C++ DistSparseMatrix::Height方法代码示例

void StackedRuizEquil
( DistSparseMatrix<Field>& A,
  DistSparseMatrix<Field>& B,
  DistMultiVec<Base<Field>>& dRowA,
  DistMultiVec<Base<Field>>& dRowB,
  DistMultiVec<Base<Field>>& dCol,
  bool progress )
{
    EL_DEBUG_CSE
    typedef Base<Field> Real;
    const Int mA = A.Height();
    const Int mB = B.Height();
    const Int n = A.Width();
    mpi::Comm comm = A.Comm();
    dRowA.SetComm( comm );
    dRowB.SetComm( comm );
    dCol.SetComm( comm );
    Ones( dRowA, mA, 1 );
    Ones( dRowB, mB, 1 );
    Ones( dCol, n, 1 );

    // TODO(poulson): Expose to control structure
    // For, simply hard-code a small number of iterations
    const Int maxIter = 4;

    DistMultiVec<Real> scales(comm), maxAbsValsB(comm);
    auto& scalesLoc = scales.Matrix();
    auto& maxAbsValsBLoc = maxAbsValsB.Matrix();
    const Int localHeight = scalesLoc.Height();
    const Int indent = PushIndent();
    for( Int iter=0; iter<maxIter; ++iter )
    {
        // Rescale the columns
        // -------------------
        ColumnMaxNorms( A, scales );
        ColumnMaxNorms( B, maxAbsValsB );
        for( Int jLoc=0; jLoc<localHeight; ++jLoc )
            scalesLoc(jLoc) = Max(scalesLoc(jLoc),maxAbsValsBLoc(jLoc));
        EntrywiseMap( scales, MakeFunction(DampScaling<Real>) );
        DiagonalScale( LEFT, NORMAL, scales, dCol );
        DiagonalSolve( RIGHT, NORMAL, scales, A );
        DiagonalSolve( RIGHT, NORMAL, scales, B );

        // Rescale the rows
        // ----------------
        RowMaxNorms( A, scales );
        EntrywiseMap( scales, MakeFunction(DampScaling<Real>) );
        DiagonalScale( LEFT, NORMAL, scales, dRowA );
        DiagonalSolve( LEFT, NORMAL, scales, A );

        RowMaxNorms( B, scales );
        EntrywiseMap( scales, MakeFunction(DampScaling<Real>) );
        DiagonalScale( LEFT, NORMAL, scales, dRowB );
        DiagonalSolve( LEFT, NORMAL, scales, B );
    }
    SetIndent( indent );
}

void SymmetricRuizEquil
( DistSparseMatrix<F>& A, 
  DistMultiVec<Base<F>>& d, 
  Int maxIter, bool progress )
{
    DEBUG_CSE
    typedef Base<F> Real;
    const Int n = A.Height();
    mpi::Comm comm = A.Comm();
    d.SetComm( comm );
    Ones( d, n, 1 );

    DistMultiVec<Real> scales(comm);
    const Int indent = PushIndent();
    for( Int iter=0; iter<maxIter; ++iter )
    {
        // Rescale the columns (and rows)
        // ------------------------------
        ColumnMaxNorms( A, scales );
        EntrywiseMap( scales, function<Real(Real)>(DampScaling<Real>) );
        EntrywiseMap( scales, function<Real(Real)>(SquareRootScaling<Real>) );
        DiagonalScale( LEFT, NORMAL, scales, d );
        SymmetricDiagonalSolve( scales, A );
    }
    SetIndent( indent );
}

void SymmetricRuizEquil
( DistSparseMatrix<Field>& A,
  DistMultiVec<Base<Field>>& d,
  Int maxIter, bool progress )
{
    EL_DEBUG_CSE
    typedef Base<Field> Real;
    const Int n = A.Height();
    const Grid& grid = A.Grid();
    d.SetGrid( grid );
    Ones( d, n, 1 );

    DistMultiVec<Real> scales(grid);
    const Int indent = PushIndent();
    for( Int iter=0; iter<maxIter; ++iter )
    {
        // Rescale the columns (and rows)
        // ------------------------------
        ColumnMaxNorms( A, scales );
        EntrywiseMap( scales, MakeFunction(DampScaling<Real>) );
        EntrywiseMap( scales, MakeFunction(SquareRootScaling<Real>) );
        DiagonalScale( LEFT, NORMAL, scales, d );
        SymmetricDiagonalSolve( scales, A );
    }
    SetIndent( indent );
}

void LAV
( const DistSparseMatrix<Real>& A,
  const DistMultiVec<Real>& b,
        DistMultiVec<Real>& x,
  const lp::affine::Ctrl<Real>& ctrl )
{
    EL_DEBUG_CSE
    const Int m = A.Height();
    const Int n = A.Width();
    const Grid& grid = A.Grid();

    DistSparseMatrix<Real> AHat(grid), G(grid);
    DistMultiVec<Real> c(grid), h(grid);

    // c := [0;1;1]
    // ============
    Zeros( c, n+2*m, 1 );
    for( Int iLoc=0; iLoc<c.LocalHeight(); ++iLoc )
        if( c.GlobalRow(iLoc) >= n )
            c.SetLocal( iLoc, 0, Real(1) );

