* Fix a couple of issues related to the recent cache friendly products

* Improve the efficiency of matrix*vector in unaligned cases
* Trivial fixes in the destructors of MatrixStorage
* Removed the matrixNorm in test/product.cpp (twice faster and
  that assumed the matrix product was ok while checking that !!)
This commit is contained in:
Gael Guennebaud 2008-07-19 00:09:01 +00:00
parent 62ec1dd616
commit 22a816ade8
9 changed files with 85 additions and 92 deletions

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@ -381,14 +381,14 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_colmajor_times_vector(
ei_pstore(&res[j OFFSET], \
ei_padd(ei_pload(&res[j OFFSET]), \
ei_padd( \
ei_padd(ei_pmul(ptmp0,ei_pload ## A0(&lhs[j OFFSET +iN0])),ei_pmul(ptmp1,ei_pload ## A13(&lhs[j OFFSET +iN1]))), \
ei_padd(ei_pmul(ptmp2,ei_pload ## A2(&lhs[j OFFSET +iN2])),ei_pmul(ptmp3,ei_pload ## A13(&lhs[j OFFSET +iN3]))) )))
ei_padd(ei_pmul(ptmp0,ei_pload ## A0(&lhs0[j OFFSET])),ei_pmul(ptmp1,ei_pload ## A13(&lhs1[j OFFSET]))), \
ei_padd(ei_pmul(ptmp2,ei_pload ## A2(&lhs2[j OFFSET])),ei_pmul(ptmp3,ei_pload ## A13(&lhs3[j OFFSET]))) )))
asm("#begin matrix_vector_product");
typedef typename ei_packet_traits<Scalar>::type Packet;
const int PacketSize = sizeof(Packet)/sizeof(Scalar);
enum { AllAligned, EvenAligned, FirstAligned, NoneAligned };
enum { AllAligned = 0, EvenAligned, FirstAligned, NoneAligned };
const int columnsAtOnce = 4;
const int peels = 2;
const int PacketAlignedMask = PacketSize-1;
@ -401,9 +401,9 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_colmajor_times_vector(
? std::min<int>( (PacketSize - ((size_t(res)/sizeof(Scalar)) & PacketAlignedMask)) & PacketAlignedMask, size)
: 0;
const int alignedSize = alignedStart + ((size-alignedStart) & ~PacketAlignedMask);
const int peeledSize = peels>1 ? alignedStart + ((alignedSize-alignedStart) & ~PeelAlignedMask) : 0;
const int peeledSize = peels>1 ? alignedStart + ((alignedSize-alignedStart) & ~PeelAlignedMask) : alignedStart;
const int alignmentStep = lhsStride % PacketSize;
const int alignmentStep = (PacketSize - lhsStride % PacketSize) & PacketAlignedMask;
int alignmentPattern = alignmentStep==0 ? AllAligned
: alignmentStep==2 ? EvenAligned
: FirstAligned;
@ -423,17 +423,17 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_colmajor_times_vector(
skipColumns = std::min(skipColumns,rhs.size());
// note that the skiped columns are processed later.
