triangle_group_limit_set/linalg.c

#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_linalg.h>
#include <memory.h>

#include "linalg.h"

#define ERROR(condition, msg, ...) if(condition){fprintf(stderr, msg, ##__VA_ARGS__); exit(1);}
#define FCMP(x, y) gsl_fcmp(x, y, 1e-10)

/*********************************************** temporary storage ********************************************************/

workspace_t *workspace_alloc(int n)
{
  workspace_t *result = (workspace_t*)malloc(sizeof(workspace_t));
  result->n = n;
  result->work_nonsymmv = gsl_eigen_nonsymmv_alloc(n);
  result->work_symmv = gsl_eigen_symmv_alloc(n);
  result->work_sv = gsl_vector_alloc(n);
  result->eval_complex = gsl_vector_complex_alloc(n);
  result->evec_complex = gsl_matrix_complex_alloc(n, n);
  result->eval_real = gsl_vector_alloc(n);
  result->evec_real = gsl_matrix_alloc(n, n);
  result->tmp = gsl_matrix_alloc(n, n);
  result->permutation = gsl_permutation_alloc(n);
  for(int i = 0; i < 20; i++)
    result->stack[i] = gsl_matrix_alloc(n, n);

  return result;
}

void workspace_free(workspace_t *workspace)
{
  gsl_eigen_nonsymmv_free(workspace->work_nonsymmv);
  gsl_eigen_symmv_free(workspace->work_symmv);
  gsl_vector_free(workspace->work_sv);
  gsl_vector_complex_free(workspace->eval_complex);
  gsl_matrix_complex_free(workspace->evec_complex);
  gsl_vector_free(workspace->eval_real);
  gsl_matrix_free(workspace->evec_real);
  gsl_matrix_free(workspace->tmp);
  gsl_permutation_free(workspace->permutation);
  for(int i = 0; i < 20; i++)
    gsl_matrix_free(workspace->stack[i]);
}

/************************************************** basic operations ********************************************************/

void invert(gsl_matrix *in, gsl_matrix *out, workspace_t *ws)
{
  int s;
  gsl_matrix_memcpy(ws->tmp, in);
  gsl_linalg_LU_decomp(ws->tmp, ws->permutation, &s);
  gsl_linalg_LU_invert(ws->tmp, ws->permutation, out);
}

void conjugate(gsl_matrix *in, gsl_matrix *conjugator, gsl_matrix *out, workspace_t *ws)
{
  invert(conjugator, out, ws);   // use out to temporarily store inverse conjugator
  gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, in, out, 0.0, ws->tmp);  // in * conjugator^{-1}
  gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, conjugator, ws->tmp, 0.0, out);
}

void multiply(gsl_matrix *a, gsl_matrix *b, gsl_matrix *out)
{
  gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, a, b, 0.0, out);
}

void multiply_right(gsl_matrix *a, gsl_matrix *b, workspace_t *ws)
{
	gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, a, b, 0.0, ws->stack[0]);
	gsl_matrix_memcpy(a, ws->stack[0]);
}

void multiply_left(gsl_matrix *a, gsl_matrix *b, workspace_t *ws)
{
	gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, a, b, 0.0, ws->stack[0]);
	gsl_matrix_memcpy(b, ws->stack[0]);
}

void multiply_many(workspace_t *ws, gsl_matrix *out, int n, ...)
{
	va_list args;
    va_start(args, n);

	gsl_matrix_set_identity(out);

	for(int i = 0; i < n; i++) {
		gsl_matrix *cur = va_arg(args, gsl_matrix *);
		multiply_right(out, cur, ws);
	}

    va_end(args);
}

void cartan_calc(gsl_matrix *g, double *mu, workspace_t *ws)
{
  gsl_matrix_memcpy(ws->tmp, g);
  gsl_linalg_SV_decomp(ws->tmp, ws->evec_real, ws->eval_real, ws->work_sv);

  for(int i = 0; i < ws->n - 1; i++)
    mu[i] = log(gsl_vector_get(ws->eval_real, i) / gsl_vector_get(ws->eval_real, i+1));
}

void initialize(gsl_matrix *g, double *data, int x, int y)
{
  gsl_matrix_view view = gsl_matrix_view_array(data, x, y);
  gsl_matrix_memcpy(g, &view.matrix);
}

void rotation_matrix(gsl_matrix *g, double *vector)
{
  double normalized[3];
  double norm = sqrt(vector[0]*vector[0] + vector[1]*vector[1] + vector[2]*vector[2]);
  for(int i = 0; i < 3; i++)
    normalized[i] = vector[i] / norm;
  gsl_matrix_set_identity(g);
  gsl_matrix_set(g, 0, 0, cos(norm));
  gsl_matrix_set(g, 0, 1, -sin(norm) * normalized[2]);
  gsl_matrix_set(g, 0, 2, +sin(norm) * normalized[1]);
  gsl_matrix_set(g, 1, 0, +sin(norm) * normalized[2]);
  gsl_matrix_set(g, 1, 1, cos(norm));
  gsl_matrix_set(g, 1, 2, -sin(norm) * normalized[0]);
  gsl_matrix_set(g, 2, 0, -sin(norm) * normalized[1]);
  gsl_matrix_set(g, 2, 1, +sin(norm) * normalized[0]);
  gsl_matrix_set(g, 2, 2, cos(norm));
  for(int i = 0; i < 3; i++)
    for(int j = 0; j < 3; j++)
      g->data[i * g->tda + j] += (1 - cos(norm)) * normalized[i] * normalized[j];
}

