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</pre><pre class='rust '>
<span class='kw'>use</span> <span class='ident'>traits</span>::<span class='ident'>operations</span>::{<span class='ident'>Transpose</span>, <span class='ident'>ApproxEq</span>};
<span class='kw'>use</span> <span class='ident'>traits</span>::<span class='ident'>structure</span>::{<span class='ident'>ColumnSlice</span>, <span class='ident'>Eye</span>, <span class='ident'>Indexable</span>, <span class='ident'>Diagonal</span>, <span class='ident'>SquareMatrix</span>, <span class='ident'>BaseFloat</span>, <span class='ident'>Cast</span>};
<span class='kw'>use</span> <span class='ident'>traits</span>::<span class='ident'>geometry</span>::<span class='ident'>Norm</span>;
<span class='kw'>use</span> <span class='ident'>std</span>::<span class='ident'>cmp</span>;
<span class='kw'>use</span> <span class='ident'>std</span>::<span class='ident'>ops</span>::{<span class='ident'>Mul</span>, <span class='ident'>Add</span>, <span class='ident'>Sub</span>};
<span class='doccomment'>/// Get the householder matrix corresponding to a reflexion to the hyperplane</span>
<span class='doccomment'>/// defined by `vector`. It can be a reflexion contained in a subspace.</span>
<span class='doccomment'>///</span>
<span class='doccomment'>/// # Arguments</span>
<span class='doccomment'>/// * `dimension` - the dimension of the space the resulting matrix operates in</span>
<span class='doccomment'>/// * `start` - the starting dimension of the subspace of the reflexion</span>
<span class='doccomment'>/// * `vector` - the vector defining the reflection.</span>
<span class='kw'>pub</span> <span class='kw'>fn</span> <span class='ident'>householder_matrix</span><span class='op'><</span><span class='ident'>N</span>, <span class='ident'>V</span>, <span class='ident'>M</span><span class='op'>></span>(<span class='ident'>dimension</span>: <span class='ident'>usize</span>, <span class='ident'>start</span>: <span class='ident'>usize</span>, <span class='ident'>vector</span>: <span class='ident'>V</span>) <span class='op'>-></span> <span class='ident'>M</span>
<span class='kw'>where</span> <span class='ident'>N</span>: <span class='ident'>BaseFloat</span>,
<span class='ident'>M</span>: <span class='ident'>Eye</span> <span class='op'>+</span> <span class='ident'>Indexable</span><span class='op'><</span>(<span class='ident'>usize</span>, <span class='ident'>usize</span>), <span class='ident'>N</span><span class='op'>></span>,
<span class='ident'>V</span>: <span class='ident'>Indexable</span><span class='op'><</span><span class='ident'>usize</span>, <span class='ident'>N</span><span class='op'>></span> {
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>qk</span> : <span class='ident'>M</span> <span class='op'>=</span> <span class='ident'>Eye</span>::<span class='ident'>new_identity</span>(<span class='ident'>dimension</span>);
<span class='kw'>let</span> <span class='ident'>subdim</span> <span class='op'>=</span> <span class='ident'>vector</span>.<span class='ident'>shape</span>();
<span class='kw'>let</span> <span class='ident'>stop</span> <span class='op'>=</span> <span class='ident'>subdim</span> <span class='op'>+</span> <span class='ident'>start</span>;
<span class='macro'>assert</span><span class='macro'>!</span>(<span class='ident'>dimension</span> <span class='op'>>=</span> <span class='ident'>stop</span>);
<span class='kw'>for</span> <span class='ident'>j</span> <span class='kw'>in</span> <span class='ident'>start</span> .. <span class='ident'>stop</span> {
<span class='kw'>for</span> <span class='ident'>i</span> <span class='kw'>in</span> <span class='ident'>start</span> .. <span class='ident'>stop</span> {
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>vv</span> <span class='op'>=</span> <span class='ident'>vector</span>.<span class='ident'>unsafe_at</span>(<span class='ident'>i</span> <span class='op'>-</span> <span class='ident'>start</span>) <span class='op'>*</span> <span class='ident'>vector</span>.<span class='ident'>unsafe_at</span>(<span class='ident'>j</span> <span class='op'>-</span> <span class='ident'>start</span>);
<span class='kw'>let</span> <span class='ident'>qkij</span> <span class='op'>=</span> <span class='ident'>qk</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>i</span>, <span class='ident'>j</span>));
<span class='ident'>qk</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>i</span>, <span class='ident'>j</span>), <span class='ident'>qkij</span> <span class='op'>-</span> <span class='ident'>vv</span> <span class='op'>-</span> <span class='ident'>vv</span>);
}
}
}
<span class='ident'>qk</span>
}
<span class='doccomment'>/// QR decomposition using Householder reflections.</span>
