Assignment On Linear Algebra: Es1101: Computational Data Analysis

Assignment
on
Linear Algebra
ES1101: COMPUTATIONAL DATA
ANALYSIS
SUBMITTED BY: Vinay Yograj Daharwal
Your name (Roll No.)
2021B.Tech062
Institute of Engineering and Technology (IET)

JK Lakshmipat University, Jaipur
December 2021
1
1. Introduction :
➢ In this assignment we are using concept of linear Algebra that include formation of matrixes ,
Eigen value and Eigen vector , finding root of characteristics equation , power method , inverse
power method, application eigen vector, And use of python to solve complex question
2. OBJECTIVE
➢ To study data and interpret the result of data, To use of python to solve real life problem and
To find Rank of teams in real life game
3. METHODOLOGY
➢ In power method we have start with an approximation , mostly we start with column matrices
of 1 and we multiple it with original and this lead to be a new matrices and we take largest
element of new matrices common that element is the approximate eigen value and left vector
is approximate eigen vector it is iteration 1 and then we multiple left vector with original
matrixes now take common and we get more approximate eigen value and vector this process
continuous till you get very approximation , the more iteration lead to most approximate eigen
value and eigen vector
4. MATHEMATICAL MODEL
➢ In this assignment we make a model to solve real life complex question using python
With power method.
5. DATA COLLECTION
➢ We take the data from world chess championship 2018
Source Wikipedia
Link - World Chess Championship 2018 - Wikipedia
6. RESULT AND DISSCUSSION

➢ We got rank of matrices that are discussed in assignment, we understand how eigen vector is
power full tool to find rank of team
7. CONCLUSION
➢ With doing this assignment we understand the use and importance of linear algebra, we also
understand how to work in team and how to collect real life data.
1. Solve the following problem using manual calculation. Suppose that there are four teams in
a league match. At the end of season, the results are as follows
Team 1 beat teams 2 and 3, but lost to team 4.
Team 2 beat team 3, but lost to teams 1 and 4.
Team 3 beat team 4, but lost to teams 1 and 2.
Team 4 beat teams 1 and 2, but lost to team 3.
(a) Form the corresponding matrix A that reflects these results, where 𝑎𝑖𝑗 = { 1 if team i
beats team j 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
(b) How small can the dominant eigenvalue for A be? How large? Explain.
(d) Find out eigen vector corresponding to most dominant eigen value using Power method
and find how the teams can be ranked using eigen vectors
Solution:
(a) 1 = Team1
2= Team 2
3= Team 3
4= Team 4
1 2 3 4
1 0 1 1 0
2 0 0 1 0
A= [ ]
3 0 0 0 1
4 1 1 0 0
(b)
Large dominant is can get from Power Method that is

1.3953369961023712 from 142th iteration from (c) and (d)
Small dominant value can also be find out by using Inverse power method where we have to
convert A to A-1 and rest thing same as Power method And then convert lambda equal to 1 by
lambda
−1 1 0 1
1 −1 0 0
A-1 = [ ]
0 1 0 0
0 0 1 0
After solving by inverse power method in python

From iteration 25
We get
⋋=2.106919340576228
So small dominant eigen value = 1/⋋=1/2.106919340576228
=0.474626618
In [1]:
import numpy as vinay
a=vinay.array([[-1,1,0,1],[1,-1,0,0],[0,1,0,0],[0,0,1,0]], dtype=float)
b=vinay.array([[1],[1],[1],[1]],dtype=float)
d=[[0],[0],[0],[0]]
count=1
c=vinay.dot(a,b)
while count<26:
c=vinay.dot(a,b)
print("Iteration No.", count)

