Row.h

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#ifndef SimTK_SIMMATRIX_SMALLMATRIX_ROW_H_
#define SimTK_SIMMATRIX_SMALLMATRIX_ROW_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) 2005-12 Stanford University and the Authors.        *
 * Authors: Michael Sherman                                                   *
 * Contributors: Peter Eastman                                                *
 *                                                                            *
 * 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/internal/common.h"
 
 
namespace SimTK {
 
// The following functions are used internally by Row.
 
namespace Impl {
 
// For those wimpy compilers that don't unroll short, constant-limit loops, 
// Peter Eastman added these recursive template implementations of 
// elementwise add, subtract, and copy. Sherm added multiply and divide.
 
template <class E1, int S1, class E2, int S2> void
conformingAdd(const Row<1,E1,S1>& r1, const Row<1,E2,S2>& r2, 
              Row<1,typename CNT<E1>::template Result<E2>::Add>& result) {
    result[0] = r1[0] + r2[0];
}
template <int N, class E1, int S1, class E2, int S2> void
conformingAdd(const Row<N,E1,S1>& r1, const Row<N,E2,S2>& r2, 
              Row<N,typename CNT<E1>::template Result<E2>::Add>& result) {
    conformingAdd(reinterpret_cast<const Row<N-1,E1,S1>&>(r1), 
                  reinterpret_cast<const Row<N-1,E2,S2>&>(r2), 
                  reinterpret_cast<Row<N-1,typename CNT<E1>::
                              template Result<E2>::Add>&>(result));
    result[N-1] = r1[N-1] + r2[N-1];
}
 
template <class E1, int S1, class E2, int S2> void
conformingSubtract(const Row<1,E1,S1>& r1, const Row<1,E2,S2>& r2, 
                   Row<1,typename CNT<E1>::template Result<E2>::Sub>& result) {
    result[0] = r1[0] - r2[0];
}
template <int N, class E1, int S1, class E2, int S2> void
conformingSubtract(const Row<N,E1,S1>& r1, const Row<N,E2,S2>& r2,
                   Row<N,typename CNT<E1>::template Result<E2>::Sub>& result) {
    conformingSubtract(reinterpret_cast<const Row<N-1,E1,S1>&>(r1), 
                       reinterpret_cast<const Row<N-1,E2,S2>&>(r2), 
                       reinterpret_cast<Row<N-1,typename CNT<E1>::
                                   template Result<E2>::Sub>&>(result));
    result[N-1] = r1[N-1] - r2[N-1];
}
 
template <class E1, int S1, class E2, int S2> void
elementwiseMultiply(const Row<1,E1,S1>& r1, const Row<1,E2,S2>& r2, 
              Row<1,typename CNT<E1>::template Result<E2>::Mul>& result) {
    result[0] = r1[0] * r2[0];
}
template <int N, class E1, int S1, class E2, int S2> void
elementwiseMultiply(const Row<N,E1,S1>& r1, const Row<N,E2,S2>& r2, 
              Row<N,typename CNT<E1>::template Result<E2>::Mul>& result) {
    elementwiseMultiply(reinterpret_cast<const Row<N-1,E1,S1>&>(r1), 
                        reinterpret_cast<const Row<N-1,E2,S2>&>(r2), 
                        reinterpret_cast<Row<N-1,typename CNT<E1>::
                                    template Result<E2>::Mul>&>(result));
    result[N-1] = r1[N-1] * r2[N-1];
}
 
template <class E1, int S1, class E2, int S2> void
elementwiseDivide(const Row<1,E1,S1>& r1, const Row<1,E2,S2>& r2, 
              Row<1,typename CNT<E1>::template Result<E2>::Dvd>& result) {
    result[0] = r1[0] / r2[0];
}
template <int N, class E1, int S1, class E2, int S2> void
elementwiseDivide(const Row<N,E1,S1>& r1, const Row<N,E2,S2>& r2, 
              Row<N,typename CNT<E1>::template Result<E2>::Dvd>& result) {
    elementwiseDivide(reinterpret_cast<const Row<N-1,E1,S1>&>(r1), 
                        reinterpret_cast<const Row<N-1,E2,S2>&>(r2), 
                        reinterpret_cast<Row<N-1,typename CNT<E1>::
                                    template Result<E2>::Dvd>&>(result));
    result[N-1] = r1[N-1] / r2[N-1];
}
 
template <class E1, int S1, class E2, int S2> void
copy(Row<1,E1,S1>& r1, const Row<1,E2,S2>& r2) {
    r1[0] = r2[0];
}
template <int N, class E1, int S1, class E2, int S2> void
copy(Row<N,E1,S1>& r1, const Row<N,E2,S2>& r2) {
    copy(reinterpret_cast<Row<N-1,E1,S1>&>(r1), 
         reinterpret_cast<const Row<N-1,E2,S2>&>(r2));
    r1[N-1] = r2[N-1];
}
 
}
 
template <int N, class ELT, int STRIDE> class Row {
    typedef ELT                                 E;
    typedef typename CNT<E>::TNeg               ENeg;
    typedef typename CNT<E>::TWithoutNegator    EWithoutNegator;
    typedef typename CNT<E>::TReal              EReal;
    typedef typename CNT<E>::TImag              EImag;
    typedef typename CNT<E>::TComplex           EComplex;
    typedef typename CNT<E>::THerm              EHerm;
    typedef typename CNT<E>::TPosTrans          EPosTrans;
    typedef typename CNT<E>::TSqHermT           ESqHermT;
    typedef typename CNT<E>::TSqTHerm           ESqTHerm;
 
    typedef typename CNT<E>::TSqrt              ESqrt;
    typedef typename CNT<E>::TAbs               EAbs;
    typedef typename CNT<E>::TStandard          EStandard;
    typedef typename CNT<E>::TInvert            EInvert;
    typedef typename CNT<E>::TNormalize         ENormalize;
 
