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## More on Vectors and Matrices

Posted by , in Programming 27 August 2011 · 593 views

My original plan for this second entry was to also add a matrix class that interacts with the vector class from my last journal entry. In the meanwhile Splinter of Chaos gave us a reason why we should use vectors in his journal and in the comments to said journal entry JTippetts brought up the problem that lots of libraries etc. use their own vector type. The "heterogeneity" of the C++ library landscape sadly isn't something we can change now, but when designing our own Vector type, we can at least take it into account.

## Integration of other (array style) Vector data

So going back to my earlier implementation of a Vector class I'd like to add a way to interact with other implementations and "raw vectors" (also known as array). Luckily everything is already based around expression templates which are easily extendable. What I want is a way to interpret raw data as a Vector compatible with my implementation. All I have to do for that is copy & paste my whole Vector class, change its name to "InPlaceVector", replace "T data[D];" with "T *const data;" and provide a constructor that takes a pointer. Then I add the following function for convenience
[source lang="cpp"]template<unsigned D, class T>InPlaceVector<T,D> MakeVector(T *const p){ return InPlaceVector<T,D>(p);}[/source]
and can apply the full functionality of my Vector class to any kind of other vector class or representation that stores its data in an array.
Example:
[source lang="cpp"]double raw_vector[3];Vector<double, 3> a(1,2,3);MakeVector<3>(raw_vector) += a;[/source]
And since the InPlaceVector object only contains a const pointer, it will once again very likely be optimized away and we don't even incur any conversion overhead.

So with that problem out of the way (sort of). We can move on to matrices.

## The Matrix

So if we are already in need of vectors chance are we also want matrices. A simple Matrix class can be obtained by just copy pasting my Vector class, replace the "operator[](unsigned)" with "operator()(unsigned, unsigned)" etc. When implementing matrices one also has to decide between row and column major order for storage. I went with column major since that makes the matrices directly usable with OpenGL. An argument for row major order would be that the native c++ matrices (double mat[3][3]) are row major and are therefore more natural in C++.
Besides the element wise operations we "inherited" from the vector class we also need matrix-vector multiplication and matrix-matrix multiplication. These could be implemented directly, but to make it more elegant we will first introduce a way to access the rows and columns like they were vector expressions. I guess this can be considered a form of the decorator pattern:
[source lang="cpp"]template<class T, unsigned D1, unsigned D2, class A>class RowVectorExpr : public VectorExpr<T, D2, RowVectorExpr<T, D1, D2, A> > {public: RowVectorExpr(const A& pa, const unsigned& i) : a(pa), index(i) { } inline T operator[](unsigned i) const { return a(index, i); }private: const A& a; const unsigned index;};template<class T, unsigned D1, unsigned D2, class A>inline RowVectorExpr<T,D1,D2,A>Row(const MatrixExpr<T,D1,D2,A> &a, const unsigned &index){ return RowVectorExpr<T,D1,D2,A>(a, index);}//similarly for columns[/source]
There are other interesting "operations" that can and should also be implemented like this like submatrix access and transposition.

Now with easy acess to rows and columns we can write the matrix-matrix multiplication as:
[source lang=cpp"]template<class T, unsigned D1, unsigned D2, class A, class B> class MatMatMulExpr : public MatrixExpr<T, D1, D2, MatMatMulExpr<T, D1, D2, A, B> > { public: MatMatMulExpr(const A& pa, const B& pb) : a(pa), b(pb) { } inline T operator()(unsigned i, unsigned j) const { return dot(Row(a,i),Column(b,j)); } private: const A& a; const B& b; }; template<class T, unsigned D1, unsigned D2, unsigned D3, class A, class B> inline MatMatMulExpr<T,D1,D3,A,B> operator*(const MatrixExpr<T,D1,D2,A> &a, const MatrixExpr<T,D2,D3,B> &b){ return MatMatMulExpr<T,D1,D3,A,B>(a, b); }//similarly for matrix-vector and vector-matrix multiplications[/source]