}
int columnBound = (rhs.size()/columnsAtOnce)*columnsAtOnce;
int columnBound = ((rhs.size()-skipColumns)/columnsAtOnce)*columnsAtOnce + skipColumns;
for (int i=skipColumns; i<columnBound; i+=columnsAtOnce)
{
Scalar tmp0 = rhs[i], tmp1 = rhs[i+1], tmp2 = rhs[i+2], tmp3 = rhs[i+3];
Packet ptmp0 = ei_pset1(tmp0), ptmp1 = ei_pset1(tmp1), ptmp2 = ei_pset1(tmp2), ptmp3 = ei_pset1(tmp3);
int iN0 = i*lhsStride, iN1 = (i+1)*lhsStride, iN2 = (i+2)*lhsStride, iN3 = (i+3)*lhsStride;
Packet ptmp0 = ei_pset1(rhs[i]), ptmp1 = ei_pset1(rhs[i+1]),
ptmp2 = ei_pset1(rhs[i+2]), ptmp3 = ei_pset1(rhs[i+3]);
const Scalar *lhs0 = lhs + i*lhsStride, *lhs1 = lhs + (i+1)*lhsStride,
*lhs2 = lhs + (i+2)*lhsStride, *lhs3 = lhs + (i+3)*lhsStride;
// process initial unaligned coeffs
for (int j=0; j<alignedStart; j++)
res[j] += tmp0 * lhs[j+iN0] + tmp1 * lhs[j+iN1] + tmp2 * lhs[j+iN2] + tmp3 * lhs[j+iN3];
res[j] += ei_pfirst(ptmp0)*lhs0[j] + ei_pfirst(ptmp1)*lhs1[j] + ei_pfirst(ptmp2)*lhs2[j] + ei_pfirst(ptmp3)*lhs3[j];
if (alignedSize>alignedStart)
{
@ -448,14 +448,29 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_colmajor_times_vector(
_EIGEN_ACCUMULATE_PACKETS(,u,,);
break;
case FirstAligned:
if (peels>1)
if(peels>1)
{
// NOTE peeling with two _EIGEN_ACCUMULATE_PACKETS() is much less efficient
// than the following code
asm("#mybegin");
Packet A00, A01, A02, A03, A10, A11, A12, A13;
for (int j = alignedStart; j<peeledSize; j+=peels*PacketSize)
{
_EIGEN_ACCUMULATE_PACKETS(,u,u,);
_EIGEN_ACCUMULATE_PACKETS(,u,u,+PacketSize);
if (peels>2) _EIGEN_ACCUMULATE_PACKETS(,u,u,+2*PacketSize);
if (peels>3) _EIGEN_ACCUMULATE_PACKETS(,u,u,+3*PacketSize);
A01 = ei_ploadu(&lhs1[j]); A11 = ei_ploadu(&lhs1[j+PacketSize]);
A02 = ei_ploadu(&lhs2[j]); A12 = ei_ploadu(&lhs2[j+PacketSize]);
A00 = ei_pload (&lhs0[j]); A10 = ei_pload (&lhs0[j+PacketSize]);
A00 = ei_pmadd(ptmp0, A00, ei_pload(&res[j]));
A10 = ei_pmadd(ptmp0, A10, ei_pload(&res[j+PacketSize]));
A00 = ei_pmadd(ptmp1, A01, A00); A10 = ei_pmadd(ptmp1, A11, A10);
A03 = ei_ploadu(&lhs3[j]); A13 = ei_ploadu(&lhs3[j+PacketSize]);
A00 = ei_pmadd(ptmp2, A02, A00); A10 = ei_pmadd(ptmp2, A12, A10);
A00 = ei_pmadd(ptmp3, A03, A00); A10 = ei_pmadd(ptmp3, A13, A10);
ei_pstore(&res[j],A00); ei_pstore(&res[j+PacketSize],A10);
}
asm("#myend");
}
for (int j = peeledSize; j<alignedSize; j+=PacketSize)
_EIGEN_ACCUMULATE_PACKETS(,u,u,);
break;
@ -468,7 +483,7 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_colmajor_times_vector(
// process remaining coeffs
for (int j=alignedSize; j<size; j++)
res[j] += tmp0 * lhs[j+iN0] + tmp1 * lhs[j+iN1] + tmp2 * lhs[j+iN2] + tmp3 * lhs[j+iN3];
res[j] += ei_pfirst(ptmp0)*lhs0[j] + ei_pfirst(ptmp1)*lhs1[j] + ei_pfirst(ptmp2)*lhs2[j] + ei_pfirst(ptmp3)*lhs3[j];
}
// process remaining first and last columns (at most columnsAtOnce-1)
@ -476,26 +491,25 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_colmajor_times_vector(
int start = columnBound;
do
{
for (int i=columnBound; i<end; i++)
for (int i=start; i<end; i++)
{
Scalar tmp0 = rhs[i];
Packet ptmp0 = ei_pset1(tmp0);
int iN0 = i*lhsStride;
Packet ptmp0 = ei_pset1(rhs[i]);
const Scalar* lhs0 = lhs + i*lhsStride;
// process first unaligned result's coeffs
for (int j=0; j<alignedStart; j++)
res[j] += tmp0 * lhs[j+iN0];
