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<h1 id="linearalgebralib">linearalgebra.lib</h1>
<p>A library of linear algebra functions. Its official prefix is <code>la</code>.</p>
<p>This library adds some new linear algebra functions:</p>
<p><code>determinant</code></p>
<p><code>minor</code></p>
<p><code>inverse</code></p>
<p><code>transpose2</code></p>
<p><code>matMul</code> matrix multiplication</p>
<p><code>identity</code></p>
<p><code>diag</code></p>
<p>How does it work? An <code>NxM</code> matrix can be flattened into a bus <code>si.bus(N*M)</code>. These buses can be passed to functions as long as <code>N</code> and sometimes <code>M</code> (if the matrix need not be square) are passed too.</p>
<h4 id="some-things-to-think-about-going-forward">Some things to think about going forward</h4>
<h5 id="implications-for-ml-in-faust">Implications for ML in Faust</h5>
<p>Next step of making a "Dense"/"Linear" layer from machine learning.
Where in the libraries should <code>ReLU</code> go?
What about 3D tensors instead of 2D matrices? Image convolutions take place on 3D tensors shaped <code>HxWxC</code>.</p>
<h5 id="design-of-matmul">Design of matMul</h5>
<p>Currently the design is <code>matMul(J, K, L, M, leftHandMat, rightHandMat)</code> where <code>leftHandMat</code> is <code>JxK</code> and <code>rightHandMat</code> is <code>LxM</code>.</p>
<p>It would also be neat to have <code>matMul(J, K, rightHandMat, L, M, leftHandMat)</code>.</p>
<p>Then a "packed" matrix could be consistently stored as a combination of a 2-channel "header" <code>N, M</code> and the values <code>si.bus(N*M)</code>.</p>
<p>This would ultimately enable <code>result = packedLeftHand : matMul(packedRightHand);</code> for the equivalent numpy code: <code>result = packedLeftHand @ packedRightHand;</code>.</p>
<h4 id="references">References</h4>
<ul>
<li><a href="https://github.com/grame-cncm/faustlibraries/blob/master/linearalgebra.lib">https://github.com/grame-cncm/faustlibraries/blob/master/linearalgebra.lib</a></li>
</ul>
<hr />
<h3 id="ladeterminant"><code>(la.)determinant</code></h3>
<p>Calculates the determinant of a bus that represents
an <code>NxN</code> matrix.</p>
<h4 id="usage">Usage</h4>
<pre><code>si.bus(N*N) : determinant(N) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: the size of each axis of the matrix.</li>
</ul>
<hr />
<h3 id="laminor"><code>(la.)minor</code></h3>
<p>An utility for finding the matrix minor when inverting a matrix.
It returns the determinant of the submatrix formed by deleting the row at
index <code>ROW</code> and column at index <code>COL</code>.
The following implementation doesn't work but looks simple.</p>
<pre><code>minor(N, ROW, COL) = par(r, N, par(c, N, select2((ROW==r)||(COL==c),_,!))) : determinant(N-1);
</code></pre>
<h4 id="usage_1">Usage</h4>
<pre><code>si.bus(N*N) : minor(N, ROW, COL) : _
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: the size of each axis of the matrix.</li>
<li><code>ROW</code>: the selected position on 0th dimension of the matrix (<code>0 <= ROW < N</code>)</li>
<li><code>COL</code>: the selected position on the 1st dimension of the matrix (<code>0 <= COL < N</code>)</li>
</ul>
<h4 id="references_1">References</h4>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Minor_(linear_algebra)#First_minor">https://en.wikipedia.org/wiki/Minor_(linear_algebra)#First_minor</a></li>
</ul>
<hr />
<h3 id="lainverse"><code>(la.)inverse</code></h3>
<p>Inverts a matrix. The incoming bus represents an <code>NxN</code> matrix.
Note, this is an unsafe operation since not all matrices are invertible.</p>
<h4 id="usage_2">Usage</h4>
<pre><code>si.bus(N*N) : inverse(N) : si.bus(N*N)
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: the size of each axis of the matrix.</li>
</ul>
<hr />
<h3 id="latranspose2"><code>(la.)transpose2</code></h3>
<p>Transposes an <code>NxM</code> matrix stored in row-major order, resulting
in an <code>MxN</code> matrix stored in row-major order.</p>
<h4 id="usage_3">Usage</h4>
<pre><code>si.bus(N*M) : transpose2(N, M) : si.bus(M*N)
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: the number of rows in the input matrix</li>
<li><code>M</code>: the number of columns in the input matrix</li>
</ul>
<hr />
<h3 id="lamatmul"><code>(la.)matMul</code></h3>
<p>Multiply a <code>JxK</code> matrix (mat1) and an <code>LxM</code> matrix (mat2) to produce a <code>JxM</code> matrix.
Note that <code>K==L</code>.
Both matrices should use row-major order.
In terms of numpy, this function is <code>mat1 @ mat2</code>.</p>
<h4 id="usage_4">Usage</h4>
<pre><code>matMul(J, K, L, M, si.bus(J*K), si.bus(L*M)) : si.bus(J*M)
</code></pre>
<p>Where:</p>
<ul>
<li><code>J</code>: the number of rows in <code>mat1</code></li>
<li><code>K</code>: the number of columns in <code>mat1</code></li>
<li><code>L</code>: the number of rows in <code>mat2</code></li>
<li><code>M</code>: the number of columns in <code>mat2</code></li>
</ul>
<hr />
<h3 id="laidentity"><code>(la.)identity</code></h3>
<p>Creates an <code>NxN</code> identity matrix.</p>
<h4 id="usage_5">Usage</h4>
<pre><code>identity(N) : si.bus(N*N)
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: The size of each axis of the identity matrix.</li>
</ul>
<hr />
<h3 id="ladiag"><code>(la.)diag</code></h3>
<p>Creates a diagonal matrix of size <code>NxN</code> with specified
values along the diagonal.</p>
<h4 id="usage_6">Usage</h4>
<pre><code>si.bus(N) : diag(N) : si.bus(N*N)
</code></pre>
<p>Where:</p>
<ul>
<li><code>N</code>: The size of each axis of the matrix.</li>
</ul></div>
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