print(c)
for i in range(4):
if c[i,0]<0:
d[i][0]= (c[i,0])*(-1)
else:
d[i][0]=c[i,0]
maxi=max(d[0][0],d[1][0],d[2][0],d[3][0])
for i in range(4):
c[i][0]=c[i][0]/maxi
print(' Dominant Eigen value of this iteration =',maxi)
print("Eigen vector of this iteration")
print(c)
print('')
print('')
count+=1
b=c
Iteration No. 1
[[1.]
[0.]
[1.]
[1.]]
Dominant Eigen value of this iteration = 1.0
Eigen vector of this iteration
[[1.]
[0.]
[1.]
[1.]]
Iteration No. 2
[[0.]
[1.]
[0.]
[1.]]
[[0.]
[1.]
[0.]
[1.]]
Iteration No. 3
[[ 2.]
[-1.]
[ 1.]
[ 0.]]
[[ 1. ]
[-0.5]
[ 0.5]
[ 0. ]]
Iteration No. 4
[[-1.5]
[ 1.5]
[-0.5]
[ 0.5]]
[[-1. ]
[ 1. ]
[-0.33333333]
[ 0.33333333]]
Iteration No. 5
[[ 2.33333333]
[-2. ]
[ 1. ]
[-0.33333333]]
[[ 1. ]
[-0.85714286]
[ 0.42857143]
[-0.14285714]]
Iteration No. 6
[[-2. ]
[ 1.85714286]
[-0.85714286]
[ 0.42857143]]
[[-1. ]
[ 0.92857143]
[-0.42857143]
[ 0.21428571]]
Iteration No. 7
[[ 2.14285714]
[-1.92857143]
[ 0.92857143]
[-0.42857143]]
[[ 1. ]
[-0.9 ]
[ 0.43333333]
[-0.2 ]]
Iteration No. 8
[[-2.1 ]
[ 1.9 ]
[-0.9 ]
[ 0.43333333]]
[[-1. ]
[ 0.9047619 ]
[-0.42857143]
[ 0.20634921]]
Iteration No. 9
[[ 2.11111111]
[-1.9047619 ]
[ 0.9047619 ]
[-0.42857143]]
[[ 1. ]
[-0.90225564]
[ 0.42857143]
[-0.20300752]]
Iteration No. 10
[[-2.10526316]
[ 1.90225564]
[-0.90225564]
[ 0.42857143]]
[[-1. ]
[ 0.90357143]
[-0.42857143]
[ 0.20357143]]
Iteration No. 11
[[ 2.10714286]
[-1.90357143]
[ 0.90357143]
[-0.42857143]]
[[ 1. ]
[-0.90338983]
[ 0.42881356]
[-0.20338983]]
Iteration No. 12
[[-2.10677966]
[ 1.90338983]
[-0.90338983]
[ 0.42881356]]
[[-1. ]
[ 0.90345937]
[-0.42880129]
[ 0.20353982]]
Iteration No. 13
[[ 2.1069992 ]
[-1.90345937]
[ 0.90345937]
[-0.42880129]]
[[ 1. ]
[-0.90339824]
[ 0.42878961]
[-0.20351279]]
Iteration No. 14
[[-2.10691103]
[ 1.90339824]
[-0.90339824]
[ 0.42878961]]
[[-1. ]
[ 0.90340703]
[-0.42877854]
[ 0.20351577]]
Iteration No. 15
[[ 2.1069228 ]
[-1.90340703]
[ 0.90340703]
[-0.42877854]]
[[ 1. ]
[-0.90340616]
[ 0.42878032]
[-0.20350938]]
Iteration No. 16
[[-2.10691553]
[ 1.90340616]
[-0.90340616]
[ 0.42878032]]
[[-1. ]
[ 0.90340886]
[-0.42878138]
[ 0.20351092]]
Iteration No. 17
[[ 2.10691978]
[-1.90340886]
[ 0.90340886]
[-0.42878138]]
[[ 1. ]
[-0.90340832]
[ 0.4287818 ]
[-0.20351102]]
Iteration No. 18
[[-2.10691934]
[ 1.90340832]
[-0.90340832]
[ 0.4287818 ]]
[[-1. ]
[ 0.90340825]
[-0.42878164]
[ 0.20351126]]
Iteration No. 19
[[ 2.10691951]
[-1.90340825]
[ 0.90340825]
[-0.42878164]]
[[ 1. ]
[-0.90340815]
[ 0.42878157]
[-0.20351116]]
Iteration No. 20
[[-2.10691931]
[ 1.90340815]
[-0.90340815]
[ 0.42878157]]
[[-1. ]
[ 0.90340818]
[-0.42878156]
[ 0.20351115]]
Iteration No. 21
[[ 2.10691933]
[-1.90340818]
[ 0.90340818]
[-0.42878156]]
[[ 1. ]
[-0.90340819]
[ 0.42878157]
[-0.20351114]]
Iteration No. 22
[[-2.10691933]
[ 1.90340819]
[-0.90340819]
[ 0.42878157]]
[[-1. ]
[ 0.90340819]
[-0.42878158]
[ 0.20351115]]
Iteration No. 23
[[ 2.10691934]
[-1.90340819]
[ 0.90340819]
[-0.42878158]]
[[ 1. ]
[-0.90340819]
[ 0.42878158]
[-0.20351115]]
Iteration No. 24
[[-2.10691934]
[ 1.90340819]
[-0.90340819]
[ 0.42878158]]
[[-1. ]
[ 0.90340819]
[-0.42878157]
[ 0.20351115]]
Iteration No. 25
[[ 2.10691934]
[-1.90340819]
[ 0.90340819]
[-0.42878157]]
[[ 1. ]
[-0.90340819]
[ 0.42878157]
[-0.20351115]]
In [ ]:

(c) let assume lambda is k

−𝑘 1 1 0
0 −𝑘 1 0
[ ]
0 0 −𝑘 1
1 1 0 −𝑘
K4-2k-1 =0
After solving equation using calculator
We got for real k ,k1= -0.4746266176

And k2=1.395336995
So ,
|k1|<|k2|
So dominant eigenvalue is k2 that is equal to 1.395336995
(d) Eigenvalue and eigenvector using Power method:

first ten iteration in copy
Rest all in python
For iteration 142 most approximate in 8 decimal place.

