cpp/src/librosa/eigen3/doc/SparseQuickReference.dox

namespace Eigen {
/** \eigenManualPage SparseQuickRefPage Quick reference guide for sparse matrices
\eigenAutoToc

<hr>

In this page, we give a quick summary of the main operations available for sparse matrices in the class SparseMatrix. First, it is recommended to read  the introductory tutorial at \ref TutorialSparse. The important point to have in mind when working on sparse matrices is how they are stored : 
i.e either row major or column major. The default is column major. Most arithmetic operations on sparse matrices will assert that they have the same storage order. 

\section SparseMatrixInit Sparse Matrix Initialization
<table class="manual">
<tr><th> Category </th> <th> Operations</th> <th>Notes</th></tr>
<tr><td>Constructor</td>
<td>
\code
  SparseMatrix<double> sm1(1000,1000); 
  SparseMatrix<std::complex<double>,RowMajor> sm2;
\endcode
</td> <td> Default is ColMajor</td> </tr>
<tr class="alt">
<td> Resize/Reserve</td>
<td> 
 \code
    sm1.resize(m,n);      // Change sm1 to a m x n matrix.
    sm1.reserve(nnz);     // Allocate room for nnz nonzeros elements.   
  \endcode 
</td>
<td> Note that when calling reserve(), it is not required that nnz is the exact number of nonzero elements in the final matrix. However, an exact estimation will avoid multiple reallocations during the insertion phase. </td>
</tr>
<tr> 
<td> Assignment </td>
<td> 
\code 
  SparseMatrix<double,Colmajor> sm1;
 // Initialize sm2 with sm1.
  SparseMatrix<double,Rowmajor> sm2(sm1), sm3;        
  // Assignment and evaluations modify the storage order.
  sm3 = sm1; 
 \endcode
</td>
<td> The copy constructor can be used to convert from a storage order to another</td>
</tr>
<tr class="alt">
<td> Element-wise Insertion</td>
<td>
\code 
// Insert a new element; 
 sm1.insert(i, j) = v_ij;  

// Update the value v_ij
 sm1.coeffRef(i,j) = v_ij;
 sm1.coeffRef(i,j) += v_ij;
 sm1.coeffRef(i,j) -= v_ij;
\endcode
</td>
<td> insert() assumes that the element does not already exist; otherwise, use coeffRef()</td>
</tr>
<tr> 
<td> Batch insertion</td>
<td>
\code
  std::vector< Eigen::Triplet<double> > tripletList;
  tripletList.reserve(estimation_of_entries);
  // -- Fill tripletList with nonzero elements...
  sm1.setFromTriplets(TripletList.begin(), TripletList.end());
\endcode
</td>
<td>A complete example is available at \link TutorialSparseFilling Triplet Insertion \endlink.</td>
</tr>
<tr class="alt"> 
<td> Constant or Random Insertion</td>
<td>
\code
sm1.setZero();
\endcode
</td>
<td>Remove all non-zero coefficients</td>
</tr>
</table>


\section SparseBasicInfos Matrix properties
Beyond the basic functions rows() and cols(), there are some useful functions that are available to easily get some informations from the matrix. 
<table class="manual">
<tr>
  <td> \code
  sm1.rows();         // Number of rows
  sm1.cols();         // Number of columns 
  sm1.nonZeros();     // Number of non zero values   
  sm1.outerSize();    // Number of columns (resp. rows) for a column major (resp. row major )
  sm1.innerSize();    // Number of rows (resp. columns) for a row major (resp. column major)
  sm1.norm();         // Euclidian norm of the matrix
  sm1.squaredNorm();  // Squared norm of the matrix
  sm1.blueNorm();
  sm1.isVector();     // Check if sm1 is a sparse vector or a sparse matrix
  sm1.isCompressed(); // Check if sm1 is in compressed form
  ...
  \endcode </td>
</tr>
</table>

\section SparseBasicOps Arithmetic operations
It is easy to perform arithmetic operations on sparse matrices provided that the dimensions are adequate and that the matrices have the same storage order. Note that the evaluation can always be done in a matrix with a different storage order. In the following, \b sm denotes a sparse matrix, \b dm a dense matrix and \b dv a dense vector.
<table class="manual">
<tr><th> Operations </th> <th> Code </th> <th> Notes </th></tr>

<tr>
  <td> add subtract </td> 
  <td> \code
  sm3 = sm1 + sm2; 
  sm3 = sm1 - sm2;
  sm2 += sm1; 
  sm2 -= sm1; \endcode
  </td>
  <td> 
  sm1 and sm2 should have the same storage order
  </td> 
</tr>

<tr class="alt"><td>
  scalar product</td><td>\code
  sm3 = sm1 * s1;   sm3 *= s1; 
  sm3 = s1 * sm1 + s2 * sm2; sm3 /= s1;\endcode
  </td>
  <td>
    Many combinations are possible if the dimensions and the storage order agree.
</tr>