    typedef typename CNT<E>::Scalar             EScalar;
    typedef typename CNT<E>::ULessScalar        EULessScalar;
    typedef typename CNT<E>::Number             ENumber;
    typedef typename CNT<E>::StdNumber          EStdNumber;
    typedef typename CNT<E>::Precision          EPrecision;
    typedef typename CNT<E>::ScalarNormSq       EScalarNormSq;
 
public:
 
    enum {
        NRows               = 1,
        NCols               = N,
        NPackedElements     = N,
        NActualElements     = N * STRIDE,   // includes trailing gap
        NActualScalars      = CNT<E>::NActualScalars * NActualElements,
        RowSpacing          = NActualElements,
        ColSpacing          = STRIDE,
        ImagOffset          = NTraits<ENumber>::ImagOffset,
        RealStrideFactor    = 1, // composite types don't change size when
                                 // cast from complex to real or imaginary
        ArgDepth            = ((int)CNT<E>::ArgDepth < (int)MAX_RESOLVED_DEPTH 
                                ? CNT<E>::ArgDepth + 1 
                                : MAX_RESOLVED_DEPTH),
        IsScalar            = 0,
        IsULessScalar       = 0,
        IsNumber            = 0,
        IsStdNumber         = 0,
        IsPrecision         = 0,
        SignInterpretation  = CNT<E>::SignInterpretation
    };
 
    typedef Row<N,E,STRIDE>                 T;
    typedef Row<N,ENeg,STRIDE>              TNeg;
    typedef Row<N,EWithoutNegator,STRIDE>   TWithoutNegator;
 
    typedef Row<N,EReal,STRIDE*CNT<E>::RealStrideFactor>         
                                            TReal;
    typedef Row<N,EImag,STRIDE*CNT<E>::RealStrideFactor>         
                                            TImag;
    typedef Row<N,EComplex,STRIDE>          TComplex;
    typedef Vec<N,EHerm,STRIDE>             THerm;
    typedef Vec<N,E,STRIDE>                 TPosTrans;
    typedef E                               TElement;
    typedef Row                             TRow;
    typedef E                               TCol;
 
    // These are the results of calculations, so are returned in new, packed
    // memory. Be sure to refer to element types here which are also packed.
    typedef Vec<N,ESqrt,1>                  TSqrt;      // Note stride
    typedef Row<N,EAbs,1>                   TAbs;       // Note stride
    typedef Row<N,EStandard,1>              TStandard;
    typedef Vec<N,EInvert,1>                TInvert;    // packed
    typedef Row<N,ENormalize,1>             TNormalize;
 
    typedef SymMat<N,ESqHermT>              TSqHermT;   // result of self outer product
    typedef EScalarNormSq                   TSqTHerm;   // result of self dot product
 
    // These recurse right down to the underlying scalar type no matter how
    // deep the elements are.
    typedef EScalar                         Scalar;
    typedef EULessScalar                    ULessScalar;
    typedef ENumber                         Number;
    typedef EStdNumber                      StdNumber;
    typedef EPrecision                      Precision;
    typedef EScalarNormSq                   ScalarNormSq;
 
    static int size() { return N; }
    static int nrow() { return 1; }
    static int ncol() { return N; }
 
 
    // Scalar norm square is sum( conjugate squares of all scalars )
    ScalarNormSq scalarNormSqr() const { 
        ScalarNormSq sum(0);
        for(int i=0;i<N;++i) sum += CNT<E>::scalarNormSqr(d[i*STRIDE]);
        return sum;
    }
 
    // sqrt() is elementwise square root; that is, the return value has the same
    // dimension as this Vec but with each element replaced by whatever it thinks
    // its square root is.
    TSqrt sqrt() const {
        TSqrt rsqrt;
        for(int i=0;i<N;++i) rsqrt[i] = CNT<E>::sqrt(d[i*STRIDE]);
        return rsqrt;
    }
 
    // abs() is elementwise absolute value; that is, the return value has the same
    // dimension as this Row but with each element replaced by whatever it thinks
    // its absolute value is.
    TAbs abs() const {
        TAbs rabs;
        for(int i=0;i<N;++i) rabs[i] = CNT<E>::abs(d[i*STRIDE]);
        return rabs;
    }
 
    TStandard standardize() const {
        TStandard rstd;
        for(int i=0;i<N;++i) rstd[i] = CNT<E>::standardize(d[i*STRIDE]);
        return rstd;
    }
 
    // Sum just adds up all the elements, getting rid of negators and
    // conjugates in the process.
    EStandard sum() const {
        E sum(0);
        for (int i=0;i<N;++i) sum += d[i*STRIDE];
        return CNT<E>::standardize(sum);
    }
 
    // This gives the resulting rowvector type when (v[i] op P) is applied to each element of v.
    // It is a row of length N, stride 1, and element types which are the regular
    // composite result of E op P. Typically P is a scalar type but it doesn't have to be.
    template <class P> struct EltResult { 
        typedef Row<N, typename CNT<E>::template Result<P>::Mul, 1> Mul;
        typedef Row<N, typename CNT<E>::template Result<P>::Dvd, 1> Dvd;
        typedef Row<N, typename CNT<E>::template Result<P>::Add, 1> Add;
        typedef Row<N, typename CNT<E>::template Result<P>::Sub, 1> Sub;
    };
 
    // This is the composite result for v op P where P is some kind of appropriately shaped
    // non-scalar type.
    template <class P> struct Result { 
        typedef MulCNTs<1,N,ArgDepth,Row,ColSpacing,RowSpacing,
            CNT<P>::NRows, CNT<P>::NCols, CNT<P>::ArgDepth,
            P, CNT<P>::ColSpacing, CNT<P>::RowSpacing> MulOp;
        typedef typename MulOp::Type Mul;
 
        typedef MulCNTsNonConforming<1,N,ArgDepth,Row,ColSpacing,RowSpacing,
            CNT<P>::NRows, CNT<P>::NCols, CNT<P>::ArgDepth,
            P, CNT<P>::ColSpacing, CNT<P>::RowSpacing> MulOpNonConforming;
        typedef typename MulOpNonConforming::Type MulNon;
 
