hw8
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@ -4,11 +4,9 @@
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#+LATEX: \setlength\parindent{0pt}
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#+LATEX: \setlength\parindent{0pt}
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#+OPTIONS: toc:nil
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#+OPTIONS: toc:nil
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TODO: Update LIZFCM org file with jacobi solve
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* Question One
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* Question One
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See ~UTEST(jacobi, solve_jacobi)~ in ~test/jacobi.t.c~ and the entry
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See ~UTEST(jacobi, solve_jacobi)~ in ~test/jacobi.t.c~ and the entry
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~Jacobi -> solve_jacobi~ in the LIZFCM API documentation.
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~Jacobi / Gauss-Siedel -> solve_jacobi~ in the LIZFCM API documentation.
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* Question Two
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* Question Two
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We cannot just perform the Jacobi algorithm on a Leslie matrix since
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We cannot just perform the Jacobi algorithm on a Leslie matrix since
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it is obviously not diagonally dominant - which is a requirement. It is
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it is obviously not diagonally dominant - which is a requirement. It is
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@ -24,14 +22,290 @@ direct solution method. It's simply the nature of the Jacobi algorithm being
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a convergent solution that determines its accuracy.
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a convergent solution that determines its accuracy.
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LU factorization also performs in order $O(n^3)$ runtime for an $n \times n$
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LU factorization also performs in order $O(n^3)$ runtime for an $n \times n$
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matrix, whereas the Jacobi algorithm runs in order $O(k n^2) = O(n^2)$ but with the
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matrix, whereas the Jacobi algorithm runs in order $O(k n^2) = O(n^2)$ on average
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con that $k$ is given by the convergence criteria, which might end up worse in
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but with the con that $k$ is given by some function on both the convergence criteria and the number of
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some cases, than LU.
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nonzero entries in the matrix - which might end up worse in some cases than the LU decomp approach.
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* Question Three
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* Question Three
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See ~UTEST(jacobi, gauss_siedel_solve)~ in ~test/jacobi.t.c~ which runs the same
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See ~UTEST(jacobi, gauss_siedel_solve)~ in ~test/jacobi.t.c~ which runs the same
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unit test as ~UTEST(jacobi, solve_jacobi)~ but using the
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unit test as ~UTEST(jacobi, solve_jacobi)~ but using the
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~Jacobi -> gauss_siedel_solve~ method as documented in the LIZFCM API reference.
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~Jacobi / Gauss-Siedel -> gauss_siedel_solve~ method as documented in the LIZFCM API reference.
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* Question Four, Five
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* Question Four, Five
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We produce the following operation counts (by hackily adding the operation count as the last element
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to the solution vector) and errors - the sum of each vector elements' absolute value away from 1.0
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using the proceeding patch and unit test.
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| N | JAC opr | JAC err | GS opr | GS err | LU opr | LU err |
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| 5 | 1622 | 0.001244 | 577 | 0.000098 | 430 | 0.000000 |
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| 6 | 2812 | 0.001205 | 775 | 0.000080 | 681 | 0.000000 |
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| 7 | 5396 | 0.001187 | 860 | 0.000178 | 1015 | 0.000000 |
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| 8 | 5618 | 0.001468 | 1255 | 0.000121 | 1444 | 0.000000 |
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| 9 | 7534 | 0.001638 | 1754 | 0.000091 | 1980 | 0.000000 |
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| 10 | 10342 | 0.001425 | 1847 | 0.000435 | 2635 | 0.000000 |
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| 11 | 12870 | 0.001595 | 2185 | 0.000368 | 3421 | 0.000000 |
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| 12 | 17511 | 0.001860 | 2912 | 0.000322 | 4350 | 0.000000 |
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| 13 | 16226 | 0.001631 | 3362 | 0.000270 | 5434 | 0.000000 |
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| 14 | 34333 | 0.001976 | 3844 | 0.000121 | 6685 | 0.000000 |
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| 15 | 38474 | 0.001922 | 4358 | 0.000311 | 8115 | 0.000000 |
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| 16 | 40405 | 0.002061 | 4904 | 0.000204 | 9736 | 0.000000 |
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| 17 | 58518 | 0.002125 | 5482 | 0.000311 | 11560 | 0.000000 |
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| 18 | 68079 | 0.002114 | 6092 | 0.000279 | 13599 | 0.000000 |
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| 19 | 95802 | 0.002159 | 6734 | 0.000335 | 15865 | 0.000000 |
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| 20 | 85696 | 0.002141 | 7408 | 0.000289 | 18370 | 0.000000 |
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| 21 | 89026 | 0.002316 | 8114 | 0.000393 | 21126 | 0.000000 |
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| 22 | 101537 | 0.002344 | 8852 | 0.000414 | 24145 | 0.000000 |
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| 23 | 148040 | 0.002323 | 9622 | 0.000230 | 27439 | 0.000000 |
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| 24 | 137605 | 0.002348 | 10424 | 0.000213 | 31020 | 0.000000 |
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| 25 | 169374 | 0.002409 | 11258 | 0.000894 | 34900 | 0.000000 |
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| 26 | 215166 | 0.002502 | 12124 | 0.000564 | 39091 | 0.000000 |
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| 27 | 175476 | 0.002616 | 13022 | 0.000535 | 43605 | 0.000000 |
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| 28 | 268454 | 0.002651 | 13952 | 0.000690 | 48454 | 0.000000 |
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| 29 | 267034 | 0.002697 | 14914 | 0.000675 | 53650 | 0.000000 |
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| 30 | 277193 | 0.002686 | 15908 | 0.000542 | 59205 | 0.000000 |
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| 31 | 336792 | 0.002736 | 16934 | 0.000390 | 65131 | 0.000000 |
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| 32 | 293958 | 0.002741 | 17992 | 0.000660 | 71440 | 0.000000 |
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| 33 | 323638 | 0.002893 | 19082 | 0.001072 | 78144 | 0.000000 |
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| 34 | 375104 | 0.003001 | 20204 | 0.001018 | 85255 | 0.000000 |
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| 35 | 436092 | 0.003004 | 21358 | 0.000912 | 92785 | 0.000000 |
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| 36 | 538143 | 0.003005 | 22544 | 0.000954 | 100746 | 0.000000 |
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| 37 | 511886 | 0.003029 | 23762 | 0.000462 | 109150 | 0.000000 |
