605 lines
21 KiB
C++
605 lines
21 KiB
C++
//=============================================================================
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//
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// Copyright (c) Kitware, Inc.
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// All rights reserved.
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// See LICENSE.txt for details.
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//
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// This software is distributed WITHOUT ANY WARRANTY; without even
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// the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
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// PURPOSE. See the above copyright notice for more information.
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//
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// Copyright 2015 National Technology & Engineering Solutions of Sandia, LLC (NTESS).
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// Copyright 2015 UT-Battelle, LLC.
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// Copyright 2015 Los Alamos National Security.
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//
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// Under the terms of Contract DE-NA0003525 with NTESS,
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// the U.S. Government retains certain rights in this software.
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// Under the terms of Contract DE-AC52-06NA25396 with Los Alamos National
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// Laboratory (LANL), the U.S. Government retains certain rights in
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// this software.
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//
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//=============================================================================
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#ifndef vtk_m_Matrix_h
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#define vtk_m_Matrix_h
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#include <vtkm/Assert.h>
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#include <vtkm/Math.h>
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#include <vtkm/TypeTraits.h>
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#include <vtkm/Types.h>
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#include <vtkm/VecTraits.h>
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namespace vtkm
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{
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/// \brief Basic Matrix type.
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///
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/// The Matrix class holds a small two dimensional array for simple linear
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/// algebra and vector operations. VTK-m provides several Matrix-based
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/// operations to assist in visualization computations.
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///
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/// A Matrix is not intended to hold very large arrays. Rather, they are a
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/// per-thread data structure to hold information like geometric transforms and
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/// tensors.
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///
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template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
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class Matrix
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{
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public:
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using ComponentType = T;
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static const vtkm::IdComponent NUM_ROWS = NumRow;
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static const vtkm::IdComponent NUM_COLUMNS = NumCol;
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VTKM_EXEC_CONT
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Matrix() {}
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VTKM_EXEC_CONT
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explicit Matrix(const ComponentType& value)
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: Components(vtkm::Vec<ComponentType, NUM_COLUMNS>(value))
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{
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}
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/// Brackets are used to reference a matrix like a 2D array (i.e.
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/// matrix[row][column]).
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///
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VTKM_EXEC_CONT
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const vtkm::Vec<ComponentType, NUM_COLUMNS>& operator[](vtkm::IdComponent rowIndex) const
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{
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VTKM_ASSERT(rowIndex >= 0);
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VTKM_ASSERT(rowIndex < NUM_ROWS);
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return this->Components[rowIndex];
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}
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/// Brackets are used to referens a matrix like a 2D array i.e.
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/// matrix[row][column].
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///
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VTKM_EXEC_CONT
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vtkm::Vec<ComponentType, NUM_COLUMNS>& operator[](vtkm::IdComponent rowIndex)
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{
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VTKM_ASSERT(rowIndex >= 0);
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VTKM_ASSERT(rowIndex < NUM_ROWS);
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return this->Components[rowIndex];
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}
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/// Parentheses are used to reference a matrix using mathematical tuple
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/// notation i.e. matrix(row,column).
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///
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VTKM_EXEC_CONT
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const ComponentType& operator()(vtkm::IdComponent rowIndex, vtkm::IdComponent colIndex) const
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{
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VTKM_ASSERT(rowIndex >= 0);
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VTKM_ASSERT(rowIndex < NUM_ROWS);
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VTKM_ASSERT(colIndex >= 0);
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VTKM_ASSERT(colIndex < NUM_COLUMNS);
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return (*this)[rowIndex][colIndex];
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}
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/// Parentheses are used to reference a matrix using mathematical tuple
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/// notation i.e. matrix(row,column).
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///
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VTKM_EXEC_CONT
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ComponentType& operator()(vtkm::IdComponent rowIndex, vtkm::IdComponent colIndex)
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{
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VTKM_ASSERT(rowIndex >= 0);
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VTKM_ASSERT(rowIndex < NUM_ROWS);
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VTKM_ASSERT(colIndex >= 0);
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VTKM_ASSERT(colIndex < NUM_COLUMNS);
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return (*this)[rowIndex][colIndex];
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}
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private:
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vtkm::Vec<vtkm::Vec<ComponentType, NUM_COLUMNS>, NUM_ROWS> Components;
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};
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/// Returns a tuple containing the given row (indexed from 0) of the given
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/// matrix.