res[j] += ei_pfirst(ptmp0) * lhs0[j];
// process aligned result's coeffs
if ((iN0 % PacketSize) == 0)
if (((i*lhsStride) % PacketSize+alignedStart) == 0)
for (int j = alignedStart;j<alignedSize;j+=PacketSize)
ei_pstore(&res[j], ei_padd(ei_pmul(ptmp0,ei_pload(&lhs[j+iN0])),ei_pload(&res[j])));
ei_pstore(&res[j], ei_pmadd(ptmp0,ei_pload(&lhs0[j]),ei_pload(&res[j])));
else
for (int j = alignedStart;j<alignedSize;j+=PacketSize)
ei_pstore(&res[j], ei_padd(ei_pmul(ptmp0,ei_ploadu(&lhs[j+iN0])),ei_pload(&res[j])));
ei_pstore(&res[j], ei_pmadd(ptmp0,ei_ploadu(&lhs0[j]),ei_pload(&res[j])));
// process remaining scalars
for (int j=alignedSize; j<size; j++)
res[j] += tmp0 * lhs[j+iN0];
res[j] += ei_pfirst(ptmp0) * lhs0[j];
}
if (skipColumns)
{
@ -507,10 +521,11 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_colmajor_times_vector(
break;
} while(true);
asm("#end matrix_vector_product");
#undef _EIGEN_ACCUMULATE_PACKETS
}
// TODO add peeling to mask unaligned load/stores
template<typename Scalar, typename ResType>
EIGEN_DONT_INLINE static void ei_cache_friendly_product_rowmajor_times_vector(
const Scalar* lhs, int lhsStride,
@ -523,10 +538,10 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_rowmajor_times_vector(
#define _EIGEN_ACCUMULATE_PACKETS(A0,A13,A2,OFFSET) {\
Packet b = ei_pload(&rhs[j]); \
ptmp0 = ei_padd(ptmp0, ei_pmul(b, ei_pload##A0 (&lhs[j+iN0]))); \
ptmp1 = ei_padd(ptmp1, ei_pmul(b, ei_pload##A13(&lhs[j+iN1]))); \
ptmp2 = ei_padd(ptmp2, ei_pmul(b, ei_pload##A2 (&lhs[j+iN2]))); \
ptmp3 = ei_padd(ptmp3, ei_pmul(b, ei_pload##A13(&lhs[j+iN3]))); }
ptmp0 = ei_pmadd(b, ei_pload##A0 (&lhs0[j]), ptmp0); \
ptmp1 = ei_pmadd(b, ei_pload##A13(&lhs1[j]), ptmp1); \
ptmp2 = ei_pmadd(b, ei_pload##A2 (&lhs2[j]), ptmp2); \
ptmp3 = ei_pmadd(b, ei_pload##A13(&lhs3[j]), ptmp3); }
asm("#begin matrix_vector_product");
typedef typename ei_packet_traits<Scalar>::type Packet;
@ -534,9 +549,9 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_rowmajor_times_vector(
enum { AllAligned, EvenAligned, FirstAligned, NoneAligned };
const int rowsAtOnce = 4;
const int peels = 2;
// const int peels = 2;
const int PacketAlignedMask = PacketSize-1;
const int PeelAlignedMask = PacketSize*peels-1;
// const int PeelAlignedMask = PacketSize*peels-1;
const bool Vectorized = sizeof(Packet) != sizeof(Scalar);
const int size = rhsSize;
@ -546,9 +561,9 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_rowmajor_times_vector(
? std::min<int>( (PacketSize - ((size_t(rhs)/sizeof(Scalar)) & PacketAlignedMask)) & PacketAlignedMask, size)
: 0;
const int alignedSize = alignedStart + ((size-alignedStart) & ~PacketAlignedMask);
const int peeledSize = peels>1 ? alignedStart + ((alignedSize-alignedStart) & ~PeelAlignedMask) : 0;
//const int peeledSize = peels>1 ? alignedStart + ((alignedSize-alignedStart) & ~PeelAlignedMask) : 0;
const int alignmentStep = lhsStride % PacketSize;
const int alignmentStep = (PacketSize - lhsStride % PacketSize) & PacketAlignedMask;
int alignmentPattern = alignmentStep==0 ? AllAligned
: alignmentStep==2 ? EvenAligned
: FirstAligned;
@ -568,19 +583,20 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_rowmajor_times_vector(
skipRows = std::min(skipRows,res.size());
// note that the skiped columns are processed later.