## Expression templates and pitfalls

So this is the point where after all the praise for expression templates I have to address a problem with them. Expression templates offer a form of lazy evaluation. This means that operations are not carried out immediately. Instead they are carried out when the result is actually needed. This has advantages when not the whole calculation is actually required. Using the tools we just introduced consider for example this:
[source lang="cpp"]Matrix<double,4,4> A, B;//fill A and B with some valuesVector<double,4> v = Row(A*B,2); //save the third row of A*B in v[/source]
In the assignment to v only the matrix elements that lie in the requested Row are actually calculated!
The downside of this is that the elements will be calculated multiple times if they are requested multiple times. This can happen when more than two matrices are multiplied in a single expression. In that case the temporary objects we avoided with the expression templates might actually be beneficial from a performance perspective. Even worse, operations where the left operand of an assignment is also part of the expression will even produce wrong results.
[source lang="cpp"]Matrix<double,4,4> A;A = A*Transpose(A); //gives wrong result![/source]
For pure Vector operations this usually isn't a problem given that the dependencies are only component wise, but for matrix and matrix-vector multiplications this has to be taken into account.
An easy way to solve both of these problems is to use an explicit evaluation function:
[source lang="cpp"]template<class T, unsigned D, class A>inline Vector<T, D> eval(const VectorExpr<T, D, A>& a){ return Vector<T, D>(a);}template<class T, unsigned D1, unsigned D2, class A>inline Matrix<T, D1, D2> eval(const MatrixExpr<T, D1, D2, A>& a){ return Matrix<T, D1, D2>(a);}[/source]
This allows us to force the execution of an expression and get a temporary from it.
[source lang="cpp"] Matrix<double,4,4> A,B,C,D;A = eval(A*Transpose(A)); //correct!A = eval(B*C)*D//less operations in exchange for a temporary object! [/source]
It would also be possible to have the expression templates detect these situations by passing "source" references through the expression tree and some template specialization magic. At the moment I'll stick with the "eval" solution.

Here is the extended Vector header and the Matrix header I'm using right now:
Vector.h