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///
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template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
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VTKM_EXEC_CONT const vtkm::Vec<T, NumCol>& MatrixGetRow(
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const vtkm::Matrix<T, NumRow, NumCol>& matrix,
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vtkm::IdComponent rowIndex)
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{
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return matrix[rowIndex];
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}
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/// Returns a tuple containing the given column (indexed from 0) of the given
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/// matrix. Might not be as efficient as the \c MatrixGetRow function.
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///
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template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
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VTKM_EXEC_CONT vtkm::Vec<T, NumRow> MatrixGetColumn(const vtkm::Matrix<T, NumRow, NumCol>& matrix,
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vtkm::IdComponent columnIndex)
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{
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vtkm::Vec<T, NumRow> columnValues;
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for (vtkm::IdComponent rowIndex = 0; rowIndex < NumRow; rowIndex++)
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{
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columnValues[rowIndex] = matrix(rowIndex, columnIndex);
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}
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return columnValues;
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}
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/// Convenience function for setting a row of a matrix.
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///
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template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
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VTKM_EXEC_CONT void MatrixSetRow(vtkm::Matrix<T, NumRow, NumCol>& matrix,
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vtkm::IdComponent rowIndex,
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const vtkm::Vec<T, NumCol>& rowValues)
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{
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matrix[rowIndex] = rowValues;
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}
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/// Convenience function for setting a column of a matrix.
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///
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template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
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VTKM_EXEC_CONT void MatrixSetColumn(vtkm::Matrix<T, NumRow, NumCol>& matrix,
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vtkm::IdComponent columnIndex,
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const vtkm::Vec<T, NumRow>& columnValues)
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{
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for (vtkm::IdComponent rowIndex = 0; rowIndex < NumRow; rowIndex++)
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{
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matrix(rowIndex, columnIndex) = columnValues[rowIndex];
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}
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}
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/// Standard matrix multiplication.
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///
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template <typename T,
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vtkm::IdComponent NumRow,
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vtkm::IdComponent NumCol,
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vtkm::IdComponent NumInternal>
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VTKM_EXEC_CONT vtkm::Matrix<T, NumRow, NumCol> MatrixMultiply(
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const vtkm::Matrix<T, NumRow, NumInternal>& leftFactor,
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const vtkm::Matrix<T, NumInternal, NumCol>& rightFactor)
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{
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vtkm::Matrix<T, NumRow, NumCol> result;
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for (vtkm::IdComponent rowIndex = 0; rowIndex < NumRow; rowIndex++)
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{
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for (vtkm::IdComponent colIndex = 0; colIndex < NumCol; colIndex++)
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{
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T sum = T(leftFactor(rowIndex, 0) * rightFactor(0, colIndex));
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for (vtkm::IdComponent internalIndex = 1; internalIndex < NumInternal; internalIndex++)
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{
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sum = T(sum + (leftFactor(rowIndex, internalIndex) * rightFactor(internalIndex, colIndex)));
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}
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result(rowIndex, colIndex) = sum;
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}
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}
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return result;
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}
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/// Standard matrix-vector multiplication.
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///
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template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
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VTKM_EXEC_CONT vtkm::Vec<T, NumRow> MatrixMultiply(
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const vtkm::Matrix<T, NumRow, NumCol>& leftFactor,
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const vtkm::Vec<T, NumCol>& rightFactor)
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{
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vtkm::Vec<T, NumRow> product;
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for (vtkm::IdComponent rowIndex = 0; rowIndex < NumRow; rowIndex++)
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{
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product[rowIndex] = vtkm::dot(vtkm::MatrixGetRow(leftFactor, rowIndex), rightFactor);
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}
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return product;
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}
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/// Standard vector-matrix multiplication
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///
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template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
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VTKM_EXEC_CONT vtkm::Vec<T, NumCol> MatrixMultiply(
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const vtkm::Vec<T, NumRow>& leftFactor,
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const vtkm::Matrix<T, NumRow, NumCol>& rightFactor)
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{
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vtkm::Vec<T, NumCol> product;
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for (vtkm::IdComponent colIndex = 0; colIndex < NumCol; colIndex++)
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{
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product[colIndex] = vtkm::dot(leftFactor, vtkm::MatrixGetColumn(rightFactor, colIndex));
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}
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return product;
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}
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/// Returns the identity matrix.