0.88171717
0.51361983
Eigen vector = [ ]
0.71667275
1.0000000
Eigen value = 1.3953369961023712
In [1]:

a=vinay.array([[0,1,1,0],[0,0,1,0],[0,0,0,1],[1,1,0,0]], dtype=float)
b=vinay.array([[1],[1],[1],[1]],dtype=float)
count=1
c=vinay.dot(a,b)
while count<143:
c=vinay.dot(a,b)
print(c)
maxi=max(c[0,0],c[1,0],c[2,0],c[3,0])
for i in range(4):
print(c)
print('')
print('')
count+=1
b=c
Iteration No. 1
[[2.]
[1.]
[1.]
[2.]]
[[1. ]
[0.5]
[0.5]
[1. ]]
Iteration No. 2
[[1. ]
[0.5]
[1. ]
[1.5]]
[[0.66666667]
[0.33333333]
[0.66666667]
[1. ]]
Iteration No. 3
[[1. ]
[0.66666667]
[1. ]
[1. ]]
[[1. ]
[0.66666667]
[1. ]
[1. ]]
Iteration No. 4
[[1.66666667]
[1. ]
[1. ]
[1.66666667]]
[[1. ]
[0.6]
[0.6]
[1. ]]
Iteration No. 5
[[1.2]
[0.6]
[1. ]
[1.6]]
[[0.75 ]
[0.375]
[0.625]
[1. ]]
Iteration No. 6
[[1. ]
[0.625]
[1. ]
[1.125]]
[[0.88888889]
[0.55555556]
[0.88888889]
[1. ]]
Iteration No. 7
[[1.44444444]
[0.88888889]
[1. ]
[1.44444444]]
[[1. ]
[0.61538462]
[0.69230769]
[1. ]]
Iteration No. 8
[[1.30769231]
[0.69230769]
[1. ]
[1.61538462]]
[[0.80952381]
[0.42857143]
[0.61904762]
[1. ]]
Iteration No. 9
[[1.04761905]
[0.61904762]
[1. ]
[1.23809524]]
[[0.84615385]
[0.5 ]
[0.80769231]
[1. ]]
Iteration No. 10
[[1.30769231]
[0.80769231]
[1. ]
[1.34615385]]
[[0.97142857]
[0.6 ]
[0.74285714]
[1. ]]
Iteration No. 11
[[1.34285714]
[0.74285714]
[1. ]
[1.57142857]]
[[0.85454545]
[0.47272727]
[0.63636364]
[1. ]]
Iteration No. 12
[[1.10909091]
[0.63636364]
[1. ]
[1.32727273]]
[[0.83561644]
[0.47945205]
[0.75342466]
[1. ]]
Iteration No. 13
[[1.23287671]
[0.75342466]
[1. ]
[1.31506849]]
[[0.9375 ]
[0.57291667]
[0.76041667]
[1. ]]
Iteration No. 14
[[1.33333333]
[0.76041667]
[1. ]
[1.51041667]]
[[0.88275862]
[0.50344828]
[0.66206897]
[1. ]]
Iteration No. 15
[[1.16551724]
[0.66206897]
[1. ]
[1.3862069 ]]
[[0.84079602]
[0.47761194]
[0.72139303]
[1. ]]
Iteration No. 16
[[1.19900498]
[0.72139303]
[1. ]
[1.31840796]]
[[0.90943396]
[0.54716981]
[0.75849057]
[1. ]]
Iteration No. 17
[[1.30566038]
[0.75849057]
[1. ]
[1.45660377]]
[[0.89637306]
[0.52072539]
[0.6865285 ]
[1. ]]
Iteration No. 18
[[1.20725389]
[0.6865285 ]
[1. ]
[1.41709845]]
[[0.85191956]
[0.48446069]
[0.70566728]
[1. ]]
Iteration No. 19
[[1.19012797]
[0.70566728]
[1. ]
[1.33638026]]
[[0.89056088]
[0.52804378]
[0.74829001]
[1. ]]
Iteration No. 20
[[1.27633379]
[0.74829001]
[1. ]
[1.41860465]]
[[0.8997107 ]
[0.52748312]
[0.70491803]
[1. ]]
Iteration No. 21
[[1.23240116]
[0.70491803]
[1. ]
[1.42719383]]
[[0.86351351]
[0.49391892]
[0.70067568]
[1. ]]
Iteration No. 22
[[1.19459459]
[0.70067568]
[1. ]
[1.35743243]]
[[0.88003982]
[0.5161772 ]
[0.73668492]
[1. ]]
Iteration No. 23
[[1.25286212]
[0.73668492]
[1. ]
[1.39621702]]
[[0.8973262 ]
[0.52762923]
[0.71622103]
[1. ]]
Iteration No. 24
[[1.24385027]
[0.71622103]
[1. ]
[1.42495544]]
[[0.87290468]
[0.50262697]
[0.70177633]
[1. ]]
Iteration No. 25
[[1.2044033 ]
[0.70177633]
[1. ]
[1.37553165]]
[[0.87559112]
[0.51018552]
[0.72699163]
[1. ]]
Iteration No. 26
[[1.23717716]
[0.72699163]
[1. ]
[1.38577665]]
[[0.89276808]
[0.52460953]
[0.72161701]
[1. ]]
Iteration No. 27
[[1.24622654]
[0.72161701]
[1. ]
[1.41737761]]
[[0.87924808]
[0.50912121]
[0.70552829]
[1. ]]
Iteration No. 28
[[1.2146495 ]
[0.70552829]
[1. ]
[1.38836929]]
[[0.87487494]
[0.50817048]
[0.72026946]
[1. ]]
Iteration No. 29
[[1.22843994]
[0.72026946]
[1. ]
[1.38304542]]
[[0.88821373]
[0.52078511]
[0.72304205]
[1. ]]
Iteration No. 30
[[1.24382716]
[0.72304205]
[1. ]
[1.40899884]]
[[0.88277373]
[0.51316015]
[0.70972379]
[1. ]]
Iteration No. 31
[[1.22288394]
[0.70972379]
[1. ]
[1.39593387]]
[[0.87603286]
[0.50842221]
[0.71636631]
[1. ]]
Iteration No. 32
[[1.22478853]
[0.71636631]
[1. ]
[1.38445507]]
[[0.88467192]
[0.51743558]
[0.72230585]
[1. ]]
Iteration No. 33
[[1.23974143]
[0.72230585]
[1. ]
[1.4021075 ]]
[[0.88419856]
[0.51515726]
[0.71321208]
[1. ]]
Iteration No. 34
[[1.22836933]
[0.71321208]
[1. ]
[1.39935582]]
[[0.87781057]
[0.50967171]
[0.71461453]
[1. ]]
Iteration No. 35
[[1.22428624]
[0.71461453]
[1. ]
[1.38748229]]
[[0.88237973]
[0.51504408]
[0.72072992]
[1. ]]
Iteration No. 36
[[1.235774 ]
[0.72072992]
[1. ]
[1.3974238 ]]
[[0.88432299]
[0.51575615]
[0.71560252]
[1. ]]
Iteration No. 37
[[1.23135867]
[0.71560252]
[1. ]
[1.40007914]]
[[0.87949219]
[0.51111577]
[0.71424534]
[1. ]]
Iteration No. 38
[[1.2253611 ]
[0.71424534]
[1. ]
[1.39060796]]
[[0.88116935]
[0.51362092]
[0.71910994]
[1. ]]
Iteration No. 39
[[1.23273086]
[0.71910994]
[1. ]
[1.39479026]]
[[0.88381091]
[0.51556851]
[0.71695367]
[1. ]]
Iteration No. 40
[[1.23252218]
[0.71695367]
[1. ]
[1.39937942]]
[[0.8807634 ]
[0.51233687]
[0.71460248]
[1. ]]
Iteration No. 41
[[1.22693935]
