354 lines
10 KiB
C
354 lines
10 KiB
C
#include <stdio.h>
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#include <stdlib.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_eigen.h>
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#include <gsl/gsl_blas.h>
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#include <gsl/gsl_linalg.h>
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#include <gsl/gsl_complex_math.h>
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#include <memory.h>
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#include <math.h>
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#include "linalg.h"
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#define ERROR(condition, msg, ...) if(condition){fprintf(stderr, msg, ##__VA_ARGS__); exit(1);}
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#define FCMP(x, y) gsl_fcmp(x, y, 1e-10)
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/*********************************************** temporary storage ********************************************************/
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workspace_t *workspace_alloc(int n)
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{
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workspace_t *result = (workspace_t*)malloc(sizeof(workspace_t));
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result->n = n;
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result->work_nonsymmv = gsl_eigen_nonsymmv_alloc(n);
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result->work_symmv = gsl_eigen_symmv_alloc(n);
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result->work_sv = gsl_vector_alloc(n);
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result->eval_complex = gsl_vector_complex_alloc(n);
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result->evec_complex = gsl_matrix_complex_alloc(n, n);
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result->eval_real = gsl_vector_alloc(n);
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result->evec_real = gsl_matrix_alloc(n, n);
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result->permutation = gsl_permutation_alloc(n);
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result->tmp_mat = malloc(MAX_TEMP_MATRICES*sizeof(gsl_matrix*));
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for(int i = 0; i < MAX_TEMP_MATRICES; i++)
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result->tmp_mat[i] = gsl_matrix_alloc(3, 3);
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result->tmp_mat_used = 0;
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result->tmp_vec = malloc(MAX_TEMP_MATRICES*sizeof(gsl_vector*));
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for(int i = 0; i < MAX_TEMP_MATRICES; i++)
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result->tmp_vec[i] = gsl_vector_alloc(3);
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result->tmp_vec_used = 0;
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return result;
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}
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void workspace_free(workspace_t *workspace)
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{
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gsl_eigen_nonsymmv_free(workspace->work_nonsymmv);
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gsl_eigen_symmv_free(workspace->work_symmv);
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gsl_vector_free(workspace->work_sv);
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gsl_vector_complex_free(workspace->eval_complex);
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gsl_matrix_complex_free(workspace->evec_complex);
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gsl_vector_free(workspace->eval_real);
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gsl_matrix_free(workspace->evec_real);
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gsl_permutation_free(workspace->permutation);
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for(int i = 0; i < MAX_TEMP_MATRICES; i++)
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gsl_matrix_free(workspace->tmp_mat[i]);
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free(workspace->tmp_mat);
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for(int i = 0; i < MAX_TEMP_VECTORS; i++)
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gsl_vector_free(workspace->tmp_vec[i]);
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free(workspace->tmp_vec);
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}
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/************************************************** basic operations ********************************************************/
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void invert(gsl_matrix *in, gsl_matrix *out, workspace_t *ws)
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{
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int s;
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_matrix_memcpy(tmp, in);
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gsl_linalg_LU_decomp(tmp, ws->permutation, &s);
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gsl_linalg_LU_invert(tmp, ws->permutation, out);
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releaseTempMatrices(ws, 1);
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}
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void solve(gsl_matrix *A, gsl_vector *b, gsl_vector *result, workspace_t *ws)
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{
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int s;
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_matrix_memcpy(tmp, A);
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gsl_linalg_LU_decomp(tmp, ws->permutation, &s);
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gsl_linalg_LU_solve(tmp, ws->permutation, b, result);
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releaseTempMatrices(ws, 1);
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}
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void conjugate(gsl_matrix *in, gsl_matrix *conjugator, gsl_matrix *out, workspace_t *ws)
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{
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gsl_matrix *tmp = getTempMatrix(ws);
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invert(conjugator, out, ws); // use out to temporarily store inverse conjugator
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gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, in, out, 0.0, tmp); // in * conjugator^{-1}
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gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, conjugator, tmp, 0.0, out);
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releaseTempMatrices(ws, 1);
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}
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void multiply(gsl_matrix *a, gsl_matrix *b, gsl_matrix *out)
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{
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gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, a, b, 0.0, out);
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}
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void multiply_right(gsl_matrix *a, gsl_matrix *b, workspace_t *ws)
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{
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, a, b, 0.0, tmp);
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gsl_matrix_memcpy(a, tmp);
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releaseTempMatrices(ws, 1);
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}
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void multiply_left(gsl_matrix *a, gsl_matrix *b, workspace_t *ws)
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{
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, a, b, 0.0, tmp);
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gsl_matrix_memcpy(b, tmp);
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releaseTempMatrices(ws, 1);
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}
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void multiply_many(workspace_t *ws, gsl_matrix *out, int n, ...)