[source lang="cpp"]/* * MathVector.h - Copyright © 2011 Jakob Progsch * * This software is provided 'as-is', without any express or implied * warranty. In no event will the authors be held liable for any damages * arising from the use of this software. * * Permission is granted to anyone to use this software for any purpose, * including commercial applications, and to alter it and redistribute it * freely, subject to the following restrictions: * * 1. The origin of this software must not be misrepresented; you must not * claim that you wrote the original software. If you use this software * in a product, an acknowledgment in the product documentation would be * appreciated but is not required. * * 2. Altered source versions must be plainly marked as such, and must not be * misrepresented as being the original software. * * 3. This notice may not be removed or altered from any source * distribution. *//* * Vector.h provides a simple static vector template class with * a basic expression template ansatz for vector operations. */ #ifndef VECTOR_H#define VECTOR_H#include <cmath>#include <algorithm>#include <ostream>template<class T, unsigned D> class Vector;//base class for all expression templatestemplate<class T, unsigned D, class A>class VectorExpr {public: inline operator const A&() const { return *static_cast<const A*>(this); }};//better use macros instead of copy pasting this stuff all over the place#define MAKE_VEC_VEC_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D, class A, class B> \class NAME : public VectorExpr<T, D, NAME<T, D, A, B> > { \public: \ NAME(const A& pa, const B& pb) : a(pa), b(pb) { } \ inline T operator[](unsigned i) const { return EXPR; } \private: \ const A& a; \ const B& b; \}; \ \template<class T, unsigned D, class A, class B> \inline NAME<T,D,A,B> \FUNCTION(const VectorExpr<T,D,A> &a, const VectorExpr<T,D,B> &b) \{ \ return NAME<T,D,A,B>(a, b); \} #define MAKE_VEC_SCAL_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D, class A> \class NAME : public VectorExpr<T, D, NAME<T, D, A> > { \public: \ NAME(const A& pa, const T& pb) : a(pa), b(pb) { } \ inline T operator[](unsigned i) const { return EXPR; } \private: \ const A& a; \ const T& b; \}; \ \template<class T, unsigned D, class A> \inline NAME<T,D,A> \FUNCTION(const VectorExpr<T,D,A> &a, const T &b) \{ \ return NAME<T,D,A>(a, b); \} #define MAKE_SCAL_VEC_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D, class A> \class NAME : public VectorExpr<T, D, NAME<T, D, A> > { \public: \ NAME(const T& pa, const A& pb) : a(pa), b(pb) { } \ inline T operator[](unsigned i) const { return EXPR; } \private: \ const T& a; \ const A& b; \}; \ \template<class T, unsigned D, class A> \inline NAME<T,D,A> \FUNCTION(const T &a, const VectorExpr<T,D,A> &b) \{ \ return NAME<T,D,A>(a, b); \} #define MAKE_VEC_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D, class A> \class NAME : public VectorExpr<T, D, NAME<T, D, A> > { \public: \ NAME(const A& pa) : a(pa) { } \ inline T operator[](unsigned i) const { return EXPR; } \private: \ const A& a; \}; \ \template<class T, unsigned D, class A> \inline NAME<T,D,A> \FUNCTION(const VectorExpr<T,D,A> &a) \{ \ return NAME<T,D,A>(a); \}//create actual functions and operatorsMAKE_VEC_VEC_EXPRESSION (EMulExpr, a[i] * b[i], multiply_elements)MAKE_VEC_VEC_EXPRESSION (EDivExpr, a[i] / b[i], divide_elements)MAKE_VEC_VEC_EXPRESSION (AddExpr, a[i] + b[i], operator+)MAKE_VEC_VEC_EXPRESSION (SubExpr, a[i] - b[i], operator-)MAKE_VEC_SCAL_EXPRESSION(DivExpr, a[i] / b, operator/)MAKE_VEC_SCAL_EXPRESSION(MulExpr1, a[i] * b, operator*)MAKE_SCAL_VEC_EXPRESSION(MulExpr2, a * b[i], operator*)MAKE_VEC_EXPRESSION (NegExpr, -a[i], operator-)//sub vector proxytemplate<class T, unsigned D, class A> class SubVectorExpr : public VectorExpr<T, D, SubVectorExpr<T, D, A> > { public: SubVectorExpr(const A& pa, const unsigned& o) : a(pa), offset(o) { } inline T operator[](unsigned i) const { return a[i+offset]; } private: const A& a; const unsigned offset; }; template<unsigned D1, class T, unsigned D2, class A> inline SubVectorExpr<T,D1,A> SubVector(const VectorExpr<T,D2,A> &a, const unsigned &o) { return SubVectorExpr<T,D1,A>(a, o); }//static size assertion