[0.71460248]
[1. ]
[1.39310027]]
[[0.8807258 ]
[0.51295839]
[0.71782342]
[1. ]]
Iteration No. 42
[[1.23078181]
[0.71782342]
[1. ]
[1.39368419]]
[[0.88311385]
[0.51505458]
[0.71752267]
[1. ]]
Iteration No. 43
[[1.23257724]
[0.71752267]
[1. ]
[1.39816843]]
[[0.88156564]
[0.51318758]
[0.71522141]
[1. ]]
Iteration No. 44
[[1.22840899]
[0.71522141]
[1. ]
[1.39475322]]
[[0.88073573]
[0.51279424]
[0.71697271]
[1. ]]
Iteration No. 45
[[1.22976695]
[0.71697271]
[1. ]
[1.39352997]]
[[0.88248332]
[0.51450111]
[0.71760208]
[1. ]]
Iteration No. 46
[[1.23210319]
[0.71760208]
[1. ]
[1.39698443]]
[[0.88197346]
[0.51367937]
[0.71582759]
[1. ]]
Iteration No. 47
[[1.22950696]
[0.71582759]
[1. ]
[1.39565283]]
[[0.88095473]
[0.51289804]
[0.71651057]
[1. ]]
Iteration No. 48
[[1.2294086 ]
[0.71651057]
[1. ]
[1.39385277]]
[[0.88202186]
[0.5140504 ]
[0.71743589]
[1. ]]
Iteration No. 49
[[1.23148629]
[0.71743589]
[1. ]
[1.39607225]]
[[0.88210785]
[0.51389596]
[0.7162953 ]
[1. ]]
Iteration No. 50
[[1.23019126]
[0.7162953 ]
[1. ]
[1.3960038 ]]
[[0.88122343]
[0.51310412]
[0.71633043]
[1. ]]
Iteration No. 51
[[1.22943455]
[0.71633043]
[1. ]
[1.39432755]]
[[0.88174012]
[0.51374616]
[0.71719159]
[1. ]]
Iteration No. 52
[[1.23093775]
[0.71719159]
[1. ]
[1.39548628]]
[[0.88208517]
[0.51393668]
[0.71659608]
[1. ]]
Iteration No. 53
[[1.23053277]
[0.71659608]
[1. ]
[1.39602185]]
[[0.88145666]
[0.51331294]
[0.71632116]
[1. ]]
Iteration No. 54
[[1.2296341 ]
[0.71632116]
[1. ]
[1.3947696 ]]
[[0.88160374]
[0.5135767 ]
[0.71696429]
[1. ]]
Iteration No. 55
[[1.23054099]
[0.71696429]
[1. ]
[1.39518044]]
[[0.88199415]
[0.51388643]
[0.71675317]
[1. ]]
Iteration No. 56
[[1.23063959]
[0.71675317]
[1. ]
[1.39588058]]
[[0.88162241]
[0.51347743]
[0.71639366]
[1. ]]
Iteration No. 57
[[1.22987109]
[0.71639366]
[1. ]
[1.39509983]]
[[0.88156493]
[0.51350709]
[0.71679458]
[1. ]]
Iteration No. 58
[[1.23030168]
[0.71679458]
[1. ]
[1.39507203]]
[[0.88189115]
[0.51380471]
[0.71680887]
[1. ]]
Iteration No. 59
[[1.23061358]
[0.71680887]
[1. ]
[1.39569586]]
[[0.88172045]
[0.51358529]
[0.71648847]
[1. ]]
Iteration No. 60
[[1.23007377]
[0.71648847]
[1. ]
[1.39530574]]
[[0.8815801 ]
[0.51349927]
[0.7166888 ]
[1. ]]
Iteration No. 61
[[1.23018807]
[0.7166888 ]
[1. ]
[1.39507936]]
[[0.88180508]
[0.51372619]
[0.7168051 ]
[1. ]]
Iteration No. 62
[[1.23053129]
[0.7168051 ]
[1. ]
[1.39553127]]
[[0.88176547]
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[0.71657298]
[1. ]]
Iteration No. 63
[[1.23021615]
[0.71657298]
[1. ]
[1.39540864]]
[[0.88161712]
[0.51352196]
[0.71663595]
[1. ]]
Iteration No. 64
[[1.23015792]
[0.71663595]
[1. ]
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[[0.88174572]
[0.51366632]
[0.71677441]
[1. ]]
Iteration No. 65
[[1.23044073]
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[1. ]
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[[0.88177592]
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Iteration No. 66
[[1.23029927]
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[[0.88165626]
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Iteration No. 67
[[1.23017328]
[0.71661934]
[1. ]
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[[0.88171179]
[0.51362823]
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Iteration No. 68
[[1.23036612]
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[1. ]
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[[0.88176795]
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Iteration No. 69
[[1.23033659]
[0.71667119]
[1. ]
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[[0.88168782]
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Iteration No. 70
[[1.23020651]
[0.71662326]
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[[0.88169714]
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Iteration No. 71
[[1.23031524]
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[[0.88175314]
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Iteration No. 72
[[1.23034438]
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[[0.88170896]
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Iteration No. 73
[[1.23024083]
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[[0.88169468]
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Iteration No. 74
[[1.23028656]
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[[0.88173836]
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Iteration No. 75
[[1.23033669]
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Iteration No. 76
[[1.23026814]
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[[0.88169845]
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Iteration No. 77