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{
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va_list args;
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va_start(args, n);
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gsl_matrix_set_identity(out);
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for(int i = 0; i < n; i++) {
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gsl_matrix *cur = va_arg(args, gsl_matrix *);
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multiply_right(out, cur, ws);
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}
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va_end(args);
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}
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void cartan_calc(gsl_matrix *g, double *mu, workspace_t *ws)
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{
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_matrix_memcpy(tmp, g);
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gsl_linalg_SV_decomp(tmp, ws->evec_real, ws->eval_real, ws->work_sv);
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for(int i = 0; i < ws->n - 1; i++)
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mu[i] = log(gsl_vector_get(ws->eval_real, i) / gsl_vector_get(ws->eval_real, i+1));
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releaseTempMatrices(ws, 1);
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}
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void initialize(gsl_matrix *g, double *data, int x, int y)
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{
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gsl_matrix_view view = gsl_matrix_view_array(data, x, y);
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gsl_matrix_memcpy(g, &view.matrix);
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}
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void rotation_matrix(gsl_matrix *g, double *vector)
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{
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double normalized[3];
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double norm = sqrt(vector[0]*vector[0] + vector[1]*vector[1] + vector[2]*vector[2]);
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for(int i = 0; i < 3; i++)
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normalized[i] = vector[i] / norm;
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gsl_matrix_set_identity(g);
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gsl_matrix_set(g, 0, 0, cos(norm));
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gsl_matrix_set(g, 0, 1, -sin(norm) * normalized[2]);
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gsl_matrix_set(g, 0, 2, +sin(norm) * normalized[1]);
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gsl_matrix_set(g, 1, 0, +sin(norm) * normalized[2]);
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gsl_matrix_set(g, 1, 1, cos(norm));
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gsl_matrix_set(g, 1, 2, -sin(norm) * normalized[0]);
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gsl_matrix_set(g, 2, 0, -sin(norm) * normalized[1]);
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gsl_matrix_set(g, 2, 1, +sin(norm) * normalized[0]);
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gsl_matrix_set(g, 2, 2, cos(norm));
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for(int i = 0; i < 3; i++)
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for(int j = 0; j < 3; j++)
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g->data[i * g->tda + j] += (1 - cos(norm)) * normalized[i] * normalized[j];
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}
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double trace(gsl_matrix *g)
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{
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return gsl_matrix_get(g, 0, 0) + gsl_matrix_get(g, 1, 1) + gsl_matrix_get(g, 2, 2);
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}
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double determinant(gsl_matrix *g, workspace_t *ws)
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{
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int s;
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double result;
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_matrix_memcpy(tmp, g);
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gsl_linalg_LU_decomp(tmp, ws->permutation, &s);
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result = gsl_linalg_LU_det(tmp, s);
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releaseTempMatrices(ws, 1);
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return result;
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}
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int eigenvectors(gsl_matrix *g, gsl_matrix *evec_real, workspace_t *ws)
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{
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gsl_matrix *g_ = getTempMatrix(ws);
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int success = 0;
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gsl_matrix_memcpy(g_, g);
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gsl_eigen_nonsymmv_params(0, ws->work_nonsymmv);
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int r = gsl_eigen_nonsymmv(g_, ws->eval_complex, ws->evec_complex, ws->work_nonsymmv);
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ERROR(r, "gsl_eigen_nonsymmv failed!\n");
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gsl_eigen_nonsymmv_sort(ws->eval_complex, ws->evec_complex, GSL_EIGEN_SORT_ABS_DESC);
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int real = 1;
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for(int i = 0; i < ws->n; i++)
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if(FCMP(GSL_IMAG(gsl_vector_complex_get(ws->eval_complex, i)), 0) != 0)
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real = 0;
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if(!real)
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goto eigenvectors_out;
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for(int i = 0; i < ws->n; i++)
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for(int j = 0; j < ws->n; j++)
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gsl_matrix_set(evec_real, i, j, GSL_REAL(gsl_matrix_complex_get(ws->evec_complex, i, j)));
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success = 1;
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eigenvectors_out:
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releaseTempMatrices(ws, 1);
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return success;
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}
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// only fills in the real eigenvectors and returns their count
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int real_eigenvectors(gsl_matrix *g, gsl_matrix *evec_real, workspace_t *ws)
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{
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gsl_matrix *g_ = getTempMatrix(ws);
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gsl_matrix_memcpy(g_, g);
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gsl_eigen_nonsymmv_params(0, ws->work_nonsymmv);
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int r = gsl_eigen_nonsymmv(g_, ws->eval_complex, ws->evec_complex, ws->work_nonsymmv);
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ERROR(r, "gsl_eigen_nonsymmv failed!\n");
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gsl_eigen_nonsymmv_sort(ws->eval_complex, ws->evec_complex, GSL_EIGEN_SORT_ABS_DESC);
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int real = 0;
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for(int i = 0; i < ws->n; i++) {