    // \hat A := [A, I, -I]
    // ====================
    Zeros( AHat, m, n+2*m );
    const Int numLocalEntriesA = A.NumLocalEntries();
    AHat.Reserve( numLocalEntriesA + 2*AHat.LocalHeight() );
    for( Int e=0; e<numLocalEntriesA; ++e )
        AHat.QueueUpdate( A.Row(e), A.Col(e), A.Value(e) );
    for( Int iLoc=0; iLoc<AHat.LocalHeight(); ++iLoc )
    {
        const Int i = AHat.GlobalRow(iLoc);
        AHat.QueueLocalUpdate( iLoc, i+n,   Real( 1) );
        AHat.QueueLocalUpdate( iLoc, i+n+m, Real(-1) );
    }
    AHat.ProcessLocalQueues();

    // G := | 0 -I  0 |
    //      | 0  0 -I |
    // ================
    Zeros( G, 2*m, n+2*m );
    G.Reserve( G.LocalHeight() );
    for( Int iLoc=0; iLoc<G.LocalHeight(); ++iLoc )
        G.QueueLocalUpdate( iLoc, G.GlobalRow(iLoc)+n, Real(-1) );
    G.ProcessLocalQueues();

    // h := | 0 |
    //      | 0 |
    // ==========
    Zeros( h, 2*m, 1 );

    // Solve the affine QP
    // ===================
    DistMultiVec<Real> xHat(grid), y(grid), z(grid), s(grid);
    LP( AHat, G, b, c, h, xHat, y, z, s, ctrl );

    // Extract x
    // =========
    x = xHat( IR(0,n), ALL );
}

void Fill( DistSparseMatrix<T>& A, T alpha )
{
    EL_DEBUG_CSE
    const Int m = A.Height();
    const Int n = A.Width();
    A.Resize( m, n );
    Zero( A );
    if( alpha != T(0) ) 
    {
        const Int localHeight = A.LocalHeight();
        A.Reserve( localHeight*n );
        for( Int iLoc=0; iLoc<localHeight; ++iLoc )
            for( Int j=0; j<n; ++j ) 
                A.QueueLocalUpdate( iLoc, j, alpha );
        A.ProcessLocalQueues();
    }
}

void Lanczos
( const DistSparseMatrix<F>& A,
        ElementalMatrix<Base<F>>& T,
        Int basisSize )
{
    DEBUG_CSE
    const Int n = A.Height();
    if( n != A.Width() )
        LogicError("A was not square");

    auto applyA =
      [&]( const DistMultiVec<F>& X, DistMultiVec<F>& Y )
      {
          Zeros( Y, n, X.Width() );
          Multiply( NORMAL, F(1), A, X, F(0), Y );
      };
    Lanczos<F>( n, applyA, T, basisSize );
}

Base<Field> ProductLanczosDecomp
( const DistSparseMatrix<Field>& A,
        DistMultiVec<Field>& V,
        AbstractDistMatrix<Base<Field>>& T,
        DistMultiVec<Field>& v,
        Int basisSize )
{
    EL_DEBUG_CSE
    const Int m = A.Height();
    const Int n = A.Width();
    mpi::Comm comm = A.Comm();

    DistMultiVec<Field> S(comm);

    // Cache the adjoint
    // -----------------
    DistSparseMatrix<Field> AAdj(comm);
    Adjoint( A, AAdj );

    if( m >= n )
    {
        auto applyA =
          [&]( const DistMultiVec<Field>& X, DistMultiVec<Field>& Y )
          {
              Zeros( S, m, X.Width() );
              Multiply( NORMAL, Field(1), A,    X, Field(0), S );
              Zeros( Y, n, X.Width() );
              Multiply( NORMAL, Field(1), AAdj, S, Field(0), Y );
          };
        return LanczosDecomp( n, applyA, V, T, v, basisSize );
    }
    else
    {
        auto applyA =
          [&]( const DistMultiVec<Field>& X, DistMultiVec<Field>& Y )
          {
              Zeros( S, n, X.Width() );
              Multiply( NORMAL, Field(1), AAdj, X, Field(0), S );
              Zeros( Y, m, X.Width() );
              Multiply( NORMAL, Field(1), A,    S, Field(0), Y );
          };
        return LanczosDecomp( m, applyA, V, T, v, basisSize );
    }
}

Base<Field> LanczosDecomp
( const DistSparseMatrix<Field>& A,
        DistMultiVec<Field>& V,
        AbstractDistMatrix<Base<Field>>& T,
        DistMultiVec<Field>& v,
        Int basisSize )
{
    EL_DEBUG_CSE
    const Int n = A.Height();
    if( n != A.Width() )
        LogicError("A was not square");

    auto applyA =
      [&]( const DistMultiVec<Field>& X, DistMultiVec<Field>& Y )
      {
          Zeros( Y, n, X.Width() );
          Multiply( NORMAL, Field(1), A, X, Field(0), Y );
      };
    return LanczosDecomp( n, applyA, V, T, v, basisSize );
}

void QP
( const DistSparseMatrix<Real>& A, 
  const DistMultiVec<Real>& B, 
        DistMultiVec<Real>& X, 
  const qp::direct::Ctrl<Real>& ctrl )
{
    DEBUG_CSE

    const Int m = A.Height();
    const Int n = A.Width();
    const Int k = B.Width();
    mpi::Comm comm = A.Comm();
    DistSparseMatrix<Real> Q(comm), AHat(comm);
    DistMultiVec<Real> bHat(comm), c(comm);