[[1.23027449]
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[[0.88172684]
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Iteration No. 78
[[1.23032347]
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[[0.8817253 ]
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Iteration No. 79
[[1.23028623]
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Iteration No. 80
[[1.23027282]
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[[0.88171935]
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Iteration No. 81
[[1.23031055]
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Iteration No. 82
[[1.23029605]
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Iteration No. 83
[[1.23027635]
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[[0.88171538]
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Iteration No. 84
[[1.23030059]
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[[0.88172403]
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Iteration No. 85
[[1.23029987]
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[[0.88171402]
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Iteration No. 86
[[1.23028155]
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[[0.88171392]
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Iteration No. 87
[[1.2302942 ]
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[[0.88172175]
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Iteration No. 88
[[1.23030006]
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[1.39534628]]
[[0.88171666]
[0.5136184 ]
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Iteration No. 89
[[1.23028638]
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[[0.88171394]
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Iteration No. 90
[[1.23029086]
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[[0.88171968]
[0.51362272]
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Iteration No. 91
[[1.23029852]
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[[0.881718 ]
[0.51362002]
[0.71666997]
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Iteration No. 92
[[1.23028999]
[0.71666997]
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[[0.88171466]
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Iteration No. 93
[[1.23028968]
[0.71667222]
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[1.39533212]]
[[0.88171817]
[0.51362124]
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Iteration No. 94
[[1.2302965 ]
[0.71667525]
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[1.39533941]]
[[0.88171845]
[0.51362073]
[0.71667151]
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Iteration No. 95
[[1.23029224]
[0.71667151]
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[1.39533918]]
[[0.88171554]
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Iteration No. 96
[[1.23028976]
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Iteration No. 97
[[1.2302947 ]
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[[0.88171837]
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Iteration No. 98
[[1.23029336]
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[[0.88171631]
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Iteration No. 99
[[1.23029042]
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[[0.88171679]
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Iteration No. 100
[[1.2302934 ]
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[[0.88171807]
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Iteration No. 101
[[1.23029372]
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Iteration No. 102
[[1.23029119]
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[[0.88171667]
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Iteration No. 103
[[1.23029261]
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Iteration No. 104
[[1.23029363]
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Iteration No. 105
[[1.23029186]
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[[0.88171672]
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Iteration No. 106
[[1.23029224]
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[[0.88171745]
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Iteration No. 107
[[1.23029336]
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[[0.88171732]
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Iteration No. 108
[[1.23029233]
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[[0.88171684]
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Iteration No. 109
[[1.23029214]
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[[0.88171726]