since the vector size ist also statictemplate<unsigned I, unsigned J>struct STATIC_DIMENSION_MISMATCH_ASSERTION;template<unsigned I>struct STATIC_DIMENSION_MISMATCH_ASSERTION<I,I> { };#define ASSERT_DIMENSION(I, J) sizeof(STATIC_DIMENSION_MISMATCH_ASSERTION<I,J>)//actual vector classtemplate<class T, unsigned D>class Vector : public VectorExpr<T, D, Vector<T,D> > {public: static const unsigned Dim = D; typedef T Type; Vector() { std::fill(data, data+Dim, T()); } Vector(const T &p1) { ASSERT_DIMENSION(1, Dim); data[0] = p1; } Vector(const T &p1, const T &p2) { ASSERT_DIMENSION(2, Dim); data[0] = p1; data[1] = p2; } Vector(const T &p1, const T &p2, const T &p3) { ASSERT_DIMENSION(3, Dim); data[0] = p1; data[1] = p2; data[2] = p3; } Vector(const T &p1, const T &p2, const T &p3, const T &p4) { ASSERT_DIMENSION(4, Dim); data[0] = p1; data[1] = p2; data[2] = p3; data[3] = p4; } T* raw() { return data; } template<class A> Vector(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] = ao[i]; } T& operator[] (unsigned i) { return data[i]; } const T& operator[] (unsigned i) const { return data[i]; } const Vector& operator*=(const T &b) { for(unsigned i = 0;i<Dim;++i) data[i] *= b; return *this; } const Vector& operator/=(const T &b) { for(unsigned i = 0;i<Dim;++i) data[i] /= b; return *this; } template<class A> const Vector& operator+=(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] += ao[i]; return *this; } template<class A> const Vector& operator-=(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] -= ao[i]; return *this; } template<class A> Vector& operator=(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] = ao[i]; return *this; } Vector& normalize() { *this /= abs(*this); return *this; }private: T data[Dim];};//InPlaceVector to use raw datatemplate<class T, unsigned D>class InPlaceVector : public VectorExpr<T, D, InPlaceVector<T,D> > {public: static const unsigned Dim = D; typedef T Type; InPlaceVector(T *const d) : data(d) { } template<class A> InPlaceVector(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] = ao[i]; } T* raw() { return data; } T& operator[] (unsigned i) { return data[i]; } const T& operator[] (unsigned i) const { return data[i]; } const InPlaceVector& operator*=(const T &b) { for(unsigned i = 0;i<Dim;++i) data[i] *= b; return *this; } const InPlaceVector& operator/=(const T &b) { for(unsigned i = 0;i<Dim;++i) data[i] /= b; return *this; } template<class A> const InPlaceVector& operator+=(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] += ao[i]; return *this; } template<class A> const InPlaceVector& operator-=(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] -= ao[i]; return *this; } template<class A> InPlaceVector& operator=(const VectorExpr<T, D, A>& a) { const A& ao ( a ); for(unsigned i = 0;i<Dim;++i) data[i] = ao[i]; return *this; } InPlaceVector& normalize() { *this /= abs(*this); return *this; }private: T *const data;};template<unsigned D, class T>InPlaceVector<T,D> MakeVector(T *const p){ return InPlaceVector<T,D>(p);}template<class T, unsigned D, class A>inline Vector<T, D> eval(const VectorExpr<T, D, A>& a){ return Vector<T, D>(a);}//"reduction" functions that don't return expression templatestemplate<class T, unsigned D, class A>inline T sum(const VectorExpr<T, D, A>& a){ const A& ao ( a ); T res = 0; for(unsigned i = 0;i<D;++i) res += ao[i]; return res;}template<class T, unsigned D, class A, class B>inline T dot(const VectorExpr<T, D, A>& a, const VectorExpr<T, D, B>& b){ return sum(multiply_elements(a, b));}template<class T, unsigned D, class A>inline T squared_norm(const VectorExpr<T, D, A>& a){ const A& ao ( a ); T res = 0; for(unsigned i = 0;i<D;++i) { T tmp = ao[i]; res += tmp*tmp; } return res;}template<class T, unsigned D, class A>inline T abs(const VectorExpr<T, D, A>& a){ return std::sqrt(squared_norm(a));}template<class T, unsigned D, class A>std::ostream& operator<<(std::ostream& out, const VectorExpr<T, D, A>& a){ const A& ao ( a ); out << '(' << ao[0]; for(unsigned i = 1;i<D;++i) { out << ", " << ao[i]; } out << ')'; return out;}#undef MAKE_VEC_VEC_EXPRESSION#undef MAKE_VEC_SCAL_EXPRESSION#undef MAKE_SCAL_VEC_EXPRESSION#undef MAKE_VEC_EXPRESSION#endif[/source]