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///
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT vtkm::Matrix<T, Size, Size> MatrixIdentity()
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{
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vtkm::Matrix<T, Size, Size> result(T(0));
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for (vtkm::IdComponent index = 0; index < Size; index++)
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{
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result(index, index) = T(1.0);
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}
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return result;
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}
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/// Fills the given matrix with the identity matrix.
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///
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT void MatrixIdentity(vtkm::Matrix<T, Size, Size>& matrix)
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{
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matrix = vtkm::MatrixIdentity<T, Size>();
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}
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/// Returns the transpose of the given matrix.
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///
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template <typename T, vtkm::IdComponent NumRows, vtkm::IdComponent NumCols>
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VTKM_EXEC_CONT vtkm::Matrix<T, NumCols, NumRows> MatrixTranspose(
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const vtkm::Matrix<T, NumRows, NumCols>& matrix)
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{
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vtkm::Matrix<T, NumCols, NumRows> result;
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for (vtkm::IdComponent index = 0; index < NumRows; index++)
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{
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vtkm::MatrixSetColumn(result, index, vtkm::MatrixGetRow(matrix, index));
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#ifdef VTKM_ICC
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// For reasons I do not really understand, the Intel compiler with with
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// optimization on is sometimes removing this for loop. It appears that the
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// optimizer sometimes does not recognize that the MatrixSetColumn function
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// has side effects. I cannot fathom any reason for this other than a bug in
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// the compiler, but unfortunately I do not know a reliable way to
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// demonstrate the problem.
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__asm__("");
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#endif
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}
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return result;
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}
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namespace detail
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{
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// Used with MatrixLUPFactor.
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT void MatrixLUPFactorFindPivot(vtkm::Matrix<T, Size, Size>& A,
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vtkm::Vec<vtkm::IdComponent, Size>& permutation,
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vtkm::IdComponent topCornerIndex,
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T& inversionParity,
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bool& valid)
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{
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vtkm::IdComponent maxRowIndex = topCornerIndex;
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T maxValue = vtkm::Abs(A(maxRowIndex, topCornerIndex));
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for (vtkm::IdComponent rowIndex = topCornerIndex + 1; rowIndex < Size; rowIndex++)
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{
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T compareValue = vtkm::Abs(A(rowIndex, topCornerIndex));
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if (maxValue < compareValue)
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{
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maxValue = compareValue;
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maxRowIndex = rowIndex;
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}
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}
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if (maxValue < vtkm::Epsilon<T>())
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{
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valid = false;
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}
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if (maxRowIndex != topCornerIndex)
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{
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// Swap rows in matrix.
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vtkm::Vec<T, Size> maxRow = vtkm::MatrixGetRow(A, maxRowIndex);
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vtkm::MatrixSetRow(A, maxRowIndex, vtkm::MatrixGetRow(A, topCornerIndex));
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vtkm::MatrixSetRow(A, topCornerIndex, maxRow);
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// Record change in permutation matrix.
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vtkm::IdComponent maxOriginalRowIndex = permutation[maxRowIndex];
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permutation[maxRowIndex] = permutation[topCornerIndex];
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permutation[topCornerIndex] = maxOriginalRowIndex;
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// Keep track of inversion parity.
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inversionParity = -inversionParity;
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}
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}
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// Used with MatrixLUPFactor
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT void MatrixLUPFactorFindUpperTriangleElements(vtkm::Matrix<T, Size, Size>& A,
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vtkm::IdComponent topCornerIndex)
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{
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// Compute values for upper triangle on row topCornerIndex
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for (vtkm::IdComponent colIndex = topCornerIndex + 1; colIndex < Size; colIndex++)
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{
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A(topCornerIndex, colIndex) /= A(topCornerIndex, topCornerIndex);
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}
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// Update the rest of the matrix for calculations on subsequent rows
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for (vtkm::IdComponent rowIndex = topCornerIndex + 1; rowIndex < Size; rowIndex++)
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{
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for (vtkm::IdComponent colIndex = topCornerIndex + 1; colIndex < Size; colIndex++)
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{
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A(rowIndex, colIndex) -= A(rowIndex, topCornerIndex) * A(topCornerIndex, colIndex);
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}
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}
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}
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/// Performs an LUP-factorization on the given matrix using Crout's method. The
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/// LU-factorization takes a matrix A and decomposes it into a lower triangular
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/// matrix L and upper triangular matrix U such that A = LU. The
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/// LUP-factorization also allows permutation of A, which makes the
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/// decomposition always possible so long as A is not singular. In addition to
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/// matrices L and U, LUP also finds permutation matrix P containing all zeros
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/// except one 1 per row and column such that PA = LU.
