BigMatrix.h

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#ifndef SimTK_SIMMATRIX_BIGMATRIX_H_
#define SimTK_SIMMATRIX_BIGMATRIX_H_
 
/* -------------------------------------------------------------------------- *
 *                       Simbody(tm): SimTKcommon                             *
 * -------------------------------------------------------------------------- *
 * This is part of the SimTK biosimulation toolkit originating from           *
 * Simbios, the NIH National Center for Physics-Based Simulation of           *
 * Biological Structures at Stanford, funded under the NIH Roadmap for        *
 * Medical Research, grant U54 GM072970. See https://simtk.org/home/simbody.  *
 *                                                                            *
 * Portions copyright (c) 2005-13 Stanford University and the Authors.        *
 * Authors: Michael Sherman                                                   *
 * Contributors:                                                              *
 *                                                                            *
 * Licensed under the Apache License, Version 2.0 (the "License"); you may    *
 * not use this file except in compliance with the License. You may obtain a  *
 * copy of the License at http://www.apache.org/licenses/LICENSE-2.0.         *
 *                                                                            *
 * Unless required by applicable law or agreed to in writing, software        *
 * distributed under the License is distributed on an "AS IS" BASIS,          *
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.   *
 * See the License for the specific language governing permissions and        *
 * limitations under the License.                                             *
 * -------------------------------------------------------------------------- */
 
#include "SimTKcommon/Scalar.h"
#include "SimTKcommon/SmallMatrix.h"
 
#include "SimTKcommon/internal/MatrixHelper.h"
#include "SimTKcommon/internal/MatrixCharacteristics.h"
 
#include <iostream>
#include <cassert>
#include <complex>
#include <cstddef>
#include <limits>
 
namespace SimTK {
    template <class ELT>    class MatrixBase;
    template <class ELT>    class VectorBase;
    template <class ELT>    class RowVectorBase;
 
    template <class ELT = Real> class MatrixView_;
    template <class ELT = Real> class Matrix_;
 
    template <class ELT = Real> class VectorView_;
    template <class ELT = Real> class Vector_;
 
    template <class ELT = Real> class RowVectorView_;
    template <class ELT = Real> class RowVector_;
 
    template <class ELT, class VECTOR_CLASS> class VectorIterator;
}
 
#include "SimTKcommon/internal/MatrixBase.h"
#include "SimTKcommon/internal/VectorBase.h"
#include "SimTKcommon/internal/RowVectorBase.h"
 
#include "SimTKcommon/internal/MatrixView_.h"
#include "SimTKcommon/internal/Matrix_.h"
 
#include "SimTKcommon/internal/VectorView_.h"
#include "SimTKcommon/internal/Vector_.h"
 
#include "SimTKcommon/internal/RowVectorView_.h"
#include "SimTKcommon/internal/RowVector_.h"
 
#include "SimTKcommon/internal/VectorIterator.h"
 
 
namespace SimTK {
 
//  ------------------------ MatrixBase definitions ----------------------------
 
template <class ELT> inline MatrixView_<ELT> 
MatrixBase<ELT>::block(int i, int j, int m, int n) const { 
    SimTK_INDEXCHECK(i,nrow()+1,"MatrixBase::block()");
    SimTK_INDEXCHECK(j,ncol()+1,"MatrixBase::block()");
    SimTK_SIZECHECK(i+m,nrow(),"MatrixBase::block()");
    SimTK_SIZECHECK(j+n,ncol(),"MatrixBase::block()");
 
    MatrixHelper<Scalar> h(MatrixCommitment(),helper,i,j,m,n);    
    return MatrixView_<ELT>(h.stealRep()); 
}
    
template <class ELT> inline MatrixView_<ELT>
MatrixBase<ELT>::updBlock(int i, int j, int m, int n) { 
    SimTK_INDEXCHECK(i,nrow()+1,"MatrixBase::updBlock()");
    SimTK_INDEXCHECK(j,ncol()+1,"MatrixBase::updBlock()");
    SimTK_SIZECHECK(i+m,nrow(),"MatrixBase::updBlock()");
    SimTK_SIZECHECK(j+n,ncol(),"MatrixBase::updBlock()");
 
    MatrixHelper<Scalar> h(MatrixCommitment(),helper,i,j,m,n);        
    return MatrixView_<ELT>(h.stealRep()); 
}
 
template <class E> inline MatrixView_<typename CNT<E>::THerm>
MatrixBase<E>::transpose() const { 
    MatrixHelper<typename CNT<Scalar>::THerm> 
        h(MatrixCommitment(),
          helper, typename MatrixHelper<typename CNT<Scalar>::THerm>::TransposeView());
    return MatrixView_<typename CNT<E>::THerm>(h.stealRep()); 
}
    
template <class E> inline MatrixView_<typename CNT<E>::THerm>
MatrixBase<E>::updTranspose() {     
    MatrixHelper<typename CNT<Scalar>::THerm> 
        h(MatrixCommitment(),
          helper, typename MatrixHelper<typename CNT<Scalar>::THerm>::TransposeView());
    return MatrixView_<typename CNT<E>::THerm>(h.stealRep()); 
}
 
template <class E> inline VectorView_<E>
MatrixBase<E>::diag() const { 
    MatrixHelper<Scalar> h(MatrixCommitment::Vector(),
                           helper, typename MatrixHelper<Scalar>::DiagonalView());
    return VectorView_<E>(h.stealRep()); 
}
    
template <class E> inline VectorView_<E>
MatrixBase<E>::updDiag() {     
    MatrixHelper<Scalar> h(MatrixCommitment::Vector(),
                           helper, typename MatrixHelper<Scalar>::DiagonalView());
    return VectorView_<E>(h.stealRep());
}
 
template <class ELT> inline VectorView_<ELT> 
MatrixBase<ELT>::col(int j) const { 
    SimTK_INDEXCHECK(j,ncol(),"MatrixBase::col()");
 
