diff options
| -rw-r--r-- | doc/software_manual.org | 46 | ||||
| -rw-r--r-- | homeworks/hw-7.org | 12 | ||||
| -rw-r--r-- | notes/Nov-27.org | 2 |
3 files changed, 60 insertions, 0 deletions
diff --git a/doc/software_manual.org b/doc/software_manual.org index 134d38d..1520431 100644 --- a/doc/software_manual.org +++ b/doc/software_manual.org @@ -1035,6 +1035,52 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance, return lambda; } #+END_SRC +*** ~least_dominant_eigenvalue~ ++ Author: Elizabeth Hunt ++ Name: ~least_dominant_eigenvalue~ ++ Location: ~src/eigen.c~ ++ Input: a pointer to an invertible matrix ~m~, an initial eigenvector guess ~v~ (that is non + zero or orthogonal to an eigenvector with the dominant eigenvalue), a ~tolerance~ and + ~max_iterations~ that act as stop conditions ++ Output: the least dominant eigenvalue with the lowest magnitude, approximated with the Inverse + Power Iteration Method +#+BEGIN_SRC c +double least_dominant_eigenvalue(Matrix_double *m, Array_double *v, + double tolerance, size_t max_iterations) { + assert(m->rows == m->cols); + assert(m->rows == v->size); + + double shift = 0.0; + Matrix_double *m_c = copy_matrix(m); + for (size_t y = 0; y < m_c->rows; ++y) + m_c->data[y]->data[y] = m_c->data[y]->data[y] - shift; + + double error = tolerance; + size_t iter = max_iterations; + double lambda = shift; + Array_double *eigenvector_1 = copy_vector(v); + + while (error >= tolerance && (--iter) > 0) { + Array_double *eigenvector_2 = solve_matrix_lu_bsubst(m_c, eigenvector_1); + Array_double *normalized_eigenvector_2 = + scale_v(eigenvector_2, 1.0 / linf_norm(eigenvector_2)); + free_vector(eigenvector_2); + eigenvector_2 = normalized_eigenvector_2; + + Array_double *mx = m_dot_v(m, eigenvector_2); + double new_lambda = + v_dot_v(mx, eigenvector_2) / v_dot_v(eigenvector_2, eigenvector_2); + + error = fabs(new_lambda - lambda); + lambda = new_lambda; + free_vector(eigenvector_1); + eigenvector_1 = eigenvector_2; + } + + return lambda; +} +#+END_SRC + *** ~leslie_matrix~ + Author: Elizabeth Hunt + Name: ~leslie_matrix~ diff --git a/homeworks/hw-7.org b/homeworks/hw-7.org index 1a5ebb0..dbdf6bb 100644 --- a/homeworks/hw-7.org +++ b/homeworks/hw-7.org @@ -12,4 +12,16 @@ See ~UTEST(eigen, leslie_matrix_dominant_eigenvalue)~ in ~test/eigen.t.c~ and the entry ~Eigen-Adjacent -> leslie_matrix~ in the LIZFCM API documentation. * Question Three +See ~UTEST(eigen, least_dominant_eigenvalue)~ in ~test/eigen.t.c~ which +finds the least dominant eigenvalue on the matrix: +\begin{bmatrix} +2 & 2 & 4 \\ +1 & 4 & 7 \\ +0 & 2 & 6 +\end{bmatrix} + +which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$ and should produce $\sqrt{17}$. + +See also the entry ~Eigen-Adjacent -> least_dominant_eigenvalue~ in the LIZFCM API +documentation. diff --git a/notes/Nov-27.org b/notes/Nov-27.org index c4d38b0..ae4ded0 100644 --- a/notes/Nov-27.org +++ b/notes/Nov-27.org @@ -14,5 +14,7 @@ x^{k + 1} \rightarrow Ax^k } x_1[i] = sum / a[i][i]; } + + err = 0.0; } #+END_SRC |