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///
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/// The result is done in place such that the lower triangular matrix, L, is
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/// stored in the lower-left triangle of A including the diagonal. The upper
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/// triangular matrix, U, is stored in the upper-right triangle of L not
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/// including the diagonal. The diagonal of U in Crout's method is all 1's (and
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/// therefore not explicitly stored).
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///
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/// The permutation matrix P is represented by the permutation vector. If
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/// permutation[i] = j then row j in the original matrix A has been moved to
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/// row i in the resulting matrices. The permutation matrix P can be
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/// represented by a matrix with p_i,j = 1 if permutation[i] = j and 0
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/// otherwise. If using LUP-factorization to compute a determinant, you also
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/// need to know the parity (whether there is an odd or even amount) of
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/// inversions. An inversion is an instance of a smaller number appearing after
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/// a larger number in permutation. Although you can compute the inversion
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/// parity after the fact, this function keeps track of it with much less
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/// compute resources. The parameter inversionParity is set to 1.0 for even
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/// parity and -1.0 for odd parity.
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///
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/// Not all matrices (specifically singular matrices) have an
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/// LUP-factorization. If the LUP-factorization succeeds, valid is set to true.
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/// Otherwise, valid is set to false and the result is indeterminant.
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///
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT void MatrixLUPFactor(vtkm::Matrix<T, Size, Size>& A,
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vtkm::Vec<vtkm::IdComponent, Size>& permutation,
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T& inversionParity,
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bool& valid)
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{
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// Initialize permutation.
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for (vtkm::IdComponent index = 0; index < Size; index++)
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{
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permutation[index] = index;
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}
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inversionParity = T(1);
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valid = true;
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for (vtkm::IdComponent rowIndex = 0; rowIndex < Size; rowIndex++)
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{
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MatrixLUPFactorFindPivot(A, permutation, rowIndex, inversionParity, valid);
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MatrixLUPFactorFindUpperTriangleElements(A, rowIndex);
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}
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}
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/// Use a previous factorization done with MatrixLUPFactor to solve the
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/// system Ax = b. Instead of A, this method takes in the LU and P
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/// matrices calculated by MatrixLUPFactor from A. The x matrix is returned.
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///
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT vtkm::Vec<T, Size> MatrixLUPSolve(
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const vtkm::Matrix<T, Size, Size>& LU,
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const vtkm::Vec<vtkm::IdComponent, Size>& permutation,
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const vtkm::Vec<T, Size>& b)
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{
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// The LUP-factorization gives us PA = LU or equivalently A = inv(P)LU.
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// Substituting into Ax = b gives us inv(P)LUx = b or LUx = Pb.
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// Now consider the intermediate vector y = Ux.
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// Substituting in the previous two equations yields Ly = Pb.
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// Solving Ly = Pb is easy because L is triangular and P is just a
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// permutation.
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vtkm::Vec<T, Size> y;
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for (vtkm::IdComponent rowIndex = 0; rowIndex < Size; rowIndex++)
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{
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y[rowIndex] = b[permutation[rowIndex]];
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// Recall that L is stored in the lower triangle of LU including diagonal.
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for (vtkm::IdComponent colIndex = 0; colIndex < rowIndex; colIndex++)
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{
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y[rowIndex] -= LU(rowIndex, colIndex) * y[colIndex];
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}
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y[rowIndex] /= LU(rowIndex, rowIndex);
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}
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// Now that we have y, we can easily solve Ux = y for x.
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vtkm::Vec<T, Size> x;
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for (vtkm::IdComponent rowIndex = Size - 1; rowIndex >= 0; rowIndex--)
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{
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// Recall that U is stored in the upper triangle of LU with the diagonal
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// implicitly all 1's.