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if(FCMP(GSL_IMAG(gsl_vector_complex_get(ws->eval_complex, i)), 0) == 0) {// real
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if(evec_real) {
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for(int j = 0; j < ws->n; j++)
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gsl_matrix_set(evec_real, j, real, GSL_REAL(gsl_matrix_complex_get(ws->evec_complex, j, i)));
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}
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real++;
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}
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}
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releaseTempMatrices(ws, 1);
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return real;
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}
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void eigensystem_symm(gsl_matrix *g, gsl_vector *eval, gsl_matrix *evec, workspace_t *ws)
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{
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gsl_matrix *g_ = getTempMatrix(ws);
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gsl_matrix_memcpy(g_, g);
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int r = gsl_eigen_symmv (g_, eval, evec, ws->work_symmv);
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ERROR(r, "gsl_eigen_symmv failed!\n");
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gsl_eigen_symmv_sort(eval, evec, GSL_EIGEN_SORT_ABS_DESC);
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releaseTempMatrices(ws, 1);
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}
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// returns number of positive directions and matrix transforming TO diagonal basis
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int diagonalize_symmetric_form(gsl_matrix *A, gsl_matrix *cob, workspace_t *ws)
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{
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gsl_matrix *A_ = getTempMatrix(ws);
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gsl_matrix_memcpy(A_, A);
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int r = gsl_eigen_symmv (A_, ws->eval_real, cob, ws->work_symmv);
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ERROR(r, "gsl_eigen_symmv failed!\n");
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gsl_eigen_symmv_sort(ws->eval_real, cob, GSL_EIGEN_SORT_VAL_ASC);
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gsl_matrix_transpose(cob);
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int positive = 0;
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for(int i = 0; i < ws->n; i++) {
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if(gsl_vector_get(ws->eval_real, i) > 0)
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positive++;
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for(int j = 0; j < ws->n; j++)
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*gsl_matrix_ptr(cob, i, j) *= sqrt(fabs(gsl_vector_get(ws->eval_real, i)));
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}
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releaseTempMatrices(ws, 1);
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return positive;
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}
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// computes a matrix in SL(3, R) which projectively transforms (e1, e2, e3, e1+e2+e3) to the 4 given vectors
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void projective_frame(gsl_vector **vertices, gsl_matrix *result, workspace_t *ws)
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{
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_vector *coeff = getTempVector(ws);
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int s;
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double det, scale;
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for(int i = 0; i < 3; i++)
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for(int j = 0; j < 3; j++)
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gsl_matrix_set(tmp, i, j, gsl_vector_get(vertices[j], i));
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gsl_linalg_LU_decomp(tmp, ws->permutation, &s);
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gsl_linalg_LU_solve(tmp, ws->permutation, vertices[3], coeff);
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det = gsl_linalg_LU_det(tmp, s);
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for(int i = 0; i < 3; i++)
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det *= gsl_vector_get(coeff, i);
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scale = 1/cbrt(det);
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for(int i = 0; i < 3; i++)
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for(int j = 0; j < 3; j++)
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gsl_matrix_set(result, i, j, scale*gsl_vector_get(vertices[j], i)*gsl_vector_get(coeff, j));
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releaseTempMatrices(ws, 1);
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releaseTempVectors(ws, 1);
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}
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void rotation_frame(gsl_matrix *rotation, gsl_matrix *result, workspace_t *ws)
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{
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gsl_matrix *tmp = getTempMatrix(ws);
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gsl_matrix *rot_basis = getTempMatrix(ws);
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gsl_matrix_memcpy(tmp, rotation);
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gsl_eigen_nonsymmv_params(0, ws->work_nonsymmv);
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int r = gsl_eigen_nonsymmv(tmp, ws->eval_complex, ws->evec_complex, ws->work_nonsymmv);
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ERROR(r, "gsl_eigen_nonsymmv failed!\n");
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double arg, minarg = 5; // greater than pi
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int minidx;
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for(int i = 0; i < 3; i++) {
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arg = gsl_complex_arg(gsl_vector_complex_get(ws->eval_complex, i));
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if(abs(arg) < minarg)
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{
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minidx = i;
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minarg = abs(arg);
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}
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}
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ERROR(FCMP(minarg, 0.0) != 0, "rotation_frame() failed! No eigenvalue was 1.\n");
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for(int i = 0; i < 3; i++) {
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gsl_complex x = gsl_matrix_complex_get(ws->evec_complex, i, (minidx+1)%3);
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gsl_complex y = gsl_matrix_complex_get(ws->evec_complex, i, (minidx+2)%3);
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gsl_complex z = gsl_matrix_complex_get(ws->evec_complex, i, minidx);
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gsl_matrix_set(result, i, 0, GSL_REAL(x)+GSL_REAL(y));
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gsl_matrix_set(result, i, 1, GSL_IMAG(x)-GSL_IMAG(y));
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gsl_matrix_set(result, i, 2, GSL_REAL(z));
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}
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releaseTempMatrices(ws, 2);
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}
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