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| 38 | 551332 | 0.003070 | 25012 | 0.000996 | 118009 | 0.000000 |
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| 39 | 592750 | 0.003110 | 26294 | 0.000989 | 127335 | 0.000000 |
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| 40 | 704208 | 0.003165 | 27608 | 0.000583 | 137140 | 0.000000 |
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#+BEGIN_SRC
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diff --git a/src/matrix.c b/src/matrix.c
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index 901a426..af5529f 100644
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--- a/src/matrix.c
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+++ b/src/matrix.c
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@@ -144,20 +144,54 @@ Array_double *solve_matrix_lu_bsubst(Matrix_double *m, Array_double *b) {
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assert(b->size == m->rows);
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assert(m->rows == m->cols);
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+ double opr = 0;
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+
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+ opr += b->size;
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Array_double *x = copy_vector(b);
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+
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+ size_t n = m->rows;
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+ opr += n * n; // (u copy)
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+ opr += n * n; // l_empty
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+ opr += n * n + n; // copy + put_identity_diagonal
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+ opr += n; // pivot check
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+ opr += m->cols;
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+ for (size_t x = 0; x < m->cols; x++) {
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+ opr += (m->rows - (x + 1));
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+ for (size_t y = x + 1; y < m->rows; y++) {
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+ opr += 1;
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+ opr += 2; // -factor
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+ opr += 4 * n; // scale, add_v, free_vector
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+ opr += 1; // -factor
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+ }
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+ }
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+ opr += n;
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Matrix_double **u_l = lu_decomp(m);
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+
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Matrix_double *u = u_l[0];
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Matrix_double *l = u_l[1];
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+ opr += n;
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+ for (int64_t row = n - 1; row >= 0; row--) {
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+ opr += 2 * (n - row);
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+ opr += 1;
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+ }
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Array_double *b_fsub = fsubst(l, b);
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+
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+ opr += n;
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+ for (size_t x = 0; x < n; x++) {
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+ opr += 2 * (x + 1);
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+ opr += 1; // /= l->data[row]->data[row]
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+ }
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x = bsubst(u, b_fsub);
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- free_vector(b_fsub);
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+ free_vector(b_fsub);
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free_matrix(u);
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free_matrix(l);
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free(u_l);
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- return x;
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+ Array_double *copy = add_element(x, opr);
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+ free_vector(x);
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+ return copy;
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}
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Matrix_double *gaussian_elimination(Matrix_double *m) {
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@@ -231,18 +265,36 @@ Array_double *jacobi_solve(Matrix_double *m, Array_double *b,
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assert(b->size == m->cols);
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size_t iter = max_iterations;
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+ double opr = 0;
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+
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+ opr += 2 * b->size; // to initialize two vectors with the same dim of b twice
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Array_double *x_k = InitArrayWithSize(double, b->size, 0.0);
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Array_double *x_k_1 =
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InitArrayWithSize(double, b->size, rand_from(0.1, 10.0));
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+ // add since these wouldn't be accounter for after the loop
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+ opr += 1; // iter decrement
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+ opr +=
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+ 3 * x_k_1->size; // 1 to perform x_k_1, x_k and 2 to perform ||x_k_1||_2
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while ((--iter) > 0 && l2_distance(x_k_1, x_k) > l2_convergence_tolerance) {
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+ opr += 1; // iter decrement
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+ opr +=
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+ 3 * x_k_1->size; // 1 to perform x_k_1, x_k and 2 to perform ||x_k_1||_2
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+
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+ opr += m->rows; // row for add oprs
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for (size_t i = 0; i < m->rows; i++) {
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double delta = 0.0;
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+
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+ opr += m->cols;
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for (size_t j = 0; j < m->cols; j++) {
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if (i == j)
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continue;
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+
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+ opr += 1;
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delta += m->data[i]->data[j] * x_k->data[j];
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}
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+
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+ opr += 2;
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x_k_1->data[i] = (b->data[i] - delta) / m->data[i]->data[i];
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}
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@@ -251,8 +303,9 @@ Array_double *jacobi_solve(Matrix_double *m, Array_double *b,
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x_k_1 = tmp;
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}
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- free_vector(x_k);
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- return x_k_1;
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+ Array_double *copy = add_element(x_k_1, opr);
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+ free_vector(x_k_1);
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+ return copy;
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}
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Array_double *gauss_siedel_solve(Matrix_double *m, Array_double *b,