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x[rowIndex] = y[rowIndex];
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for (vtkm::IdComponent colIndex = rowIndex + 1; colIndex < Size; colIndex++)
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{
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x[rowIndex] -= LU(rowIndex, colIndex) * x[colIndex];
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}
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}
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return x;
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}
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} // namespace detail
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/// Solve the linear system Ax = b for x. If a single solution is found, valid
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/// is set to true, false otherwise.
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///
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT vtkm::Vec<T, Size> SolveLinearSystem(const vtkm::Matrix<T, Size, Size>& A,
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const vtkm::Vec<T, Size>& b,
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bool& valid)
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{
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// First, we will make an LUP-factorization to help us.
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vtkm::Matrix<T, Size, Size> LU = A;
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vtkm::Vec<vtkm::IdComponent, Size> permutation;
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T inversionParity; // Unused.
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detail::MatrixLUPFactor(LU, permutation, inversionParity, valid);
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// Next, use the decomposition to solve the system.
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return detail::MatrixLUPSolve(LU, permutation, b);
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}
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/// Find and return the inverse of the given matrix. If the matrix is singular,
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/// the inverse will not be correct and valid will be set to false.
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///
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT vtkm::Matrix<T, Size, Size> MatrixInverse(const vtkm::Matrix<T, Size, Size>& A,
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bool& valid)
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{
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// First, we will make an LUP-factorization to help us.
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vtkm::Matrix<T, Size, Size> LU = A;
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vtkm::Vec<vtkm::IdComponent, Size> permutation;
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T inversionParity; // Unused
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detail::MatrixLUPFactor(LU, permutation, inversionParity, valid);
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// We will use the decomposition to solve AX = I for X where X is
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// clearly the inverse of A. Our solve method only works for vectors,
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// so we solve for one column of invA at a time.
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vtkm::Matrix<T, Size, Size> invA;
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vtkm::Vec<T, Size> ICol(T(0));
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for (vtkm::IdComponent colIndex = 0; colIndex < Size; colIndex++)
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{
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ICol[colIndex] = 1;
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vtkm::Vec<T, Size> invACol = detail::MatrixLUPSolve(LU, permutation, ICol);
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ICol[colIndex] = 0;
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vtkm::MatrixSetColumn(invA, colIndex, invACol);
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}
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return invA;
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}
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/// Compute the determinant of a matrix.
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///
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template <typename T, vtkm::IdComponent Size>
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VTKM_EXEC_CONT T MatrixDeterminant(const vtkm::Matrix<T, Size, Size>& A)
|
|
{
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|
// First, we will make an LUP-factorization to help us.
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|
vtkm::Matrix<T, Size, Size> LU = A;
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|
vtkm::Vec<vtkm::IdComponent, Size> permutation;
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|
T inversionParity;
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|
bool valid;
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detail::MatrixLUPFactor(LU, permutation, inversionParity, valid);
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|
|
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// If the matrix is singular, the factorization is invalid, but in that
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// case we know that the determinant is 0.
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|
if (!valid)
|
|
{
|
|
return 0;
|
|
}
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|
|
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// The determinant is equal to the product of the diagonal of the L matrix,
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|
// possibly negated depending on the parity of the inversion. The
|
|
// inversionParity variable is set to 1.0 and -1.0 for even and odd parity,
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|
// respectively. This sign determines whether the product should be negated.
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|
T product = inversionParity;
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|
for (vtkm::IdComponent index = 0; index < Size; index++)
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|
{
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|
product *= LU(index, index);
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|
}
|
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return product;
|
|
}
|
|
|
|
// Specializations for common small determinants.
