User:Kevindunn

Bonus question [1]

Use the pseudo code developed in the course notes to write a MATLAB or Python function that implements Gauss elimination, without pivoting. The function should take \(A\) and \(b\) as inputs, and return vector \(x\).

Use your function to solve this set of equations from question 1, and print the update \(A\) matrix and \(b\) vector after every pivot step.

Solution

A function that implements the Gauss elimination without pivoting is provided below.

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MATLAB code

Python code

<span class="k">function</span><span class="w"> </span>x <span class="p">=</span><span class="w"> </span><span class="nf">gauss</span><span class="p">(</span>A,b<span class="p">)</span><span class="w"></span> <span class="c">% This function performs the Gauss elimination without pivoting</span> <span class="c">% </span> <span class="c">% x = GAUSS(A, b)</span> <span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">]</span> <span class="p">=</span> <span class="nb">size</span><span class="p">(</span><span class="n">A</span><span class="p">);</span> <span class="c">% Check for zero diagonal elements</span> <span class="k">if</span> <span class="n">any</span><span class="p">(</span><span class="nb">diag</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">==</span><span class="mi">0</span><span class="p">)</span> <span class="n">error</span><span class="p">(</span><span class="s">'Division by zero will occur; pivoting not supported'</span><span class="p">)</span> <span class="k">end</span> <span class="c">% Forward elimination</span> <span class="k">for</span> <span class="n">row</span><span class="p">=</span><span class="mi">1</span><span class="p">:</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span> <span class="k">for</span> <span class="nb">i</span><span class="p">=</span><span class="n">row</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span><span class="n">n</span> <span class="nb">factor</span> <span class="p">=</span> <span class="n">A</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="n">row</span><span class="p">)</span> <span class="o">/</span> <span class="n">A</span><span class="p">(</span><span class="n">row</span><span class="p">,</span><span class="n">row</span><span class="p">);</span> <span class="k">for</span> <span class="nb">j</span> <span class="p">=</span> <span class="n">row</span><span class="p">:</span><span class="n">n</span> <span class="n">A</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="nb">j</span><span class="p">)</span> <span class="p">=</span> <span class="n">A</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="nb">j</span><span class="p">)</span> <span class="o">-</span> <span class="nb">factor</span><span class="o">*</span><span class="n">A</span><span class="p">(</span><span class="n">row</span><span class="p">,</span><span class="nb">j</span><span class="p">);</span> <span class="k">end</span> <span class="n">b</span><span class="p">(</span><span class="nb">i</span><span class="p">)</span> <span class="p">=</span> <span class="n">b</span><span class="p">(</span><span class="nb">i</span><span class="p">)</span> <span class="o">-</span> <span class="nb">factor</span><span class="o">*</span><span class="n">b</span><span class="p">(</span><span class="n">row</span><span class="p">);</span> <span class="k">end</span> <span class="n">A_and_b</span> <span class="p">=</span> <span class="p">[</span><span class="n">A</span> <span class="n">b</span><span class="p">]</span> <span class="k">end</span> <span class="c">% Backward substitution</span> <span class="n">x</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="p">=</span> <span class="n">b</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">A</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">n</span><span class="p">);</span> <span class="k">for</span> <span class="n">row</span> <span class="p">=</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span><span class="mi">1</span> <span class="n">sums</span> <span class="p">=</span> <span class="n">b</span><span class="p">(</span><span class="n">row</span><span class="p">);</span> <span class="k">for</span> <span class="nb">j</span> <span class="p">=</span> <span class="n">row</span><span class="o">+</span><span class="mi">1</span><span class="p">:</span> <span class="n">n</span> <span class="n">sums</span> <span class="p">=</span> <span class="n">sums</span> <span class="o">-</span> <span class="n">A</span><span class="p">(</span><span class="n">row</span><span class="p">,</span><span class="nb">j</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="p">(</span><span class="nb">j</span><span class="p">);</span> <span class="k">end</span> <span class="n">x</span><span class="p">(</span><span class="n">row</span><span class="p">)</span> <span class="p">=</span> <span class="n">sums</span> <span class="o">/</span> <span class="n">A</span><span class="p">(</span><span class="n">row</span><span class="p">,</span><span class="n">row</span><span class="p">);</span> <span class="k">end</span> <span class="c">%Now run this in the command window or an M-file:</span> <span class="c">% >> A = [1 1 1 1;1 -2 -2 -2;1 4 -4 1;1 -5 -5 -3];</span> <span class="c">% >> b = [0 4 2 -4]';</span> <span class="c">% >> x = gauss(A,b)</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span> <span class="k">def</span> <span class="nf">forward_elimination</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span> <span class="sd">"""</span> <span class="sd"> Calculates the forward part of Gaussian elimination.</span> <span class="sd"> """</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">row</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span> <span class="n">factor</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">row</span><span class="p">]</span> <span class="o">/</span> <span class="n">A</span><span class="p">[</span><span class="n">row</span><span class="p">,</span><span class="n">row</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">row</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">factor</span> <span class="o">*</span> <span class="n">A</span><span class="p">[</span><span class="n">row</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">factor</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="n">row</span><span class="p">]</span> <span class="k">print</span><span class="p">(</span><span class="s">'A = </span><span class="se">\n</span><span class="si">%s</span><span class="s"> and b = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">b</span><span class="p">))</span> <span class="k">return</span> <span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="k">def</span> <span class="nf">back_substitution</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span> <span class="sd">""""</span> <span class="sd"> Does back substitution, returns the Gauss result.</span> <span class="sd"> """</span> <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="n">x</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">a</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span> <span class="n">sums</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="n">row</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">row</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span> <span class="n">sums</span> <span class="o">=</span> <span class="n">sums</span> <span class="o">-</span> <span class="n">a</span><span class="p">[</span><span class="n">row</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="n">x</span><span class="p">[</span><span class="n">row</span><span class="p">]</span> <span class="o">=</span> <span class="n">sums</span> <span class="o">/</span> <span class="n">a</span><span class="p">[</span><span class="n">row</span><span class="p">,</span><span class="n">row</span><span class="p">]</span> <span class="k">return</span> <span class="n">x</span> <span class="k">def</span> <span class="nf">gauss</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span> <span class="sd">"""</span> <span class="sd"> This function performs Gauss elimination without pivoting.</span> <span class="sd"> """</span> <span class="n">n</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c"># Check for zero diagonal elements</span> <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">==</span><span class="mi">0</span><span class="p">):</span> <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">((</span><span class="s">'Division by zero will occur; '</span> <span class="s">'pivoting currently not supported'</span><span class="p">))</span> <span class="n">A</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">forward_elimination</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">return</span> <span class="n">back_substitution</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="c"># Main program starts here</span> <span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">'__main__'</span><span class="p">:</span> <span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">]])</span> <span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">])</span> <span class="n">x</span> <span class="o">=</span> <span class="n">gauss</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">print</span><span class="p">(</span><span class="s">'Gauss result is x = </span><span class="se">\n</span><span class="s"> </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="n">x</span><span class="p">)</span>