<span class='doccomment'>///</span>
<span class='doccomment'>/// # Arguments</span>
<span class='doccomment'>/// * `m` - matrix to decompose</span>
<span class='kw'>pub</span> <span class='kw'>fn</span> <span class='ident'>qr</span><span class='op'><</span><span class='ident'>N</span>, <span class='ident'>V</span>, <span class='ident'>M</span><span class='op'>></span>(<span class='ident'>m</span>: <span class='kw-2'>&</span><span class='ident'>M</span>) <span class='op'>-></span> (<span class='ident'>M</span>, <span class='ident'>M</span>)
<span class='kw'>where</span> <span class='ident'>N</span>: <span class='ident'>BaseFloat</span>,
<span class='ident'>V</span>: <span class='ident'>Indexable</span><span class='op'><</span><span class='ident'>usize</span>, <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Norm</span><span class='op'><</span><span class='ident'>N</span><span class='op'>></span>,
<span class='ident'>M</span>: <span class='ident'>Copy</span> <span class='op'>+</span> <span class='ident'>Eye</span> <span class='op'>+</span> <span class='ident'>ColumnSlice</span><span class='op'><</span><span class='ident'>V</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Transpose</span> <span class='op'>+</span> <span class='ident'>Indexable</span><span class='op'><</span>(<span class='ident'>usize</span>, <span class='ident'>usize</span>), <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span>
<span class='ident'>Mul</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>M</span><span class='op'>></span> {
<span class='kw'>let</span> (<span class='ident'>rows</span>, <span class='ident'>cols</span>) <span class='op'>=</span> <span class='ident'>m</span>.<span class='ident'>shape</span>();
<span class='macro'>assert</span><span class='macro'>!</span>(<span class='ident'>rows</span> <span class='op'>>=</span> <span class='ident'>cols</span>);
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>q</span> : <span class='ident'>M</span> <span class='op'>=</span> <span class='ident'>Eye</span>::<span class='ident'>new_identity</span>(<span class='ident'>rows</span>);
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>r</span> <span class='op'>=</span> <span class='op'>*</span><span class='ident'>m</span>;
<span class='kw'>for</span> <span class='ident'>ite</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>cmp</span>::<span class='ident'>min</span>(<span class='ident'>rows</span> <span class='op'>-</span> <span class='number'>1</span>, <span class='ident'>cols</span>) {
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>v</span> <span class='op'>=</span> <span class='ident'>r</span>.<span class='ident'>column_slice</span>(<span class='ident'>ite</span>, <span class='ident'>ite</span>, <span class='ident'>rows</span>);
<span class='kw'>let</span> <span class='ident'>alpha</span> <span class='op'>=</span>
<span class='kw'>if</span> <span class='kw'>unsafe</span> { <span class='ident'>v</span>.<span class='ident'>unsafe_at</span>(<span class='ident'>ite</span>) } <span class='op'>>=</span> ::<span class='ident'>zero</span>() {
<span class='op'>-</span><span class='ident'>Norm</span>::<span class='ident'>norm</span>(<span class='kw-2'>&</span><span class='ident'>v</span>)
}
<span class='kw'>else</span> {
<span class='ident'>Norm</span>::<span class='ident'>norm</span>(<span class='kw-2'>&</span><span class='ident'>v</span>)
};
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>x</span> <span class='op'>=</span> <span class='ident'>v</span>.<span class='ident'>unsafe_at</span>(<span class='number'>0</span>);
<span class='ident'>v</span>.<span class='ident'>unsafe_set</span>(<span class='number'>0</span>, <span class='ident'>x</span> <span class='op'>-</span> <span class='ident'>alpha</span>);
}
<span class='kw'>if</span> <span class='op'>!</span>::<span class='ident'>is_zero</span>(<span class='kw-2'>&</span><span class='ident'>v</span>.<span class='ident'>normalize_mut</span>()) {
<span class='kw'>let</span> <span class='ident'>qk</span>: <span class='ident'>M</span> <span class='op'>=</span> <span class='ident'>householder_matrix</span>(<span class='ident'>rows</span>, <span class='ident'>ite</span>, <span class='ident'>v</span>);
<span class='ident'>r</span> <span class='op'>=</span> <span class='ident'>qk</span> <span class='op'>*</span> <span class='ident'>r</span>;
<span class='ident'>q</span> <span class='op'>=</span> <span class='ident'>q</span> <span class='op'>*</span> <span class='ident'>Transpose</span>::<span class='ident'>transpose</span>(<span class='kw-2'>&</span><span class='ident'>qk</span>);
}
}
(<span class='ident'>q</span>, <span class='ident'>r</span>)
}
