#+TITLE: Homework 8 #+AUTHOR: Elizabeth Hunt #+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} #+LATEX: \setlength\parindent{0pt} #+OPTIONS: toc:nil * Question One See ~UTEST(jacobi, solve_jacobi)~ in ~test/jacobi.t.c~ and the entry ~Jacobi / Gauss-Siedel -> solve_jacobi~ in the LIZFCM API documentation. * Question Two We cannot just perform the Jacobi algorithm on a Leslie matrix since it is obviously not diagonally dominant - which is a requirement. It is certainly not always the case, but, if a Leslie matrix $L$ is invertible, we can first perform gaussian elimination on $L$ augmented with $n_{k+1}$ to obtain $n_k$ with the Jacobi method. See ~UTEST(jacobi, leslie_solve)~ in ~test/jacobi.t.c~ for an example wherein this method is tested on a Leslie matrix to recompute a given initial population distribution. In terms of accuracy, an LU factorization and back substitution approach will always be as correct as possible within the limits of computation; it's a direct solution method. It's simply the nature of the Jacobi algorithm being a convergent solution that determines its accuracy. LU factorization also performs in order $O(n^3)$ runtime for an $n \times n$ matrix, whereas the Jacobi algorithm runs in order $O(k n^2) = O(n^2)$ on average but with the con that $k$ is given by some function on both the convergence criteria and the number of nonzero entries in the matrix - which might end up worse in some cases than the LU decomp approach. * Question Three See ~UTEST(jacobi, gauss_siedel_solve)~ in ~test/jacobi.t.c~ which runs the same unit test as ~UTEST(jacobi, solve_jacobi)~ but using the ~Jacobi / Gauss-Siedel -> gauss_siedel_solve~ method as documented in the LIZFCM API reference. * Question Four, Five We produce the following operation counts (by hackily adding the operation count as the last element to the solution vector) and errors - the sum of each vector elements' absolute value away from 1.0 using the proceeding patch and unit test. | N | JAC opr | JAC err | GS opr | GS err | LU opr | LU err | | 5 | 1622 | 0.001244 | 577 | 0.000098 | 430 | 0.000000 | | 6 | 2812 | 0.001205 | 775 | 0.000080 | 681 | 0.000000 | | 7 | 5396 | 0.001187 | 860 | 0.000178 | 1015 | 0.000000 | | 8 | 5618 | 0.001468 | 1255 | 0.000121 | 1444 | 0.000000 | | 9 | 7534 | 0.001638 | 1754 | 0.000091 | 1980 | 0.000000 | | 10 | 10342 | 0.001425 | 1847 | 0.000435 | 2635 | 0.000000 | | 11 | 12870 | 0.001595 | 2185 | 0.000368 | 3421 | 0.000000 | | 12 | 17511 | 0.001860 | 2912 | 0.000322 | 4350 | 0.000000 | | 13 | 16226 | 0.001631 | 3362 | 0.000270 | 5434 | 0.000000 | | 14 | 34333 | 0.001976 | 3844 | 0.000121 | 6685 | 0.000000 | | 15 | 38474 | 0.001922 | 4358 | 0.000311 | 8115 | 0.000000 | | 16 | 40405 | 0.002061 | 4904 | 0.000204 | 9736 | 0.000000 | | 17 | 58518 | 0.002125 | 5482 | 0.000311 | 11560 | 0.000000 | | 18 | 68079 | 0.002114 | 6092 | 0.000279 | 13599 | 0.000000 | | 19 | 95802 | 0.002159 | 6734 | 0.000335 | 15865 | 0.000000 | | 20 | 85696 | 0.002141 | 7408 | 0.000289 | 18370 | 0.000000 | | 21 | 89026 | 0.002316 | 8114 | 0.000393 | 21126 | 0.000000 | | 22 | 101537 | 0.002344 | 8852 | 0.000414 | 24145 | 0.000000 | | 23 | 148040 | 0.002323 | 9622 | 0.000230 | 27439 | 0.000000 | | 24 | 137605 | 0.002348 | 10424 | 0.000213 | 31020 | 0.000000 | | 25 | 169374 | 0.002409 | 11258 | 0.000894 | 34900 | 0.000000 | | 26 | 215166 | 0.002502 | 12124 | 0.000564 | 39091 | 0.000000 | | 27 | 175476 | 0.002616 | 13022 | 0.000535 | 43605 | 0.000000 | | 28 | 268454 | 0.002651 | 13952 | 0.000690 | 48454 | 0.000000 | | 29 | 267034 | 0.002697 | 14914 | 0.000675 | 53650 | 0.000000 | | 30 | 277193 | 0.002686 | 15908 | 0.000542 | 59205 | 0.000000 | | 31 | 336792 | 0.002736 | 16934 | 0.000390 | 65131 | 0.000000 | | 32 | 293958 | 0.002741 | 17992 | 0.000660 | 71440 | 0.000000 | | 33 | 323638 | 0.002893 | 19082 | 0.001072 | 78144 | 0.000000 | | 34 | 375104 | 0.003001 | 20204 | 0.001018 | 85255 | 0.000000 | | 35 | 436092 | 0.003004 | 21358 | 0.000912 | 92785 | 0.000000 | | 36 | 538143 | 0.003005 | 22544 | 0.000954 | 100746 | 0.000000 | | 37 | 511886 | 0.003029 | 23762 | 0.000462 | 109150 | 0.000000 | | 38 | 551332 | 0.003070 | 25012 | 0.000996 | 118009 | 0.000000 | | 39 | 592750 | 0.003110 | 26294 | 0.000989 | 127335 | 0.000000 | | 40 | 704208 | 0.003165 | 27608 | 0.000583 | 137140 | 0.000000 | #+BEGIN_SRC diff --git a/src/matrix.c b/src/matrix.c index 901a426..af5529f 100644 --- a/src/matrix.c +++ b/src/matrix.c @@ -144,20 +144,54 @@ Array_double *solve_matrix_lu_bsubst(Matrix_double *m, Array_double *b) { assert(b->size == m->rows); assert(m->rows == m->cols); + double opr = 0; + + opr += b->size; Array_double *x = copy_vector(b); + + size_t n = m->rows; + opr += n * n; // (u copy) + opr += n * n; // l_empty + opr += n * n + n; // copy + put_identity_diagonal + opr += n; // pivot check + opr += m->cols; + for (size_t x = 0; x < m->cols; x++) { + opr += (m->rows - (x + 1)); + for (size_t y = x + 1; y < m->rows; y++) { + opr += 1; + opr += 2; // -factor + opr += 4 * n; // scale, add_v, free_vector + opr += 1; // -factor + } + } + opr += n; Matrix_double **u_l = lu_decomp(m); + Matrix_double *u = u_l[0]; Matrix_double *l = u_l[1]; + opr += n; + for (int64_t row = n - 1; row >= 0; row--) { + opr += 2 * (n - row); + opr += 1; + } Array_double *b_fsub = fsubst(l, b); + + opr += n; + for (size_t x = 0; x < n; x++) { + opr += 2 * (x + 1); + opr += 1; // /= l->data[row]->data[row] + } x = bsubst(u, b_fsub); - free_vector(b_fsub); + free_vector(b_fsub); free_matrix(u); free_matrix(l); free(u_l); - return x; + Array_double *copy = add_element(x, opr); + free_vector(x); + return copy; } Matrix_double *gaussian_elimination(Matrix_double *m) { @@ -231,18 +265,36 @@ Array_double *jacobi_solve(Matrix_double *m, Array_double *b, assert(b->size == m->cols); size_t iter = max_iterations; + double opr = 0; + + opr += 2 * b->size; // to initialize two vectors with the same dim of b twice Array_double *x_k = InitArrayWithSize(double, b->size, 0.0); Array_double *x_k_1 = InitArrayWithSize(double, b->size, rand_from(0.1, 10.0)); + // add since these wouldn't be accounter for after the loop + opr += 1; // iter decrement + opr += + 3 * x_k_1->size; // 1 to perform x_k_1, x_k and 2 to perform ||x_k_1||_2 while ((--iter) > 0 && l2_distance(x_k_1, x_k) > l2_convergence_tolerance) { + opr += 1; // iter decrement + opr += + 3 * x_k_1->size; // 1 to perform x_k_1, x_k and 2 to perform ||x_k_1||_2 + + opr += m->rows; // row for add oprs for (size_t i = 0; i < m->rows; i++) { double delta = 0.0; + + opr += m->cols; for (size_t j = 0; j < m->cols; j++) { if (i == j) continue; + + opr += 1; delta += m->data[i]->data[j] * x_k->data[j]; } + + opr += 2; x_k_1->data[i] = (b->data[i] - delta) / m->data[i]->data[i]; } @@ -251,8 +303,9 @@ Array_double *jacobi_solve(Matrix_double *m, Array_double *b, x_k_1 = tmp; } - free_vector(x_k); - return x_k_1; + Array_double *copy = add_element(x_k_1, opr); + free_vector(x_k_1); + return copy; } Array_double *gauss_siedel_solve(Matrix_double *m, Array_double *b, @@ -262,30 +315,48 @@ Array_double *gauss_siedel_solve(Matrix_double *m, Array_double *b, assert(b->size == m->cols); size_t iter = max_iterations; + double opr = 0; + + opr += 2 * b->size; // to initialize two vectors with the same dim of b twice Array_double *x_k = InitArrayWithSize(double, b->size, 0.0); Array_double *x_k_1 = InitArrayWithSize(double, b->size, rand_from(0.1, 10.0)); while ((--iter) > 0) { + opr += 1; // iter decrement + + opr += x_k->size; // copy oprs for (size_t i = 0; i < x_k->size; i++) x_k->data[i] = x_k_1->data[i]; + opr += m->rows; // row for add oprs for (size_t i = 0; i < m->rows; i++) { double delta = 0.0; + + opr += m->cols; for (size_t j = 0; j < m->cols; j++) { if (i == j) continue; + + opr += 1; delta += m->data[i]->data[j] * x_k_1->data[j]; } + + opr += 2; x_k_1->data[i] = (b->data[i] - delta) / m->data[i]->data[i]; } + opr += + 3 * x_k_1->size; // 1 to perform x_k_1, x_k and 2 to perform ||x_k_1||_2 if (l2_distance(x_k_1, x_k) <= l2_convergence_tolerance) break; } free_vector(x_k); - return x_k_1; + + Array_double *copy = add_element(x_k_1, opr); + free_vector(x_k_1); + return copy; } #+END_SRC And this unit test: #+BEGIN_SRC c UTEST(hw_8, p4_5) { printf("| N | JAC opr | JAC err | GS opr | GS err | LU opr | LU err | \n"); for (size_t i = 5; i < 100; i++) { Matrix_double *m = generate_ddm(i); double oprs[3] = {0.0, 0.0, 0.0}; double errs[3] = {0.0, 0.0, 0.0}; Array_double *b_1 = InitArrayWithSize(double, m->rows, 1.0); Array_double *b = m_dot_v(m, b_1); double tolerance = 0.001; size_t max_iter = 400; // JACOBI { Array_double *solution_with_opr_count = jacobi_solve(m, b, tolerance, max_iter); Array_double *solution = slice_element(solution_with_opr_count, solution_with_opr_count->size - 1); for (size_t i = 0; i < solution->size; i++) errs[0] += fabs(solution->data[i] - 1.0); oprs[0] = solution_with_opr_count->data[solution_with_opr_count->size - 1]; free_vector(solution); free_vector(solution_with_opr_count); } // GAUSS-SIEDEL { Array_double *solution_with_opr_count = gauss_siedel_solve(m, b, tolerance, max_iter); Array_double *solution = slice_element(solution_with_opr_count, solution_with_opr_count->size - 1); for (size_t i = 0; i < solution->size; i++) errs[1] += fabs(solution->data[i] - 1.0); oprs[1] = solution_with_opr_count->data[solution_with_opr_count->size - 1]; free_vector(solution); free_vector(solution_with_opr_count); } // LU-BSUBST { Array_double *solution_with_opr_count = solve_matrix_lu_bsubst(m, b); Array_double *solution = slice_element(solution_with_opr_count, solution_with_opr_count->size - 1); for (size_t i = 0; i < solution->size; i++) errs[2] += fabs(solution->data[i] - 1.0); oprs[2] = solution_with_opr_count->data[solution_with_opr_count->size - 1]; free_vector(solution); free_vector(solution_with_opr_count); } free_matrix(m); free_vector(b_1); free_vector(b); printf("| %zu | %f | %f | %f | %f | %f | %f | \n", i, oprs[0], errs[0], oprs[1], errs[1], oprs[2], errs[2]); } } #+END_SRC