    MatrixHelper<Scalar> h(MatrixCommitment::Vector(),
                           helper,0,j,nrow(),1);    
    return VectorView_<ELT>(h.stealRep()); 
}
    
template <class ELT> inline VectorView_<ELT>
MatrixBase<ELT>::updCol(int j) {
    SimTK_INDEXCHECK(j,ncol(),"MatrixBase::updCol()");
 
    MatrixHelper<Scalar> h(MatrixCommitment::Vector(),
                           helper,0,j,nrow(),1);        
    return VectorView_<ELT>(h.stealRep()); 
}
 
template <class ELT> inline RowVectorView_<ELT> 
MatrixBase<ELT>::row(int i) const { 
    SimTK_INDEXCHECK(i,nrow(),"MatrixBase::row()");
 
    MatrixHelper<Scalar> h(MatrixCommitment::RowVector(),
                           helper,i,0,1,ncol());    
    return RowVectorView_<ELT>(h.stealRep()); 
}
    
template <class ELT> inline RowVectorView_<ELT>
MatrixBase<ELT>::updRow(int i) { 
    SimTK_INDEXCHECK(i,nrow(),"MatrixBase::updRow()");
 
    MatrixHelper<Scalar> h(MatrixCommitment::RowVector(),
                           helper,i,0,1,ncol());        
    return RowVectorView_<ELT>(h.stealRep()); 
}
 
// M = diag(v) * M; v must have nrow() elements.
// That is, M[i] *= v[i].
template <class ELT> template <class EE> inline MatrixBase<ELT>& 
MatrixBase<ELT>::rowScaleInPlace(const VectorBase<EE>& v) {
    assert(v.nrow() == nrow());
    for (int i=0; i < nrow(); ++i)
        (*this)[i] *= v[i];
    return *this;
}
 
template <class ELT> template <class EE> inline void
MatrixBase<ELT>::rowScale(const VectorBase<EE>& v, typename MatrixBase<ELT>::template EltResult<EE>::Mul& out) const {
    assert(v.nrow() == nrow());
    out.resize(nrow(), ncol());
    for (int j=0; j<ncol(); ++j)
        for (int i=0; i<nrow(); ++i)
           out(i,j) = (*this)(i,j) * v[i];
}
 
// M = M * diag(v); v must have ncol() elements
// That is, M(i) *= v[i]
template <class ELT> template <class EE>  inline MatrixBase<ELT>& 
MatrixBase<ELT>::colScaleInPlace(const VectorBase<EE>& v) {
    assert(v.nrow() == ncol());
    for (int j=0; j < ncol(); ++j)
        (*this)(j) *= v[j];
    return *this;
}
 
template <class ELT> template <class EE> inline void
MatrixBase<ELT>::colScale(const VectorBase<EE>& v, typename MatrixBase<ELT>::template EltResult<EE>::Mul& out) const {
    assert(v.nrow() == ncol());
    out.resize(nrow(), ncol());
    for (int j=0; j<ncol(); ++j)
        for (int i=0; i<nrow(); ++i)
           out(i,j) = (*this)(i,j) * v[j];
}
 
 
// M(i,j) *= r[i]*c[j]; r must have nrow() elements; c must have ncol() elements
template <class ELT> template <class ER, class EC> inline MatrixBase<ELT>& 
MatrixBase<ELT>::rowAndColScaleInPlace(const VectorBase<ER>& r, const VectorBase<EC>& c) {
    assert(r.nrow()==nrow() && c.nrow()==ncol());
    for (int j=0; j<ncol(); ++j)
        for (int i=0; i<nrow(); ++i)
            (*this)(i,j) *= (r[i]*c[j]);
    return *this;
}
 
template <class ELT> template <class ER, class EC> inline void
MatrixBase<ELT>::rowAndColScale(
    const VectorBase<ER>& r, 
    const VectorBase<EC>& c,
    typename EltResult<typename VectorBase<ER>::template EltResult<EC>::Mul>::Mul& 
                          out) const
{
    assert(r.nrow()==nrow() && c.nrow()==ncol());
    out.resize(nrow(), ncol());
    for (int j=0; j<ncol(); ++j)
        for (int i=0; i<nrow(); ++i)
            out(i,j) = (*this)(i,j) * (r[i]*c[j]);
}
 
// M(i,j) = s
template <class ELT> template <class S> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseAssign(const S& s) {
    for (int j=0; j<ncol(); ++j)
    for (int i=0; i<nrow(); ++i)
        (*this)(i,j) = s;
    return *this;
}
 
// Set M(i,j) = M(i,j)^-1.
template <class ELT> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseInvertInPlace() {
    const int nr=nrow(), nc=ncol();
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i) {
            ELT& e = updElt(i,j);
            e = CNT<ELT>::invert(e);
        }
    return *this;
}
 
template <class ELT> inline void
MatrixBase<ELT>::elementwiseInvert(MatrixBase<typename CNT<E>::TInvert>& out) const {
    const int nr=nrow(), nc=ncol();
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) = CNT<ELT>::invert((*this)(i,j));
}
 
// M(i,j) += s
template <class ELT> template <class S> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseAddScalarInPlace(const S& s) {
    for (int j=0; j<ncol(); ++j)
        for (int i=0; i<nrow(); ++i)
            (*this)(i,j) += s;
    return *this;
}
 
template <class ELT> template <class S> inline void 
MatrixBase<ELT>::elementwiseAddScalar(
    const S& s,
    typename MatrixBase<ELT>::template EltResult<S>::Add& out) const
{
    const int nr=nrow(), nc=ncol();
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) = (*this)(i,j) + s;
}
 