<span class='doccomment'>/// Eigendecomposition of a square symmetric matrix using the qr algorithm</span>
<span class='kw'>pub</span> <span class='kw'>fn</span> <span class='ident'>eigen_qr</span><span class='op'><</span><span class='ident'>N</span>, <span class='ident'>V</span>, <span class='ident'>VS</span>, <span class='ident'>M</span><span class='op'>></span>(<span class='ident'>m</span>: <span class='kw-2'>&</span><span class='ident'>M</span>, <span class='ident'>eps</span>: <span class='kw-2'>&</span><span class='ident'>N</span>, <span class='ident'>niter</span>: <span class='ident'>usize</span>) <span class='op'>-></span> (<span class='ident'>M</span>, <span class='ident'>V</span>)
<span class='kw'>where</span> <span class='ident'>N</span>: <span class='ident'>BaseFloat</span>,
<span class='ident'>V</span>: <span class='ident'>Mul</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>V</span><span class='op'>></span>,
<span class='ident'>VS</span>: <span class='ident'>Indexable</span><span class='op'><</span><span class='ident'>usize</span>, <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Norm</span><span class='op'><</span><span class='ident'>N</span><span class='op'>></span>,
<span class='ident'>M</span>: <span class='ident'>Indexable</span><span class='op'><</span>(<span class='ident'>usize</span>, <span class='ident'>usize</span>), <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>SquareMatrix</span><span class='op'><</span><span class='ident'>N</span>, <span class='ident'>V</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Add</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>M</span><span class='op'>></span> <span class='op'>+</span>
<span class='ident'>Sub</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>M</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>ColumnSlice</span><span class='op'><</span><span class='ident'>VS</span><span class='op'>></span> <span class='op'>+</span>
<span class='ident'>ApproxEq</span><span class='op'><</span><span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Copy</span> {
<span class='kw'>let</span> (<span class='kw-2'>mut</span> <span class='ident'>eigenvectors</span>, <span class='kw-2'>mut</span> <span class='ident'>eigenvalues</span>) <span class='op'>=</span> <span class='ident'>hessenberg</span>(<span class='ident'>m</span>);
<span class='comment'>// Allocate arrays for Givens rotation components</span>
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>c</span> <span class='op'>=</span> <span class='ident'>Vec</span>::<span class='op'><</span><span class='ident'>N</span><span class='op'>></span>::<span class='ident'>with_capacity</span>(::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() <span class='op'>-</span> <span class='number'>1</span>);
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>s</span> <span class='op'>=</span> <span class='ident'>Vec</span>::<span class='op'><</span><span class='ident'>N</span><span class='op'>></span>::<span class='ident'>with_capacity</span>(::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() <span class='op'>-</span> <span class='number'>1</span>);
<span class='kw'>if</span> ::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() <span class='op'>==</span> <span class='number'>1</span> {
<span class='kw'>return</span> (<span class='ident'>eigenvectors</span>, <span class='ident'>eigenvalues</span>.<span class='ident'>diagonal</span>());
}
<span class='kw'>unsafe</span> {
<span class='ident'>c</span>.<span class='ident'>set_len</span>(::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() <span class='op'>-</span> <span class='number'>1</span>);
<span class='ident'>s</span>.<span class='ident'>set_len</span>(::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() <span class='op'>-</span> <span class='number'>1</span>);
}
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>iter</span> <span class='op'>=</span> <span class='number'>0</span>;
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>curdim</span> <span class='op'>=</span> ::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() <span class='op'>-</span> <span class='number'>1</span>;
<span class='kw'>for</span> _ <span class='kw'>in</span> <span class='number'>0</span> .. ::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() {
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>stop</span> <span class='op'>=</span> <span class='bool-val'>false</span>;
<span class='kw'>while</span> <span class='op'>!</span><span class='ident'>stop</span> <span class='op'>&&</span> <span class='ident'>iter</span> <span class='op'><</span> <span class='ident'>niter</span> {
<span class='kw'>let</span> <span class='ident'>lambda</span>;
