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update the big linear algebra table (fixes, add notes and definitions)

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Gael Guennebaud 2010-06-30 10:37:23 +02:00
parent a06cd0fb13
commit 1b8277fc2a
2 changed files with 36 additions and 10 deletions
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2
doc/C02_TutorialMatrixArithmetic.dox
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@ -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:

44
doc/TopicLinearAlgebraDecompositions.dox
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@ -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>
*/