double trace(gsl_matrix *g)
{
  return gsl_matrix_get(g, 0, 0) + gsl_matrix_get(g, 1, 1) + gsl_matrix_get(g, 2, 2);
}

double determinant(gsl_matrix *g, workspace_t *ws)
{
  int s;
  gsl_matrix_memcpy(ws->tmp, g);
  gsl_linalg_LU_decomp(ws->tmp, ws->permutation, &s);
  return gsl_linalg_LU_det(ws->tmp, s);
}

int jordan_calc(gsl_matrix *g, double *mu, workspace_t *ws)
{
  gsl_eigen_nonsymmv_params(1, ws->work_nonsymmv);
  gsl_eigen_nonsymmv(g, ws->eval_complex, ws->evec_complex, ws->work_nonsymmv);
  gsl_eigen_nonsymmv_sort(ws->eval_complex, ws->evec_complex, GSL_EIGEN_SORT_ABS_DESC);

  int real = 1;
  for(int i = 0; i < 4; i++)
    if(FCMP(GSL_IMAG(gsl_vector_complex_get(ws->eval_complex, i)), 0) != 0)
      real = 0;

  if(!real)
	  return 0; // non-real eigenvalues!

  for(int i = 0; i < ws->n - 1; i++) {
    mu[i] = log(GSL_REAL(gsl_vector_complex_get(ws->eval_complex, i)) / GSL_REAL(gsl_vector_complex_get(ws->eval_complex, i+1)));
  }

  return 1;
}

int eigenvectors(gsl_matrix *g, gsl_matrix *evec_real, workspace_t *ws)
{
	gsl_matrix_memcpy(ws->stack[0], g);
	gsl_eigen_nonsymmv_params(0, ws->work_nonsymmv);
	int r = gsl_eigen_nonsymmv(ws->stack[0], ws->eval_complex, ws->evec_complex, ws->work_nonsymmv);
	ERROR(r, "gsl_eigen_nonsymmv failed!\n");

	gsl_eigen_nonsymmv_sort(ws->eval_complex, ws->evec_complex, GSL_EIGEN_SORT_ABS_DESC);

	int real = 1;
	for(int i = 0; i < ws->n; i++)
		if(FCMP(GSL_IMAG(gsl_vector_complex_get(ws->eval_complex, i)), 0) != 0)
			real = 0;

	if(!real)
		return 0; // non-real eigenvalues!

	for(int i = 0; i < ws->n; i++)
	  for(int j = 0; j < ws->n; j++)
		  gsl_matrix_set(evec_real, i, j, GSL_REAL(gsl_matrix_complex_get(ws->evec_complex, i, j)));

	return 1;
}

// only fills in the real eigenvectors and returns their count
int real_eigenvectors(gsl_matrix *g, gsl_matrix *evec_real, workspace_t *ws)
{
	gsl_matrix_memcpy(ws->stack[0], g);
	gsl_eigen_nonsymmv_params(0, ws->work_nonsymmv);
	int r = gsl_eigen_nonsymmv(ws->stack[0], ws->eval_complex, ws->evec_complex, ws->work_nonsymmv);
	ERROR(r, "gsl_eigen_nonsymmv failed!\n");

	gsl_eigen_nonsymmv_sort(ws->eval_complex, ws->evec_complex, GSL_EIGEN_SORT_ABS_DESC);

	int real = 0;

	for(int i = 0; i < ws->n; i++) {
		if(FCMP(GSL_IMAG(gsl_vector_complex_get(ws->eval_complex, i)), 0) == 0) {// real
			for(int j = 0; j < ws->n; j++)
				gsl_matrix_set(evec_real, j, real, GSL_REAL(gsl_matrix_complex_get(ws->evec_complex, j, i)));
			real++;
		}
	}

	return real;
}

void eigensystem_symm(gsl_matrix *g, gsl_vector *eval, gsl_matrix *evec, workspace_t *ws)
{
	gsl_matrix_memcpy(ws->stack[0], g);
	int r = gsl_eigen_symmv (ws->stack[0], eval, evec, ws->work_symmv);
	ERROR(r, "gsl_eigen_symmv failed!\n");

	gsl_eigen_symmv_sort(eval, evec, GSL_EIGEN_SORT_ABS_DESC);
}

// returns number of positive directions and matrix transforming TO diagonal basis
int diagonalize_symmetric_form(gsl_matrix *A, gsl_matrix *cob, workspace_t *ws)
{
	gsl_matrix_memcpy(ws->stack[0], A);
	int r = gsl_eigen_symmv (ws->stack[0], ws->eval_real, cob, ws->work_symmv);
	ERROR(r, "gsl_eigen_symmv failed!\n");

	gsl_eigen_symmv_sort(ws->eval_real, cob, GSL_EIGEN_SORT_VAL_ASC);

	gsl_matrix_transpose(cob);

	int positive = 0;

	for(int i = 0; i < ws->n; i++) {
		if(gsl_vector_get(ws->eval_real, i) > 0)
			positive++;

		for(int j = 0; j < ws->n; j++)
			*gsl_matrix_ptr(cob, i, j) *= sqrt(fabs(gsl_vector_get(ws->eval_real, i)));
	}

	return positive;

//	printf("Eigenvalues: %.10f, %.10f, %.10f\n", gsl_vector_get(ws->eval_real, 0), gsl_vector_get(ws->eval_real, 1), gsl_vector_get(ws->eval_real, 2));

//	return 0;
}