[0.51361998]
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Iteration No. 110
[[1.23029307]
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[[0.88171736]
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Iteration No. 111
[[1.2302926 ]
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[[0.88171696]
[0.51361961]
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Iteration No. 112
[[1.23029219]
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[1.39533658]]
[[0.88171715]
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Iteration No. 113
[[1.23029282]
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[[0.88171733]
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Iteration No. 114
[[1.23029272]
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[[0.88171707]
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Iteration No. 115
[[1.2302923 ]
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[[0.8817171 ]
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Iteration No. 116
[[1.23029265]
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[[0.88171728]
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Iteration No. 117
[[1.23029275]
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[[0.88171714]
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Iteration No. 118
[[1.23029241]
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Iteration No. 119
[[1.23029256]
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[1.39533686]]
[[0.88171723]
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Iteration No. 120
[[1.23029272]
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[[0.88171718]
[0.51361982]
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Iteration No. 121
[[1.2302925 ]
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Iteration No. 122
[[1.23029252]
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Iteration No. 123
[[1.23029268]
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[[0.88171719]
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[0.71667271]
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Iteration No. 124
[[1.23029256]
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[[0.88171712]
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[0.71667273]
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Iteration No. 125
[[1.23029251]
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[[0.88171717]
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Iteration No. 126
[[1.23029264]
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[[0.88171719]
[0.51361985]
[0.71667274]
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Iteration No. 127
[[1.23029259]
[0.71667274]
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[0.71667272]
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Iteration No. 128
[[1.23029253]
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[[0.88171716]
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Iteration No. 129
[[1.23029261]
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[1.39533699]]
[[0.88171719]
[0.51361985]
[0.71667275]
[1. ]]
Iteration No. 130
[[1.2302926 ]
[0.71667275]
[1. ]
[1.39533704]]
[[0.88171715]
[0.51361982]
[0.71667273]
[1. ]]
Iteration No. 131
[[1.23029254]
[0.71667273]
[1. ]
[1.39533697]]
[[0.88171715]
[0.51361982]
[0.71667276]
[1. ]]
Iteration No. 132
[[1.23029258]
[0.71667276]
[1. ]
[1.39533698]]
[[0.88171718]
[0.51361985]
[0.71667276]
[1. ]]
Iteration No. 133
[[1.2302926 ]
[0.71667276]
[1. ]
[1.39533702]]
[[0.88171716]
[0.51361982]
[0.71667273]
[1. ]]
Iteration No. 134
[[1.23029256]
[0.71667273]
[1. ]
[1.39533699]]
[[0.88171715]
[0.51361982]
[0.71667275]
[1. ]]
Iteration No. 135
[[1.23029257]
[0.71667275]
[1. ]
[1.39533698]]
[[0.88171717]
[0.51361984]
[0.71667276]
[1. ]]
Iteration No. 136
[[1.2302926 ]
[0.71667276]
[1. ]
[1.39533701]]
[[0.88171717]
[0.51361983]
[0.71667274]
[1. ]]
Iteration No. 137
[[1.23029257]
[0.71667274]
[1. ]
[1.395337 ]]
[[0.88171716]
[0.51361982]
[0.71667275]
[1. ]]
Iteration No. 138
[[1.23029257]
[0.71667275]
[1. ]
[1.39533698]]
[[0.88171717]
[0.51361983]
[0.71667276]
[1. ]]
Iteration No. 139
[[1.23029259]
[0.71667276]
[1. ]
[1.395337 ]]
[[0.88171717]
[0.51361983]
[0.71667275]
[1. ]]
Iteration No. 140
[[1.23029258]
[0.71667275]
[1. ]
[1.395337 ]]
[[0.88171716]
[0.51361982]
[0.71667275]
[1. ]]
Iteration No. 141
[[1.23029257]
[0.71667275]
[1. ]
[1.39533698]]
[[0.88171717]
[0.51361983]
[0.71667275]
[1. ]]
Iteration No. 142
[[1.23029259]
[0.71667275]
[1. ]
[1.395337 ]]
[[0.88171717]
[0.51361983]
[0.71667275]
[1. ]]
From iteration 142 we got eigen vector

