doc updates/improvements

doc/TopicLinearAlgebraDecompositions.dox

@@ -112,27 +112,15 @@ namespace Eigen {
     <tr><td colspan="9">\n Singular values and eigenvalues decompositions</td></tr>
 
     <tr>
-        <td>SVD</td>
-        <td>-</td>
-        <td>Average</td>
-        <td>Good</td>
-        <td>Yes</td>
-        <td>Singular values/vectors, least squares</td>
-        <td>Yes</td>
-        <td>Average</td>
-        <td>-</td>
-    </tr>
-
-    <tr>
-        <td>JacobiSVD</td>
+        <td>JacobiSVD (two-sided)</td>
         <td>-</td>
         <td>Slow (but fast for small matrices)</td>
-        <td>Proven</td>
+        <td>Excellent-Proven<sup><a href="#note3">3</a></sup></td>
         <td>Yes</td>
         <td>Singular values/vectors, least squares</td>
-        <td>-</td>
+        <td>Yes (and does least squares)</td>
         <td>Excellent</td>
-        <td>-</td>
+        <td>R-SVD</td>
     </tr>
 
     <tr>
@@ -251,6 +239,7 @@ namespace Eigen {
 <ul>
 <li><a name="note1">\b 1: </a>There exist two variants of the LDLT algorithm. Eigen's one produces a pure diagonal D matrix, and therefore it cannot handle indefinite matrices, unlike Lapack's one which produces a block diagonal D matrix.</li>
 <li><a name="note2">\b 2: </a>Eigenvalues, SVD and Schur decompositions rely on iterative algorithms. Their convergence speed depends on how well the eigenvalues are separated.</li>
+<li><a name="note3">\b 3: </a>Our JacobiSVD is two-sided, making for proven and optimal precision for square matrices. For non-square matrices, we have to use a QR preconditioner first. The default choice, ColPivHouseholderQR, is already very reliable, but if you want it to be proven, use FullPivHouseholderQR instead.
 </ul>
 
 \section TopicLinAlgTerminology Terminology