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>a</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>curdim</span> <span class='op'>-</span> <span class='number'>1</span>, <span class='ident'>curdim</span> <span class='op'>-</span> <span class='number'>1</span>));
<span class='kw'>let</span> <span class='ident'>b</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>curdim</span> <span class='op'>-</span> <span class='number'>1</span>, <span class='ident'>curdim</span>));
<span class='kw'>let</span> <span class='ident'>c</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>curdim</span>, <span class='ident'>curdim</span> <span class='op'>-</span> <span class='number'>1</span>));
<span class='kw'>let</span> <span class='ident'>d</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>curdim</span>, <span class='ident'>curdim</span>));
<span class='kw'>let</span> <span class='ident'>trace</span> <span class='op'>=</span> <span class='ident'>a</span> <span class='op'>+</span> <span class='ident'>d</span>;
<span class='kw'>let</span> <span class='ident'>determinant</span> <span class='op'>=</span> <span class='ident'>a</span> <span class='op'>*</span> <span class='ident'>d</span> <span class='op'>-</span> <span class='ident'>b</span> <span class='op'>*</span> <span class='ident'>c</span>;
<span class='kw'>let</span> <span class='ident'>constquarter</span>: <span class='ident'>N</span> <span class='op'>=</span> <span class='ident'>Cast</span>::<span class='ident'>from</span>(<span class='number'>0.25f64</span>);
<span class='kw'>let</span> <span class='ident'>consthalf</span>: <span class='ident'>N</span> <span class='op'>=</span> <span class='ident'>Cast</span>::<span class='ident'>from</span>(<span class='number'>0.5f64</span>);
<span class='kw'>let</span> <span class='ident'>e</span> <span class='op'>=</span> (<span class='ident'>constquarter</span> <span class='op'>*</span> <span class='ident'>trace</span> <span class='op'>*</span> <span class='ident'>trace</span> <span class='op'>-</span> <span class='ident'>determinant</span>).<span class='ident'>sqrt</span>();
<span class='kw'>let</span> <span class='ident'>lambda1</span> <span class='op'>=</span> <span class='ident'>consthalf</span> <span class='op'>*</span> <span class='ident'>trace</span> <span class='op'>+</span> <span class='ident'>e</span>;
<span class='kw'>let</span> <span class='ident'>lambda2</span> <span class='op'>=</span> <span class='ident'>consthalf</span> <span class='op'>*</span> <span class='ident'>trace</span> <span class='op'>-</span> <span class='ident'>e</span>;
<span class='kw'>if</span> (<span class='ident'>lambda1</span> <span class='op'>-</span> <span class='ident'>d</span>).<span class='ident'>abs</span>() <span class='op'><</span> (<span class='ident'>lambda2</span> <span class='op'>-</span> <span class='ident'>d</span>).<span class='ident'>abs</span>() {
<span class='ident'>lambda</span> <span class='op'>=</span> <span class='ident'>lambda1</span>;
}
<span class='kw'>else</span> {
<span class='ident'>lambda</span> <span class='op'>=</span> <span class='ident'>lambda2</span>;
}
}
<span class='comment'>// Shift matrix</span>
<span class='kw'>for</span> <span class='ident'>k</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>curdim</span> <span class='op'>+</span> <span class='number'>1</span> {
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>a</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>k</span>, <span class='ident'>k</span>));
<span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>k</span>, <span class='ident'>k</span>), <span class='ident'>a</span> <span class='op'>-</span> <span class='ident'>lambda</span>);
}
}
<span class='comment'>// Givens rotation from left</span>
<span class='kw'>for</span> <span class='ident'>k</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>curdim</span> {
<span class='kw'>let</span> <span class='ident'>x_i</span> <span class='op'>=</span> <span class='kw'>unsafe</span> { <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>k</span>, <span class='ident'>k</span>)) };
<span class='kw'>let</span> <span class='ident'>x_j</span> <span class='op'>=</span> <span class='kw'>unsafe</span> { <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>k</span> <span class='op'>+</span> <span class='number'>1</span>, <span class='ident'>k</span>)) };
<span class='kw'>let</span> <span class='ident'>ctmp</span>;
<span class='kw'>let</span> <span class='ident'>stmp</span>;
<span class='kw'>if</span> <span class='ident'>x_j</span>.<span class='ident'>abs</span>() <span class='op'><</span> <span class='op'>*</span><span class='ident'>eps</span> {
