### Project Description

**Implementing an algorithm in python**

Let the test cases be called Case 1 – Case 5, respective to their order in the assignment.

Case 1

We expect the program to cease prior to approximation attempt due to a 0 in the diagonal.

After we run the program, we see the output file stopped under the guise of “cannot proceed”,

i.e. it properly stopped due to divide by zero error.

Case 2, Case 3, Case 4, Case 5

We expect all of these sequences to converge very quickly. After running our program, we see

that after a single iteration, the program ceased since the error of x was less that the tolerance

allowed, i.e. the sequence represented by x converged for each case.

For n in the range of 1000-10000, we expect our program to scale quick easily. There should be

no change in how the data is processed or the number of iterations it will take, assume the

same number of non-zero elements in the matrix. However, as the number of non-zero

elements increase, the time required per each iteration will increase.

.

This code covers all the test cases mentioned

**• A small matrix A having a 0 on the diagonal;**

**• A small diagonally dominant matrix A;**

**• A small matrix A such that**

**? A has no 0 on the main diagonal**

**? A is not diagonally dominant**

**? all of the eigenvalues of C have absolute value < 1.**

**What do you expect to happen in this case?**

**• A small matrix A such that**

**? A has no 0 on the main diagonal**

**? A is not diagonally dominant**

**? one or more of the eigenvalues of C have absolute value > 1.**

**What do you expect to happen in this case?**

**• A small matrix A such that**

**? A has no 0 on the main diagonal ? A is not diagonally dominant**

**? all of CtC’s eigenvalues2 have absolute value < 1.**

**[That is, the spectral radius 0 = rs(CtC) < 1. This means ?C?2 := ?rs(CtC) < 1.]**

**What do you expect to happen in this case?**

**• A large (say n = 1000–10000) sparse diagonally dominant matrix.**

**And also help me answer the question**

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### Project Details

**Date**December 4, 2016**Tags**Programming, Python Programming