#### Qr Eigen Values

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/**
* @file
* \brief Compute real eigen values and eigen vectors of a symmetric matrix
* using [QR decomposition](https://en.wikipedia.org/wiki/QR_decomposition)
* method.
* \author [Krishna Vedala](https://github.com/kvedala)
*/
#include <cassert>
#include <cmath>
#include <cstdlib>
#include <ctime>
#include <iostream>
#ifdef _OPENMP
#include <omp.h>
#endif

#include "./qr_decompose.h"
using qr_algorithm::operator<<;

#define LIMS 9 /**< limit of range of matrix values */

/**
* create a symmetric square matrix of given size with random elements. A
* symmetric square matrix will *always* have real eigen values.
*
* \param[out] A matrix to create (must be pre-allocated in memory)
*/
void create_matrix(std::valarray<std::valarray<double>> *A) {
int i, j, tmp, lim2 = LIMS >> 1;
int N = A->size();

#ifdef _OPENMP
#pragma omp for
#endif
for (i = 0; i < N; i++) {
A[0][i][i] = (std::rand() % LIMS) - lim2;
for (j = i + 1; j < N; j++) {
tmp = (std::rand() % LIMS) - lim2;
A[0][i][j] = tmp;  // summetrically distribute random values
A[0][j][i] = tmp;
}
}
}

/**
* Perform multiplication of two matrices.
* * R2 must be equal to C1
* * Resultant matrix size should be R1xC2
* \param[in] A first matrix to multiply
* \param[in] B second matrix to multiply
* \param[out] OUT output matrix (must be pre-allocated)
* \returns pointer to resultant matrix
*/
void mat_mul(const std::valarray<std::valarray<double>> &A,
const std::valarray<std::valarray<double>> &B,
std::valarray<std::valarray<double>> *OUT) {
int R1 = A.size();
int C1 = A[0].size();
int R2 = B.size();
int C2 = B[0].size();
if (C1 != R2) {
perror("Matrix dimensions mismatch!");
return;
}

for (int i = 0; i < R1; i++) {
for (int j = 0; j < C2; j++) {
OUT[0][i][j] = 0.f;
for (int k = 0; k < C1; k++) {
OUT[0][i][j] += A[i][k] * B[k][j];
}
}
}
}

namespace qr_algorithm {
/** Compute eigen values using iterative shifted QR decomposition algorithm as
* follows:
* 1. Use last diagonal element of A as eigen value approximation \f$c\f$
* 2. Shift diagonals of matrix \f$A' = A - cI\f$
* 3. Decompose matrix \f$A'=QR\f$
* 4. Compute next approximation \f$A'_1 = RQ \f$
* 5. Shift diagonals back \f$A_1 = A'_1 + cI\f$
* 6. Termination condition check: last element below diagonal is almost 0
*   1. If not 0, go back to step 1 with the new approximation \f$A_1\f$
*   2. If 0, continue to step 7
* 7. Save last known \f$c\f$ as the eigen value.
* 8. Are all eigen values found?
*   1. If not, remove last row and column of \f$A_1\f$ and go back to step 1.
*   2. If yes, stop.
*
* \note The matrix \f$A\f$ gets modified
*
* \param[in,out] A matrix to compute eigen values for
* \param[in] print_intermediates (optional) whether to print intermediate A, Q
* and R matrices (default = false)
*/
std::valarray<double> eigen_values(std::valarray<std::valarray<double>> *A,
bool print_intermediates = false) {
int rows = A->size();
int columns = rows;
int counter = 0, num_eigs = rows - 1;
double last_eig = 0;

std::valarray<std::valarray<double>> Q(rows);
std::valarray<std::valarray<double>> R(columns);

/* number of eigen values = matrix size */
std::valarray<double> eigen_vals(rows);
for (int i = 0; i < rows; i++) {
Q[i] = std::valarray<double>(columns);
R[i] = std::valarray<double>(columns);
}

/* continue till all eigen values are found */
while (num_eigs > 0) {
/* iterate with QR decomposition */
while (std::abs(A[0][num_eigs][num_eigs - 1]) >
std::numeric_limits<double>::epsilon()) {
// initial approximation = last diagonal element
last_eig = A[0][num_eigs][num_eigs];
for (int i = 0; i < rows; i++) {
A[0][i][i] -= last_eig; /* A - cI */
}

qr_decompose(*A, &Q, &R);

if (print_intermediates) {
std::cout << *A << "\n";
std::cout << Q << "\n";
std::cout << R << "\n";
printf("-------------------- %d ---------------------\n",
++counter);
}