<span class='ident'>ctmp</span> <span class='op'>=</span> <span class='ident'>N</span>::<span class='ident'>one</span>();
<span class='ident'>stmp</span> <span class='op'>=</span> <span class='ident'>N</span>::<span class='ident'>zero</span>();
}
<span class='kw'>else</span> <span class='kw'>if</span> <span class='ident'>x_i</span>.<span class='ident'>abs</span>() <span class='op'><</span> <span class='op'>*</span><span class='ident'>eps</span> {
<span class='ident'>ctmp</span> <span class='op'>=</span> <span class='ident'>N</span>::<span class='ident'>zero</span>();
<span class='ident'>stmp</span> <span class='op'>=</span> <span class='op'>-</span><span class='ident'>N</span>::<span class='ident'>one</span>();
}
<span class='kw'>else</span> {
<span class='kw'>let</span> <span class='ident'>r</span> <span class='op'>=</span> <span class='ident'>x_i</span>.<span class='ident'>hypot</span>(<span class='ident'>x_j</span>);
<span class='ident'>ctmp</span> <span class='op'>=</span> <span class='ident'>x_i</span> <span class='op'>/</span> <span class='ident'>r</span>;
<span class='ident'>stmp</span> <span class='op'>=</span> <span class='op'>-</span><span class='ident'>x_j</span> <span class='op'>/</span> <span class='ident'>r</span>;
}
<span class='ident'>c</span>[<span class='ident'>k</span>] <span class='op'>=</span> <span class='ident'>ctmp</span>;
<span class='ident'>s</span>[<span class='ident'>k</span>] <span class='op'>=</span> <span class='ident'>stmp</span>;
<span class='kw'>for</span> <span class='ident'>j</span> <span class='kw'>in</span> <span class='ident'>k</span> .. (<span class='ident'>curdim</span> <span class='op'>+</span> <span class='number'>1</span>) {
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>a</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>k</span>, <span class='ident'>j</span>));
<span class='kw'>let</span> <span class='ident'>b</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>k</span> <span class='op'>+</span> <span class='number'>1</span>, <span class='ident'>j</span>));
<span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>k</span>, <span class='ident'>j</span>), <span class='ident'>ctmp</span> <span class='op'>*</span> <span class='ident'>a</span> <span class='op'>-</span> <span class='ident'>stmp</span> <span class='op'>*</span> <span class='ident'>b</span>);
<span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>k</span> <span class='op'>+</span> <span class='number'>1</span>, <span class='ident'>j</span>), <span class='ident'>stmp</span> <span class='op'>*</span> <span class='ident'>a</span> <span class='op'>+</span> <span class='ident'>ctmp</span> <span class='op'>*</span> <span class='ident'>b</span>);
}
}
}
<span class='comment'>// Givens rotation from right applied to eigenvalues</span>
<span class='kw'>for</span> <span class='ident'>k</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>curdim</span> {
<span class='kw'>for</span> <span class='ident'>i</span> <span class='kw'>in</span> <span class='number'>0</span> .. (<span class='ident'>k</span> <span class='op'>+</span> <span class='number'>2</span>) {
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>a</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>i</span>, <span class='ident'>k</span>));
<span class='kw'>let</span> <span class='ident'>b</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>i</span>, <span class='ident'>k</span> <span class='op'>+</span> <span class='number'>1</span>));
<span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>i</span>, <span class='ident'>k</span>), <span class='ident'>c</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>a</span> <span class='op'>-</span> <span class='ident'>s</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>b</span>);
<span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>i</span>, <span class='ident'>k</span> <span class='op'>+</span> <span class='number'>1</span>), <span class='ident'>s</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>a</span> <span class='op'>+</span> <span class='ident'>c</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>b</span>);
}
}
}
<span class='comment'>// Shift back</span>
<span class='kw'>for</span> <span class='ident'>k</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>curdim</span> <span class='op'>+</span> <span class='number'>1</span> {
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>a</span> <span class='op'>=</span> <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>k</span>, <span class='ident'>k</span>));
<span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>k</span>, <span class='ident'>k</span>), <span class='ident'>a</span> <span class='op'>+</span> <span class='ident'>lambda</span>);
}
}
