> #Ok to post
> #Timothy Nasralla, HW6, 9/27/21 #Question 1: Given the information about the
> survival rates and fertility of a group of people, set up a recurrence in
> human language then express in maple rotation a REC and use GrowthCe to find
> the growth constants.
> read "/Users/tan88/OneDrive - Rutgers University/Documents/mw5.txt"
> Help5()
           RecToSeq(INI,REC,N), GrowthC(INI,REC,K) , GrowthCe(REC)

           LeslieMod(SUR,FER): e.g. LeslieMod([9/10,9/10],[0,1,1]);

           LeslieMat(SUR,FER); e.g. LeslieMat([9/10,9/10],[0,1,1]);

> with(LinearAlgebra) #m0(t) = 0.1m0(t-1) + 1.2m1(t-1) + 0.9m2(t-1) + 0.1m3(t-1)
> #m1(t) = 0.95m0(t-1), m2(t) = 0.97m1(t-1), m3(t) = 0.9m2(t-1) #In order to
> have the first equation entirely in terms of m0. m0(t) = 0.1*m0(t-1) +
> 1.2*.95*m0(t-2) + 0.9*0.97*0.95*m0(t-3) + 0.1*0.9*0.97*0.95m0(t-4)
> 
[&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]


> #m0(t) = 0.1*m0(t-1) + 1.14m0*(t-2) + 0.82935*m0(t-3) + 0.082935*m0(t-4)
> REC := [0.1, 1.14, 0.82935, 0.082935]

> GrowthCe(REC)
                                 1.385732629

> #Question 2, find the LeslieMat of the previously given example and compare
> the growth factors.
> SUR := [.95, .97, .9]; FER := [.1, 1.2, .9, .1]


> LeslieMat(SUR,FER)

> Eigenvalues(%)

> #The largest eigenvalue, 1.3857... is equivalent to the growth constant found
> in question 1.

> #Question 3: Find the transition matrix of the given Markov chain and solve
> for P^1000 to find the "page ranks".
> #Part A, find the transition matrix P. P := Matrix([[.5, 1/6, 1/6, 1/6], [.2,
> .4, .2, .2], [7/30, 7/30, .3, 7/30], [4/15, 4/15, 4/15, .2]])

> #Find P^1000 and analyze the information.
> evalf(P^1000)

> #Since the rows are identical we are able to determine the page ranks.
> #.3152 of the surfers stay on Site 1, .2627 of the surfers stay on Site 2,
> .2251 of the surfers stay on Site 3, and .1970 of the #surfers stay on Site 4.
> #Therefore, the page ranks (or popularity of the sites) would be, in
> decreasing order: S1, S2, S3, S4.