 
        typedef DvdCNTs<1,N,ArgDepth,Row,ColSpacing,RowSpacing,
            CNT<P>::NRows, CNT<P>::NCols, CNT<P>::ArgDepth,
            P, CNT<P>::ColSpacing, CNT<P>::RowSpacing> DvdOp;
        typedef typename DvdOp::Type Dvd;
 
        typedef AddCNTs<1,N,ArgDepth,Row,ColSpacing,RowSpacing,
            CNT<P>::NRows, CNT<P>::NCols, CNT<P>::ArgDepth,
            P, CNT<P>::ColSpacing, CNT<P>::RowSpacing> AddOp;
        typedef typename AddOp::Type Add;
 
        typedef SubCNTs<1,N,ArgDepth,Row,ColSpacing,RowSpacing,
            CNT<P>::NRows, CNT<P>::NCols, CNT<P>::ArgDepth,
            P, CNT<P>::ColSpacing, CNT<P>::RowSpacing> SubOp;
        typedef typename SubOp::Type Sub;
    };
 
    // Shape-preserving element substitution (always packed)
    template <class P> struct Substitute {
        typedef Row<N,P> Type;
    };
 
    // Default construction initializes to NaN when debugging but
    // is left uninitialized otherwise.
    Row(){ 
    #ifndef NDEBUG
        setToNaN();
    #endif
    }
 
    // It's important not to use the default copy constructor or copy
    // assignment because the compiler doesn't understand that we may
    // have noncontiguous storage and will try to copy the whole array.
    Row(const Row& src) {
        Impl::copy(*this, src);
    }
    Row& operator=(const Row& src) {    // no harm if src and 'this' are the same
        Impl::copy(*this, src);
        return *this;
    }
 
    // We want an implicit conversion from a Row of the same length
    // and element type but with a different stride.
    template <int SS> Row(const Row<N,E,SS>& src) {
        Impl::copy(*this, src);
    }
 
    // We want an implicit conversion from a Row of the same length
    // and *negated* element type, possibly with a different stride.
    template <int SS> Row(const Row<N,ENeg,SS>& src) {
        Impl::copy(*this, src);
    }
 
    // Construct a Row from a Row of the same length, with any
    // stride. Works as long as the element types are compatible.
    template <class EE, int SS> explicit Row(const Row<N,EE,SS>& vv) {
        Impl::copy(*this, vv);
    }
 
    // Construction using an element assigns to each element.
    explicit Row(const E& e)
      { for (int i=0;i<N;++i) d[i*STRIDE]=e; }
 
    // Construction using a negated element assigns to each element.
    explicit Row(const ENeg& ne)
      { for (int i=0;i<N;++i) d[i*STRIDE]=ne; }
 
    // Given an int, turn it into a suitable floating point number
    // and then feed that to the above single-element constructor.
    explicit Row(int i) 
      { new (this) Row(E(Precision(i))); }
 
    // A bevy of constructors for Rows up to length 6.
    Row(const E& e0,const E& e1)
      { assert(N==2);(*this)[0]=e0;(*this)[1]=e1; }
    Row(const E& e0,const E& e1,const E& e2)
      { assert(N==3);(*this)[0]=e0;(*this)[1]=e1;(*this)[2]=e2; }
    Row(const E& e0,const E& e1,const E& e2,const E& e3)
      { assert(N==4);(*this)[0]=e0;(*this)[1]=e1;(*this)[2]=e2;(*this)[3]=e3; }
    Row(const E& e0,const E& e1,const E& e2,const E& e3,const E& e4)
      { assert(N==5);(*this)[0]=e0;(*this)[1]=e1;(*this)[2]=e2;
        (*this)[3]=e3;(*this)[4]=e4; }
    Row(const E& e0,const E& e1,const E& e2,const E& e3,const E& e4,const E& e5)
      { assert(N==6);(*this)[0]=e0;(*this)[1]=e1;(*this)[2]=e2;
        (*this)[3]=e3;(*this)[4]=e4;(*this)[5]=e5; }
    Row(const E& e0,const E& e1,const E& e2,const E& e3,const E& e4,const E& e5,const E& e6)
      { assert(N==7);(*this)[0]=e0;(*this)[1]=e1;(*this)[2]=e2;
        (*this)[3]=e3;(*this)[4]=e4;(*this)[5]=e5;(*this)[6]=e6; }
    Row(const E& e0,const E& e1,const E& e2,const E& e3,const E& e4,const E& e5,const E& e6,const E& e7)
      { assert(N==8);(*this)[0]=e0;(*this)[1]=e1;(*this)[2]=e2;
        (*this)[3]=e3;(*this)[4]=e4;(*this)[5]=e5;(*this)[6]=e6;(*this)[7]=e7; }
    Row(const E& e0,const E& e1,const E& e2,const E& e3,const E& e4,const E& e5,const E& e6,const E& e7,const E& e8)
      { assert(N==9);(*this)[0]=e0;(*this)[1]=e1;(*this)[2]=e2;
        (*this)[3]=e3;(*this)[4]=e4;(*this)[5]=e5;(*this)[6]=e6;(*this)[7]=e7;(*this)[8]=e8; }
 
    // Construction from a pointer to anything assumes we're pointing
    // at an element list of the right length.
    template <class EE> explicit Row(const EE* p)
      { assert(p); for(int i=0;i<N;++i) d[i*STRIDE]=p[i]; }
    template <class EE> Row& operator=(const EE* p)
      { assert(p); for(int i=0;i<N;++i) d[i*STRIDE]=p[i]; return *this; }
 
    // Conforming assignment ops.
    template <class EE, int SS> Row& operator=(const Row<N,EE,SS>& vv) {
        Impl::copy(*this, vv);
        return *this;
    }
    template <class EE, int SS> Row& operator+=(const Row<N,EE,SS>& r)
      { for(int i=0;i<N;++i) d[i*STRIDE] += r[i]; return *this; }
    template <class EE, int SS> Row& operator+=(const Row<N,negator<EE>,SS>& r)
      { for(int i=0;i<N;++i) d[i*STRIDE] -= -(r[i]); return *this; }
    template <class EE, int SS> Row& operator-=(const Row<N,EE,SS>& r)
      { for(int i=0;i<N;++i) d[i*STRIDE] -= r[i]; return *this; }
    template <class EE, int SS> Row& operator-=(const Row<N,negator<EE>,SS>& r)
      { for(int i=0;i<N;++i) d[i*STRIDE] += -(r[i]); return *this; }
 