<tr>
  <td> %Sparse %Product </td>
  <td> \code
  sm3 = sm1 * sm2;
  dm2 = sm1 * dm1;
  dv2 = sm1 * dv1;
  \endcode </td>
  <td>
  </td>
</tr> 

<tr class='alt'>
  <td> transposition, adjoint</td>
  <td> \code
  sm2 = sm1.transpose();
  sm2 = sm1.adjoint();
  \endcode </td>
  <td>
  Note that the transposition change the storage order. There is no support for transposeInPlace().
  </td>
</tr> 
<tr>
<td> Permutation </td>
<td> 
\code 
perm.indices();      // Reference to the vector of indices
sm1.twistedBy(perm); // Permute rows and columns
sm2 = sm1 * perm;    // Permute the columns
sm2 = perm * sm1;    // Permute the columns
\endcode 
</td>
<td> 

</td>
</tr>
<tr>
  <td>
  Component-wise ops
  </td>
  <td>\code 
  sm1.cwiseProduct(sm2);
  sm1.cwiseQuotient(sm2);
  sm1.cwiseMin(sm2);
  sm1.cwiseMax(sm2);
  sm1.cwiseAbs();
  sm1.cwiseSqrt();
  \endcode</td>
  <td>
  sm1 and sm2 should have the same storage order
  </td>
</tr>
</table>

\section sparseotherops Other supported operations
<table class="manual">
<tr><th style="min-width:initial"> Code </th> <th> Notes</th> </tr>
<tr><td colspan="2">Sub-matrices</td></tr>
<tr>
<td> 
\code 
  sm1.block(startRow, startCol, rows, cols); 
  sm1.block(startRow, startCol); 
  sm1.topLeftCorner(rows, cols); 
  sm1.topRightCorner(rows, cols);
  sm1.bottomLeftCorner( rows, cols);
  sm1.bottomRightCorner( rows, cols);
  \endcode
</td><td>
Contrary to dense matrices, here <strong>all these methods are read-only</strong>.\n
See \ref TutorialSparse_SubMatrices and below for read-write sub-matrices.
</td>
</tr>
<tr class="alt"><td colspan="2"> Range </td></tr>
<tr class="alt">
<td> 
\code 
  sm1.innerVector(outer);           // RW
  sm1.innerVectors(start, size);    // RW
  sm1.leftCols(size);               // RW
  sm2.rightCols(size);              // RO because sm2 is row-major
  sm1.middleRows(start, numRows);   // RO because sm1 is column-major
  sm1.middleCols(start, numCols);   // RW
  sm1.col(j);                       // RW
\endcode
</td>
<td>
A inner vector is either a row (for row-major) or a column (for column-major).\n
As stated earlier, for a read-write sub-matrix (RW), the evaluation can be done in a matrix with different storage order.
</td>
</tr>
<tr><td colspan="2"> Triangular and selfadjoint views</td></tr>
<tr>
<td> 
\code
  sm2 = sm1.triangularview<Lower>();
  sm2 = sm1.selfadjointview<Lower>();
\endcode
</td>
<td> Several combination between triangular views and blocks views are possible
\code 
  \endcode </td>
</tr>
<tr class="alt"><td colspan="2">Triangular solve </td></tr>
<tr class="alt">
<td> 
\code 
 dv2 = sm1.triangularView<Upper>().solve(dv1);
 dv2 = sm1.topLeftCorner(size, size)
          .triangularView<Lower>().solve(dv1);
\endcode 
</td>
<td> For general sparse solve, Use any suitable module described at \ref TopicSparseSystems </td>
</tr>
<tr><td colspan="2"> Low-level API</td></tr>
<tr>
<td>
\code
sm1.valuePtr();      // Pointer to the values
sm1.innerIndextr();  // Pointer to the indices.
sm1.outerIndexPtr(); // Pointer to the beginning of each inner vector
\endcode
</td>
<td>
If the matrix is not in compressed form, makeCompressed() should be called before.\n
Note that these functions are mostly provided for interoperability purposes with external libraries.\n
A better access to the values of the matrix is done by using the InnerIterator class as described in \link TutorialSparse the Tutorial Sparse \endlink section</td>
</tr>
<tr class="alt"><td colspan="2">Mapping external buffers</td></tr>
<tr class="alt">
<td>
\code
int outerIndexPtr[cols+1];
int innerIndices[nnz];
double values[nnz];
Map<SparseMatrix<double> > sm1(rows,cols,nnz,outerIndexPtr, // read-write
                               innerIndices,values);
Map<const SparseMatrix<double> > sm2(...);                  // read-only
\endcode
</td>
<td>As for dense matrices, class Map<SparseMatrixType> can be used to see external buffers as an %Eigen's SparseMatrix object. </td>
</tr>
</table>
*/
}