<span class='comment'>// Givens rotation from right applied to eigenvectors</span>
<span class='kw'>for</span> <span class='ident'>k</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>curdim</span> {
<span class='kw'>for</span> <span class='ident'>i</span> <span class='kw'>in</span> <span class='number'>0</span> .. ::<span class='ident'>dimension</span>::<span class='op'><</span><span class='ident'>M</span><span class='op'>></span>() {
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>a</span> <span class='op'>=</span> <span class='ident'>eigenvectors</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>i</span>, <span class='ident'>k</span>));
<span class='kw'>let</span> <span class='ident'>b</span> <span class='op'>=</span> <span class='ident'>eigenvectors</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>i</span>, <span class='ident'>k</span> <span class='op'>+</span> <span class='number'>1</span>));
<span class='ident'>eigenvectors</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>i</span>, <span class='ident'>k</span>), <span class='ident'>c</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>a</span> <span class='op'>-</span> <span class='ident'>s</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>b</span>);
<span class='ident'>eigenvectors</span>.<span class='ident'>unsafe_set</span>((<span class='ident'>i</span>, <span class='ident'>k</span> <span class='op'>+</span> <span class='number'>1</span>), <span class='ident'>s</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>a</span> <span class='op'>+</span> <span class='ident'>c</span>[<span class='ident'>k</span>] <span class='op'>*</span> <span class='ident'>b</span>);
}
}
}
<span class='ident'>iter</span> <span class='op'>=</span> <span class='ident'>iter</span> <span class='op'>+</span> <span class='number'>1</span>;
<span class='ident'>stop</span> <span class='op'>=</span> <span class='bool-val'>true</span>;
<span class='kw'>for</span> <span class='ident'>j</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>curdim</span> {
<span class='comment'>// Check last row</span>
<span class='kw'>if</span> <span class='kw'>unsafe</span> { <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>curdim</span>, <span class='ident'>j</span>)) }.<span class='ident'>abs</span>() <span class='op'>>=</span> <span class='op'>*</span><span class='ident'>eps</span> {
<span class='ident'>stop</span> <span class='op'>=</span> <span class='bool-val'>false</span>;
<span class='kw'>break</span>;
}
<span class='comment'>// Check last column</span>
<span class='kw'>if</span> <span class='kw'>unsafe</span> { <span class='ident'>eigenvalues</span>.<span class='ident'>unsafe_at</span>((<span class='ident'>j</span>, <span class='ident'>curdim</span>)) }.<span class='ident'>abs</span>() <span class='op'>>=</span> <span class='op'>*</span><span class='ident'>eps</span> {
<span class='ident'>stop</span> <span class='op'>=</span> <span class='bool-val'>false</span>;
<span class='kw'>break</span>;
}
}
}
<span class='kw'>if</span> <span class='ident'>stop</span> {
<span class='kw'>if</span> <span class='ident'>curdim</span> <span class='op'>></span> <span class='number'>1</span> {
<span class='ident'>curdim</span> <span class='op'>=</span> <span class='ident'>curdim</span> <span class='op'>-</span> <span class='number'>1</span>;
}
<span class='kw'>else</span> {
<span class='kw'>break</span>;
}
}
}
(<span class='ident'>eigenvectors</span>, <span class='ident'>eigenvalues</span>.<span class='ident'>diagonal</span>())
}
<span class='doccomment'>/// Cholesky decomposition G of a square symmetric positive definite matrix A, such that A = G * G^T</span>
<span class='doccomment'>///</span>
<span class='doccomment'>/// # Arguments</span>
<span class='doccomment'>/// * `m` - square symmetric positive definite matrix to decompose</span>
<span class='kw'>pub</span> <span class='kw'>fn</span> <span class='ident'>cholesky</span><span class='op'><</span><span class='ident'>N</span>, <span class='ident'>V</span>, <span class='ident'>VS</span>, <span class='ident'>M</span><span class='op'>></span>(<span class='ident'>m</span>: <span class='kw-2'>&</span><span class='ident'>M</span>) <span class='op'>-></span> <span class='prelude-ty'>Result</span><span class='op'><</span><span class='ident'>M</span>, <span class='kw-2'>&</span><span class='lifetime'>'static</span> <span class='ident'>str</span><span class='op'>></span>
<span class='kw'>where</span> <span class='ident'>N</span>: <span class='ident'>BaseFloat</span>,
<span class='ident'>V</span>: <span class='ident'>Mul</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>V</span><span class='op'>></span>,