// M(i,j) -= s
template <class ELT> template <class S> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseSubtractScalarInPlace(const S& s) {
    for (int j=0; j<ncol(); ++j)
        for (int i=0; i<nrow(); ++i)
            (*this)(i,j) -= s;
    return *this;
}
 
template <class ELT> template <class S> inline void 
MatrixBase<ELT>::elementwiseSubtractScalar(
    const S& s,
    typename MatrixBase<ELT>::template EltResult<S>::Sub& out) const
{
    const int nr=nrow(), nc=ncol();
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) = (*this)(i,j) - s;
}
 
// M(i,j) = s - M(i,j)
template <class ELT> template <class S> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseSubtractFromScalarInPlace(const S& s) {
    const int nr=nrow(), nc=ncol();
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i) {
            ELT& e = updElt(i,j);
            e = s - e;
        }
    return *this;
}
 
template <class ELT> template <class S> inline void 
MatrixBase<ELT>::elementwiseSubtractFromScalar(
    const S& s,
    typename MatrixBase<S>::template EltResult<ELT>::Sub& out) const
{
    const int nr=nrow(), nc=ncol();
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) = s - (*this)(i,j);
}
 
// M(i,j) *= R(i,j); R must have same dimensions as this
template <class ELT> template <class EE> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseMultiplyInPlace(const MatrixBase<EE>& r) {
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            (*this)(i,j) *= r(i,j);
    return *this;
}
 
template <class ELT> template <class EE> inline void 
MatrixBase<ELT>::elementwiseMultiply(
    const MatrixBase<EE>& r, 
    typename MatrixBase<ELT>::template EltResult<EE>::Mul& out) const
{
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) = (*this)(i,j) * r(i,j);
}
 
// M(i,j) = R(i,j) * M(i,j); R must have same dimensions as this
template <class ELT> template <class EE> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseMultiplyFromLeftInPlace(const MatrixBase<EE>& r) {
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i) {
            ELT& e = updElt(i,j);
            e = r(i,j) * e;
        }
    return *this;
}
 
template <class ELT> template <class EE> inline void 
MatrixBase<ELT>::elementwiseMultiplyFromLeft(
    const MatrixBase<EE>& r, 
    typename MatrixBase<EE>::template EltResult<ELT>::Mul& out) const
{
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) =  r(i,j) * (*this)(i,j);
}
 
// M(i,j) /= R(i,j); R must have same dimensions as this
template <class ELT> template <class EE> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseDivideInPlace(const MatrixBase<EE>& r) {
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            (*this)(i,j) /= r(i,j);
    return *this;
}
 
template <class ELT> template <class EE> inline void 
MatrixBase<ELT>::elementwiseDivide(
    const MatrixBase<EE>& r,
    typename MatrixBase<ELT>::template EltResult<EE>::Dvd& out) const
{
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) = (*this)(i,j) / r(i,j);
}
// M(i,j) = R(i,j) / M(i,j); R must have same dimensions as this
template <class ELT> template <class EE> inline MatrixBase<ELT>& 
MatrixBase<ELT>::elementwiseDivideFromLeftInPlace(const MatrixBase<EE>& r) {
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i) {
            ELT& e = updElt(i,j);
            e = r(i,j) / e;
        }
    return *this;
}
 
template <class ELT> template <class EE> inline void 
MatrixBase<ELT>::elementwiseDivideFromLeft(
    const MatrixBase<EE>& r,
    typename MatrixBase<EE>::template EltResult<ELT>::Dvd& out) const
{
    const int nr=nrow(), nc=ncol();
    assert(r.nrow()==nr && r.ncol()==nc);
    out.resize(nr,nc);
    for (int j=0; j<nc; ++j)
        for (int i=0; i<nr; ++i)
            out(i,j) = r(i,j) / (*this)(i,j);
}
 
/*
template <class ELT> inline MatrixView_< typename CNT<ELT>::TReal > 
MatrixBase<ELT>::real() const { 
    if (!CNT<ELT>::IsComplex) { // known at compile time
        return MatrixView_< typename CNT<ELT>::TReal >( // this is just ELT
            MatrixHelper(helper,0,0,nrow(),ncol()));    // a view of the whole matrix
    }
    // Elements are complex -- helper uses underlying precision (real) type.
    MatrixHelper<Precision> h(helper,typename MatrixHelper<Precision>::RealView);    
    return MatrixView_< typename CNT<ELT>::TReal >(h); 
}
*/
 
 
//  ----------------------------------------------------------------------------
 
// + and - allow mixed element types, but will fail to compile if the elements aren't
// compatible. At run time these will fail if the dimensions are incompatible.
template <class E1, class E2>
Matrix_<typename CNT<E1>::template Result<E2>::Add>
operator+(const MatrixBase<E1>& l, const MatrixBase<E2>& r) {
    return Matrix_<typename CNT<E1>::template Result<E2>::Add>(l) += r;
}
 
template <class E>
Matrix_<E> operator+(const MatrixBase<E>& l, const typename CNT<E>::T& r) {
    return Matrix_<E>(l) += r;
}
 
template <class E>
Matrix_<E> operator+(const typename CNT<E>::T& l, const MatrixBase<E>& r) {
    return Matrix_<E>(r) += l;
}
 
template <class E1, class E2>
Matrix_<typename CNT<E1>::template Result<E2>::Sub>
operator-(const MatrixBase<E1>& l, const MatrixBase<E2>& r) {
    return Matrix_<typename CNT<E1>::template Result<E2>::Sub>(l) -= r;
}
 
template <class E>
Matrix_<E> operator-(const MatrixBase<E>& l, const typename CNT<E>::T& r) {
    return Matrix_<E>(l) -= r;
}
 
template <class E>
Matrix_<E> operator-(const typename CNT<E>::T& l, const MatrixBase<E>& r) {
    Matrix_<E> temp(r.nrow(), r.ncol());
    temp = l;
    return (temp -= r);
}
 