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|
|
|
template <typename T>
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|
VTKM_EXEC_CONT T MatrixDeterminant(const vtkm::Matrix<T, 1, 1>& A)
|
|
{
|
|
return A(0, 0);
|
|
}
|
|
|
|
template <typename T>
|
|
VTKM_EXEC_CONT T MatrixDeterminant(const vtkm::Matrix<T, 2, 2>& A)
|
|
{
|
|
return A(0, 0) * A(1, 1) - A(1, 0) * A(0, 1);
|
|
}
|
|
|
|
template <typename T>
|
|
VTKM_EXEC_CONT T MatrixDeterminant(const vtkm::Matrix<T, 3, 3>& A)
|
|
{
|
|
return A(0, 0) * A(1, 1) * A(2, 2) + A(1, 0) * A(2, 1) * A(0, 2) + A(2, 0) * A(0, 1) * A(1, 2) -
|
|
A(0, 0) * A(2, 1) * A(1, 2) - A(1, 0) * A(0, 1) * A(2, 2) - A(2, 0) * A(1, 1) * A(0, 2);
|
|
}
|
|
|
|
//---------------------------------------------------------------------------
|
|
// Implementations of traits for matrices.
|
|
|
|
/// Tag used to identify 2 dimensional types (matrices). A TypeTraits class
|
|
/// will typedef this class to DimensionalityTag.
|
|
///
|
|
struct TypeTraitsMatrixTag
|
|
{
|
|
};
|
|
|
|
template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
|
|
struct TypeTraits<vtkm::Matrix<T, NumRow, NumCol>>
|
|
{
|
|
using NumericTag = typename TypeTraits<T>::NumericTag;
|
|
using DimensionalityTag = vtkm::TypeTraitsMatrixTag;
|
|
};
|
|
|
|
/// A matrix has vector traits to implement component-wise operations.
|
|
///
|
|
template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
|
|
struct VecTraits<vtkm::Matrix<T, NumRow, NumCol>>
|
|
{
|
|
private:
|
|
using MatrixType = vtkm::Matrix<T, NumRow, NumCol>;
|
|
|
|
public:
|
|
using ComponentType = T;
|
|
static const vtkm::IdComponent NUM_COMPONENTS = NumRow * NumCol;
|
|
using HasMultipleComponents = vtkm::VecTraitsTagMultipleComponents;
|
|
using IsSizeStatic = vtkm::VecTraitsTagSizeStatic;
|
|
|
|
VTKM_EXEC_CONT
|
|
static vtkm::IdComponent GetNumberOfComponents(const MatrixType&) { return NUM_COMPONENTS; }
|
|
|
|
VTKM_EXEC_CONT
|
|
static const ComponentType& GetComponent(const MatrixType& matrix, vtkm::IdComponent component)
|
|
{
|
|
vtkm::IdComponent colIndex = component % NumCol;
|
|
vtkm::IdComponent rowIndex = component / NumCol;
|
|
return matrix(rowIndex, colIndex);
|
|
}
|
|
VTKM_EXEC_CONT
|
|
static ComponentType& GetComponent(MatrixType& matrix, vtkm::IdComponent component)
|
|
{
|
|
vtkm::IdComponent colIndex = component % NumCol;
|
|
vtkm::IdComponent rowIndex = component / NumCol;
|
|
return matrix(rowIndex, colIndex);
|
|
}
|
|
VTKM_EXEC_CONT
|
|
static void SetComponent(MatrixType& matrix, vtkm::IdComponent component, T value)
|
|
{
|
|
GetComponent(matrix, component) = value;
|
|
}
|
|
};
|
|
|
|
} // namespace vtkm
|
|
|
|
//---------------------------------------------------------------------------
|
|
// Basic comparison operators.
|
|
|
|
template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
|
|
VTKM_EXEC_CONT bool operator==(const vtkm::Matrix<T, NumRow, NumCol>& a,
|
|
const vtkm::Matrix<T, NumRow, NumCol>& b)
|
|
{
|
|
for (vtkm::IdComponent colIndex = 0; colIndex < NumCol; colIndex++)
|
|
{
|
|
for (vtkm::IdComponent rowIndex = 0; rowIndex < NumRow; rowIndex++)
|
|
{
|
|
if (a(rowIndex, colIndex) != b(rowIndex, colIndex))
|
|
return false;
|
|
}
|
|
}
|
|
return true;
|
|
}
|
|
template <typename T, vtkm::IdComponent NumRow, vtkm::IdComponent NumCol>
|
|
VTKM_EXEC_CONT bool operator!=(const vtkm::Matrix<T, NumRow, NumCol>& a,
|
|
const vtkm::Matrix<T, NumRow, NumCol>& b)
|
|
{
|
|
return !(a == b);
|
|
}
|
|
|
|
#endif //vtk_m_Matrix_h
|