<span class='ident'>VS</span>: <span class='ident'>Indexable</span><span class='op'><</span><span class='ident'>usize</span>, <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Norm</span><span class='op'><</span><span class='ident'>N</span><span class='op'>></span>,
<span class='ident'>M</span>: <span class='ident'>Indexable</span><span class='op'><</span>(<span class='ident'>usize</span>, <span class='ident'>usize</span>), <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>SquareMatrix</span><span class='op'><</span><span class='ident'>N</span>, <span class='ident'>V</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Add</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>M</span><span class='op'>></span> <span class='op'>+</span>
<span class='ident'>Sub</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>M</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>ColumnSlice</span><span class='op'><</span><span class='ident'>VS</span><span class='op'>></span> <span class='op'>+</span>
<span class='ident'>ApproxEq</span><span class='op'><</span><span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Copy</span> {
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>out</span> <span class='op'>=</span> <span class='ident'>m</span>.<span class='ident'>clone</span>().<span class='ident'>transpose</span>();
<span class='kw'>if</span> <span class='op'>!</span><span class='ident'>ApproxEq</span>::<span class='ident'>approx_eq</span>(<span class='kw-2'>&</span><span class='ident'>out</span>, <span class='kw-2'>&</span><span class='ident'>m</span>) {
<span class='kw'>return</span> <span class='prelude-val'>Err</span>(<span class='string'>"Cholesky: Input matrix is not symmetric"</span>);
}
<span class='kw'>for</span> <span class='ident'>i</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>out</span>.<span class='ident'>nrows</span>() {
<span class='kw'>for</span> <span class='ident'>j</span> <span class='kw'>in</span> <span class='number'>0</span> .. (<span class='ident'>i</span> <span class='op'>+</span> <span class='number'>1</span>) {
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>sum</span>: <span class='ident'>N</span> <span class='op'>=</span> <span class='ident'>out</span>[(<span class='ident'>i</span>, <span class='ident'>j</span>)];
<span class='kw'>for</span> <span class='ident'>k</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>j</span> {
<span class='ident'>sum</span> <span class='op'>=</span> <span class='ident'>sum</span> <span class='op'>-</span> <span class='ident'>out</span>[(<span class='ident'>i</span>, <span class='ident'>k</span>)] <span class='op'>*</span> <span class='ident'>out</span>[(<span class='ident'>j</span>, <span class='ident'>k</span>)];
}
<span class='kw'>if</span> <span class='ident'>i</span> <span class='op'>></span> <span class='ident'>j</span> {
<span class='ident'>out</span>[(<span class='ident'>i</span>, <span class='ident'>j</span>)] <span class='op'>=</span> <span class='ident'>sum</span> <span class='op'>/</span> <span class='ident'>out</span>[(<span class='ident'>j</span>, <span class='ident'>j</span>)];
}
<span class='kw'>else</span> <span class='kw'>if</span> <span class='ident'>sum</span> <span class='op'>></span> <span class='ident'>N</span>::<span class='ident'>zero</span>() {
<span class='ident'>out</span>[(<span class='ident'>i</span>, <span class='ident'>i</span>)] <span class='op'>=</span> <span class='ident'>sum</span>.<span class='ident'>sqrt</span>();
}
<span class='kw'>else</span> {
<span class='kw'>return</span> <span class='prelude-val'>Err</span>(<span class='string'>"Cholesky: Input matrix is not positive definite to machine precision"</span>);
}
}
}
<span class='kw'>for</span> <span class='ident'>i</span> <span class='kw'>in</span> <span class='number'>0</span> .. <span class='ident'>out</span>.<span class='ident'>nrows</span>() {
<span class='kw'>for</span> <span class='ident'>j</span> <span class='kw'>in</span> <span class='ident'>i</span> <span class='op'>+</span> <span class='number'>1</span> .. <span class='ident'>out</span>.<span class='ident'>ncols</span>() {
<span class='ident'>out</span>[(<span class='ident'>i</span>, <span class='ident'>j</span>)] <span class='op'>=</span> <span class='ident'>N</span>::<span class='ident'>zero</span>();
}
}
<span class='kw'>return</span> <span class='prelude-val'>Ok</span>(<span class='ident'>out</span>);
}
<span class='doccomment'>/// Hessenberg</span>
<span class='doccomment'>/// Returns the matrix m in Hessenberg form and the corresponding similarity transformation</span>
<span class='doccomment'>///</span>
<span class='doccomment'>/// # Arguments</span>
<span class='doccomment'>/// * `m` - matrix to transform</span>
<span class='doccomment'>///</span>