// Scalar multiply and divide. You might wish the scalar could be
// a templatized type "E2", but that would create horrible ambiguities since
// E2 would match not only scalar types but everything else including
// matrices.
template <class E> Matrix_<E>
operator*(const MatrixBase<E>& l, const typename CNT<E>::StdNumber& r) 
  { return Matrix_<E>(l)*=r; }
 
template <class E> Matrix_<E>
operator*(const typename CNT<E>::StdNumber& l, const MatrixBase<E>& r) 
  { return Matrix_<E>(r)*=l; }
 
template <class E> Matrix_<E>
operator/(const MatrixBase<E>& l, const typename CNT<E>::StdNumber& r) 
  { return Matrix_<E>(l)/=r; }
 
// Handle ints explicitly.
template <class E> Matrix_<E>
operator*(const MatrixBase<E>& l, int r) 
  { return Matrix_<E>(l)*= typename CNT<E>::StdNumber(r); }
 
template <class E> Matrix_<E>
operator*(int l, const MatrixBase<E>& r) 
  { return Matrix_<E>(r)*= typename CNT<E>::StdNumber(l); }
 
template <class E> Matrix_<E>
operator/(const MatrixBase<E>& l, int r) 
  { return Matrix_<E>(l)/= typename CNT<E>::StdNumber(r); }
 
 
 
template <class E1, class E2>
Vector_<typename CNT<E1>::template Result<E2>::Add>
operator+(const VectorBase<E1>& l, const VectorBase<E2>& r) {
    return Vector_<typename CNT<E1>::template Result<E2>::Add>(l) += r;
}
template <class E>
Vector_<E> operator+(const VectorBase<E>& l, const typename CNT<E>::T& r) {
    return Vector_<E>(l) += r;
}
template <class E>
Vector_<E> operator+(const typename CNT<E>::T& l, const VectorBase<E>& r) {
    return Vector_<E>(r) += l;
}
template <class E1, class E2>
Vector_<typename CNT<E1>::template Result<E2>::Sub>
operator-(const VectorBase<E1>& l, const VectorBase<E2>& r) {
    return Vector_<typename CNT<E1>::template Result<E2>::Sub>(l) -= r;
}
template <class E>
Vector_<E> operator-(const VectorBase<E>& l, const typename CNT<E>::T& r) {
    return Vector_<E>(l) -= r;
}
template <class E>
Vector_<E> operator-(const typename CNT<E>::T& l, const VectorBase<E>& r) {
    Vector_<E> temp(r.size());
    temp = l;
    return (temp -= r);
}
 
// Scalar multiply and divide.
 
template <class E> Vector_<E>
operator*(const VectorBase<E>& l, const typename CNT<E>::StdNumber& r) 
  { return Vector_<E>(l)*=r; }
 
template <class E> Vector_<E>
operator*(const typename CNT<E>::StdNumber& l, const VectorBase<E>& r) 
  { return Vector_<E>(r)*=l; }
 
template <class E> Vector_<E>
operator/(const VectorBase<E>& l, const typename CNT<E>::StdNumber& r) 
  { return Vector_<E>(l)/=r; }
 
// Handle ints explicitly
template <class E> Vector_<E>
operator*(const VectorBase<E>& l, int r) 
  { return Vector_<E>(l)*= typename CNT<E>::StdNumber(r); }
 
template <class E> Vector_<E>
operator*(int l, const VectorBase<E>& r) 
  { return Vector_<E>(r)*= typename CNT<E>::StdNumber(l); }
 
template <class E> Vector_<E>
operator/(const VectorBase<E>& l, int r) 
  { return Vector_<E>(l)/= typename CNT<E>::StdNumber(r); }
 
// These are fancier "scalars"; whether they are allowed depends on
// whether the element type and the CNT are compatible.
 
// Vector * Vec
template <class E1, int M, class E2, int S> 
Vector_<typename CNT<E1>::template Result< Vec<M,E2,S> >::Mul>
operator*(const VectorBase<E1>& v, const Vec<M,E2,S>& s) {
    Vector_<typename CNT<E1>::template Result< Vec<M,E2,S> >::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// Vec * Vector
template <class E1, int M, class E2, int S> 
Vector_<typename Vec<M,E2,S>::template Result<E1>::Mul>
operator*(const Vec<M,E2,S>& s, const VectorBase<E1>& v) {
    Vector_<typename Vec<M,E2,S>::template Result<E1>::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
// Vector * Row
template <class E1, int N, class E2, int S> 
Vector_<typename CNT<E1>::template Result< Row<N,E2,S> >::Mul>
operator*(const VectorBase<E1>& v, const Row<N,E2,S>& s) {
    Vector_<typename CNT<E1>::template Result< Row<N,E2,S> >::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// Row * Vector
template <class E1, int N, class E2, int S> 
Vector_<typename Row<N,E2,S>::template Result<E1>::Mul>
operator*(const Row<N,E2,S>& s, const VectorBase<E1>& v) {
    Vector_<typename Row<N,E2,S>::template Result<E1>::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
// Vector * Mat
template <class E1, int M, int N, class E2, int S1, int S2> 
Vector_<typename CNT<E1>::template Result< Mat<M,N,E2,S1,S2> >::Mul>
operator*(const VectorBase<E1>& v, const Mat<M,N,E2,S1,S2>& s) {
    Vector_<typename CNT<E1>::template Result< Mat<M,N,E2,S1,S2> >::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// Mat * Vector
template <class E1, int M, int N, class E2, int S1, int S2> 
Vector_<typename Mat<M,N,E2,S1,S2>::template Result<E1>::Mul>
operator*(const Mat<M,N,E2,S1,S2>& s, const VectorBase<E1>& v) {
    Vector_<typename Mat<M,N,E2,S1,S2>::template Result<E1>::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
// Vector * SymMat
template <class E1, int M, class E2, int S> 
Vector_<typename CNT<E1>::template Result< SymMat<M,E2,S> >::Mul>
operator*(const VectorBase<E1>& v, const SymMat<M,E2,S>& s) {
    Vector_<typename CNT<E1>::template Result< SymMat<M,E2,S> >::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// SymMat * Vector
template <class E1, int M, class E2, int S> 
Vector_<typename SymMat<M,E2,S>::template Result<E1>::Mul>
operator*(const SymMat<M,E2,S>& s, const VectorBase<E1>& v) {
    Vector_<typename SymMat<M,E2,S>::template Result<E1>::Mul> res(v.nrow());
    for (int i=0; i < v.nrow(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
 
