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 & p\mathbf{1}_n = \mathbf{1}_n\\
 a^\mathrm{T}pa > 0 \text{ for all } a ∈ ℝ^{n}\backslash\{\mathbf{0}_n\}
 \bigr\},
-\end{aligned}\]</p><p>where <span>$\mathbf{1}_n$</span> and <span>$\mathbr{0}_n$</span> are the vectors of length <span>$n$</span> containing ones and zeros, respectively. More details about this manifold can be found in [<a href="../misc/references.html#DouikHassibi:2019">DH19</a>].</p><p><strong>Constructor</strong></p><pre><code class="nohighlight hljs">MultinomialSymmetricPositiveDefinite(n)</code></pre><p>Generate the manifold of matrices <span>$\mathbb R^{n×n}$</span> that are symmetric, positive definite, and doubly stochastic.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/5079a1989300a1ddc223fd0f2a83494097d5743d/src/manifolds/MultinomialSymmetricPositiveDefinite.jl#L1-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Random.rand!-Tuple{Random.AbstractRNG, MultinomialSymmetricPositiveDefinite, AbstractMatrix}" href="#Random.rand!-Tuple{Random.AbstractRNG, MultinomialSymmetricPositiveDefinite, AbstractMatrix}"><code>Random.rand!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">Random.rand!(
+\end{aligned}\]</p><p>where <span>$\mathbf{1}_n$</span> and <span>$\mathbr{0}_n$</span> are the vectors of length <span>$n$</span> containing ones and zeros, respectively. More details about this manifold can be found in [<a href="../misc/references.html#DouikHassibi:2019">DH19</a>].</p><p><strong>Constructor</strong></p><pre><code class="nohighlight hljs">MultinomialSymmetricPositiveDefinite(n)</code></pre><p>Generate the manifold of matrices <span>$\mathbb R^{n×n}$</span> that are symmetric, positive definite, and doubly stochastic.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/2e29737cfc990a23b131d3d0c11e71f16b2e7a9e/src/manifolds/MultinomialSymmetricPositiveDefinite.jl#L1-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Random.rand!-Tuple{Random.AbstractRNG, MultinomialSymmetricPositiveDefinite, AbstractMatrix}" href="#Random.rand!-Tuple{Random.AbstractRNG, MultinomialSymmetricPositiveDefinite, AbstractMatrix}"><code>Random.rand!</code></a> — <span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">Random.rand!(
     rng::AbstractRNG,
     M::MultinomialSymmetricPositiveDefinite,
     p::AbstractMatrix,
-)</code></pre><p>Generate a random point on <a href="multinomialsymmetricpositivedefinite.html#Manifolds.MultinomialSymmetricPositiveDefinite"><code>MultinomialSymmetricPositiveDefinite</code></a> manifold. The steps are as follows:</p><ol><li>Generate a random <a href="https://en.wikipedia.org/wiki/Totally_positive_matrix">totally positive matrix</a>  a. Construct a vector <code>L</code> of <code>n</code> random positive increasing real numbers.  b. Construct the <a href="https://en.wikipedia.org/wiki/Vandermonde_matrix">Vandermonde matrix</a>     <code>V</code> based on the sequence <code>L</code>.  c. Perform LU factorization of <code>V</code> in such way that both L and U components have     positive elements.  d. Convert the LU factorization into LDU factorization by taking the diagonal of U     and dividing U by it, <code>V=LDU</code>.  e. Construct a new matrix <code>R = UDL</code> which is totally positive.</li><li>Project the totally positive matrix <code>R</code> onto the manifold of <a href="multinomialdoublystochastic.html#Manifolds.MultinomialDoubleStochastic"><code>MultinomialDoubleStochastic</code></a> matrices.</li><li>Symmetrize the projected matrix and return the result.</li></ol><p>This method roughly follows the procedure described in https://math.stackexchange.com/questions/2773460/how-to-generate-a-totally-positive-matrix-randomly-using-software-like-maple</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/5079a1989300a1ddc223fd0f2a83494097d5743d/src/manifolds/MultinomialSymmetricPositiveDefinite.jl#L68-L91">source</a></section></article><h2 id="Literature"><a class="docs-heading-anchor" href="#Literature">Literature</a><a id="Literature-1"></a><a class="docs-heading-anchor-permalink" href="#Literature" title="Permalink"></a></h2><div class="citation noncanonical"><dl><dt>[DH19]</dt><dd><div>A. Douik and B. Hassibi. <em>Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry</em>. <a href="https://doi.org/10.1109/tsp.2019.2946024">IEEE Transactions on Signal Processing <strong>67</strong>, 5761–5774</a> (2019), <a href="https://arxiv.org/abs/1802.02628">arXiv:1802.02628</a>.</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="multinomialsymmetric.html">« Multinomial symmetric matrices</a><a class="docs-footer-nextpage" href="oblique.html">Oblique manifold »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Thursday 24 October 2024 04:14">Thursday 24 October 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+)</code></pre><p>Generate a random point on <a href="multinomialsymmetricpositivedefinite.html#Manifolds.MultinomialSymmetricPositiveDefinite"><code>MultinomialSymmetricPositiveDefinite</code></a> manifold. The steps are as follows:</p><ol><li>Generate a random <a href="https://en.wikipedia.org/wiki/Totally_positive_matrix">totally positive matrix</a>  a. Construct a vector <code>L</code> of <code>n</code> random positive increasing real numbers.  b. Construct the <a href="https://en.wikipedia.org/wiki/Vandermonde_matrix">Vandermonde matrix</a>     <code>V</code> based on the sequence <code>L</code>.  c. Perform LU factorization of <code>V</code> in such way that both L and U components have     positive elements.  d. Convert the LU factorization into LDU factorization by taking the diagonal of U     and dividing U by it, <code>V=LDU</code>.  e. Construct a new matrix <code>R = UDL</code> which is totally positive.</li><li>Project the totally positive matrix <code>R</code> onto the manifold of <a href="multinomialdoublystochastic.html#Manifolds.MultinomialDoubleStochastic"><code>MultinomialDoubleStochastic</code></a> matrices.</li><li>Symmetrize the projected matrix and return the result.</li></ol><p>This method roughly follows the procedure described in https://math.stackexchange.com/questions/2773460/how-to-generate-a-totally-positive-matrix-randomly-using-software-like-maple</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/2e29737cfc990a23b131d3d0c11e71f16b2e7a9e/src/manifolds/MultinomialSymmetricPositiveDefinite.jl#L68-L91">source</a></section></article><h2 id="Literature"><a class="docs-heading-anchor" href="#Literature">Literature</a><a id="Literature-1"></a><a class="docs-heading-anchor-permalink" href="#Literature" title="Permalink"></a></h2><div class="citation noncanonical"><dl><dt>[DH19]</dt><dd><div>A. Douik and B. Hassibi. <em>Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry</em>. <a href="https://doi.org/10.1109/tsp.2019.2946024">IEEE Transactions on Signal Processing <strong>67</strong>, 5761–5774</a> (2019), <a href="https://arxiv.org/abs/1802.02628">arXiv:1802.02628</a>.</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="multinomialsymmetric.html">« Multinomial symmetric matrices</a><a class="docs-footer-nextpage" href="oblique.html">Oblique manifold »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Thursday 24 October 2024 08:23">Thursday 24 October 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>