# linear_algebra

#Homework 5 2.)

The LU decomposition is excuted by the functions `lu_tridiag.m`

.
The function `solve_tridiag.m`

solves the x vector in the equation Ax = B
The `test_array.m`

function then creates the matrixs that will be decomposed.

3.)

The table below compares the error between the created code and AN\b

size of A | norm(error) |
---|---|

3 | 2.341457211866148746537419356 |

4 | 1.5089317413322578254053496494 |

5 | 2.718353701243559239486557999 |

6 | 2.9952458816401601460199799476 |

7 | 2.1975878506014523416922656907 |

8 | 2.2726826857500208589613066579 |

9 | 1.2565685195978946175898727233 |

10 | 0.0000000000000001092738104726 |

#Spring Mass System ![This file shows the caclations to arrive at a final matrix.](Spring Mass System.PNG)

![Through calculating in the command prompt these values are porduced.](Eigen Values.PNG)

The table below shows the values of lamda and omega

lamda | omega |
---|---|

40.5220 | 6.3657 |

14.4090 | 3.7959 |

2.5690 | 1.6028 |

#Curvature of Slender Columns ![This file shows the calculations for the Curvature.](Curvature of Slender Column.PNG)

The table below expresses the values produced in the column.

# of segments | smallest | largest | number of eigenvalues |
---|---|---|---|

5 | 0.382 | 3.618 | 4 |

6 | 0.385 | 5.374 | 5 |

10 | 0.39 | 15.6 | 9 |

If the segment length (dx) approaches 0, then it means that the number of segments continues to grow to infinity. Thus, the number of eigenvalues will also apporach infinity. This matches the trend of the values given above.