    // Conforming binary ops with 'this' on left, producing new packed result.
    // Cases: r=r+r, r=r-r, s=r*v r=r*m
 
    template <class EE, int SS> Row<N,typename CNT<E>::template Result<EE>::Add>
    conformingAdd(const Row<N,EE,SS>& r) const {
        Row<N,typename CNT<E>::template Result<EE>::Add> result;
        Impl::conformingAdd(*this, r, result);
        return result;
    }
 
    template <class EE, int SS> Row<N,typename CNT<E>::template Result<EE>::Sub>
    conformingSubtract(const Row<N,EE,SS>& r) const {
        Row<N,typename CNT<E>::template Result<EE>::Sub> result;
        Impl::conformingSubtract(*this, r, result);
        return result;
    }
 
    template <class EE, int SS> typename CNT<E>::template Result<EE>::Mul
    conformingMultiply(const Vec<N,EE,SS>& r) const {
        return (*this)*r;
    }
 
    template <int MatNCol, class EE, int CS, int RS> 
    Row<MatNCol,typename CNT<E>::template Result<EE>::Mul>
    conformingMultiply(const Mat<N,MatNCol,EE,CS,RS>& m) const {
        Row<MatNCol,typename CNT<E>::template Result<EE>::Mul> result;
        for (int j=0;j<N;++j) result[j] = conformingMultiply(m(j));
        return result;
    }
 
    template <class EE, int SS> Row<N,typename CNT<E>::template Result<EE>::Mul>
    elementwiseMultiply(const Row<N,EE,SS>& r) const {
        Row<N,typename CNT<E>::template Result<EE>::Mul> result;
        Impl::elementwiseMultiply(*this, r, result);
        return result;
    }
 
    template <class EE, int SS> Row<N,typename CNT<E>::template Result<EE>::Dvd>
    elementwiseDivide(const Row<N,EE,SS>& r) const {
        Row<N,typename CNT<E>::template Result<EE>::Dvd> result;
        Impl::elementwiseDivide(*this, r, result);
        return result;
    }
 
    const E& operator[](int i) const { assert(0 <= i && i < N); return d[i*STRIDE]; }
    E&       operator[](int i)         { assert(0 <= i && i < N); return d[i*STRIDE]; }
    const E& operator()(int i) const { return (*this)[i]; }
    E&       operator()(int i)         { return (*this)[i]; }
 
    ScalarNormSq normSqr() const { return scalarNormSqr(); }
    typename CNT<ScalarNormSq>::TSqrt 
        norm() const { return CNT<ScalarNormSq>::sqrt(scalarNormSqr()); }
 
    // If the elements of this Row are scalars, the result is what you get by
    // dividing each element by the norm() calculated above. If the elements are
    // *not* scalars, then the elements are *separately* normalized. That means
    // you will get a different answer from Row<2,Row3>::normalize() than you
    // would from a Row<6>::normalize() containing the same scalars.
    //
    // Normalize returns a row of the same dimension but in new, packed storage
    // and with a return type that does not include negator<> even if the original
    // Row<> does, because we can eliminate the negation here almost for free.
    // But we can't standardize (change conjugate to complex) for free, so we'll retain
    // conjugates if there are any.
    TNormalize normalize() const {
        if (CNT<E>::IsScalar) {
            return castAwayNegatorIfAny() / (int(SignInterpretation)*norm());
        } else {
            TNormalize elementwiseNormalized;
            for (int j=0; j<N; ++j) 
                elementwiseNormalized[j] = CNT<E>::normalize((*this)[j]);
            return elementwiseNormalized;
        }
    }
 
    TInvert invert() const {assert(false); return TInvert();} // TODO default inversion
 
    const Row&   operator+() const { return *this; }
    const TNeg&  operator-() const { return negate(); }
    TNeg&        operator-()       { return updNegate(); }
    const THerm& operator~() const { return transpose(); }
    THerm&       operator~()       { return updTranspose(); }
 
    const TNeg&  negate() const { return *reinterpret_cast<const TNeg*>(this); }
    TNeg&        updNegate()    { return *reinterpret_cast<TNeg*>(this); }
 
    const THerm& transpose()    const { return *reinterpret_cast<const THerm*>(this); }
    THerm&       updTranspose()       { return *reinterpret_cast<THerm*>(this); }
 
    const TPosTrans& positionalTranspose() const
        { return *reinterpret_cast<const TPosTrans*>(this); }
    TPosTrans&       updPositionalTranspose()
        { return *reinterpret_cast<TPosTrans*>(this); }
 
    const TReal& real() const { return *reinterpret_cast<const TReal*>(this); }
    TReal&       real()       { return *reinterpret_cast<      TReal*>(this); }
 
    // Had to contort these routines to get them through VC++ 7.net
    const TImag& imag()    const { 
        const int offs = ImagOffset;
        const EImag* p = reinterpret_cast<const EImag*>(this);
        return *reinterpret_cast<const TImag*>(p+offs);
    }
    TImag& imag() { 
        const int offs = ImagOffset;
        EImag* p = reinterpret_cast<EImag*>(this);
        return *reinterpret_cast<TImag*>(p+offs);
    }
 
    const TWithoutNegator& castAwayNegatorIfAny() const {return *reinterpret_cast<const TWithoutNegator*>(this);}
    TWithoutNegator&       updCastAwayNegatorIfAny()    {return *reinterpret_cast<TWithoutNegator*>(this);}
 
 
    // These are elementwise binary operators, (this op ee) by default but 
    // (ee op this) if 'FromLeft' appears in the name. The result is a packed 
    // Row<N> but the element type may change. These are mostly used to 
    // implement global operators. We call these "scalar" operators but 
    // actually the "scalar" can be a composite type.
 
