What are the ingredients of a general analytic solution?

What are the ingredients of a general analytic solution?

What is the analytical solution?

It's a formula for finding any iterate without scrolling on Excel.  For a system of two linear recursion equations where the coefficient matrix has two distinct eigevalues the formula for the solution is:

 $\overrightarrow{p}_n=c_1\times\left(\lambda_1\right)^n\times\overrightarrow{v}_1+c_2\times\left(\lambda_2\right)^n\times\overrightarrow{v}_2+\overrightarrow{p}_{star}$  

Step 1

 A system of DDS is first transformed into a matrix-vector form.

Example:

 $a_n=2a_{n-1}+3b_{n-1}+2$ 

 $b_n=4a_{n-1}+2b_{n-1}+5$ 

Step 2 Find

 $\overrightarrow{p}_{star},\$  and the eigenvalues and eigenvectors

 $\lambda_{1\ }=\ eigenvalue\ number\ 1$ 

 $\overrightarrow{v}_1=\ eigenvector\ number\ 1$ 

 $\lambda_2=eigenvalue\ number\ 2$ 

 $\overrightarrow{v}_2=eigenvector\ number\ 2$  

Step 3 Find  $c_{1\ }$   and  $c_2$  

 $\overrightarrow{p}_n=c_1\times\left(\lambda_1\right)^n\times\overrightarrow{v}_1+c_2\times\left(\lambda_2\right)^n\times\overrightarrow{v}_2+\overrightarrow{p}_{star}$  

For a system of 2 linear recursion equations the solution formula is given above with all notation defined in the previous slide.  We need  $\overrightarrow{p}_0$ to find  $c_1$   and  $c_{2\ }$ .

 $\overrightarrow{p}_0$  must be given

 $a_0=1,\ b_0=2$  

Multiple Select

The general analytic solution to a system of 2 linear recursion equations was given by the following:

• The eigenvalues can only be found if we know the coefficient matrix A.

Multiple Select

The solution to a linear system of 2 recursion equations is given by the following:

• The equilibrium vector is attracting

• The equilibrium vector is repelling

• We need to know the eigenvalues to determine if the equilibrium vector is attracting or repelling.

Multiple Select

A solution to a linear system of 2 equations is given by the following.

• If we had information about the vector

  $\overrightarrow{d}$ and we knew $A$ , we could find $\overrightarrow{p}_{star}$
  

Multiple Select

If I plotted the recursion equations in Excel, the graphs would be

• Convergent

• Divergent