 
template <class E1, class E2>
RowVector_<typename CNT<E1>::template Result<E2>::Add>
operator+(const RowVectorBase<E1>& l, const RowVectorBase<E2>& r) {
    return RowVector_<typename CNT<E1>::template Result<E2>::Add>(l) += r;
}
template <class E>
RowVector_<E> operator+(const RowVectorBase<E>& l, const typename CNT<E>::T& r) {
    return RowVector_<E>(l) += r;
}
template <class E>
RowVector_<E> operator+(const typename CNT<E>::T& l, const RowVectorBase<E>& r) {
    return RowVector_<E>(r) += l;
}
template <class E1, class E2>
RowVector_<typename CNT<E1>::template Result<E2>::Sub>
operator-(const RowVectorBase<E1>& l, const RowVectorBase<E2>& r) {
    return RowVector_<typename CNT<E1>::template Result<E2>::Sub>(l) -= r;
}
template <class E>
RowVector_<E> operator-(const RowVectorBase<E>& l, const typename CNT<E>::T& r) {
    return RowVector_<E>(l) -= r;
}
template <class E>
RowVector_<E> operator-(const typename CNT<E>::T& l, const RowVectorBase<E>& r) {
    RowVector_<E> temp(r.size());
    temp = l;
    return (temp -= r);
}
 
// Scalar multiply and divide 
 
template <class E> RowVector_<E>
operator*(const RowVectorBase<E>& l, const typename CNT<E>::StdNumber& r) 
  { return RowVector_<E>(l)*=r; }
 
template <class E> RowVector_<E>
operator*(const typename CNT<E>::StdNumber& l, const RowVectorBase<E>& r) 
  { return RowVector_<E>(r)*=l; }
 
template <class E> RowVector_<E>
operator/(const RowVectorBase<E>& l, const typename CNT<E>::StdNumber& r) 
  { return RowVector_<E>(l)/=r; }
 
// Handle ints explicitly.
template <class E> RowVector_<E>
operator*(const RowVectorBase<E>& l, int r) 
  { return RowVector_<E>(l)*= typename CNT<E>::StdNumber(r); }
 
template <class E> RowVector_<E>
operator*(int l, const RowVectorBase<E>& r) 
  { return RowVector_<E>(r)*= typename CNT<E>::StdNumber(l); }
 
template <class E> RowVector_<E>
operator/(const RowVectorBase<E>& l, int r) 
  { return RowVector_<E>(l)/= typename CNT<E>::StdNumber(r); }
 
 
// These are fancier "scalars"; whether they are allowed depends on
// whether the element type and the CNT are compatible.
 
// RowVector * Vec
template <class E1, int M, class E2, int S> 
RowVector_<typename CNT<E1>::template Result< Vec<M,E2,S> >::Mul>
operator*(const RowVectorBase<E1>& v, const Vec<M,E2,S>& s) {
    RowVector_<typename CNT<E1>::template Result< Vec<M,E2,S> >::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// Vec * RowVector
template <class E1, int M, class E2, int S> 
RowVector_<typename Vec<M,E2,S>::template Result<E1>::Mul>
operator*(const Vec<M,E2,S>& s, const RowVectorBase<E1>& v) {
    RowVector_<typename Vec<M,E2,S>::template Result<E1>::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
// RowVector * Row
template <class E1, int N, class E2, int S> 
RowVector_<typename CNT<E1>::template Result< Row<N,E2,S> >::Mul>
operator*(const RowVectorBase<E1>& v, const Row<N,E2,S>& s) {
    RowVector_<typename CNT<E1>::template Result< Row<N,E2,S> >::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// Row * RowVector
template <class E1, int N, class E2, int S> 
RowVector_<typename Row<N,E2,S>::template Result<E1>::Mul>
operator*(const Row<N,E2,S>& s, const RowVectorBase<E1>& v) {
    RowVector_<typename Row<N,E2,S>::template Result<E1>::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
// RowVector * Mat
template <class E1, int M, int N, class E2, int S1, int S2> 
RowVector_<typename CNT<E1>::template Result< Mat<M,N,E2,S1,S2> >::Mul>
operator*(const RowVectorBase<E1>& v, const Mat<M,N,E2,S1,S2>& s) {
    RowVector_<typename CNT<E1>::template Result< Mat<M,N,E2,S1,S2> >::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// Mat * RowVector
template <class E1, int M, int N, class E2, int S1, int S2> 
RowVector_<typename Mat<M,N,E2,S1,S2>::template Result<E1>::Mul>
operator*(const Mat<M,N,E2,S1,S2>& s, const RowVectorBase<E1>& v) {
    RowVector_<typename Mat<M,N,E2,S1,S2>::template Result<E1>::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
// RowVector * SymMat
template <class E1, int M, class E2, int S> 
RowVector_<typename CNT<E1>::template Result< SymMat<M,E2,S> >::Mul>
operator*(const RowVectorBase<E1>& v, const SymMat<M,E2,S>& s) {
    RowVector_<typename CNT<E1>::template Result< SymMat<M,E2,S> >::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = v[i]*s; 
    return res;
}
 
// SymMat * RowVector
template <class E1, int M, class E2, int S> 
RowVector_<typename SymMat<M,E2,S>::template Result<E1>::Mul>
operator*(const SymMat<M,E2,S>& s, const RowVectorBase<E1>& v) {
    RowVector_<typename SymMat<M,E2,S>::template Result<E1>::Mul> res(v.ncol());
    for (int i=0; i < v.ncol(); ++i)
        res[i] = s*v[i]; 
    return res;
}
 
 
 
 
    // TODO: these should use LAPACK!
 
