.3405   -0.4969    0.1100   -0.2794   -0.0988   -0.4833   -0.2591    0.0511    0.1847   -0.0653    0.2228    0.8301   -0.2984   -0.3558    0.0165   -0.2370   -0.6907    0.2652   -0.1265    0.5201    0.2595   -0.0564   -0.0607    0.2148   -0.3203    0.6818    0.2200    0.3102    0.1624   -0.6066   -0.1417   -0.5199   -0.0470    0.4529   -0.0837   -0.0451   -0.0130   -0.2541    0.2927   -0.6362   -0.3530    0.0403   -0.4965    0.0128    0.2046   -0.1039    0.0020   -0.4202    0.0626    0.0358    0.1774    0.1421    0.1188   -0.3085    0.0430    0.6889    0.4511    0.1087   -0.1101    0.0199    0.1185   -0.4467   -0.3304   -0.3669   -0.2889   -0.0427   -0.2554    0.2232   -0.4714    0.0591   -0.8916    0.0384    0.8119   -0.4654    0.0275    0.4534    0.5067    0.4254    0.1965   -0.0465    0.1590    0.0948   -0.3789    0.7445    0.3424    0.7057    0.2175    0.4618   -0.0338    0.3395    0.3454    0.2369    0.0656   -0.0117    0.0355    0.0070  Columns 15 through 21 0.1967   0.3148   -0.5118    0.2138    0.3815    0.1757   -0.1566   -0.0740   -0.4876    0.3242    0.2377    0.6960   -0.3547   -0.2809    0.5027    0.3439   -0.4392    0.4407    0.4467   -0.2238    0.0113    0.4234    0.0264    0.4876   -0.1787   -0.6035   -0.4938    0.5242   -0.5711   -0.4444    0.5852   -0.3694   -0.4718   -0.0075    0.1030   -0.0147    0.1962   -0.3945    0.2584   -0.2840   -0.0934    0.5762   -0.6901   -0.1062   -0.0682    0.6136   -0.0112   -0.4823    0.1044   -0.1613   -0.2685    0.2151   -0.1046    0.1116    0.2814   -0.5517    0.6510    0.0619    0.1365   -0.5677   -0.2098    0.4622   -0.0916   -0.0281    0.3353   -0.0249    0.1357   -0.6337   -0.0754   -0.4523    0.1312   -0.2915    0.0560    0.6482   -0.4068    0.0049   -0.0797   -0.3868    0.0472   -0.4748   -0.2024    0.0489    0.6645    0.2158   -0.0254    0.2754   -0.0445   -0.6555    0.3137    0.2549   -0.0380    0.1354   -0.4360    0.4632    0.2069    0.4264    0.4379   -0.1048    0.4631    0.0528    0.3153    0.4966   -0.1929    0.4877   -0.0775    0.3008    0.3339    0.1011   -0.3204    0.0702   -0.4535    0.0327   -0.5316    0.0496   -0.2146   -0.5656    0.3476   -0.3663    0.1244   -0.0099   -0.0585    0.0209    0.0625    0.1453    0.2494    0.5424   -0.4442    0.8011    0.5128    0.1953    0.1092    0.2300   -0.0676    0.2406   -0.0463    0.3521   -0.0926    0.3938   -0.4512   -0.0239    0.4294   -0.6797   -0.4899   -0.3040   -0.0311   -0.4214    0.0555   -0.1907    0.8641   -0.0921   -0.1931   -0.3858   -0.6497   -0.8716    0.0544   -0.4313   -0.5134   -0.1206   -0.3892   -0.1072   -1.1739   -0.2979   -0.4386   -0.6743    0.0168   -0.4334    0.2936    0.0807    0.0028    0.0292    0.1164    0.0391    0.1465    0.0412    0.7305  Columns 22 through 25 0.0899   0.0936    0.0623    0.0217   -0.5221    0.2493   -0.3957   -0.2705   -0.1824   -0.0158   -0.1362   -0.0686   -0.3959   -0.3624   -0.3425   -0.1427    0.0045   -0.0592   -0.0024    0.0032    0.4130   -0.5544    0.4536    0.4766    0.0084   -0.0465    0.1170    0.0978    0.5185    0.4731    0.6375    0.3262    0.1777    0.0535    0.1013    0.0412    0.0046    0.6759   -0.1540   -0.6076    0.1712    0.0581   -0.0295   -0.1125   -0.0107   -0.3894   -0.6625   -0.4834   -0.1358    0.0068   -0.1388   -0.0785   -0.0878   -0.4979   -0.3378    0.6620   -0.1527   -0.0158   -0.0724    0.1091   -0.7006    0.3082    0.3890    0.5702   -0.0658   -0.0549    0.0939    0.1042   -0.3361    0.1250    0.7810   -0.6639   -0.1537   -0.1956    0.1911   -0.1148    0.8963   -0.3379    0.0107   -0.5547    0.2329    0.1050   -0.0145   -0.1062    0.4842   -0.1724   -0.8312    0.5913   -0.6003   -1.0808    0.5228   -0.2720   -0.3742    0.4020   -0.2697    0.4131    0.4167    0.7838   -0.4912    0.3054

DD = 1.0e+008 * Columns 1 through 7 0.0017        0         0         0         0         0         0         0    0.0267         0         0         0         0         0         0         0    0.0702         0         0         0         0         0         0         0    0.1154         0         0         0         0         0         0         0    0.2452         0         0         0         0         0         0         0    0.3962         0         0         0         0         0         0         0    0.5060         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 8 through 14 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    0.5581         0         0         0         0         0         0         0    0.7073         0         0         0         0         0         0         0    0.9847         0         0         0         0         0         0         0    1.4801         0         0         0         0         0         0         0    1.6623         0         0         0         0         0         0         0    1.7262         0         0         0         0         0         0         0    2.0577         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 15 through 21 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    2.1074         0         0         0         0         0         0         0    2.3728         0         0         0         0         0         0         0    2.7262         0         0         0         0         0         0         0    2.9417         0         0         0         0         0         0         0    3.0589         0         0         0         0         0         0         0    3.0752         0         0         0         0         0         0         0    3.6051         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0  Columns 22 through 25 0        0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    3.7647         0         0         0         0    3.8339         0         0         0         0    4.3103         0         0         0         0    4.7170