}
int rowBound = (res.size()/rowsAtOnce)*rowsAtOnce;
int rowBound = ((res.size()-skipRows)/rowsAtOnce)*rowsAtOnce + skipRows;
for (int i=skipRows; i<rowBound; i+=rowsAtOnce)
{
Scalar tmp0 = Scalar(0), tmp1 = Scalar(0), tmp2 = Scalar(0), tmp3 = Scalar(0);
Packet ptmp0 = ei_pset1(Scalar(0)), ptmp1 = ei_pset1(Scalar(0)), ptmp2 = ei_pset1(Scalar(0)), ptmp3 = ei_pset1(Scalar(0));
int iN0 = i*lhsStride, iN1 = (i+1)*lhsStride, iN2 = (i+2)*lhsStride, iN3 = (i+3)*lhsStride;
const Scalar *lhs0 = lhs + i*lhsStride, *lhs1 = lhs + (i+1)*lhsStride,
*lhs2 = lhs + (i+2)*lhsStride, *lhs3 = lhs + (i+3)*lhsStride;
// process initial unaligned coeffs
for (int j=0; j<alignedStart; j++)
{
Scalar b = rhs[j];
tmp0 += b * lhs[j+iN0]; tmp1 += b * lhs[j+iN1]; tmp2 += b * lhs[j+iN2]; tmp3 += b * lhs[j+iN3];
tmp0 += b*lhs0[j]; tmp1 += b*lhs1[j]; tmp2 += b*lhs2[j]; tmp3 += b*lhs3[j];
}
if (alignedSize>alignedStart)
@ -614,9 +630,9 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_rowmajor_times_vector(
for (int j=alignedSize; j<size; j++)
{
Scalar b = rhs[j];
tmp0 += b * lhs[j+iN0]; tmp1 += b * lhs[j+iN1]; tmp2 += b * lhs[j+iN2]; tmp3 += b * lhs[j+iN3];
tmp0 += b*lhs0[j]; tmp1 += b*lhs1[j]; tmp2 += b*lhs2[j]; tmp3 += b*lhs3[j];
}
res[i] = tmp0; res[i+1] = tmp1; res[i+2] = tmp2; res[i+3] = tmp3;
res[i] += tmp0; res[i+1] += tmp1; res[i+2] += tmp2; res[i+3] += tmp3;
}
// process remaining first and last rows (at most columnsAtOnce-1)
@ -624,30 +640,30 @@ EIGEN_DONT_INLINE static void ei_cache_friendly_product_rowmajor_times_vector(
int start = rowBound;
do
{
for (int i=rowBound; i<end; i++)
for (int i=start; i<end; i++)
{
Scalar tmp0 = Scalar(0);
Packet ptmp0 = ei_pset1(tmp0);
int iN0 = i*lhsStride;
const Scalar* lhs0 = lhs + i*lhsStride;
// process first unaligned result's coeffs
for (int j=0; j<alignedStart; j++)
tmp0 += rhs[j] * lhs[j+iN0];
tmp0 += rhs[j] * lhs0[j];
if (alignedSize>alignedStart)
{
// process aligned rhs coeffs
if (iN0 % PacketSize==0)
if ((i*lhsStride+alignedStart) % PacketSize==0)
for (int j = alignedStart;j<alignedSize;j+=PacketSize)
ptmp0 = ei_padd(ptmp0, ei_pmul(ei_pload(&rhs[j]), ei_pload(&lhs[j+iN0])));
ptmp0 = ei_pmadd(ei_pload(&rhs[j]), ei_pload(&lhs0[j]), ptmp0);
else
for (int j = alignedStart;j<alignedSize;j+=PacketSize)
ptmp0 = ei_padd(ptmp0, ei_pmul(ei_pload(&rhs[j]), ei_ploadu(&lhs[j+iN0])));
ptmp0 = ei_pmadd(ei_pload(&rhs[j]), ei_ploadu(&lhs0[j]), ptmp0);
