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implement two levels of blocking in PartialLU => high speedup

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This commit is contained in:
Gael Guennebaud 2009-08-04 11:28:02 +02:00
parent 912da9fade
commit 4bec101470
2 changed files with 142 additions and 84 deletions
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219
Eigen/src/LU/PartialLU.h
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@ -201,98 +201,153 @@ PartialLU<MatrixType>::PartialLU(const MatrixType& matrix)
compute(matrix);
}
/** \internal performs the LU decomposition in place of the matrix \a lu.
* In addition, this function returns the row transpositions in the
* vector \a row_transpositions which must have a size equal to the number
* of columns of the matrix \a lu, and an integer \a nb_transpositions
* which returns the actual number of transpositions.
*/
template<typename MatrixType, typename IntVector>
void ei_lu_unblocked(MatrixType& lu, IntVector& row_transpositions, int& nb_transpositions)
{
const int rows = lu.rows();
const int size = std::min(lu.rows(),lu.cols());
nb_transpositions = 0;
for(int k = 0; k < size; ++k)
{
int row_of_biggest_in_col;
lu.block(k,k,rows-k,1).cwise().abs().maxCoeff(&row_of_biggest_in_col);
row_of_biggest_in_col += k;
row_transpositions.coeffRef(k) = row_of_biggest_in_col;
if(k != row_of_biggest_in_col)
{
lu.row(k).swap(lu.row(row_of_biggest_in_col));
++nb_transpositions;
}
if(k<rows-1)
{
lu.col(k).end(rows-k-1) /= lu.coeff(k,k);
for(int col = k + 1; col < size; ++col)
lu.col(col).end(rows-k-1) -= lu.col(k).end(rows-k-1) * lu.coeff(k,col);
}
}
}
/** This is the blocked version of ei_lu_unblocked() */
template<typename MatrixType, typename IntVector>
void ei_lu_blocked(MatrixType& lu, IntVector& row_transpositions, int& nb_transpositions)
template<typename Scalar, int StorageOrder>
struct ei_partial_lu_impl
{
const int size = lu.rows();
// automatically adjust the number of subdivisions to the size
// of the matrix so that there is enough sub blocks:
int blockSize = size/8;
blockSize = (blockSize/16)*16;
blockSize = std::min(std::max(blockSize,8), 256);
// if the matrix is too small, no blocking:
if(size<32)
blockSize = size;
nb_transpositions = 0;
for(int k = 0; k < size; k+=blockSize)
{
int bs = std::min(size-k,blockSize);
int ps = size - k;
int rs = size - k - bs;
// partition the matrix:
// A00 | A01 | A02
// lu = A10 | A11 | A12
// A20 | A21 | A22
Block<MatrixType,Dynamic,Dynamic> A_0(lu,0,0,size,k);
Block<MatrixType,Dynamic,Dynamic> A11_21(lu,k,k,ps,bs);
Block<MatrixType,Dynamic,Dynamic> A_2(lu,0,k+bs,size,rs);
Block<MatrixType,Dynamic,Dynamic> A11(lu,k,k,bs,bs);
Block<MatrixType,Dynamic,Dynamic> A12(lu,k,k+bs,bs,rs);
Block<MatrixType,Dynamic,Dynamic> A21(lu,k+bs,k,rs,bs);
Block<MatrixType,Dynamic,Dynamic> A22(lu,k+bs,k+bs,rs,rs);
// FIXME add a stride to Map, so that the following mapping becomes easier,
// another option would be to create an expression being able to automatically
// warp any Map, Matrix, and Block expressions as a unique type, but since that's exactly
// a Map + stride, why not adding a stride to Map, and convenient ctors from a Matrix,
// and Block.
typedef Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > MapLU;
typedef Block<MapLU, Dynamic, Dynamic> MatrixType;
typedef Block<MatrixType,Dynamic,Dynamic> BlockType;
VectorBlock<IntVector,Dynamic> row_transpositions_in_panel(row_transpositions,k,bs);
int nb_transpositions_in_panel;
ei_lu_unblocked(A11_21, row_transpositions_in_panel, nb_transpositions_in_panel);
nb_transpositions_in_panel += nb_transpositions_in_panel;
// update permutations and apply them to A10
for(int i=k;i<k+bs; ++i)
/** \internal performs the LU decomposition in-place of the matrix \a lu
* using an unblocked algorithm.
*
* In addition, this function returns the row transpositions in the
* vector \a row_transpositions which must have a size equal to the number
* of columns of the matrix \a lu, and an integer \a nb_transpositions
* which returns the actual number of transpositions.
*/
static void unblocked_lu(MatrixType& lu, int* row_transpositions, int& nb_transpositions)
{
const int rows = lu.rows();
const int size = std::min(lu.rows(),lu.cols());
nb_transpositions = 0;
for(int k = 0; k < size; ++k)
{
int piv = (row_transpositions.coeffRef(i) += k);
A_0.row(i).swap(A_0.row(piv));
}
int row_of_biggest_in_col;
lu.block(k,k,rows-k,1).cwise().abs().maxCoeff(&row_of_biggest_in_col);
row_of_biggest_in_col += k;
if(rs)
{
// apply permutations to A_2
for(int i=k;i<k+bs; ++i)
