linear_algebra

lu_tridiag.m is a function that takes three input vectors, x, y, and z as the diagonal entries of a tridiagonal matrix with y as the main diagonal, x above, and z below and outputs the diagonal vectors of the tridiagonal matrix's LU-decomposition. This output comes in the form of ud, the main diagonal of the upper triangular matrix, U, uo, the offset diagonal of U, and lo, the offset diagonal of the lower triangular matrix, L, which has a main diagonal of ones. solve_tridiag.m is a funtion that takes the output of lu_tridiag.m, (ud, uo, and lo) and solves for the solution to the equation Ax = b, where A = LU. Comparison of backslash operator and solve_tridiag.m

| size of A | norm(error) |

| 3 | 2.1204e-15 |

| 4 | NaN |

| 5 | Inf |

| 6 | 3.1241e-15 |

| 7 | 2.1243e-15 |

| 8 | NaN |

| 9 | 1.6932e-15 |

| 10 | 3.6074e-15 |

These values were generated by the modified test_arrays.m

Eigenvalues: 40.4522, 12.9987, -0.9509

Natural Frequencies: 6.3602, 3.6054

These numbers were generated by Eigens.m

| ------------- | ------- | -------- | ---------------- |

| 5 | 3.6180 | 0.3820 | 4 |

| 6 | 3.7321 | 0.2679 | 5 |

| 10 | 3.9021 | 0.0979 | 9 |

As the segment length decrease towards 0, the number of eigenvalues increases. Eventually the number of eigenvalues would approach infinity.

These values were generated by PoleEig.m

ngs13002/linear_algebra

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