tmp0 += ei_predux(ptmp0);
}
// process remaining scalars
for (int j=alignedSize; j<size; j++)
tmp0 += rhs[j] * lhs[j+iN0];
tmp0 += rhs[j] * lhs0[j];
res[i] += tmp0;
}
if (skipRows)

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@ -117,7 +117,7 @@ template <typename Scalar, typename Packet, int LoadMode> inline void ei_pstoret
if(LoadMode == Aligned)
ei_pstore(to, from);
else
ei_pstoreu(to, from);
ei_pstoreu(to, from);
}
#endif // EIGEN_DUMMY_PACKET_MATH_H

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@ -66,6 +66,8 @@ template<typename MatrixType, int Alignment> class Map
inline int rows() const { return m_rows.value(); }
inline int cols() const { return m_cols.value(); }
inline int stride() const { return this->innerSize(); }
inline const Scalar& coeff(int row, int col) const
{
if(Flags & RowMajorBit)
@ -156,40 +158,6 @@ template<typename MatrixType, int Alignment> class Map
const ei_int_if_dynamic<ColsAtCompileTime> m_cols;
};
/** Constructor copying an existing array of data. Only useful for dynamic-size matrices:
* for fixed-size matrices, it is redundant to pass the \a rows and \a cols parameters.
* \param data The array of data to copy
* \param rows The number of rows of the matrix to construct
* \param cols The number of columns of the matrix to construct
*
* \sa Matrix(const Scalar *), Matrix::map(const Scalar *, int, int)
*/
template<typename _Scalar, int _Rows, int _Cols, int _MaxRows, int _MaxCols, unsigned int _Flags>
inline Matrix<_Scalar, _Rows, _Cols, _MaxRows, _MaxCols, _Flags>
::Matrix(const Scalar *data, int rows, int cols)
: m_storage(rows*cols, rows, cols)
{
*this = Map<Matrix>(data, rows, cols);
}
/** Constructor copying an existing array of data. Only useful for dynamic-size vectors:
* for fixed-size vectors, it is redundant to pass the \a size parameter.
*
* \only_for_vectors
*
* \param data The array of data to copy
* \param size The size of the vector to construct
*
* \sa Matrix(const Scalar *), Matrix::map(const Scalar *, int)
*/
template<typename _Scalar, int _Rows, int _Cols, int _MaxRows, int _MaxCols, unsigned int _Flags>
inline Matrix<_Scalar, _Rows, _Cols, _MaxRows, _MaxCols, _Flags>
::Matrix(const Scalar *data, int size)
: m_storage(size, RowsAtCompileTime == 1 ? 1 : size, ColsAtCompileTime == 1 ? 1 : size)
{
*this = Map<Matrix>(data, size);
}
/** Constructor copying an existing array of data.
* Only for fixed-size matrices and vectors.