Matrix.h
[source lang="cpp"]/* * MathMatrix.h - Copyright © 2011 Jakob Progsch * * This software is provided 'as-is', without any express or implied * warranty. In no event will the authors be held liable for any damages * arising from the use of this software. * * Permission is granted to anyone to use this software for any purpose, * including commercial applications, and to alter it and redistribute it * freely, subject to the following restrictions: * * 1. The origin of this software must not be misrepresented; you must not * claim that you wrote the original software. If you use this software * in a product, an acknowledgment in the product documentation would be * appreciated but is not required. * * 2. Altered source versions must be plainly marked as such, and must not be * misrepresented as being the original software. * * 3. This notice may not be removed or altered from any source * distribution. *//* * Matrix.h provides a simple static Matrix template class with * a basic expression template ansatz for Matrix operations. */ #ifndef Matrix_H#define Matrix_H#include "Vector.h"template<class T, unsigned D1, unsigned D2> class Matrix;//base class for all expression templatestemplate<class T, unsigned D1, unsigned D2, class A>class MatrixExpr {public: inline operator const A&() const { return *static_cast<const A*>(this); }};//better use macros instead of copy pasting this stuff all over the place#define MAKE_MAT_MAT_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D1, unsigned D2, class A, class B> \class NAME : public MatrixExpr<T, D1, D2, NAME<T, D1, D2, A, B> > { \public: \ NAME(const A& pa, const B& pb) : a(pa), b(pb) { } \ inline T operator()(unsigned i, unsigned j) const { return EXPR; } \private: \ const A& a; \ const B& b; \}; \ \template<class T, unsigned D1, unsigned D2, class A, class B> \inline NAME<T,D1,D2,A,B> \FUNCTION(const MatrixExpr<T,D1,D2,A> &a, const MatrixExpr<T,D1,D2,B> &b)\{ \ return NAME<T,D1,D2,A,B>(a, b); \} #define MAKE_MAT_SCAL_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D1, unsigned D2, class A> \class NAME : public MatrixExpr<T, D1, D2, NAME<T, D1, D2, A> > { \public: \ NAME(const A& pa, const T& pb) : a(pa), b(pb) { } \ inline T operator()(unsigned i, unsigned j) const { return EXPR; } \private: \ const A& a; \ const T& b; \}; \ \template<class T, unsigned D1, unsigned D2, class A> \inline NAME<T,D1,D2,A> \FUNCTION(const MatrixExpr<T,D1,D2,A> &a, const T &b) \{ \ return NAME<T,D1,D2,A>(a, b); \} #define MAKE_SCAL_MAT_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D1, unsigned D2, class A> \class NAME : public MatrixExpr<T, D1, D2, NAME<T, D1, D2, A> > { \public: \ NAME(const T& pa, const A& pb) : a(pa), b(pb) { } \ inline T operator()(unsigned i, unsigned j) const { return EXPR; } \private: \ const T& a; \ const A& b; \}; \ \template<class T, unsigned D1, unsigned D2, class A> \inline NAME<T,D1,D2,A> \FUNCTION(const T &a, const MatrixExpr<T,D1,D2,A> &b) \{ \ return NAME<T,D1,D2,A>(a, b); \} #define MAKE_MAT_EXPRESSION(NAME, EXPR, FUNCTION) \template<class T, unsigned D1, unsigned D2, class A> \class NAME : public MatrixExpr<T, D1, D2, NAME<T, D1, D2, A> > { \public: \ NAME(const A& pa) : a(pa) { } \ inline T operator()(unsigned i, unsigned j) const { return EXPR; } \private: \ const A& a; \}; \ \template<class T, unsigned D1, unsigned D2, class A> \inline NAME<T,D1,D2,A> \FUNCTION(const MatrixExpr<T,D1,D2,A> &a) \{ \ return NAME<T,D1,D2,A>(a); \}//create actual functions and operatorsMAKE_MAT_MAT_EXPRESSION (MATEMulExpr, a(i,j) * b(i,j), multiply_elements)MAKE_MAT_MAT_EXPRESSION (MATEDivExpr, a(i,j) / b(i,j), divide_elements)MAKE_MAT_MAT_EXPRESSION (MATAddExpr, a(i,j) + b(i,j), operator+)MAKE_MAT_MAT_EXPRESSION (MATSubExpr, a(i,j) - b(i,j), operator-)MAKE_MAT_SCAL_EXPRESSION(MATDivExpr, a(i,j) / b, operator/)MAKE_MAT_SCAL_EXPRESSION(MATMulExpr1, a(i,j) * b, operator*)MAKE_SCAL_MAT_EXPRESSION(MATMulExpr2, a * b(i,j), operator*)MAKE_MAT_EXPRESSION (MATNegExpr, -a(i,j), operator-)//transpositiontemplate<class T, unsigned D1, unsigned D2, class A> class TransExpr : public