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@@ -262,30 +315,48 @@ Array_double *gauss_siedel_solve(Matrix_double *m, Array_double *b,
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assert(b->size == m->cols);
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size_t iter = max_iterations;
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+ double opr = 0;
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+
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+ opr += 2 * b->size; // to initialize two vectors with the same dim of b twice
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Array_double *x_k = InitArrayWithSize(double, b->size, 0.0);
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Array_double *x_k_1 =
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InitArrayWithSize(double, b->size, rand_from(0.1, 10.0));
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while ((--iter) > 0) {
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+ opr += 1; // iter decrement
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+
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+ opr += x_k->size; // copy oprs
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for (size_t i = 0; i < x_k->size; i++)
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x_k->data[i] = x_k_1->data[i];
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+ opr += m->rows; // row for add oprs
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for (size_t i = 0; i < m->rows; i++) {
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double delta = 0.0;
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+
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+ opr += m->cols;
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for (size_t j = 0; j < m->cols; j++) {
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if (i == j)
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continue;
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+
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+ opr += 1;
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delta += m->data[i]->data[j] * x_k_1->data[j];
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}
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+
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+ opr += 2;
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x_k_1->data[i] = (b->data[i] - delta) / m->data[i]->data[i];
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}
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+ opr +=
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+ 3 * x_k_1->size; // 1 to perform x_k_1, x_k and 2 to perform ||x_k_1||_2
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if (l2_distance(x_k_1, x_k) <= l2_convergence_tolerance)
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break;
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}
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free_vector(x_k);
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- return x_k_1;
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+
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+ Array_double *copy = add_element(x_k_1, opr);
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+ free_vector(x_k_1);
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+ return copy;
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}
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#+END_SRC
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And this unit test:
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#+BEGIN_SRC c
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UTEST(hw_8, p4_5) {
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printf("| N | JAC opr | JAC err | GS opr | GS err | LU opr | LU err | \n");
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for (size_t i = 5; i < 100; i++) {
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Matrix_double *m = generate_ddm(i);
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double oprs[3] = {0.0, 0.0, 0.0};
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double errs[3] = {0.0, 0.0, 0.0};
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Array_double *b_1 = InitArrayWithSize(double, m->rows, 1.0);
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Array_double *b = m_dot_v(m, b_1);
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double tolerance = 0.001;
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size_t max_iter = 400;
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// JACOBI
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{
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Array_double *solution_with_opr_count =
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jacobi_solve(m, b, tolerance, max_iter);
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Array_double *solution = slice_element(solution_with_opr_count,
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solution_with_opr_count->size - 1);
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for (size_t i = 0; i < solution->size; i++)
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errs[0] += fabs(solution->data[i] - 1.0);
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oprs[0] =
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solution_with_opr_count->data[solution_with_opr_count->size - 1];
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free_vector(solution);
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free_vector(solution_with_opr_count);
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}
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// GAUSS-SIEDEL
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{
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Array_double *solution_with_opr_count =
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gauss_siedel_solve(m, b, tolerance, max_iter);
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Array_double *solution = slice_element(solution_with_opr_count,
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solution_with_opr_count->size - 1);
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for (size_t i = 0; i < solution->size; i++)
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errs[1] += fabs(solution->data[i] - 1.0);
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oprs[1] =
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solution_with_opr_count->data[solution_with_opr_count->size - 1];
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free_vector(solution);
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free_vector(solution_with_opr_count);
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}
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// LU-BSUBST
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{
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Array_double *solution_with_opr_count = solve_matrix_lu_bsubst(m, b);
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Array_double *solution = slice_element(solution_with_opr_count,
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solution_with_opr_count->size - 1);
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for (size_t i = 0; i < solution->size; i++)
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errs[2] += fabs(solution->data[i] - 1.0);
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oprs[2] =
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solution_with_opr_count->data[solution_with_opr_count->size - 1];
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free_vector(solution);
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free_vector(solution_with_opr_count);
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}
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free_matrix(m);
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free_vector(b_1);
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free_vector(b);
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printf("| %zu | %f | %f | %f | %f | %f | %f | \n", i, oprs[0], errs[0],
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oprs[1], errs[1], oprs[2], errs[2]);
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}
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}
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#+END_SRC
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BIN
homeworks/hw-8.pdf
Normal file
BIN
homeworks/hw-8.pdf
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Binary file not shown.
