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refine docs for chol()

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conrad
2021-07-06 11:34:22 +10:00
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@@ -12136,7 +12136,7 @@ See also:
</table>
<ul>
<li>
Cholesky decomposition of symmetric/hermitian positive-definite matrix <i>X</i> into triangular matrix <i>R</i>, with an optional permutation matrix/vector <i>P</i>
Cholesky decomposition of symmetric/hermitian matrix <i>X</i> into triangular matrix <i>R</i>, with an optional permutation vector/matrix <i>P</i>
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@@ -12148,7 +12148,15 @@ The optional argument <i>layout</i> is either <code>"upper"</code> or <code>"low
</li>
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<li>
Forms 5 to 7 also produce a permutation matrix/vector <i>P</i>
Forms 1 to 4 require <i>X</i> to be positive definite
</li>
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<li>
Forms 5 to 7 require <i>X</i> to be positive semi-definite
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Forms 5 to 7 also provide a permutation vector/matrix <i>P</i> with type <a href="#Col">uvec</a> or <a href="#Mat">umat</a>
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@@ -12161,9 +12169,9 @@ The decomposition has the following form:
<li>forms 1 to 4, with <i>layout = "upper"</i>: <i>X = R.t() * R</i></li>
<li>forms 1 to 4, with <i>layout = "lower"</i>: <i>X = R * R.t()</i></li>
<li>forms 5 to 7, with <i>layout = "upper"</i> and <i>P_mode = "matrix"</i>: <i>X = P * R.t() * R * P.t()</i></li>
<li>forms 5 to 7, with <i>layout = "upper"</i> and <i>P_mode = "vector"</i>: <i>X = Y(P,P)</i>, where <i>Y = R.t() * R</i> and <i>Y(P,P)</i> is a <a href="#submat">non-contiguous view</a> of <i>Y</i></li>
<li>forms 5 to 7, with <i>layout = "lower"</i> and <i>P_mode = "matrix"</i>: <i>X = P * R * R.t() * P.t()</i></li>
<li>forms 5 to 7, with <i>layout = "upper"</i> and <i>P_mode = "vector"</i>: <i>X(P,P) = R.t() * R</i>, where <i>X(P,P)</i> is a <a href="#submat">non-contiguous view</a> of <i>X</i></li>
<li>forms 5 to 7, with <i>layout = "lower"</i> and <i>P_mode = "vector"</i>: <i>X(P,P) = R * R.t()</i>, where <i>X(P,P)</i> is a <a href="#submat">non-contiguous view</a> of <i>X</i></li>
<li>forms 5 to 7, with <i>layout = "lower"</i> and <i>P_mode = "vector"</i>: <i>X = Y(P,P)</i>, where <i>Y = R * R.t()</i> and <i>Y(P,P)</i> is a <a href="#submat">non-contiguous view</a> of <i>Y</i></li>
</ul>
</li>
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@@ -12178,11 +12186,19 @@ The decomposition has the following form:
Examples:
<ul>
<pre>
mat X(5, 5, fill::randu);
mat Y = X.t()*X;
mat A(5, 5, fill::randu);
mat B = A.t()*A;
mat R1 = chol(Y);
mat R2 = chol(Y, "lower");
mat R1 = chol(B);
mat R2 = chol(B, "lower");
mat R;
uvec P_vec;
umat P_mat;
chol(R, P_vec, B);
chol(R, P_mat, B, "lower", "matrix");
</pre>
</ul>
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@@ -12194,8 +12210,9 @@ See also:
<li><a href="#lu">lu()</a></li>
<li><a href="#qr">qr()</a></li>
<li><a href="#is_sympd">.is_sympd()</a></li>
<li><a href="http://mathworld.wolfram.com/CholeskyDecomposition.html">Cholesky decomposition in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Cholesky decomposition in Wikipedia</a></li>
<li><a href="https://mathworld.wolfram.com/CholeskyDecomposition.html">Cholesky decomposition in MathWorld</a></li>
<li><a href="https://en.wikipedia.org/wiki/Cholesky_decomposition">Cholesky decomposition in Wikipedia</a></li>
<li><a href="https://en.wikipedia.org/wiki/Definite_matrix">Definite matrix in Wikipedia</a></li>
</ul>
</li>
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