    //TODO: consider converting 'e' to Standard Numbers as precalculation and 
    // changing return type appropriately.
    template <class EE> Row<N, typename CNT<E>::template Result<EE>::Mul>
    scalarMultiply(const EE& e) const {
        Row<N, typename CNT<E>::template Result<EE>::Mul> result;
        for (int j=0; j<N; ++j) result[j] = (*this)[j] * e;
        return result;
    }
    template <class EE> Row<N, typename CNT<EE>::template Result<E>::Mul>
    scalarMultiplyFromLeft(const EE& e) const {
        Row<N, typename CNT<EE>::template Result<E>::Mul> result;
        for (int j=0; j<N; ++j) result[j] = e * (*this)[j];
        return result;
    }
 
    // TODO: should precalculate and store 1/e, while converting to Standard 
    // Numbers. Note that return type should change appropriately.
    template <class EE> Row<N, typename CNT<E>::template Result<EE>::Dvd>
    scalarDivide(const EE& e) const {
        Row<N, typename CNT<E>::template Result<EE>::Dvd> result;
        for (int j=0; j<N; ++j) result[j] = (*this)[j] / e;
        return result;
    }
    template <class EE> Row<N, typename CNT<EE>::template Result<E>::Dvd>
    scalarDivideFromLeft(const EE& e) const {
        Row<N, typename CNT<EE>::template Result<E>::Dvd> result;
        for (int j=0; j<N; ++j) result[j] = e / (*this)[j];
        return result;
    }
 
    template <class EE> Row<N, typename CNT<E>::template Result<EE>::Add>
    scalarAdd(const EE& e) const {
        Row<N, typename CNT<E>::template Result<EE>::Add> result;
        for (int j=0; j<N; ++j) result[j] = (*this)[j] + e;
        return result;
    }
    // Add is commutative, so no 'FromLeft'.
 
    template <class EE> Row<N, typename CNT<E>::template Result<EE>::Sub>
    scalarSubtract(const EE& e) const {
        Row<N, typename CNT<E>::template Result<EE>::Sub> result;
        for (int j=0; j<N; ++j) result[j] = (*this)[j] - e;
        return result;
    }
    template <class EE> Row<N, typename CNT<EE>::template Result<E>::Sub>
    scalarSubtractFromLeft(const EE& e) const {
        Row<N, typename CNT<EE>::template Result<E>::Sub> result;
        for (int j=0; j<N; ++j) result[j] = e - (*this)[j];
        return result;
    }
 
    // Generic assignments for any element type not listed explicitly, including scalars.
    // These are done repeatedly for each element and only work if the operation can
    // be performed leaving the original element type.
    template <class EE> Row& operator =(const EE& e) {return scalarEq(e);}
    template <class EE> Row& operator+=(const EE& e) {return scalarPlusEq(e);}
    template <class EE> Row& operator-=(const EE& e) {return scalarMinusEq(e);}
    template <class EE> Row& operator*=(const EE& e) {return scalarTimesEq(e);}
    template <class EE> Row& operator/=(const EE& e) {return scalarDivideEq(e);}
 
    // Generalized scalar assignment & computed assignment methods. These will work
    // for any assignment-compatible element, not just scalars.
    template <class EE> Row& scalarEq(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] = ee; return *this; }
    template <class EE> Row& scalarPlusEq(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] += ee; return *this; }
    template <class EE> Row& scalarMinusEq(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] -= ee; return *this; }
    template <class EE> Row& scalarMinusEqFromLeft(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] = ee - d[i*STRIDE]; return *this; }
    template <class EE> Row& scalarTimesEq(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] *= ee; return *this; }
    template <class EE> Row& scalarTimesEqFromLeft(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] = ee * d[i*STRIDE]; return *this; }
    template <class EE> Row& scalarDivideEq(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] /= ee; return *this; }
    template <class EE> Row& scalarDivideEqFromLeft(const EE& ee)
      { for(int i=0;i<N;++i) d[i*STRIDE] = ee / d[i*STRIDE]; return *this; }
 
 
    // Specialize for int to avoid warnings and ambiguities.
    Row& scalarEq(int ee)       {return scalarEq(Precision(ee));}
    Row& scalarPlusEq(int ee)   {return scalarPlusEq(Precision(ee));}
    Row& scalarMinusEq(int ee)  {return scalarMinusEq(Precision(ee));}
    Row& scalarTimesEq(int ee)  {return scalarTimesEq(Precision(ee));}
    Row& scalarDivideEq(int ee) {return scalarDivideEq(Precision(ee));}
    Row& scalarMinusEqFromLeft(int ee)  {return scalarMinusEqFromLeft(Precision(ee));}
    Row& scalarTimesEqFromLeft(int ee)  {return scalarTimesEqFromLeft(Precision(ee));}
    Row& scalarDivideEqFromLeft(int ee) {return scalarDivideEqFromLeft(Precision(ee));}
 
    void setToNaN() {
        (*this) = CNT<ELT>::getNaN();
    }
 
    void setToZero() {
        (*this) = ELT(0);
    }
 
    template <int NN>
    const Row<NN,ELT,STRIDE>& getSubRow(int j) const {
        assert(0 <= j && j + NN <= N);
        return Row<NN,ELT,STRIDE>::getAs(&(*this)[j]);
    }
    template <int NN>
    Row<NN,ELT,STRIDE>& updSubRow(int j) {
        assert(0 <= j && j + NN <= N);
        return Row<NN,ELT,STRIDE>::updAs(&(*this)[j]);
    }
 
    template <int NN>
    static const Row& getSubRow(const Row<NN,ELT,STRIDE>& r, int j) {
        assert(0 <= j && j + N <= NN);
        return getAs(&r[j]);
    }
    template <int NN>
    static Row& updSubRow(Row<NN,ELT,STRIDE>& r, int j) {
        assert(0 <= j && j + N <= NN);
        return updAs(&r[j]);
    }
 