344
homeworks/hw-8.tex
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344
homeworks/hw-8.tex
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@ -0,0 +1,344 @@
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% Created 2023-12-09 Sat 21:43
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% Intended LaTeX compiler: pdflatex
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\documentclass[11pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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|
\usepackage{longtable}
|
||||||
|
\usepackage{wrapfig}
|
||||||
|
\usepackage{rotating}
|
||||||
|
\usepackage[normalem]{ulem}
|
||||||
|
\usepackage{amsmath}
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage{capt-of}
|
||||||
|
\usepackage{hyperref}
|
||||||
|
\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
|
||||||
|
\author{Elizabeth Hunt}
|
||||||
|
\date{\today}
|
||||||
|
\title{Homework 7}
|
||||||
|
\hypersetup{
|
||||||
|
pdfauthor={Elizabeth Hunt},
|
||||||
|
pdftitle={Homework 7},
|
||||||
|
pdfkeywords={},
|
||||||
|
pdfsubject={},
|
||||||
|
pdfcreator={Emacs 28.2 (Org mode 9.7-pre)},
|
||||||
|
pdflang={English}}
|
||||||
|
\begin{document}
|
||||||
|
|
||||||
|
\maketitle
|
||||||
|
\setlength\parindent{0pt}
|
||||||
|
|
||||||
|
\section{Question One}
|
||||||
|
\label{sec:orgb6d5cda}
|
||||||
|
See \texttt{UTEST(jacobi, solve\_jacobi)} in \texttt{test/jacobi.t.c} and the entry
|
||||||
|
\texttt{Jacobi / Gauss-Siedel -> solve\_jacobi} in the LIZFCM API documentation.
|
||||||
|
\section{Question Two}
|
||||||
|
\label{sec:org9786314}
|
||||||
|
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 \texttt{UTEST(jacobi, leslie\_solve)}
|
||||||
|
in \texttt{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.
|
||||||
|
|
||||||
|
\section{Question Three}
|
||||||
|
\label{sec:org0ea87d0}
|
||||||
|
See \texttt{UTEST(jacobi, gauss\_siedel\_solve)} in \texttt{test/jacobi.t.c} which runs the same
|
||||||
|
unit test as \texttt{UTEST(jacobi, solve\_jacobi)} but using the
|
||||||
|
\texttt{Jacobi / Gauss-Siedel -> gauss\_siedel\_solve} method as documented in the LIZFCM API reference.
|
||||||
|
|
||||||
|
\section{Question Four, Five}
|
||||||
|
\label{sec:org8eea2ae}
|
||||||
|
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.
