lizfcm/homeworks/hw-9.tex

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% Created 2023-12-11 Mon 19:24
% Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{longtable}
\usepackage{wrapfig}
\usepackage{rotating}
\usepackage[normalem]{ulem}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{capt-of}
\usepackage{hyperref}
\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
\author{Elizabeth Hunt}
\date{\today}
\title{Homework 9}
\hypersetup{
pdfauthor={Elizabeth Hunt},
pdftitle={Homework 9},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 28.2 (Org mode 9.7-pre)},
pdflang={English}}
\begin{document}
\maketitle
\setlength\parindent{0pt}
\section{Question One}
\label{sec:org69bed2d}
With a \texttt{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 \texttt{0.189s} total runtime, while the below
parallel implementation runs in about \texttt{0.066s} after the cpu cache has filled on the first run.
\begin{verbatim}
#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{verbatim}
\section{Question Two}
\label{sec:orgbeace25}
I only see lowerings in performance (likely due to overhead) on my machine using OpenMP until
\texttt{matrix\_dimension} becomes quite large, about \texttt{300} in testing. At \texttt{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 \texttt{0.174} seconds to \texttt{.052}.
\begin{verbatim}
#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{verbatim}
\section{Question Three}
\label{sec:org33439e0}
\url{https://static.simponic.xyz/lizfcm.pdf}
\end{document}