// Dot product
template <class E1, class E2> 
typename CNT<E1>::template Result<E2>::Mul
operator*(const RowVectorBase<E1>& r, const VectorBase<E2>& v) {
    assert(r.ncol() == v.nrow());
    typename CNT<E1>::template Result<E2>::Mul sum(0);
    for (int j=0; j < r.ncol(); ++j)
        sum += r(j) * v[j];
    return sum;
}
 
template <class E1, class E2> 
Vector_<typename CNT<E1>::template Result<E2>::Mul>
operator*(const MatrixBase<E1>& m, const VectorBase<E2>& v) {
    assert(m.ncol() == v.nrow());
    Vector_<typename CNT<E1>::template Result<E2>::Mul> res(m.nrow());
    for (int i=0; i< m.nrow(); ++i)
        res[i] = m[i]*v;
    return res;
}
 
template <class E1, class E2> 
Matrix_<typename CNT<E1>::template Result<E2>::Mul>
operator*(const MatrixBase<E1>& m1, const MatrixBase<E2>& m2) {
    assert(m1.ncol() == m2.nrow());
    Matrix_<typename CNT<E1>::template Result<E2>::Mul> 
        res(m1.nrow(),m2.ncol());
 
    for (int j=0; j < res.ncol(); ++j)
        for (int i=0; i < res.nrow(); ++i)
            res(i,j) = m1[i] * m2(j);
 
    return res;
}
 
 
// This "private" static method is used to implement VectorView's 
// fillVectorViewFromStream() and Vector's readVectorFromStream() 
// namespace-scope static methods, which are in turn used to implement 
// VectorView's and 
// Vector's stream extraction operators ">>". This method has to be in the 
// header file so that we don't need to pass streams through the API, but it 
// is not intended for use by users and has no Doxygen presence, unlike 
// fillArrayFromStream() and readArrayFromStream() and (more commonly)
// the extraction operators.
template <class T> static inline 
std::istream& readVectorFromStreamHelper
   (std::istream& in, bool isFixedSize, Vector_<T>& out)
{
    // If already failed, bad, or eof, set failed bit and return without 
    // touching the Vector.
    if (!in.good()) {in.setstate(std::ios::failbit); return in;}
 
    // If the passed-in Vector isn't resizeable, then we have to treat it as
    // a fixed size VectorView regardless of the setting of the isFixedSize
    // argument.
    if (!out.isResizeable())
        isFixedSize = true; // might be overriding the argument here
 
    // numRequired will be ignored unless isFixedSize==true.
    const int numRequired = isFixedSize ? out.size() : 0;
 
    if (!isFixedSize)
        out.clear(); // We're going to replace the entire contents of the Array.
 
    // Skip initial whitespace. If that results in eof this may be a successful
    // read of a 0-length, unbracketed Vector. That is OK for either a
    // variable-length Vector or a fixed-length VectorView of length zero.
    std::ws(in); if (in.fail()) return in;
    if (in.eof()) {
        if (isFixedSize && numRequired != 0)
            in.setstate(std::ios_base::failbit); // zero elements not OK
        return in;
    }
    
    // Here the stream is good and the next character is non-white.
    assert(in.good());
 
    // Use this for raw i/o (peeks and gets).
    typename       std::iostream::int_type ch;
#ifndef NDEBUG
    const typename std::iostream::int_type EOFch = 
        std::iostream::traits_type::eof();
#endif
 
    // First we'll look for the optional "~". If found, the brackets become
    // required.
    bool tildeFound = false;
    ch = in.peek(); if (in.fail()) return in;
    assert(ch != EOFch); // we already checked above
    if ((char)ch == '~') {
        tildeFound = true;
        in.get(); // absorb the tilde
        // Eat whitespace after the tilde to see what's next.
        if (in.good()) std::ws(in);
        // If we hit eof after the tilde we don't like the formatting.
        if (!in.good()) {in.setstate(std::ios_base::failbit); return in;}
    }
 
    // Here the stream is good, the next character is non-white, and we
    // might have seen a tilde.
    assert(in.good());
 
    // Now see if the sequence is bare or surrounded by (), or [].
    bool lookForCloser = true;
    char openBracket, closeBracket;
    ch = in.peek(); if (in.fail()) return in;
    assert(ch != EOFch); // we already checked above
 
    openBracket = (char)ch;
    if      (openBracket=='(') {in.get(); closeBracket = ')';}
    else if (openBracket=='[') {in.get(); closeBracket = ']';}
    else lookForCloser = false;
 
    // If we found a tilde, the opening bracket was mandatory. If we didn't
    // find one then we reject the formatting.
    if (tildeFound && !lookForCloser)
    {   in.setstate(std::ios_base::failbit); return in;}
 
    // If lookForCloser is true, then closeBracket contains the terminating
    // delimiter, otherwise we're not going to quit until eof.
 
    // Eat whitespace after the opening bracket to see what's next.
    if (in.good()) std::ws(in);
 
    // If we're at eof now it must be because the open bracket was the
    // last non-white character in the stream, which is an error.
    if (!in.good()) {
        if (in.eof()) {
            assert(lookForCloser); // or we haven't read anything that could eof
            in.setstate(std::ios::failbit);
        }
        return in;
    }
 
    // istream is good and next character is non-white; ready to read first
    // value or terminator.
 
