#+TITLE: Homework 9
#+AUTHOR: Elizabeth Hunt
#+LATEX_HEADER: \notindent \notag  \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt}
#+OPTIONS: toc:nil

* Question One

With a ~matrix_dimension~ set to 700, I consistently see about a 3x improvement in performance on my
10-thread machine. The serial implementation gives an average ~0.189s~ total runtime, while the below
parallel implementation runs in about ~0.066s~ after the cpu cache has filled on the first run.

#+BEGIN_SRC c
#include <math.h>
#include <omp.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>

#define matrix_dimension 700

int n = matrix_dimension;
float sum;

int main() {
  float A[n][n];
  float x0[n];
  float b[n];
  float x1[n];
  float res[n];

  srand((unsigned int)(time(NULL)));

  // not worth parallellization - rand() is not thread-safe
  for (int i = 0; i < n; i++) {
    for (int j = 0; j < n; j++) {
      A[i][j] = ((float)rand() / (float)(RAND_MAX) * 5.0);
    }
    x0[i] = ((float)rand() / (float)(RAND_MAX) * 5.0);
  }

#pragma omp parallel for private(sum)
  for (int i = 0; i < n; i++) {
    sum = 0.0;
    for (int j = 0; j < n; j++) {
      sum += fabs(A[i][j]);
    }
    A[i][i] += sum;
  }

#pragma omp parallel for private(sum)
  for (int i = 0; i < n; i++) {
    sum = 0.0;
    for (int j = 0; j < n; j++) {
      sum += A[i][j];
    }
    b[i] = sum;
  }

  float tol = 0.0001;
  float error = 10.0 * tol;
  int maxiter = 100;
  int iter = 0;

  while (error > tol && iter < maxiter) {
#pragma omp parallel for
    for (int i = 0; i < n; i++) {
      float temp_sum = b[i];
      for (int j = 0; j < n; j++) {
        temp_sum -= A[i][j] * x0[j];
      }
      res[i] = temp_sum;
      x1[i] = x0[i] + res[i] / A[i][i];
    }

    sum = 0.0;
#pragma omp parallel for reduction(+ : sum)
    for (int i = 0; i < n; i++) {
      float val = x1[i] - x0[i];
      sum += val * val;
    }
    error = sqrt(sum);

#pragma omp parallel for
    for (int i = 0; i < n; i++) {
      x0[i] = x1[i];
    }

    iter++;
  }

  for (int i = 0; i < n; i++)
    printf("x[%d] = %6f \t res[%d] = %6f\n", i, x1[i], i, res[i]);

  return 0;
}

#+END_SRC

* Question Two

I only see lowerings in performance (likely due to overhead) on my machine using OpenMP until
~matrix_dimension~ becomes quite large, about ~300~ in testing. At ~matrix_dimension=1000~, I see another
about 3x improvement in total runtime (including initialization & I/O which was untouched, so, even further
improvements could be made) on my 10-thread machine; from around ~0.174~ seconds to ~.052~.

#+BEGIN_SRC c
  #include <math.h>
  #include <stdio.h>
  #include <stdlib.h>
  #include <time.h>

  #ifdef _OPENMP
  #include <omp.h>
  #else
  #define omp_get_num_threads() 0
  #define omp_set_num_threads(int) 0
  #define omp_get_thread_num() 0
  #endif

  #define matrix_dimension 1000

  int n = matrix_dimension;
  float ynrm;

  int main() {
    float A[n][n];
    float v0[n];
    float v1[n];
    float y[n];
    //
    // create a matrix
    //
    // not worth parallellization - rand() is not thread-safe
    srand((unsigned int)(time(NULL)));
    float a = 5.0;
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
        A[i][j] = ((float)rand() / (float)(RAND_MAX)*a);
      }
      v0[i] = ((float)rand() / (float)(RAND_MAX)*a);
    }
    //
    // modify the diagonal entries for diagonal dominance
    // --------------------------------------------------
    //
    for (int i = 0; i < n; i++) {
      float sum = 0.0;
      for (int j = 0; j < n; j++) {
        sum = sum + fabs(A[i][j]);
      }
      A[i][i] = A[i][i] + sum;
    }
    //
    // generate a vector of ones
    // -------------------------
    //
    for (int j = 0; j < n; j++) {
      v0[j] = 1.0;
    }
    //
    // power iteration test
    // --------------------
    //
    float tol = 0.0000001;
    float error = 10.0 * tol;
    float lam1, lam0;
    int maxiter = 100;
    int iter = 0;

    while (error > tol && iter < maxiter) {
  #pragma omp parallel for
      for (int i = 0; i < n; i++) {
        y[i] = 0;
        for (int j = 0; j < n; j++) {
          y[i] = y[i] + A[i][j] * v0[j];
        }
      }

      ynrm = 0.0;
  #pragma omp parallel for reduction(+ : ynrm)
      for (int i = 0; i < n; i++) {
        ynrm += y[i] * y[i];
      }
      ynrm = sqrt(ynrm);

  #pragma omp parallel for
      for (int i = 0; i < n; i++) {
        v1[i] = y[i] / ynrm;
      }

  #pragma omp parallel for
      for (int i = 0; i < n; i++) {
        y[i] = 0.0;
        for (int j = 0; j < n; j++) {
          y[i] += A[i][j] * v1[j];
        }
      }

      lam1 = 0.0;
  #pragma omp parallel for reduction(+ : lam1)
      for (int i = 0; i < n; i++) {
        lam1 += v1[i] * y[i];
      }

      error = fabs(lam1 - lam0);
      lam0 = lam1;

  #pragma omp parallel for
      for (int i = 0; i < n; i++) {
        v0[i] = v1[i];
      }

      iter++;
    }

    printf("in %d iterations, eigenvalue = %f\n", iter, lam1);
  }
#+END_SRC

* Question Three
[[https://static.simponic.xyz/lizfcm.pdf]]