    Row<N-1,ELT,1> drop1(int p) const {
        assert(0 <= p && p < N);
        Row<N-1,ELT,1> out;
        int nxt=0;
        for (int i=0; i<N-1; ++i, ++nxt) {
            if (nxt==p) ++nxt;  // skip the loser
            out[i] = (*this)[nxt];
        }
        return out;
    }
 
    template <class EE> Row<N+1,ELT,1> append1(const EE& v) const {
        Row<N+1,ELT,1> out;
        Row<N,ELT,1>::updAs(&out[0]) = (*this);
        out[N] = v;
        return out;
    }
 
 
    template <class EE> Row<N+1,ELT,1> insert1(int p, const EE& v) const {
        assert(0 <= p && p <= N);
        if (p==N) return append1(v);
        Row<N+1,ELT,1> out;
        int nxt=0;
        for (int i=0; i<N; ++i, ++nxt) {
            if (i==p) out[nxt++] = v;
            out[nxt] = (*this)[i];
        }
        return out;
    }
 
    static const Row& getAs(const ELT* p)  {return *reinterpret_cast<const Row*>(p);}
    static Row&       updAs(ELT* p)        {return *reinterpret_cast<Row*>(p);}
 
    static Row<N,ELT,1> getNaN() { return Row<N,ELT,1>(CNT<ELT>::getNaN()); }
 
    bool isNaN() const {
        for (int j=0; j<N; ++j)
            if (CNT<ELT>::isNaN((*this)[j]))
                return true;
        return false;
    }
 
    bool isInf() const {
        bool seenInf = false;
        for (int j=0; j<N; ++j) {
            const ELT& e = (*this)[j];
            if (!CNT<ELT>::isFinite(e)) {
                if (!CNT<ELT>::isInf(e)) 
                    return false; // something bad was found
                seenInf = true; 
            }
        }
        return seenInf;
    }
 
    bool isFinite() const {
        for (int j=0; j<N; ++j)
            if (!CNT<ELT>::isFinite((*this)[j]))
                return false;
        return true;
    }
 
    static double getDefaultTolerance() {return CNT<ELT>::getDefaultTolerance();}
 
    template <class E2, int CS2>
    bool isNumericallyEqual(const Row<N,E2,CS2>& r, double tol) const {
        for (int j=0; j<N; ++j)
            if (!CNT<ELT>::isNumericallyEqual((*this)(j), r(j), tol))
                return false;
        return true;
    }
 
    template <class E2, int CS2>
    bool isNumericallyEqual(const Row<N,E2,CS2>& r) const {
        const double tol = std::max(getDefaultTolerance(),r.getDefaultTolerance());
        return isNumericallyEqual(r, tol);
    }
 
    bool isNumericallyEqual
       (const ELT& e,
        double     tol = getDefaultTolerance()) const 
    {
        for (int j=0; j<N; ++j)
            if (!CNT<ELT>::isNumericallyEqual((*this)(j), e, tol))
                return false;
        return true;
    }
private:
    ELT d[NActualElements];    // data
};
 
// Global operators involving two rows.    //
//   v+v, v-v, v==v, v!=v                  //
 
// v3 = v1 + v2 where all v's have the same length N. 
template <int N, class E1, int S1, class E2, int S2> inline
typename Row<N,E1,S1>::template Result< Row<N,E2,S2> >::Add
operator+(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) { 
    return Row<N,E1,S1>::template Result< Row<N,E2,S2> >
        ::AddOp::perform(l,r);
}
 
// v3 = v1 - v2, similar to +
template <int N, class E1, int S1, class E2, int S2> inline
typename Row<N,E1,S1>::template Result< Row<N,E2,S2> >::Sub
operator-(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) { 
    return Row<N,E1,S1>::template Result< Row<N,E2,S2> >
        ::SubOp::perform(l,r);
}
 
template <int N, class E1, int S1, class E2, int S2> inline bool
operator==(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) { 
    for (int i=0; i < N; ++i) if (l[i] != r[i]) return false;
    return true;
}
template <int N, class E1, int S1, class E2, int S2> inline bool
operator!=(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) {return !(l==r);} 
 
template <int N, class E1, int S1, class E2, int S2> inline bool
operator<(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) 
{   for (int i=0; i < N; ++i) if (l[i] >= r[i]) return false;
    return true; }
template <int N, class E1, int S1, class E2> inline bool
operator<(const Row<N,E1,S1>& v, const E2& e) 
{   for (int i=0; i < N; ++i) if (v[i] >= e) return false;
    return true; }
 
template <int N, class E1, int S1, class E2, int S2> inline bool
operator>(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) 
{   for (int i=0; i < N; ++i) if (l[i] <= r[i]) return false;
    return true; }
template <int N, class E1, int S1, class E2> inline bool
operator>(const Row<N,E1,S1>& v, const E2& e) 
{   for (int i=0; i < N; ++i) if (v[i] <= e) return false;
    return true; }
 
template <int N, class E1, int S1, class E2, int S2> inline bool
operator<=(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) 
{   for (int i=0; i < N; ++i) if (l[i] > r[i]) return false;
    return true; }
template <int N, class E1, int S1, class E2> inline bool
operator<=(const Row<N,E1,S1>& v, const E2& e) 
{   for (int i=0; i < N; ++i) if (v[i] > e) return false;
    return true; }
 
template <int N, class E1, int S1, class E2, int S2> inline bool
operator>=(const Row<N,E1,S1>& l, const Row<N,E2,S2>& r) 
{   for (int i=0; i < N; ++i) if (l[i] < r[i]) return false;
    return true; }
template <int N, class E1, int S1, class E2> inline bool
operator>=(const Row<N,E1,S1>& v, const E2& e) 
{   for (int i=0; i < N; ++i) if (v[i] < e) return false;
    return true; }
 
// Global operators involving a row and a scalar. //
 
// I haven't been able to figure out a nice way to templatize for the
// built-in reals without introducing a lot of unwanted type matches
// as well. So we'll just grind them out explicitly here.
 