    // We need to figure out whether the elements are space- or comma-
    // separated and then insist on consistency.
    bool commaOK = true, commaRequired = false;
    bool terminatorSeen = false;
    int nextIndex = 0;
    while (true) {
        char c;
 
        // Here at the top of this loop, we have already successfully read 
        // n=nextIndex values of type T. For fixed-size reads, it might be
        // the case that n==numRequired already, but we still may need to
        // look for a closing bracket before we can declare victory.
        // The stream is good() (not at eof) but it might be the case that 
        // there is nothing but white space left; we don't know yet because
        // if we have satisfied the fixed-size count and are not expecting
        // a terminator then we should quit without absorbing the trailing
        // white space.
        assert(in.good());
 
        // Look for closing bracket before trying to read value.
        if (lookForCloser) {
            // Eat white space to find the closing bracket.
            std::ws(in); if (!in.good()) break; // eof?
            ch = in.peek(); assert(ch != EOFch);
            if (!in.good()) break;
            c = (char)ch;
            if (c == closeBracket) {   
                in.get(); // absorb the closing bracket
                terminatorSeen = true; 
                break; 
            }
            // next char not a closing bracket; fall through
        }
 
        // We didn't look or didn't find a closing bracket. The istream is good 
        // but we might be looking at white space.
 
        // If we already got all the elements we want, break for final checks.
        if (isFixedSize && (nextIndex == numRequired))
            break; // that's a full count.
 
        // Look for comma before value, except the first time.
        if (commaOK && nextIndex != 0) {
            // Eat white space to find the comma.
            std::ws(in); if (!in.good()) break; // eof?
            ch = in.peek(); assert(ch != EOFch);
            if (!in.good()) break;
            c = (char)ch;
            if (c == ',') {
                in.get(); // absorb comma
                commaRequired = true; // all commas from now on
            } else { // next char not a comma
                if (commaRequired) // bad, e.g.: v1, v2, v3 v4 
                {   in.setstate(std::ios::failbit); break; }
                else commaOK = false; // saw: v1 v2 (no commas now)
            }
            if (!in.good()) break; // might be eof
        }
 
        // No closing bracket yet; don't have enough elements; skipped comma 
        // if any; istream is good; might be looking at white space.
        assert(in.good());
 
        // Now read in an element of type T.
        // The extractor T::operator>>() will ignore leading white space.
        if (!isFixedSize && nextIndex >= out.size())
        {
            // grow output vector geometrically (resized to actual size after loop)
            out.resizeKeep(out.size() == 0 ? 1 : 2*out.size());
        }
        in >> out[nextIndex]; if (in.fail()) break;
        ++nextIndex;
 
        if (!in.good()) break; // might be eof
    }
    out.resizeKeep(nextIndex);  // cut off trailing elements
 
    // We will get here under a number of circumstances:
    //  - the fail bit is set in the istream, or
    //  - we reached eof
    //  - we saw a closing brace
    //  - we got all the elements we wanted (for a fixed-size read)
    // Note that it is possible that we consumed everything except some
    // trailing white space (meaning we're not technically at eof), but
    // for consistency with built-in operator>>()'s we won't try to absorb
    // that trailing white space.
 
    if (!in.fail()) {
        if (lookForCloser && !terminatorSeen)
            in.setstate(std::ios::failbit); // missing terminator
 
        if (isFixedSize && nextIndex != numRequired)
            in.setstate(std::ios::failbit); // wrong number of values
    }
 
    return in;
}
 
 
 
//------------------------------------------------------------------------------
//                          RELATED GLOBAL OPERATORS
//------------------------------------------------------------------------------
// These are logically part of the Matrix_<T> class but are not actually 
// class members; that is, they are in the SimTK namespace.
 
template <class E> inline void
writeUnformatted(std::ostream& o, const VectorBase<E>& v) {
    const int sz = v.size();
    for (int i=0; i < sz; ++i) {
        if (i != 0) o << " ";
        writeUnformatted(o, v[i]);
    }
} 
template <class E> inline void
writeUnformatted(std::ostream& o, const VectorView_<E>& v) 
{   writeUnformatted(o, static_cast< const VectorBase<E> >(v)); }
 
template <class E> inline void
writeUnformatted(std::ostream& o, const Vector_<E>& v) 
{   writeUnformatted(o, static_cast< const VectorBase<E> >(v)); }
 
template <class E> inline void
writeUnformatted(std::ostream& o, const RowVectorBase<E>& v) 
{   writeUnformatted(o, ~v); }
 
template <class E> inline void
writeUnformatted(std::ostream& o, const RowVectorView_<E>& v) 
{   writeUnformatted(o, static_cast< const RowVectorBase<E> >(v)); }
 
template <class E> inline void
writeUnformatted(std::ostream& o, const RowVector_<E>& v) 
{   writeUnformatted(o, static_cast< const RowVectorBase<E> >(v)); }
 
template <class E> inline void
writeUnformatted(std::ostream& o, const MatrixBase<E>& v) {
    const int nr = v.nrow();
    for (int i=0; i < nr; ++i) {
        if (i != 0) o << std::endl;
        writeUnformatted(o, v[i]);
    }
}
template <class E> inline void
writeUnformatted(std::ostream& o, const MatrixView_<E>& v) 
{   writeUnformatted(o, static_cast< const MatrixBase<E> >(v)); }
 
template <class E> inline void
writeUnformatted(std::ostream& o, const Matrix_<E>& v) 
{   writeUnformatted(o, static_cast< const MatrixBase<E> >(v)); }
 
template <class E> inline bool
readUnformatted(std::istream& in, VectorView_<E>& v) {
    for (int i=0; i < v.size(); ++i)
        if (!readUnformatted(in, v[i])) return false;
    return true;
}
 
template <class E> inline bool
readUnformatted(std::istream& in, Vector_<E>& v) {
    if (!v.isResizeable())
        return readUnformatted(in, v.updAsVectorView());
 
