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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Thanks for the reproducible example!<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Here is the solution:
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><br>
library(Matrix)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Omega <- Matrix(c(1,0.8,0.8,1),2,2)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">invSqrtV <- Diagonal(2,1)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Omega.M <- as(as.matrix(Omega),"dgCMatrix")<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-bottom:12.0pt"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">invSqrtV.M <- as.vector(diag(invSqrtV))<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">partialsolveCS <- '<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">using namespace Rcpp;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">using namespace Eigen;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">const SparseMatrix<double> Omega(as<SparseMatrix<double> >(Omegas));<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">const VectorXd invSqrtV(as<VectorXd>(invSqrtVs));<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">MatrixXd mat = invSqrtV.asDiagonal();<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Eigen::SimplicialLLT<SparseMatrix<double> > solver;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">MatrixXd L = solver.compute(Omega).matrixL().solve(mat);<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">return wrap(L.sparseView());'<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">my.partialsolveCS <- cxxfunction(signature(Omegas = "dgCMatrix", invSqrtVs = "numeric"),
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> body = partialsolveCS, plugin = "RcppEigen")<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">my.partial.solveR <- function(Omega,invSqrtV){ solve(t(chol(Omega)), invSqrtV)}<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">my_partialsolveCS(Omega.M, invSqrtV.M)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> my.partial.solveR(Omega, invSqrtV)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Few notes:<o:p></o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo2"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">1.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">The previous solution suffers from taking the cholesky decomp twice. Once on omega and then again on L matrix from Omega. The solver.info() returned
false since it could not do it. Anything further on was then zero’d out.<o:p></o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo2"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">2.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">The new approach uses the requirement that after the sparse solver has broken apart the matrix, it can act upon a matrix / vector that is dense.
<br>
(e.g. Ax = b => x = A^(-1)b, where A is sparse and b is dense)<o:p></o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo2"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">3.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">We will exploit the property that the second matrix is diagonal based by passing in a vector instead of an actual matrix and gaining the solve benefit
of having the matrix registered as a diagonal matrix within eigen.<o:p></o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo2"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">4.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">The unfortunate side effect is the return of a matrix that is dense. We can easily switch between types using:<o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:1.0in;text-indent:-.25in;mso-list:l0 level2 lfo2">
<![if !supportLists]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">a.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">SparseMatrix => MatrixXd: MatrixXd changed_to_dense = MatrixXd(sparse_matrix_to_change)<o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:1.0in;text-indent:-.25in;mso-list:l0 level2 lfo2">
<![if !supportLists]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">b.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">MatrixXd => SparseMatrix: SparseMatrix<double> changed_to_sparse = dense_matrix_to_change.sparseView();<o:p></o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo2"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">5.<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">We shouldn’t lose lots of speed using this approach due to how Eigen library is built (e.g.
<a href="http://eigen.tuxfamily.org/dox/TopicInsideEigenExample.html">http://eigen.tuxfamily.org/dox/TopicInsideEigenExample.html</a> )<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Misc:<o:p></o:p></span></b></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">I’d highly encourage you in the future to use the sourceCpp form as it allows for comments and easy of reuse.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal" style="line-height:11.25pt;background:white;word-break:break-all">
<span style="font-size:10.0pt;font-family:"Lucida Console";color:blue">Rcpp::sourceCpp('path/to/file/eigen_sparse_solve.cpp')</span><span style="font-size:10.0pt;font-family:"Lucida Console";color:black"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><br>
#include <RcppEigen.h><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">using namespace Rcpp;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">// [[Rcpp::depends(RcppEigen)]]<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">using Eigen::Map;
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">using Eigen::SparseMatrix;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">using Eigen::LLT;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">// [[Rcpp::export]]<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Eigen::MatrixXd my_partialsolveCS( Eigen::SparseMatrix<double> Omega, Eigen::VectorXd invSqrtV){<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> // If we know invSqrtV is a diagonal, we'll use this to our advantage<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> Eigen::MatrixXd mat = invSqrtV.asDiagonal();<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> // Set up sparse solver.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> // Note, sparse solvers require one matrix to be sparse (omega) and the other matrix or vector to be dense.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> Eigen::SimplicialLLT<SparseMatrix<double> > solver;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> // Access the Lower Cholemsky result and use it to solve the system.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> Eigen::MatrixXd L = solver.compute(Omega).matrixL().solve(mat);<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> // Convert to sparse matrix and return.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> return L.sparseView();<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">}<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">/*** R<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"># Runs the test code on compile<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">library(Matrix)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Omega <- Matrix(c(1,0.8,0.8,1),2,2)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">invSqrtV <- Diagonal(2,1)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Omega.M <- as(as.matrix(Omega),"dgCMatrix")<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">invSqrtV.M <- as.vector(diag(invSqrtV))<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">my.partial.solveR <- function(Omega,invSqrtV){ solve(t(chol(Omega)), invSqrtV)}<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">my.partial.solveR(Omega, invSqrtV)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">my_partialsolveCS(Omega.M, invSqrtV.M)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">*/<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><br>
<br>
Sincerely,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-bottom:12.0pt"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">JJB<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-bottom:12.0pt"><o:p> </o:p></p>
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