A_2.row(i).swap(A_2.row(row_transpositions.coeff(i)));
row_transpositions[k] = row_of_biggest_in_col;
// A12 = A11^-1 A12
A11.template triangularView<UnitLowerTriangular>().solveInPlace(A12);
if(k != row_of_biggest_in_col)
{
lu.row(k).swap(lu.row(row_of_biggest_in_col));
++nb_transpositions;
}
A22 -= A21 * A12;
if(k<rows-1)
{
lu.col(k).end(rows-k-1) /= lu.coeff(k,k);
// TODO implement a fast rank one update routine
for(int col = k + 1; col < size; ++col)
lu.col(col).end(rows-k-1) -= lu.col(k).end(rows-k-1) * lu.coeff(k,col);
}
}
}
/** \internal performs the LU decomposition in-place of the matrix represented
* by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
* recursive, blocked algorithm.
*
* In addition, this function returns the row transpositions in the
* vector \a row_transpositions which must have a size equal to the number
* of columns of the matrix \a lu, and an integer \a nb_transpositions
* which returns the actual number of transpositions.
*
* \note This very low level interface using pointers, etc. is to:
* 1 - reduce the number of instanciations to the strict minimum
* 2 - avoid infinite recursion of the instanciations with Block<Block<Block<...> > >
*/
static void blocked_lu(int rows, int cols, Scalar* lu_data, int luStride, int* row_transpositions, int& nb_transpositions, int maxBlockSize=256)
{
MapLU lu1(lu_data,StorageOrder==RowMajor?rows:luStride,StorageOrder==RowMajor?luStride:cols);
MatrixType lu(lu1,0,0,rows,cols);
const int size = std::min(rows,cols);
// if the matrix is too small, no blocking:
if(size<=16)
{
unblocked_lu(lu, row_transpositions, nb_transpositions);
return;
}
// automatically adjust the number of subdivisions to the size
// of the matrix so that there is enough sub blocks:
int blockSize;
{
blockSize = size/8;
blockSize = (blockSize/16)*16;
blockSize = std::min(std::max(blockSize,8), maxBlockSize);
}
nb_transpositions = 0;
for(int k = 0; k < size; k+=blockSize)
{
int bs = std::min(size-k,blockSize);
int ps = size - k;
int rs = size - k - bs;
// partition the matrix:
// A00 | A01 | A02
// lu = A10 | A11 | A12
// A20 | A21 | A22
BlockType A_0(lu,0,0,size,k);
BlockType A_2(lu,0,k+bs,size,rs);
BlockType A11(lu,k,k,bs,bs);
BlockType A12(lu,k,k+bs,bs,rs);
BlockType A21(lu,k+bs,k,rs,bs);
BlockType A22(lu,k+bs,k+bs,rs,rs);
int nb_transpositions_in_panel;
// recursively calls the blocked LU algorithm with a very small
// blocking size:
blocked_lu(ps, bs, &lu.coeffRef(k,k), luStride,
row_transpositions+k, nb_transpositions_in_panel, 16);
nb_transpositions_in_panel += nb_transpositions_in_panel;
// update permutations and apply them to A10
for(int i=k;i<k+bs; ++i)
{
int piv = (row_transpositions[i] += k);
A_0.row(i).swap(A_0.row(piv));
}
if(rs)
{
// apply permutations to A_2
for(int i=k;i<k+bs; ++i)
A_2.row(i).swap(A_2.row(row_transpositions[i]));
// A12 = A11^-1 A12
A11.template triangularView<UnitLowerTriangular>().solveInPlace(A12);
A22 -= A21 * A12;
}
}
}
};
/** \internal performs the LU decomposition with partial pivoting in-place.
*/
template<typename MatrixType, typename IntVector>
void ei_partial_lu_inplace(MatrixType& lu, IntVector& row_transpositions, int& nb_transpositions)
{
ei_assert(lu.cols() == row_transpositions.size());
ei_assert((&row_transpositions.coeffRef(1)-&row_transpositions.coeffRef(0)) == 1);
ei_partial_lu_impl
<typename MatrixType::Scalar, MatrixType::Flags&RowMajorBit?RowMajor:ColMajor>
::blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0,0), lu.stride(), &row_transpositions.coeffRef(0), nb_transpositions);
}
template<typename MatrixType>
@ -307,7 +362,7 @@ void PartialLU<MatrixType>::compute(const MatrixType& matrix)
IntColVectorType rows_transpositions(size);
int nb_transpositions;
ei_lu_blocked(m_lu, rows_transpositions, nb_transpositions);
ei_partial_lu_inplace(m_lu, rows_transpositions, nb_transpositions);
m_det_p = (nb_transpositions%2) ? -1 : 1;
for(int k = 0; k < size; ++k) m_p.coeffRef(k) = k;

7
test/inverse.cpp
View File

@ -81,13 +81,16 @@ template<typename MatrixType> void inverse(const MatrixType& m)
void test_inverse()
{
int s;
for(int i = 0; i < g_repeat; i++) {
CALL_SUBTEST( inverse(Matrix<double,1,1>()) );
CALL_SUBTEST( inverse(Matrix2d()) );
CALL_SUBTEST( inverse(Matrix3f()) );
CALL_SUBTEST( inverse(Matrix4f()) );
CALL_SUBTEST( inverse(MatrixXf(72,72)) );
CALL_SUBTEST( inverse(MatrixXcd(56,56)) );
s = ei_random<int>(50,320);
CALL_SUBTEST( inverse(MatrixXf(s,s)) );
s = ei_random<int>(25,100);
CALL_SUBTEST( inverse(MatrixXcd(s,s)) );
}
// test some tricky cases for 4x4 matrices