// new approximation A' = R * Q
mat_mul(R, Q, A);

for (int i = 0; i < rows; i++) {
A[0][i][i] += last_eig; /* A + cI */
}
}

/* store the converged eigen value */
eigen_vals[num_eigs] = last_eig;
// A[0][num_eigs][num_eigs];
if (print_intermediates) {
std::cout << "========================\n";
std::cout << "Eigen value: " << last_eig << ",\n";
std::cout << "========================\n";
}

num_eigs--;
rows--;
columns--;
}
eigen_vals[0] = A[0][0][0];

if (print_intermediates) {
std::cout << Q << "\n";
std::cout << R << "\n";
}

return eigen_vals;
}

}  // namespace qr_algorithm

/**
* test function to compute eigen values of a 2x2 matrix
* \f[\begin{bmatrix}
* 5 & 7\\
* 7 & 11
* \end{bmatrix}\f]
* which are approximately, {15.56158, 0.384227}
*/
void test1() {
std::valarray<std::valarray<double>> X = {{5, 7}, {7, 11}};
double y[] = {15.56158, 0.384227};  // corresponding y-values

std::cout << "------- Test 1 -------" << std::endl;
std::valarray<double> eig_vals = qr_algorithm::eigen_values(&X);

for (int i = 0; i < 2; i++) {
std::cout << i + 1 << "/2 Checking for " << y[i] << " --> ";
bool result = false;
for (int j = 0; j < 2 && !result; j++) {
if (std::abs(y[i] - eig_vals[j]) < 0.1) {
result = true;
std::cout << "(" << eig_vals[j] << ") ";
}
}
assert(result);  // ensure that i^th expected eigen value was computed
std::cout << "found\n";
}
std::cout << "Test 1 Passed\n\n";
}

/**
* test function to compute eigen values of a 2x2 matrix
* \f[\begin{bmatrix}
* -4& 4& 2& 0& -3\\
* 4& -4& 4& -3& -1\\
* 2& 4& 4& 3& -3\\
* 0& -3& 3& -1&-1\\
* -3& -1& -3& -3& 0
* \end{bmatrix}\f]
* which are approximately, {9.27648, -9.26948, 2.0181, -1.03516, -5.98994}
*/
void test2() {
std::valarray<std::valarray<double>> X = {{-4, 4, 2, 0, -3},
{4, -4, 4, -3, -1},
{2, 4, 4, 3, -3},
{0, -3, 3, -1, -3},
{-3, -1, -3, -3, 0}};
double y[] = {9.27648, -9.26948, 2.0181, -1.03516,
-5.98994};  // corresponding y-values

std::cout << "------- Test 2 -------" << std::endl;
std::valarray<double> eig_vals = qr_algorithm::eigen_values(&X);

std::cout << X << "\n"
<< "Eigen values: " << eig_vals << "\n";

for (int i = 0; i < 5; i++) {
std::cout << i + 1 << "/5 Checking for " << y[i] << " --> ";
bool result = false;
for (int j = 0; j < 5 && !result; j++) {
if (std::abs(y[i] - eig_vals[j]) < 0.1) {
result = true;
std::cout << "(" << eig_vals[j] << ") ";
}
}
assert(result);  // ensure that i^th expected eigen value was computed
std::cout << "found\n";
}
std::cout << "Test 2 Passed\n\n";
}

/**
* main function
*/
int main(int argc, char **argv) {
int mat_size = 5;
if (argc == 2) {
mat_size = atoi(argv[1]);
} else {  // if invalid input argument is given run tests
test1();
test2();
std::cout << "Usage: ./qr_eigen_values [mat_size]\n";
return 0;
}

if (mat_size < 2) {
fprintf(stderr, "Matrix size should be > 2\n");
return -1;
}

// initialize random number generator
std::srand(std::time(nullptr));

int i, rows = mat_size, columns = mat_size;

std::valarray<std::valarray<double>> A(rows);

for (int i = 0; i < rows; i++) {
A[i] = std::valarray<double>(columns);
}

/* create a random matrix */
create_matrix(&A);

std::cout << A << "\n";

clock_t t1 = clock();
std::valarray<double> eigen_vals = qr_algorithm::eigen_values(&A);
double dtime = static_cast<double>(clock() - t1) / CLOCKS_PER_SEC;

std::cout << "Eigen vals: ";
for (i = 0; i < mat_size; i++) std::cout << eigen_vals[i] << "\t";
std::cout << "\nTime taken to compute: " << dtime << " sec\n";

return 0;
}