// SCALAR MULTIPLY
 
// v = v*real, real*v 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<float>::Mul
operator*(const Row<N,E,S>& l, const float& r)
  { return Row<N,E,S>::template Result<float>::MulOp::perform(l,r); }
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<float>::Mul
operator*(const float& l, const Row<N,E,S>& r) {return r*l;}
 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<double>::Mul
operator*(const Row<N,E,S>& l, const double& r)
  { return Row<N,E,S>::template Result<double>::MulOp::perform(l,r); }
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<double>::Mul
operator*(const double& l, const Row<N,E,S>& r) {return r*l;}
 
// v = v*int, int*v -- just convert int to v's precision float
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<typename CNT<E>::Precision>::Mul
operator*(const Row<N,E,S>& l, int r) {return l * (typename CNT<E>::Precision)r;}
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<typename CNT<E>::Precision>::Mul
operator*(int l, const Row<N,E,S>& r) {return r * (typename CNT<E>::Precision)l;}
 
// Complex, conjugate, and negator are all easy to templatize.
 
// v = v*complex, complex*v
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Mul
operator*(const Row<N,E,S>& l, const std::complex<R>& r)
  { return Row<N,E,S>::template Result<std::complex<R> >::MulOp::perform(l,r); }
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Mul
operator*(const std::complex<R>& l, const Row<N,E,S>& r) {return r*l;}
 
// v = v*conjugate, conjugate*v (convert conjugate->complex)
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Mul
operator*(const Row<N,E,S>& l, const conjugate<R>& r) {return l*(std::complex<R>)r;}
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Mul
operator*(const conjugate<R>& l, const Row<N,E,S>& r) {return r*(std::complex<R>)l;}
 
// v = v*negator, negator*v: convert negator to standard number
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<typename negator<R>::StdNumber>::Mul
operator*(const Row<N,E,S>& l, const negator<R>& r) {return l * (typename negator<R>::StdNumber)(R)r;}
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<typename negator<R>::StdNumber>::Mul
operator*(const negator<R>& l, const Row<N,E,S>& r) {return r * (typename negator<R>::StdNumber)(R)l;}
 
 
// SCALAR DIVIDE. This is a scalar operation when the scalar is on the right,
// but when it is on the left it means scalar * pseudoInverse(row), which is 
// a vec.
 
// v = v/real, real/v 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<float>::Dvd
operator/(const Row<N,E,S>& l, const float& r)
  { return Row<N,E,S>::template Result<float>::DvdOp::perform(l,r); }
template <int N, class E, int S> inline
typename CNT<float>::template Result<Row<N,E,S> >::Dvd
operator/(const float& l, const Row<N,E,S>& r)
  { return CNT<float>::template Result<Row<N,E,S> >::DvdOp::perform(l,r); }
 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<double>::Dvd
operator/(const Row<N,E,S>& l, const double& r)
  { return Row<N,E,S>::template Result<double>::DvdOp::perform(l,r); }
template <int N, class E, int S> inline
typename CNT<double>::template Result<Row<N,E,S> >::Dvd
operator/(const double& l, const Row<N,E,S>& r)
  { return CNT<double>::template Result<Row<N,E,S> >::DvdOp::perform(l,r); }
 
// v = v/int, int/v -- just convert int to v's precision float
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<typename CNT<E>::Precision>::Dvd
operator/(const Row<N,E,S>& l, int r) {return l / (typename CNT<E>::Precision)r;}
template <int N, class E, int S> inline
typename CNT<typename CNT<E>::Precision>::template Result<Row<N,E,S> >::Dvd
operator/(int l, const Row<N,E,S>& r) {return (typename CNT<E>::Precision)l / r;}
 
 
// Complex, conjugate, and negator are all easy to templatize.
 
// v = v/complex, complex/v
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Dvd
operator/(const Row<N,E,S>& l, const std::complex<R>& r)
  { return Row<N,E,S>::template Result<std::complex<R> >::DvdOp::perform(l,r); }
template <int N, class E, int S, class R> inline
typename CNT<std::complex<R> >::template Result<Row<N,E,S> >::Dvd
operator/(const std::complex<R>& l, const Row<N,E,S>& r)
  { return CNT<std::complex<R> >::template Result<Row<N,E,S> >::DvdOp::perform(l,r); }
 
// v = v/conjugate, conjugate/v (convert conjugate->complex)
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Dvd
operator/(const Row<N,E,S>& l, const conjugate<R>& r) {return l/(std::complex<R>)r;}
template <int N, class E, int S, class R> inline
typename CNT<std::complex<R> >::template Result<Row<N,E,S> >::Dvd
operator/(const conjugate<R>& l, const Row<N,E,S>& r) {return (std::complex<R>)l/r;}
 
// v = v/negator, negator/v: convert negator to number
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<typename negator<R>::StdNumber>::Dvd
operator/(const Row<N,E,S>& l, const negator<R>& r) {return l/(typename negator<R>::StdNumber)(R)r;}
template <int N, class E, int S, class R> inline
typename CNT<R>::template Result<Row<N,E,S> >::Dvd
operator/(const negator<R>& l, const Row<N,E,S>& r) {return (typename negator<R>::StdNumber)(R)l/r;}
 
 
// Add and subtract are odd as scalar ops. They behave as though the
// scalar stands for a vector each of whose elements is that scalar,
// and then a normal vector add or subtract is done.
 
// SCALAR ADD
 
// v = v+real, real+v 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<float>::Add
operator+(const Row<N,E,S>& l, const float& r)
  { return Row<N,E,S>::template Result<float>::AddOp::perform(l,r); }
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<float>::Add
operator+(const float& l, const Row<N,E,S>& r) {return r+l;}
 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<double>::Add
operator+(const Row<N,E,S>& l, const double& r)
  { return Row<N,E,S>::template Result<double>::AddOp::perform(l,r); }
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<double>::Add
operator+(const double& l, const Row<N,E,S>& r) {return r+l;}
 
// v = v+int, int+v -- just convert int to v's precision float
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<typename CNT<E>::Precision>::Add
operator+(const Row<N,E,S>& l, int r) {return l + (typename CNT<E>::Precision)r;}
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<typename CNT<E>::Precision>::Add
operator+(int l, const Row<N,E,S>& r) {return r + (typename CNT<E>::Precision)l;}
 
// Complex, conjugate, and negator are all easy to templatize.
 