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\begin{tabular}{rrrrrrr}
|
||||||
|
N & JAC opr & JAC err & GS opr & GS err & LU opr & LU err\\[0pt]
|
||||||
|
5 & 1622 & 0.001244 & 577 & 0.000098 & 430 & 0.000000\\[0pt]
|
||||||
|
6 & 2812 & 0.001205 & 775 & 0.000080 & 681 & 0.000000\\[0pt]
|
||||||
|
7 & 5396 & 0.001187 & 860 & 0.000178 & 1015 & 0.000000\\[0pt]
|
||||||
|
8 & 5618 & 0.001468 & 1255 & 0.000121 & 1444 & 0.000000\\[0pt]
|
||||||
|
9 & 7534 & 0.001638 & 1754 & 0.000091 & 1980 & 0.000000\\[0pt]
|
||||||
|
10 & 10342 & 0.001425 & 1847 & 0.000435 & 2635 & 0.000000\\[0pt]
|
||||||
|
11 & 12870 & 0.001595 & 2185 & 0.000368 & 3421 & 0.000000\\[0pt]
|
||||||
|
12 & 17511 & 0.001860 & 2912 & 0.000322 & 4350 & 0.000000\\[0pt]
|
||||||
|
13 & 16226 & 0.001631 & 3362 & 0.000270 & 5434 & 0.000000\\[0pt]
|
||||||
|
14 & 34333 & 0.001976 & 3844 & 0.000121 & 6685 & 0.000000\\[0pt]
|
||||||
|
15 & 38474 & 0.001922 & 4358 & 0.000311 & 8115 & 0.000000\\[0pt]
|
||||||
|
16 & 40405 & 0.002061 & 4904 & 0.000204 & 9736 & 0.000000\\[0pt]
|
||||||
|
17 & 58518 & 0.002125 & 5482 & 0.000311 & 11560 & 0.000000\\[0pt]
|
||||||
|
18 & 68079 & 0.002114 & 6092 & 0.000279 & 13599 & 0.000000\\[0pt]
|
||||||
|
19 & 95802 & 0.002159 & 6734 & 0.000335 & 15865 & 0.000000\\[0pt]
|
||||||
|
20 & 85696 & 0.002141 & 7408 & 0.000289 & 18370 & 0.000000\\[0pt]
|
||||||
|
21 & 89026 & 0.002316 & 8114 & 0.000393 & 21126 & 0.000000\\[0pt]
|
||||||
|
22 & 101537 & 0.002344 & 8852 & 0.000414 & 24145 & 0.000000\\[0pt]
|
||||||
|
23 & 148040 & 0.002323 & 9622 & 0.000230 & 27439 & 0.000000\\[0pt]
|
||||||
|
24 & 137605 & 0.002348 & 10424 & 0.000213 & 31020 & 0.000000\\[0pt]
|
||||||
|
25 & 169374 & 0.002409 & 11258 & 0.000894 & 34900 & 0.000000\\[0pt]
|
||||||
|
26 & 215166 & 0.002502 & 12124 & 0.000564 & 39091 & 0.000000\\[0pt]
|
||||||
|
27 & 175476 & 0.002616 & 13022 & 0.000535 & 43605 & 0.000000\\[0pt]
|
||||||
|
28 & 268454 & 0.002651 & 13952 & 0.000690 & 48454 & 0.000000\\[0pt]
|
||||||
|
29 & 267034 & 0.002697 & 14914 & 0.000675 & 53650 & 0.000000\\[0pt]
|
||||||
|
30 & 277193 & 0.002686 & 15908 & 0.000542 & 59205 & 0.000000\\[0pt]
|
||||||
|
31 & 336792 & 0.002736 & 16934 & 0.000390 & 65131 & 0.000000\\[0pt]
|
||||||
|
32 & 293958 & 0.002741 & 17992 & 0.000660 & 71440 & 0.000000\\[0pt]
|
||||||
|
33 & 323638 & 0.002893 & 19082 & 0.001072 & 78144 & 0.000000\\[0pt]
|
||||||
|
34 & 375104 & 0.003001 & 20204 & 0.001018 & 85255 & 0.000000\\[0pt]
|
||||||
|
35 & 436092 & 0.003004 & 21358 & 0.000912 & 92785 & 0.000000\\[0pt]
|
||||||
|
36 & 538143 & 0.003005 & 22544 & 0.000954 & 100746 & 0.000000\\[0pt]
|
||||||
|
37 & 511886 & 0.003029 & 23762 & 0.000462 & 109150 & 0.000000\\[0pt]
|
||||||
|
38 & 551332 & 0.003070 & 25012 & 0.000996 & 118009 & 0.000000\\[0pt]
|
||||||
|
39 & 592750 & 0.003110 & 26294 & 0.000989 & 127335 & 0.000000\\[0pt]
|
||||||
|
40 & 704208 & 0.003165 & 27608 & 0.000583 & 137140 & 0.000000\\[0pt]
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
\begin{verbatim}
|
||||||
|
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{verbatim}
|
||||||
|
|
||||||
|
|
||||||
|
And this unit test:
|
||||||
|
\begin{verbatim}
|
||||||
|
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{verbatim}
|
||||||
|
\end{document}
|
||||||
Loading…
Reference in New Issue
Block a user