MatrixExpr<T, D1, D2, TransExpr<T, D1, D2, A> > { public: TransExpr(const A& pa) : a(pa) { } inline T operator()(unsigned i, unsigned j) const { return a(j,i); } private: const A& a; }; template<class T, unsigned D1, unsigned D2, class A> inline TransExpr<T,D2,D1,A> Transpose(const MatrixExpr<T,D1,D2,A> &a) { return TransExpr<T,D2,D1,A>(a); }//Row, Column vector and submatrix proxiestemplate<class T, unsigned D1, unsigned D2, class A> class RowVectorExpr : public VectorExpr<T, D2, RowVectorExpr<T, D1, D2, A> > { public: RowVectorExpr(const A& pa, const unsigned& i) : a(pa), index(i) { } inline T operator[](unsigned i) const { return a(index, i); } private: const A& a; const unsigned index; }; template<class T, unsigned D1, unsigned D2, class A> inline RowVectorExpr<T,D1,D2,A> Row(const MatrixExpr<T,D1,D2,A> &a, const unsigned &index) { return RowVectorExpr<T,D1,D2,A>(a, index); } template<class T, unsigned D1, unsigned D2, class A> class ColumnVectorExpr : public VectorExpr<T, D1, ColumnVectorExpr<T, D1, D2, A> > { public: ColumnVectorExpr(const A& pa, const unsigned& i) : a(pa), index(i) { } inline T operator[](unsigned i) const { return a(i, index); } private: const A& a; const unsigned index; }; template<class T, unsigned D1, unsigned D2, class A> inline ColumnVectorExpr<T,D1,D2,A> Column(const MatrixExpr<T,D1,D2,A> &a, const unsigned &index) { return ColumnVectorExpr<T,D1,D2,A>(a, index); }template<class T, unsigned D1, unsigned D2, class A> class SubMatrixExpr : public MatrixExpr<T, D1, D2, SubMatrixExpr<T, D1, D2, A> > { public: SubMatrixExpr(const A& pa, const unsigned& i, const unsigned& j) : a(pa), offseti(i), offsetj(j) { } inline T operator()(unsigned i, unsigned j) const { return a(i+offseti, j+offsetj); } private: const A& a; const unsigned offseti, offsetj; }; template<unsigned D3, unsigned D4, class T, unsigned D1, unsigned D2, class A> inline SubMatrixExpr<T,D3,D4,A> SubMatrix(const MatrixExpr<T,D1,D2,A> &a, const unsigned &i, const unsigned &j) { return SubMatrixExpr<T,D3,D4,A>(a, i, j); }//matrix-vector and vector-matrix multiplicationstemplate<class T, unsigned D1, unsigned D2, class A, class B> class MatVecMulExpr : public VectorExpr<T, D1, MatVecMulExpr<T, D1, D2, A, B> > { public: MatVecMulExpr(const A& pa, const B& pb) : a(pa), b(pb) { } inline T operator[](unsigned i) const { return dot(Row(a, i), b); } private: const A& a; const B& b; }; template<class T, unsigned D1, unsigned D2, class A, class B> inline MatVecMulExpr<T,D1,D2,A,B> operator*(const MatrixExpr<T,D1,D2,A> &a, const VectorExpr<T,D2,B> &b) { return MatVecMulExpr<T,D1,D2,A,B>(a, b); }template<class T, unsigned D1, unsigned D2, class A, class B> class VecMatMulExpr : public VectorExpr<T, D2, VecMatMulExpr<T, D1, D2, A, B> > { public: VecMatMulExpr(const A& pa, const B& pb) : a(pa), b(pb) { } inline T operator[](unsigned i) const { return dot(Column(a, i), b); } private: const A& a; const B& b; }; template<class T, unsigned D1, unsigned D2, class A, class B> inline VecMatMulExpr<T,D1,D2,A,B> operator*(const VectorExpr<T,D1,B> &b, const MatrixExpr<T,D1,D2,A> &a) { return VecMatMulExpr<T,D1,D2,A,B>(a, b); }template<class T, unsigned D1, unsigned D2, class A, class B> class MatMatMulExpr : public MatrixExpr<T, D1, D2, MatMatMulExpr<T, D1, D2, A, B> > { public: MatMatMulExpr(const A& pa, const B& pb) : a(pa), b(pb) { } inline T operator()(unsigned i, unsigned j) const { return dot(Row(a,i),Column(b,j)); } private: const A& a; const B& b; }; template<class T, unsigned D1, unsigned D2, unsigned D3, class A, class B> inline MatMatMulExpr<T,D1,D3,A,B> operator*(const MatrixExpr<T,D1,D2,A> &a, const MatrixExpr<T,D2,D3,B> &b){ return MatMatMulExpr<T,D1,D3,A,B>(a, b); }template<class T, unsigned D1, unsigned D2> class Identity : public MatrixExpr<T, D1, D2, Identity<T, D1, D2> > { public: inline T operator()(unsigned i, unsigned j) const { return i==j?1:0; } };//actual Matrix classtemplate<class T, unsigned D1, unsigned D2>class Matrix : public MatrixExpr<T, D1, D2, Matrix<T,D1,D2> > {public: static const unsigned Dim1 = D1; static const unsigned Dim2 = D2; typedef T