// v = v+complex, complex+v
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Add
operator+(const Row<N,E,S>& l, const std::complex<R>& r)
  { return Row<N,E,S>::template Result<std::complex<R> >::AddOp::perform(l,r); }
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Add
operator+(const std::complex<R>& l, const Row<N,E,S>& r) {return r+l;}
 
// v = v+conjugate, conjugate+v (convert conjugate->complex)
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Add
operator+(const Row<N,E,S>& l, const conjugate<R>& r) {return l+(std::complex<R>)r;}
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Add
operator+(const conjugate<R>& l, const Row<N,E,S>& r) {return r+(std::complex<R>)l;}
 
// v = v+negator, negator+v: convert negator to standard number
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<typename negator<R>::StdNumber>::Add
operator+(const Row<N,E,S>& l, const negator<R>& r) {return l + (typename negator<R>::StdNumber)(R)r;}
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<typename negator<R>::StdNumber>::Add
operator+(const negator<R>& l, const Row<N,E,S>& r) {return r + (typename negator<R>::StdNumber)(R)l;}
 
// SCALAR SUBTRACT -- careful, not commutative.
 
// v = v-real, real-v 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<float>::Sub
operator-(const Row<N,E,S>& l, const float& r)
  { return Row<N,E,S>::template Result<float>::SubOp::perform(l,r); }
template <int N, class E, int S> inline
typename CNT<float>::template Result<Row<N,E,S> >::Sub
operator-(const float& l, const Row<N,E,S>& r)
  { return CNT<float>::template Result<Row<N,E,S> >::SubOp::perform(l,r); }
 
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<double>::Sub
operator-(const Row<N,E,S>& l, const double& r)
  { return Row<N,E,S>::template Result<double>::SubOp::perform(l,r); }
template <int N, class E, int S> inline
typename CNT<double>::template Result<Row<N,E,S> >::Sub
operator-(const double& l, const Row<N,E,S>& r)
  { return CNT<double>::template Result<Row<N,E,S> >::SubOp::perform(l,r); }
 
// v = v-int, int-v // just convert int to v's precision float
template <int N, class E, int S> inline
typename Row<N,E,S>::template Result<typename CNT<E>::Precision>::Sub
operator-(const Row<N,E,S>& l, int r) {return l - (typename CNT<E>::Precision)r;}
template <int N, class E, int S> inline
typename CNT<typename CNT<E>::Precision>::template Result<Row<N,E,S> >::Sub
operator-(int l, const Row<N,E,S>& r) {return (typename CNT<E>::Precision)l - r;}
 
 
// Complex, conjugate, and negator are all easy to templatize.
 
// v = v-complex, complex-v
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Sub
operator-(const Row<N,E,S>& l, const std::complex<R>& r)
  { return Row<N,E,S>::template Result<std::complex<R> >::SubOp::perform(l,r); }
template <int N, class E, int S, class R> inline
typename CNT<std::complex<R> >::template Result<Row<N,E,S> >::Sub
operator-(const std::complex<R>& l, const Row<N,E,S>& r)
  { return CNT<std::complex<R> >::template Result<Row<N,E,S> >::SubOp::perform(l,r); }
 
// v = v-conjugate, conjugate-v (convert conjugate->complex)
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<std::complex<R> >::Sub
operator-(const Row<N,E,S>& l, const conjugate<R>& r) {return l-(std::complex<R>)r;}
template <int N, class E, int S, class R> inline
typename CNT<std::complex<R> >::template Result<Row<N,E,S> >::Sub
operator-(const conjugate<R>& l, const Row<N,E,S>& r) {return (std::complex<R>)l-r;}
 
// v = v-negator, negator-v: convert negator to standard number
template <int N, class E, int S, class R> inline
typename Row<N,E,S>::template Result<typename negator<R>::StdNumber>::Sub
operator-(const Row<N,E,S>& l, const negator<R>& r) {return l-(typename negator<R>::StdNumber)(R)r;}
template <int N, class E, int S, class R> inline
typename CNT<R>::template Result<Row<N,E,S> >::Sub
operator-(const negator<R>& l, const Row<N,E,S>& r) {return (typename negator<R>::StdNumber)(R)l-r;}
 
 
// Row I/O
template <int N, class E, int S, class CHAR, class TRAITS> inline
std::basic_ostream<CHAR,TRAITS>&
operator<<(std::basic_ostream<CHAR,TRAITS>& o, const Row<N,E,S>& v) {
    o << "[" << v[0]; for(int i=1;i<N;++i) o<<','<<v[i]; o<<']'; return o;
}
 
template <int N, class E, int S, class CHAR, class TRAITS> inline
std::basic_istream<CHAR,TRAITS>&
operator>>(std::basic_istream<CHAR,TRAITS>& is, Row<N,E,S>& v) {
    CHAR openBracket, closeBracket;
    is >> openBracket; if (is.fail()) return is;
    if (openBracket==CHAR('('))
        closeBracket = CHAR(')');
    else if (openBracket==CHAR('['))
        closeBracket = CHAR(']');
    else {
        closeBracket = CHAR(0);
        is.unget(); if (is.fail()) return is;
    }
 
    for (int i=0; i < N; ++i) {
        is >> v[i];
        if (is.fail()) return is;
        if (i != N-1) {
            CHAR c; is >> c; if (is.fail()) return is;
            if (c != ',') is.unget();
            if (is.fail()) return is;
        }
    }
 
    // Get the closing bracket if there was an opening one. If we don't
    // see the expected character we'll set the fail bit in the istream.
    if (closeBracket != CHAR(0)) {
        CHAR closer; is >> closer; if (is.fail()) return is;
        if (closer != closeBracket) {
            is.unget(); if (is.fail()) return is;
            is.setstate( std::ios::failbit );
        }
    }
 
    return is;
}
 
} //namespace SimTK
 
 
#endif //SimTK_SIMMATRIX_SMALLMATRIX_ROW_H_