<span class='doccomment'>/// # Returns</span>
<span class='doccomment'>/// * First return value `q` - Similarity matrix p such that q * h * q^T = m</span>
<span class='doccomment'>/// * Second return value `h` - Matrix m in Hessenberg form</span>
<span class='kw'>pub</span> <span class='kw'>fn</span> <span class='ident'>hessenberg</span><span class='op'><</span><span class='ident'>N</span>, <span class='ident'>V</span>, <span class='ident'>M</span><span class='op'>></span>(<span class='ident'>m</span>: <span class='kw-2'>&</span><span class='ident'>M</span>) <span class='op'>-></span> (<span class='ident'>M</span>, <span class='ident'>M</span>)
<span class='kw'>where</span> <span class='ident'>N</span>: <span class='ident'>BaseFloat</span>,
<span class='ident'>V</span>: <span class='ident'>Indexable</span><span class='op'><</span><span class='ident'>usize</span>, <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Norm</span><span class='op'><</span><span class='ident'>N</span><span class='op'>></span>,
<span class='ident'>M</span>: <span class='ident'>Copy</span> <span class='op'>+</span> <span class='ident'>Eye</span> <span class='op'>+</span> <span class='ident'>ColumnSlice</span><span class='op'><</span><span class='ident'>V</span><span class='op'>></span> <span class='op'>+</span> <span class='ident'>Transpose</span> <span class='op'>+</span> <span class='ident'>Indexable</span><span class='op'><</span>(<span class='ident'>usize</span>, <span class='ident'>usize</span>), <span class='ident'>N</span><span class='op'>></span> <span class='op'>+</span>
<span class='ident'>Mul</span><span class='op'><</span><span class='ident'>M</span>, <span class='ident'>Output</span> <span class='op'>=</span> <span class='ident'>M</span><span class='op'>></span> {
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>h</span> <span class='op'>=</span> <span class='ident'>m</span>.<span class='ident'>clone</span>();
<span class='kw'>let</span> (<span class='ident'>rows</span>, <span class='ident'>cols</span>) <span class='op'>=</span> <span class='ident'>h</span>.<span class='ident'>shape</span>();
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>q</span> : <span class='ident'>M</span> <span class='op'>=</span> <span class='ident'>Eye</span>::<span class='ident'>new_identity</span>(<span class='ident'>cols</span>);
<span class='kw'>if</span> <span class='ident'>cols</span> <span class='op'><=</span> <span class='number'>2</span> {
<span class='kw'>return</span> (<span class='ident'>q</span>, <span class='ident'>h</span>);
}
<span class='kw'>for</span> <span class='ident'>ite</span> <span class='kw'>in</span> <span class='number'>0</span> .. (<span class='ident'>cols</span> <span class='op'>-</span> <span class='number'>2</span>) {
<span class='kw'>let</span> <span class='kw-2'>mut</span> <span class='ident'>v</span> <span class='op'>=</span> <span class='ident'>h</span>.<span class='ident'>column_slice</span>(<span class='ident'>ite</span>, <span class='ident'>ite</span> <span class='op'>+</span> <span class='number'>1</span>, <span class='ident'>rows</span>);
<span class='kw'>let</span> <span class='ident'>alpha</span> <span class='op'>=</span> <span class='ident'>Norm</span>::<span class='ident'>norm</span>(<span class='kw-2'>&</span><span class='ident'>v</span>);
<span class='kw'>unsafe</span> {
<span class='kw'>let</span> <span class='ident'>x</span> <span class='op'>=</span> <span class='ident'>v</span>.<span class='ident'>unsafe_at</span>(<span class='number'>0</span>);
<span class='ident'>v</span>.<span class='ident'>unsafe_set</span>(<span class='number'>0</span>, <span class='ident'>x</span> <span class='op'>-</span> <span class='ident'>alpha</span>);
}
<span class='kw'>if</span> <span class='op'>!</span>::<span class='ident'>is_zero</span>(<span class='kw-2'>&</span><span class='ident'>v</span>.<span class='ident'>normalize_mut</span>()) {
<span class='kw'>let</span> <span class='ident'>p</span>: <span class='ident'>M</span> <span class='op'>=</span> <span class='ident'>householder_matrix</span>(<span class='ident'>rows</span>, <span class='ident'>ite</span> <span class='op'>+</span> <span class='number'>1</span>, <span class='ident'>v</span>);
<span class='ident'>q</span> <span class='op'>=</span> <span class='ident'>q</span> <span class='op'>*</span> <span class='ident'>p</span>;
<span class='ident'>h</span> <span class='op'>=</span> <span class='ident'>p</span> <span class='op'>*</span> <span class='ident'>h</span> <span class='op'>*</span> <span class='ident'>p</span>;
}
}
<span class='kw'>return</span> (<span class='ident'>q</span>, <span class='ident'>h</span>);
}
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