Type; Matrix() { std::fill(data, data+Dim1*Dim2, T()); } template<class A> Matrix(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) = ao(i,j); } T* raw() { return data; } T& operator() (unsigned i, unsigned j) { return data[i+j*Dim1]; } const T& operator() (unsigned i, unsigned j) const { return data[i+j*Dim1]; } const Matrix& operator*=(const T &b) { for(unsigned i = 0;i<Dim1*Dim2;++i) data[i] *= b; return *this; } const Matrix& operator/=(const T &b) { for(unsigned i = 0;i<Dim1*Dim2;++i) data[i] /= b; return *this; } template<class A> const Matrix& operator+=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) += ao(i,j); return *this; } template<class A> const Matrix& operator-=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) -= ao(i,j); return *this; } template<class A> const Matrix& operator*=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); *this = eval(*this * a); return *this; } template<class A> Matrix& operator=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) = ao(i,j); return *this; } private: T data[Dim1*Dim2];};//InPlaceMatrix to use raw data COLUMN MAJOR!template<class T, unsigned D1, unsigned D2>class InPlaceMatrix : public MatrixExpr<T, D1, D2, InPlaceMatrix<T,D1,D2> > {public: static const unsigned Dim1 = D1; static const unsigned Dim2 = D2; typedef T Type; InPlaceMatrix(T *const d) : data(d) { } template<class A> InPlaceMatrix(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) = ao(i,j); } T* raw() { return data; } T& operator() (unsigned i, unsigned j) { return data[i+j*Dim1]; } const T& operator() (unsigned i, unsigned j) const { return data[i+j*Dim1]; } const InPlaceMatrix& operator*=(const T &b) { for(unsigned i = 0;i<Dim1*Dim2;++i) data[i] *= b; return *this; } const InPlaceMatrix& operator/=(const T &b) { for(unsigned i = 0;i<Dim1*Dim2;++i) data[i] /= b; return *this; } template<class A> const InPlaceMatrix& operator+=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) += ao(i,j); return *this; } template<class A> const InPlaceMatrix& operator-=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) -= ao(i,j); return *this; } template<class A> const InPlaceMatrix& operator*=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); *this = eval(*this * a); return *this; } template<class A> InPlaceMatrix& operator=(const MatrixExpr<T, D1, D2, A>& a) { const A& ao ( a ); for(unsigned j = 0;j<Dim2;++j) for(unsigned i = 0;i<Dim1;++i) operator()(i,j) = ao(i,j); return *this; } private: T *const data;};template<unsigned D1, unsigned D2, class T>InPlaceMatrix<T,D1,D2> MakeMatrix(T *const p){ return InPlaceMatrix<T,D1,D2>(p);}template<class T, unsigned D1, unsigned D2, class A>inline Matrix<T, D1, D2> eval(const MatrixExpr<T, D1, D2, A>& a){ return Matrix<T, D1, D2>(a);}template<class T, unsigned D1, unsigned D2, class A>std::ostream& operator<<(std::ostream& out, const MatrixExpr<T, D1, D2, A>& a){ const A& ao ( a ); for(unsigned i = 0;i<D1;++i) { out << '|' << ao(i, 0); for(unsigned j = 1;j<D2;++j) out << ", " << ao(i, j); out << "|\n"; } return out;}#undef MAKE_MAT_MAT_EXPRESSION#undef MAKE_MAT_SCAL_EXPRESSION#undef MAKE_SCAL_MAT_EXPRESSION#undef MAKE_MAT_EXPRESSION#endif[/source]

__Homer__
My first impression is to run away covering my eyes.
In my mind, vectors (tuples) and matrices are very simple structures, with the kind of code involved that mainly gets inlined - can you explain the benefits of templating everything into obfuscurity?
I tried to explain this in my first entry about vectors here. The expression template approach gets rid of unnecessary temporary objects and enforces "loop fusion" by combining multiple operations into a single composite operation, so it's mostly for performance. Also apart from the "eval"-thing you don't have to deal with the complexity of all the template stuff when you are using these vector and matrix classes. Vectors and matrices can of course be implemented in more straight forward ways. But since they are typically used all over the place I think it's worth it to make them as efficient and sophisticated as possible.

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