    Array_<E,int> a;
    if (!readUnformatted(in,a)) return false;
    v.resize(a.size());
    for (int i=0; i<a.size(); ++i)
        v[i] = a[i];
    return true;
}
 
template <class E> inline bool
readUnformatted(std::istream& in, RowVectorView_<E>& v) 
{   VectorView_<E> vt(~v);
    return readUnformatted<E>(in, vt); }
 
template <class E> inline bool
readUnformatted(std::istream& in, RowVector_<E>& v) 
{   Vector_<E> vt(~v);
    return readUnformatted<E>(in, vt); }
 
template <class E> inline bool
readUnformatted(std::istream& in, MatrixView_<E>& v) { 
    for (int row=0; row < v.nrow(); ++row) {
        RowVectorView_<E> oneRow(v[row]);
        if (!readUnformatted<E>(in, oneRow)) return false;
    }
    return true;
}
 
template <class E> inline bool
fillUnformatted(std::istream& in, Matrix_<E>& v) {
    return readUnformatted<E>(in, v.updAsMatrixView());
}
 
template <class E> inline bool
readUnformatted(std::istream& in, Matrix_<E>& v) {
    SimTK_ASSERT_ALWAYS(!"implemented", 
        "SimTK::readUnformatted(istream, Matrix) is not implemented; try"
        " SimTK::fillUnformatted(istream, Matrix) instead.");
    return false;
}
  
template <class T> inline std::ostream&
operator<<(std::ostream& o, const VectorBase<T>& v)
{   o << "~["; 
    if (v.size()) {
        o << v[0];
        for (int i=1; i < v.size(); ++i) o << " " << v[i];
    }
    return o << "]"; 
}
 
template <class T> inline std::ostream&
operator<<(std::ostream& o, const RowVectorBase<T>& v)
{   o << "["; 
    if (v.size()) {
        o << v[0];
        for (int i=1; i < v.size(); ++i) o << " " << v[i];
    }
    return o << "]"; 
}
 
template <class T> inline std::ostream&
operator<<(std::ostream& o, const MatrixBase<T>& m) {
    for (int i=0;i<m.nrow();++i)
        o << std::endl << m[i];
    if (m.nrow()) o << std::endl;
    return o; 
}
 
 
template <class T> static inline 
std::istream& readVectorFromStream(std::istream& in, Vector_<T>& out)
{   return readVectorFromStreamHelper<T>(in, false /*variable sizez*/, out); }
 
 
 
template <class T> static inline 
std::istream& fillVectorFromStream(std::istream& in, Vector_<T>& out)
{   return readVectorFromStreamHelper<T>(in, true /*fixed size*/, out); }
 
template <class T> static inline 
std::istream& fillVectorViewFromStream(std::istream& in, VectorView_<T>& out)
{   return readVectorFromStreamHelper<T>(in, true /*fixed size*/, out); }
 
 
template <class T> inline
std::istream& operator>>(std::istream& in, Vector_<T>& out) 
{   return readVectorFromStream<T>(in, out); }
 
template <class T> inline
std::istream& operator>>(std::istream& in, VectorView_<T>& out) 
{   return fillVectorViewFromStream<T>(in, out); }  // End of Matrix serialization.
 
// Friendly abbreviations for vectors and matrices with scalar elements.
typedef Vector_<Real>           Vector;
 
typedef Matrix_<Real>           Matrix;
 
typedef RowVector_<Real>        RowVector;
typedef Vector_<Complex>        ComplexVector;
typedef Matrix_<Complex>        ComplexMatrix;
typedef RowVector_<Complex>     ComplexRowVector;
 
typedef VectorView_<Real>       VectorView;
typedef MatrixView_<Real>       MatrixView;
typedef RowVectorView_<Real>    RowVectorView;
 
typedef VectorView_<Complex>    ComplexVectorView;
typedef MatrixView_<Complex>    ComplexMatrixView;
typedef RowVectorView_<Complex> ComplexRowVectorView;
 
typedef Vector_<float>          fVector;
typedef Vector_<double>         dVector;
typedef Vector_<fComplex>       fComplexVector;
typedef Vector_<dComplex>       dComplexVector;
 
typedef VectorView_<float>      fVectorView;
typedef VectorView_<double>     dVectorView;
typedef VectorView_<fComplex>   fComplexVectorView;
typedef VectorView_<dComplex>   dComplexVectorView;
 
typedef RowVector_<float>       fRowVector;
typedef RowVector_<double>      dRowVector;
typedef RowVector_<fComplex>    fComplexRowVector;
typedef RowVector_<dComplex>    dComplexRowVector;
 
typedef RowVectorView_<float>   fRowVectorView;
typedef RowVectorView_<double>  dRowVectorView;
typedef RowVectorView_<fComplex> fComplexRowVectorView;
typedef RowVectorView_<dComplex> dComplexRowVectorView;
 
typedef Matrix_<float>          fMatrix;
typedef Matrix_<double>         dMatrix;
typedef Matrix_<fComplex>       fComplexMatrix;
typedef Matrix_<dComplex>       dComplexMatrix;
 
typedef MatrixView_<float>      fMatrixView;
typedef MatrixView_<double>     dMatrixView;
typedef MatrixView_<fComplex>   fComplexMatrixView;
typedef MatrixView_<dComplex>   dComplexMatrixView;
} //namespace SimTK
 
#endif //SimTK_SIMMATRIX_BIGMATRIX_H_