0.88171717
0.51361983
[ ]
0.71667275
1.0000000
So, Rank 1 : Team 4
Rank 2: Team 1
Rank 3 : Team 3
Rank 4: Team 2
Q2. Identify one from the following categories based on discussion within your team:
a) Sports tournaments (select any one sport and tournament)
b) Economics
c) Entertainment
d) Politics or Policies
e) Sustainable Development Goals (select one among 17 SDGs defined by UN)
Solution:
(a) World chess championship 2018

Source = Wikipedia
Q3.Collect real data of the tournaments in which one team played or can be compared with all other
teams based on the defined parameter.
Solution
FC SM SK DL VK AG WS LA
FC 1 1 0.5 1 1.5 1.5 1.5 2
SM 1 1 1.5 0.5 1.5 1.5 1 1
SK 1.5 0.5 1 1 1.5 1 1.5 1
DL 1 1.5 1 1 1 1 1 1
0.5 0.5 0.5 1 1 1 1 2
VK
0.5 0.5 1 1 1 1 1.5 1
AG
0.5 1 0.5 1 1 0.5 1 1.5
WS [0 1 1 1 0 1 0.5 1 ]
LA
Here,
FC=Fabiano Caruana
SM=Shakhriyar Mamedyarov
SK= Sergey kharjakin
DL=Ding Liren
VK= Vladimir Kramnik
AG=Alexander Grischuk
WS= Wesley So
LA= Levon Aronian
4. Define the rank of teams. You should be able to use Python for Mathematical calculation,
wherever needed.
Solution : from iteration 12 we get