Now we use this function to solve the system of equations in Question 1. The commented lines in the MATLAB code show how the function is used. The results are: \(x =[1.3333, 3.1667,1.5000,-6.0000]\). Also, the \(A\) and \(b\) (denoted as \(C=[A\quad b]\) below) are updated as:

\[\begin{split}C^1= \left[ \begin{array}{ccccc} 1 & 1 & 1 & 1 & 0\\ 0 & -3 & -3 & -3 & 4\\ 0 & 3 & -5 & 0 & 2\\ 0 & -6 & -6 & -4 & -4 \end{array} \right]\end{split}\]

\[\begin{split}C^2= \left[ \begin{array}{ccccc} 1 & 1 & 1 & 1 & 0\\ 0 & -3 & -3 & -3 & 4\\ 0 & 0 & -8 & -3 & 6\\ 0 & 0 & 0 & 2 & -12 \end{array} \right]\end{split}\]

\[\begin{split}C^3= \left[ \begin{array}{ccccc} 1 & 1 & 1 & 1 & 0\\ 0 & -3 & -3 & -3 & 4\\ 0 & 0 & -8 & -3 & 6\\ 0 & 0 & 0 & 2 & -12 \end{array} \right]\end{split}\]

The final result for \(x = [1.333, 3.167, 1.5, -6]\).