> #OK to Post Homework
> #Jeton Hida, September 27, 2021, Assignment 6
> 
> #Question 1 #0 y.o, 1 y.o, 2 y.o, 3y.o #P(S) 0 -> 1 = .95, P(S) 1 -> 2 = .97,
> P(S) 2 -> 3 = .9, S=Survival #Fertility rate 0 y.o = .1, 1 y.o = 1.2, 2 y.o =
> .9, 3 y.o = .1 #n3(t)=.9*n2(t-1), n3(t-1) = .9 *(.97*(.95*n0(t-4)))
> #n2(t)=.97*n1(t-1), n2(t-1) = .97*(.95*n0(t-3)), n2(t-2) = .97*(.95*n0(t-4))
> #n1(t)=.95*n0(t), n1(t-1) = .95*n0(t-2), n1(t-2)=.95*n0(t-3), n1(t-3) =
> .95*n0(t-4) 
#n0(t)=.1*n0(t-1)+1.2*(.95*n0(t-2))+.9*(.97*(.95*n0(t-3)))+.1*(.9*(.97*(.95*n0(t-4)))) #the number of 0 y.o olds at time t
> #REC:=[.1, 1.2*.95, .9*.97*.95, .1*.9*.97*.95]
> read "/Users/jeton/Desktop/Math 336/M5.txt"
> GrowthCe([.1, 1.2*.95, .9*.97*.95, .1*.9*.97*.95])
                                 1.385732629
> #Question 2 with(LinearAlgebra)
[&x, Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, BezoutMatrix, 

  BidiagonalForm, BilinearForm, CARE, CharacteristicMatrix, 

  CharacteristicPolynomial, Column, ColumnDimension, ColumnOperation, 

  ColumnSpace, CompanionMatrix, CompressedSparseForm, ConditionNumber, 

  ConstantMatrix, ConstantVector, Copy, CreatePermutation, CrossProduct, DARE, 

  DeleteColumn, DeleteRow, Determinant, Diagonal, DiagonalMatrix, Dimension, 

  Dimensions, DotProduct, EigenConditionNumbers, Eigenvalues, Eigenvectors, 

  Equal, ForwardSubstitute, FrobeniusForm, FromCompressedSparseForm, 

  FromSplitForm, GaussianElimination, GenerateEquations, GenerateMatrix, 

  Generic, GetResultDataType, GetResultShape, GivensRotationMatrix, 

  GramSchmidt, HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm, 

  HilbertMatrix, HouseholderMatrix, IdentityMatrix, IntersectionBasis, 

  IsDefinite, IsOrthogonal, IsSimilar, IsUnitary, JordanBlockMatrix, 

  JordanForm, KroneckerProduct, LA_Main, LUDecomposition, LeastSquares, 

  LinearSolve, LyapunovSolve, Map, Map2, MatrixAdd, MatrixExponential, 

  MatrixFunction, MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixPower, 

  MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial, Minor, 

  Modular, Multiply, NoUserValue, Norm, Normalize, NullSpace, 

  OuterProductMatrix, Permanent, Pivot, PopovForm, ProjectionMatrix, 

  QRDecomposition, RandomMatrix, RandomVector, Rank, RationalCanonicalForm, 

  ReducedRowEchelonForm, Row, RowDimension, RowOperation, RowSpace, 

  ScalarMatrix, ScalarMultiply, ScalarVector, SchurForm, SingularValues, 

  SmithForm, SplitForm, StronglyConnectedBlocks, SubMatrix, SubVector, 

  SumBasis, SylvesterMatrix, SylvesterSolve, ToeplitzMatrix, Trace, Transpose, 

  TridiagonalForm, UnitVector, VandermondeMatrix, VectorAdd, VectorAngle, 

  VectorMatrixMultiply, VectorNorm, VectorScalarMultiply, ZeroMatrix, 

  ZeroVector, Zip]
> LM:=Matrix([[.1, 1.2, .9, .1], [.95, 0, 0, 0], [0, .97, 0, 0], [0, 0, .9, 0]])

> Eigenvalues(LM)
Vector[column](4, [1.385732628853638 + 0.*I, -0.58335151608636 + 0.40312587748502526*I, -0.58335151608636 - 0.40312587748502526*I, -0.11902959668091699 + 0.*I])

> #Largest eigenvalue agrees with growth constant found above!
> #Question 3 #4 sites 1,2,3,4 denoted by S1,S2,S3,S4 respectively
> #Prob of staying at S1 = .5, Prob at staying at S2 = .4, Prob at staying at S3
> = .3, Prob at staying at S4 = .2
> #Prob of 1->2 = Prob of 1->3 = Prob of 1->4 = .167 #Prob of 2->1 = Prob of
> 2->3 = Prob of 2->4 = .2 #Prob of 3->1 = Prob of 3->2 = Prob of 3->4 = .233
> #Prob of 4->1 = Prob of 4->2 = Prob of 4->3 = .267 #P= Transition Matrix
> 
> P:=Matrix([[.5,.167,.167,.167],[.2,.4,.2,.2],[.233,.233,.3,.233],[.267,.267,.2
> 67,.2]])

> evalf(evalm(P&^1000))
Matrix(4, 4, [[0.4201023560, 0.3502853665, 0.3003672119, 0.2626555046], [0.4195034329, 0.3497859786, 0.2999389904, 0.2622810472], [0.4190752116, 0.3494289238, 0.2996328183, 0.2620133156], [0.4199342499, 0.3501451981, 0.3002470185, 0.2625504016]])

> #Around 42% of people stay on S1, 34.9% of people stay on S2, 29.9% of people
> stay on S3, 26.2% of people stay on S4 #Page Rank would now be from most
> popular to least popular S1,S2,S3,S4