[[1. ]
[0.94319696]
[0.94086389]
[0.9052184 ]
[0.73335755]
[0.76522272]
[0.70775759]
[0.57636516]]
So, Rank 1 = Fabiano Caruana

Rank 2 = Shakhriyar Mamedyarov
Rank 3 = Sergey kharjakin
Rank 4= Ding Liren
Rank 5= Alexander Grischuk
Rank 6 = Vladimir Kramnik
Rank 7= Wesley So
Rank 8= Levon Aronian

In [2]:
a=vinay.array([[1,1,0.5,1,1.5,1.5,1.5,2],[1,1,1.5,0.5,1.5,1.5,1,1],[1.5,0.5,1,1,1.5,
b=vinay.array([[1],[1],[1],[1],[1],[1],[1],[1]],dtype=float)
count=1
c=vinay.dot(a,b)
print(a)
while count<14:
c=vinay.dot(a,b)

print(c)
maxi=max(c[0,0],c[1,0],c[2,0],c[3,0],c[4,0],c[5,0],c[6,0],c[7,0])
for i in range(8):
print(c)
print('')
print('')
count+=1
b=c
[[1. 1. 0.5 1. 1.5 1.5 1.5 2. ]

[1. 1. 1.5 0.5 1.5 1.5 1. 1. ]
[1.5 0.5 1. 1. 1.5 1. 1.5 1. ]
[1. 1.5 1. 1. 1. 1. 1. 1. ]
[0.5 0.5 0.5 1. 1. 1. 1. 2. ]
[0.5 0.5 1. 1. 1. 1. 1.5 1. ]
[0.5 1. 0.5 1. 1. 0.5 1. 1.5]
[0. 1. 1. 1. 0. 1. 0.5 1. ]]
Iteration No. 1
[[10. ]
[ 9. ]
[ 9. ]
[ 8.5]
[ 7.5]
[ 7.5]
[ 7. ]
[ 5.5]]
[[1. ]
[0.9 ]
[0.9 ]
[0.85]
[0.75]
[0.75]
[0.7 ]
[0.55]]
Iteration No. 2
[[7.6 ]
[7.175]
[7.175]
[6.85 ]
[5.55 ]
[5.8 ]
[5.35 ]
[4.3 ]]
[[1. ]
[0.94407895]
[0.94407895]
[0.90131579]
[0.73026316]
[0.76315789]
[0.70394737]
[0.56578947]]
Iteration No. 3
[[7.74506579]
[7.32072368]
[7.29769737]
[7.02467105]
[5.67434211]
[5.93256579]
[5.48190789]
[4.47039474]]
[[1. ]
[0.9452113 ]
[0.94223827]
[0.90698662]
[0.73263963]
[0.76598004]
[0.70779359]
[0.57719261]]
Iteration No. 4
[[7.78732215]
[7.3449777 ]
[7.325653 ]
[7.0506477 ]
[5.71150987]
[5.95933319]
[5.5125292 ]
[4.49150563]]
[[1. ]
[0.94319685]
[0.94071529]
[0.9054008 ]
[0.73343696]
[0.7652609 ]
[0.70788508]
[0.57677152]]
Iteration No. 5
[[7.78237275]
[7.33967358]
[7.32173 ]
[7.04426583]
[5.70748285]
[5.95501152]
[5.50806507]
[4.4852879 ]]
[[1. ]
[0.94311514]
[0.94080947]
[0.90515657]
[0.73338595]
[0.76519228]
[0.70776166]
[0.57633938]]
Iteration No. 6
[[7.78086505]
[7.33887602]
[7.32077669]
[7.04331803]
[5.70613753]
[5.95408372]
[5.50692927]
[4.48449367]]
[[1. ]
[0.94319539]
[0.94086925]
[0.90521015]
[0.73335516]
[0.76522131]
[0.70775283]
[0.57634899]]
Iteration No. 7
[[7.78103208]
[7.33907086]
[7.32090938]
[7.04355077]
[5.70626974]
[5.9542318 ]
[5.50708229]
[4.4847215 ]]
[[1. ]
[0.94320018]
[0.94086611]
[0.90522063]
[0.73335641]
[0.76522391]
[0.7077573 ]
[0.57636589]]
Iteration No. 8
[[7.78109208]
[7.33910333]
[7.3209472 ]
[7.04359052]
[5.70632318]
[5.954269 ]
[5.50712837]
[4.48475537]]
[[1. ]
[0.94319708]
[0.94086371]
[0.90521876]
[0.73335762]
[0.76522279]
[0.70775777]
[0.5763658 ]]
Iteration No. 9
[[7.78108657]
[7.33909622]
[7.32094269]
[7.04358208]
[5.70631894]
[5.95426388]
[5.50712318]
[4.48474703]]
[[1. ]
[0.94319683]
[0.9408638 ]
[0.90521832]
[0.7333576 ]
[0.76522268]
[0.7077576 ]
[0.57636514]]
Iteration No. 10
[[7.78108414]
[7.33909484]
[7.32094115]
[7.04358038]
[5.70631678]
[5.95426235]
[5.50712129]
[4.48474557]]
[[1. ]
[0.94319695]
[0.9408639 ]
[0.90521838]
[0.73335755]
[0.76522272]
[0.70775758]
[0.57636513]]
Iteration No. 11
[[7.78108431]
[7.3390951 ]
[7.3209413 ]
[7.04358068]
[5.70631691]
[5.95426252]
[5.50712146]
[4.48474587]]
[[1. ]
[0.94319696]
[0.9408639 ]
[0.9052184 ]
[0.73335755]
[0.76522272]
[0.70775759]
[0.57636516]]
Iteration No. 12
[[7.78108441]
[7.33909516]
[7.32094136]
[7.04358075]
[5.706317 ]
[5.95426258]
[5.50712154]
[4.48474593]]
[[1. ]
[0.94319696]
[0.94086389]
[0.9052184 ]
[0.73335755]
[0.76522272]
[0.70775759]
[0.57636516]]
Iteration No. 13
[[7.7810844 ]
[7.33909515]
[7.32094135]
[7.04358074]
[5.70631699]
[5.95426258]
[5.50712154]
[4.48474592]]
[[1. ]
[0.94319696]
[0.94086389]
[0.9052184 ]
[0.73335755]
[0.76522272]
[0.70775759]
[0.57636516]]

Assignment On Linear Algebra: Es1101: Computational Data Analysis

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SUBMITTED BY: Vinay Yograj Daharwal

Your name (Roll No.)

Institute of Engineering and Technology (IET)

6. RESULT AND DISSCUSSION

Large dominant is can get from Power Method that is

After solving by inverse power method in python

print("Iteration No.", count)

print(' Dominant Eigen value of this iteration =',maxi)

print("Eigen vector of this iteration")

Dominant Eigen value of this iteration = 1.0

Eigen vector of this iteration

Dominant Eigen value of this iteration = 1.0

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.0

Eigen vector of this iteration

Dominant Eigen value of this iteration = 1.5

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.3333333333333335

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.0

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.142857142857143

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.111111111111111

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1052631578947367

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.107142857142857

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.106779661016949

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1069991954947707

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.106911034746086

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.106922798115259

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1069155341476002

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.106919779546846

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.106919335774768

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1069195114773396

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.106919309640416

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1069193334866405

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1069193336027556

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1069193426293147

Eigen vector of this iteration

Dominant Eigen value of this iteration = 2.1069193409552347

Eigen vector of this iteration