From 1b8277fc2a2675253b8bd49b468cda84a3bf099d Mon Sep 17 00:00:00 2001
From: Gael Guennebaud <g.gael@free.fr>
Date: Wed, 30 Jun 2010 10:37:23 +0200
Subject: update the big linear algebra table (fixes, add notes and
 definitions)

---
 doc/C02_TutorialMatrixArithmetic.dox     |  2 +-
 doc/TopicLinearAlgebraDecompositions.dox | 44 +++++++++++++++++++++++++-------
 2 files changed, 36 insertions(+), 10 deletions(-)

diff --git a/doc/C02_TutorialMatrixArithmetic.dox b/doc/C02_TutorialMatrixArithmetic.dox
index 323cc550b..7e5615d6a 100644
--- a/doc/C02_TutorialMatrixArithmetic.dox
+++ b/doc/C02_TutorialMatrixArithmetic.dox
@@ -152,7 +152,7 @@ Example: \include tut_arithmetic_redux_basic.cpp
 Output: \include tut_arithmetic_redux_basic.out
 </td></tr></table>
 
-The \em trace of a matrix, as returned by the function \c trace(), is the sum of the diagonal coefficients and can also be computed as efficiently using <tt>a.diagonal().sum()</tt>, as we see later on.
+The \em trace of a matrix, as returned by the function \c trace(), is the sum of the diagonal coefficients and can also be computed as efficiently using <tt>a.diagonal().sum()</tt>, as we will see later on.
 
 There also exist variants of the \c minCoeff and \c maxCoeff functions returning the coordinates of the respective coefficient via the arguments:
 
diff --git a/doc/TopicLinearAlgebraDecompositions.dox b/doc/TopicLinearAlgebraDecompositions.dox
index adaa6cf35..0a8d89b2e 100644
--- a/doc/TopicLinearAlgebraDecompositions.dox
+++ b/doc/TopicLinearAlgebraDecompositions.dox
@@ -99,7 +99,7 @@ namespace Eigen {
 
     <tr>
         <td>LDLT</td>
-        <td>Positive or negative semidefinite</td>
+        <td>Positive or negative semidefinite<sup><a href="#note1">1</a></sup></td>
         <td>Very fast</td>
         <td>Good</td>
         <td>-</td>
@@ -109,6 +109,8 @@ namespace Eigen {
         <td><em>Soon: blocking</em></td>
     </tr>
 
+    <tr><td colspan="8">\n Singular values and eigenvalues decompositions</td></tr>
+
     <tr>
         <td>SVD</td>
         <td>-</td>
@@ -136,7 +138,7 @@ namespace Eigen {
     <tr>
         <td>SelfAdjointEigenSolver</td>
         <td>Self-adjoint</td>
-        <td>Fast, depends on condition number</td>
+        <td>Fast-average<sup><a href="#note2">2</a></sup></td>
         <td>Good</td>
         <td>Yes</td>
         <td>Eigenvalues/vectors</td>
@@ -148,7 +150,7 @@ namespace Eigen {
     <tr>
         <td>ComplexEigenSolver</td>
         <td>Square</td>
-        <td>Slow, depends on condition number</td>
+        <td>Slow-very slow<sup><a href="#note2">2</a></sup></td>
         <td>Depends on condition number</td>
         <td>Yes</td>
         <td>Eigenvalues/vectors</td>
@@ -160,7 +162,7 @@ namespace Eigen {
     <tr>
         <td>EigenSolver</td>
         <td>Square and real</td>
-        <td>Average, depends on condition number</td>
+        <td>Average-slow<sup><a href="#note2">2</a></sup></td>
         <td>Depends on condition number</td>
         <td>Yes</td>
         <td>Eigenvalues/vectors</td>
@@ -172,7 +174,7 @@ namespace Eigen {
     <tr>
         <td>GeneralizedSelfAdjointEigenSolver</td>
         <td>Square</td>
-        <td>Fast, depends on condition number</td>
+        <td>Fast-average<sup><a href="#note2">2</a></sup></td>
         <td>Depends on condition number</td>
         <td>-</td>
         <td>Generalized eigenvalues/vectors</td>
@@ -181,10 +183,12 @@ namespace Eigen {
         <td>-</td>
     </tr>
 
+    <tr><td colspan="8">\n Helper decompositions</td></tr>
+
     <tr>
         <td>RealSchur</td>
         <td>Square and real</td>
-        <td>Average, depends on condition number</td>
+        <td>Average-slow<sup><a href="#note2">2</a></sup></td>
         <td>Depends on condition number</td>
         <td>Yes</td>
         <td>-</td>
@@ -196,7 +200,7 @@ namespace Eigen {
     <tr>
         <td>ComplexSchur</td>
         <td>Square and real</td>
-        <td>Slow, depends on condition number</td>
+        <td>Slow-very slow<sup><a href="#note2">2</a></sup></td>
         <td>Depends on condition number</td>
         <td>Yes</td>
         <td>-</td>
@@ -231,7 +235,7 @@ namespace Eigen {
 
     <tr>
         <td>HessenbergDecomposition</td>
-        <td>-</td>
+        <td>Square</td>
         <td>Average</td>
         <td>Good</td>
         <td>-</td>
@@ -243,10 +247,32 @@ namespace Eigen {
 
 </table>
 
+\b Notes:
+<ul>
+<li><a name="note1">\b 1: </a>There exist a couple of variants of the LDLT algorithm. Eigen's one produces a pure diagonal matrix, and therefore it cannot handle indefinite matrix, unlike Lapack's one which produces a block diagonal matrix.</li>
+<li><a name="note2">\b 2: </a>Eigenvalues and Schur decompositions rely on iterative algorithms. Their convergence speed depends on how the eigenvalues are well separated.</li>
+</ul>
+
 \section TopicLinAlgTerminology Terminology
 
-TODO explain selfadjoint, positive definite/semidefinite, blocking, unrollers,  ....
+<dl>
+  <dt><b>Selfadjoint</b></dt>
+    <dd>For a real matrix, selfadjoint is a synonym for symmetric. For a complex matrix, selfadjoint is a synonym for \em hermitian.
+        More generally, a matrix \f$ A \f$ is selfadjoint if and only if it is equal to its adjoint \f$ A^* \f$. The adjoint is also called the \em conjugate \em transpose. </dd>
+  <dt><b>Positive/negative definite</b></dt>
+    <dd>A selfadjoint matrix \f$ A \f$ is positive definite if \f$ v^* A v > 0 \f$ for any non zero vector \f$ v \f$.
+        In the same vein, it is negative definite if \f$ v^* A v < 0 \f$ for any non zero vector \f$ v \f$ </dd>
+  <dt><b>Positive/negative semidefinite</b></dt>
+    <dd>A selfadjoint matrix \f$ A \f$ is positive semi-definite if \f$ v^* A v \ge 0 \f$ for any non zero vector \f$ v \f$.
+        In the same vein, it is negative semi-definite if \f$ v^* A v \le 0 \f$ for any non zero vector \f$ v \f$ </dd>
 
+  <dt><b>Blocking</b></dt>
+    <dd>Means the algorithm can work per block, whence guarantying a good scaling of the performance for large matrices.</dd>
+  <dt><b>Meta-unroller</b></dt>
+    <dd>Means the algorithm is automatically and explicitly unrolled for very small fixed size matrices.</dd>
+  <dt><b></b></dt>
+    <dd></dd>
+</dl>
 
 */
 
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