* \param data The array of data to copy

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@ -319,8 +319,7 @@ class Matrix : public MatrixBase<Matrix<_Scalar, _Rows, _Cols, _MaxRows, _MaxCol
m_storage.data()[2] = z;
m_storage.data()[3] = w;
}
Matrix(const Scalar *data, int rows, int cols);
Matrix(const Scalar *data, int size);
explicit Matrix(const Scalar *data);
/** Constructor copying the value of the expression \a other */

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@ -161,7 +161,7 @@ template<typename T> class ei_matrix_storage<T, Dynamic, Dynamic, Dynamic>
public:
inline ei_matrix_storage(int size, int rows, int cols)
: m_data(ei_aligned_malloc<T>(size)), m_rows(rows), m_cols(cols) {}
inline ~ei_matrix_storage() { delete[] m_data; }
inline ~ei_matrix_storage() { ei_aligned_free(m_data); }
inline void swap(ei_matrix_storage& other)
{ std::swap(m_data,other.m_data); std::swap(m_rows,other.m_rows); std::swap(m_cols,other.m_cols); }
inline int rows(void) const {return m_rows;}
@ -187,7 +187,7 @@ template<typename T, int _Rows> class ei_matrix_storage<T, Dynamic, _Rows, Dynam
int m_cols;
public:
inline ei_matrix_storage(int size, int, int cols) : m_data(ei_aligned_malloc<T>(size)), m_cols(cols) {}
inline ~ei_matrix_storage() { delete[] m_data; }
inline ~ei_matrix_storage() { ei_aligned_free(m_data); }
inline void swap(ei_matrix_storage& other) { std::swap(m_data,other.m_data); std::swap(m_cols,other.m_cols); }
inline static int rows(void) {return _Rows;}
inline int cols(void) const {return m_cols;}
@ -211,7 +211,7 @@ template<typename T, int _Cols> class ei_matrix_storage<T, Dynamic, Dynamic, _Co
int m_rows;
public:
inline ei_matrix_storage(int size, int rows, int) : m_data(ei_aligned_malloc<T>(size)), m_rows(rows) {}
inline ~ei_matrix_storage() { delete[] m_data; }
inline ~ei_matrix_storage() { ei_aligned_free(m_data); }
inline void swap(ei_matrix_storage& other) { std::swap(m_data,other.m_data); std::swap(m_rows,other.m_rows); }
inline int rows(void) const {return m_rows;}
inline static int cols(void) {return _Cols;}

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@ -94,8 +94,8 @@ template<typename Lhs, typename Rhs> struct ei_product_mode
: Lhs::MaxColsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD
&& ( Lhs::MaxRowsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD
|| Rhs::MaxColsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD )
&& (!(Rhs::IsVectorAtCompileTime && (Lhs::Flags&RowMajorBit) && (!Lhs::Flags&DirectAccessBit)))
&& (!(Lhs::IsVectorAtCompileTime && (!Rhs::Flags&RowMajorBit) && (!Rhs::Flags&DirectAccessBit)))
&& (!(Rhs::IsVectorAtCompileTime && (Lhs::Flags&RowMajorBit) && (!(Lhs::Flags&DirectAccessBit))))
&& (!(Lhs::IsVectorAtCompileTime && (!(Rhs::Flags&RowMajorBit)) && (!(Rhs::Flags&DirectAccessBit))))
? CacheFriendlyProduct
: NormalProduct };
};

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@ -189,5 +189,14 @@ inline __m128i ei_preduxp(const __m128i* vecs)
return _mm_add_epi32(tmp0, tmp2);
}
#if (defined __GNUC__)
template <> inline __m128 ei_pmadd(const __m128& a, const __m128& b, const __m128& c)
{
__m128 res = b;
asm("mulps %[a], %[b] \n\taddps %[c], %[b]" : [b] "+x" (res) : [a] "x" (a), [c] "x" (c));
return res;
}
#endif
#endif // EIGEN_PACKET_MATH_SSE_H

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@ -38,7 +38,6 @@ template<typename VectorType> void tmap(const VectorType& m)
VectorType ma1 = Map<VectorType>(array1, size);
VectorType ma2 = Map<VectorType, Aligned>(array2, size);
VERIFY_IS_APPROX(ma1, ma2);
VERIFY_IS_APPROX(ma1, VectorType(array2, size));
ei_aligned_free(array1);
ei_aligned_free(array2);
}

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@ -23,12 +23,14 @@
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#include "main.h"
#include <Eigen/Array>
#include <Eigen/QR>
template<typename Derived1, typename Derived2>
bool areNotApprox(const MatrixBase<Derived1>& m1, const MatrixBase<Derived2>& m2, typename Derived1::RealScalar epsilon = precision<typename Derived1::RealScalar>())
{
return !((m1-m2).matrixNorm() < epsilon * std::max(m1.matrixNorm(), m2.matrixNorm()));
return !((m1-m2).cwise().abs2().maxCoeff() < epsilon * epsilon
* std::max(m1.cwise().abs2().maxCoeff(), m2.cwise().abs2().maxCoeff()));
}
template<typename MatrixType> void product(const MatrixType& m)