q5 hw7
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@ -1096,7 +1096,44 @@ double least_dominant_eigenvalue(Matrix_double *m, Array_double *v,
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return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations);
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}
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#+END_SRC
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*** ~partition_find_eigenvalues~
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+ Author: Elizabeth Hunt
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+ Name: ~partition_find_eigenvalues~
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+ Location: ~src/eigen.c~
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+ Input: a pointer to an invertible matrix ~m~, a matrix whose rows correspond to initial
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eigenvector guesses at each "partition" which is computed from a uniform distribution
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between the number of rows this "guess matrix" has and the distance between the least
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dominant eigenvalue and the most dominant. Additionally, a ~max_iterations~ and a ~tolerance~
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that act as stop conditions.
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+ Output: a vector of ~doubles~ corresponding to the "nearest" eigenvalue at the midpoint of
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each partition, via the given guess of that partition.
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#+BEGIN_SRC c
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Array_double *partition_find_eigenvalues(Matrix_double *m,
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Matrix_double *guesses,
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double tolerance,
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size_t max_iterations) {
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assert(guesses->rows >=
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2); // we need at least, the most and least dominant eigenvalues
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double end = dominant_eigenvalue(m, guesses->data[guesses->rows - 1],
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tolerance, max_iterations);
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double begin =
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least_dominant_eigenvalue(m, guesses->data[0], tolerance, max_iterations);
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double delta = (end - begin) / guesses->rows;
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Array_double *eigenvalues = InitArrayWithSize(double, guesses->rows, 0.0);
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for (size_t i = 0; i < guesses->rows; i++) {
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double box_midpoint = ((delta * i) + (delta * (i + 1))) / 2;
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double nearest_eigenvalue = shift_inverse_power_eigenvalue(
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m, guesses->data[i], box_midpoint, tolerance, max_iterations);
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eigenvalues->data[i] = nearest_eigenvalue;
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}
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return eigenvalues;
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}
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#+END_SRC
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*** ~leslie_matrix~
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+ Author: Elizabeth Hunt
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+ Name: ~leslie_matrix~
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@ -41,3 +41,21 @@ With the initial guess: $[0.5, 1.0, 0.75]$.
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See also the entry ~Eigen-Adjacent -> shift_inverse_power_eigenvalue~ in the LIZFCM API
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documentation.
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* Question Five
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See ~UTEST(eigen, partition_find_eigenvalues)~ in ~test/eigen.t.c~ which
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finds the eigenvalues in a partition of 10 on the matrix:
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\begin{bmatrix}
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2 & 2 & 4 \\
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1 & 4 & 7 \\
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0 & 2 & 6
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\end{bmatrix}
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which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$, and should produce all three from
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the partitions when given the guesses $[0.5, 1.0, 0.75]$ from the questions above.
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See also the entry ~Eigen-Adjacent -> partition_find_eigenvalues~ in the LIZFCM API
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documentation.
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* Question Six
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@ -82,6 +82,10 @@ extern double shift_inverse_power_eigenvalue(Matrix_double *m, Array_double *v,
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extern double least_dominant_eigenvalue(Matrix_double *m, Array_double *v,
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double tolerance,
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size_t max_iterations);
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extern Array_double *partition_find_eigenvalues(Matrix_double *m,
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Matrix_double *guesses,
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double tolerance,
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size_t max_iterations);
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extern Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
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Array_double *age_class_offspring);
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#endif // LIZFCM_H
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26
src/eigen.c
26
src/eigen.c
@ -84,6 +84,32 @@ double shift_inverse_power_eigenvalue(Matrix_double *m, Array_double *v,
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return lambda;
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}
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Array_double *partition_find_eigenvalues(Matrix_double *m,
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Matrix_double *guesses,
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double tolerance,
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size_t max_iterations) {
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assert(guesses->rows >=
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2); // we need at least, the most and least dominant eigenvalues
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double end = dominant_eigenvalue(m, guesses->data[guesses->rows - 1],
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tolerance, max_iterations);
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double begin =
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least_dominant_eigenvalue(m, guesses->data[0], tolerance, max_iterations);
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double delta = (end - begin) / guesses->rows;
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Array_double *eigenvalues = InitArrayWithSize(double, guesses->rows, 0.0);
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for (size_t i = 0; i < guesses->rows; i++) {
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double box_midpoint = ((delta * i) + (delta * (i + 1))) / 2;
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double nearest_eigenvalue = shift_inverse_power_eigenvalue(
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m, guesses->data[i], box_midpoint, tolerance, max_iterations);
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eigenvalues->data[i] = nearest_eigenvalue;
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}
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return eigenvalues;
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}
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double least_dominant_eigenvalue(Matrix_double *m, Array_double *v,
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double tolerance, size_t max_iterations) {
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return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations);
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@ -1,4 +1,5 @@
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#include "lizfcm.test.h"
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#include <math.h>
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Matrix_double *eigen_test_matrix() {
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// produces a matrix that has eigenvalues [5 + sqrt{17}, 2, 5 - sqrt{17}]
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@ -69,6 +70,40 @@ UTEST(eigen, shifted_eigenvalue) {
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EXPECT_NEAR(approx_middle_eigenvalue, expected_middle_eigenvalue, tolerance);
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}
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UTEST(eigen, partition_find_eigenvalues) {
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Matrix_double *m = eigen_test_matrix();
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double least_dominant_eigenvalue = 0.87689; // 5 - sqrt{17}
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double dominant_eigenvalue = 9.12311; // 5 + sqrt{17}
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double expected_middle_eigenvalue = 2.0;
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double expected_eigenvalues[3] = {least_dominant_eigenvalue,
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expected_middle_eigenvalue,
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dominant_eigenvalue};
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size_t partitions = 10;
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Matrix_double *guesses = InitMatrixWithSize(double, partitions, 3, 0.0);
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for (size_t y = 0; y < guesses->rows; y++) {
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free_vector(guesses->data[y]);
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guesses->data[y] = InitArray(double, {0.5, 1.0, 0.75});
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}
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double tolerance = 0.0001;
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uint64_t max_iterations = 64;
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int eigenvalues_found[3] = {false, false, false};
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Array_double *partition_eigenvalues =
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partition_find_eigenvalues(m, guesses, tolerance, max_iterations);
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for (size_t i = 0; i < partition_eigenvalues->size; i++)
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for (size_t eigenvalue_i = 0; eigenvalue_i < 3; eigenvalue_i++)
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if (fabs(partition_eigenvalues->data[i] - expected_eigenvalues[i]) <=
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tolerance)
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eigenvalues_found[eigenvalue_i] = true;
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for (size_t eigenvalue_i = 0; eigenvalue_i < 3; eigenvalue_i++)
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EXPECT_TRUE(eigenvalues_found[eigenvalue_i]);
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}
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UTEST(eigen, leslie_matrix_dominant_eigenvalue) {
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Array_double *felicity = InitArray(double, {0.0, 1.5, 0.8});
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Array_double *survivor_ratios = InitArray(double, {0.8, 0.55});
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@ -1,7 +1,8 @@
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#include "lizfcm.h"
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#include "utest.h"
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#ifndef LIZFCM_TEST_H
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#define LIZFCM_TEST_H
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#include "lizfcm.h"
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#include "utest.h"
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#endif // LIZFCM_TEST_H
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