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    <blockquote> <p>For my understanding, for a "A,b = create_matrix_1(4,4)" the matrix A is 16*16; b is 16*1</p> </blockquote> <p>Not necessarily: in the code above we don't have the definitions of the functions <code>lil_matrix</code> and <code>zeros</code>. The statements before the <code>for</code> loop only say that <code>A</code> is equal to the value returned by the function <code>lil_matrix</code> when the values <code>(n*m, n*m)</code> are passed to it as a tuple. On the other hand <code>b</code> is equal to the value returned by the function <code>zeros</code>, when you pass to it <code>n*m</code>.</p> <p>Then there is the loop section. Basically, there are 2 nested loops, i.e. there are two iterators, i and j, that are incremented respectively from 0 to n-1 and from 0 to m-1; so, assuming n = 4 and m = 4:</p> <p>1) We start by iterating through all values of <code>j</code>, having <code>i=0</code>:</p> <pre><code>Iteration Value of index 1 0+0*4 2 0+1*4 3 0+2*4 4 0+3*4 </code></pre> <p>2) Now that all <code>j</code> values, from <code>j=0</code> to <code>j=m-1</code>, have been cycled through, we increment <code>i</code>, and again we iterate through all values of <code>j</code>:</p> <pre><code>5 1+0*4 6 1+1*4 7 1+2*4 8 1+3*4 </code></pre> <p>3) Then we repeat the same cycle for <code>i=2</code>, <code>i=3</code> etc., until <code>i=n-1</code>.</p> <p>Now we get to the <code>if</code> and <code>else</code> blocks. For each of the cycles described above, if we have a boundary value, i.e. either <code>i</code> or <code>j</code> are at the first or last iteration of the cycle (e.g. either <code>i</code> or <code>j</code> are <code>= 0</code>, or <code>i=n-1</code> or <code>j=m-1</code>) then the values of <code>A</code> and <code>b</code> are not any more the ones described at the beginning of this answer, but are changed according to the two lines after the line starting with <code>if</code>. If instead the values of <code>i</code> or <code>j</code> are not boundary values, but are interior nodes, the values of <code>A</code> and <code>b</code> are changed according to the lines after the <code>else</code> statement. </p> <p>Both the statements in the <code>if</code> and in the <code>else</code> section only change some of the elements of <code>A</code> and <code>b</code>, which are arrays. Looking at the code, <code>A</code> must be a bidimensional array (matrix), as you pointed out correctly in your question, and <code>b</code> is a one dimensional array (vector). (In Python, by the way, arrays are actually lists, but I will use here the general term array). Both in the <code>if</code> and <code>else</code> block we have the <code>index</code> keyword: e.g. <code>b[index]</code>means that the value of <code>b</code> that we are modifying is the Nth value of <code>b</code>, where <code>index</code> = N. For <code>A</code> we have two values in the square brackets, since <code>A</code> is a matrix. By the way there is in my opinion an error in the syntax since when you select a single value in a matrix you don't use the comma, e.g. you don't write <code>A[x, y]</code> but <code>A[x][y]</code> (the comma can only be used when you define the matrix, not when you select one of its values). However both <code>A</code> and <code>b</code> are created in the first place using functions not included in this code so we don't know much more than that. </p> <p>Other than <code>lil_matrix</code> and <code>zeros</code>, the other function which you need to investigate, because its definition is not included in this code, is <code>linspace</code>: this is used at the beginning to create the variables <code>x</code> and <code>y</code> which then are used in the <code>if</code> block.</p>
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