    Herk( LOWER, ADJOINT, Real(1), A, Q );
    MakeHermitian( LOWER, Q );
    Zeros( AHat, 0, n );
    Zeros( bHat, 0, 1 );
    Zeros( X,    n, k );

    DistMultiVec<Real> q(comm), y(comm), z(comm);
    auto& qLoc = q.Matrix();
    auto& XLoc = X.Matrix();
    auto& BLoc = B.LockedMatrix();
    for( Int j=0; j<k; ++j )
    {
        auto xLoc = XLoc( ALL, IR(j) );
        auto bLoc = BLoc( ALL, IR(j) );

        Zeros( c, n, 1 );
        Zeros( q, m, 1 );
        qLoc = bLoc;
        Multiply( ADJOINT, Real(-1), A, q, Real(0), c );

        Zeros( q, n, 1 );
        qLoc = xLoc;
        El::QP( Q, AHat, bHat, c, q, y, z, ctrl );
        xLoc = qLoc;
    }
}

void RowMaxNorms( const DistSparseMatrix<F>& A, DistMultiVec<Base<F>>& norms )
{
    DEBUG_CSE
    typedef Base<F> Real;
    const Int localHeight = A.LocalHeight();
    const F* valBuf = A.LockedValueBuffer();
    const Int* offsetBuf = A.LockedOffsetBuffer();

    norms.SetComm( A.Comm() );
    norms.Resize( A.Height(), 1 );
    auto& normsLoc = norms.Matrix();
    for( Int iLoc=0; iLoc<localHeight; ++iLoc )
    {
        Real rowMax = 0;
        const Int offset = offsetBuf[iLoc];
        const Int numConn = offsetBuf[iLoc+1] - offset;
        for( Int e=offset; e<offset+numConn; ++e )
            rowMax = Max(rowMax,Abs(valBuf[e]));
        normsLoc(iLoc) = rowMax;
    }
}

void RowTwoNorms( const DistSparseMatrix<F>& A, DistMultiVec<Base<F>>& norms )
{
    DEBUG_CSE
    typedef Base<F> Real;
    const Int localHeight = A.LocalHeight();
    const F* valBuf = A.LockedValueBuffer();
    const Int* offsetBuf = A.LockedOffsetBuffer();

    norms.SetComm( A.Comm() );
    norms.Resize( A.Height(), 1 );
    auto& normLoc = norms.Matrix();
    for( Int iLoc=0; iLoc<localHeight; ++iLoc )
    {
        Real scale = 0;
        Real scaledSquare = 1;
        const Int offset = offsetBuf[iLoc];
        const Int numConn = offsetBuf[iLoc+1] - offset;
        for( Int e=offset; e<offset+numConn; ++e )
            UpdateScaledSquare( valBuf[e], scale, scaledSquare );
        normLoc(iLoc) = scale*Sqrt(scaledSquare);
    }
}

void GetMappedDiagonal
( const DistSparseMatrix<T>& A,
        DistMultiVec<S>& d,
        function<S(const T&)> func,
        Int offset )
{
    EL_DEBUG_CSE
    const Int m = A.Height();
    const Int n = A.Width();
    const T* valBuf = A.LockedValueBuffer();
    const Int* colBuf = A.LockedTargetBuffer();

    if( m != n )
        LogicError("DistSparseMatrix GetMappedDiagonal assumes square matrix");
    if( offset != 0 )
        LogicError("DistSparseMatrix GetMappedDiagonal assumes offset=0");

    d.SetGrid( A.Grid() );
    d.Resize( El::DiagonalLength(m,n,offset), 1 );
    Fill( d, S(1) );

    S* dBuf = d.Matrix().Buffer();
    const Int dLocalHeight = d.LocalHeight();
    for( Int iLoc=0; iLoc<dLocalHeight; ++iLoc )
    {
        const Int i = d.GlobalRow(iLoc);
        const Int thisOff = A.RowOffset(iLoc);
        const Int nextOff = A.RowOffset(iLoc+1);
        auto it = std::lower_bound( colBuf+thisOff, colBuf+nextOff, i );
        if( *it == i )
        {
            const Int e = it-colBuf;
            dBuf[iLoc] = func(valBuf[e]);
        }
        else
            dBuf[iLoc] = func(0);
    }
}

void IPM
( const DistSparseMatrix<Real>& A,
  const DistMultiVec<Real>& b, 
        Real lambda,
        DistMultiVec<Real>& x,
  const qp::affine::Ctrl<Real>& ctrl )
{
    DEBUG_CSE
    const Int m = A.Height();
    const Int n = A.Width();
    mpi::Comm comm = A.Comm();
    DistSparseMatrix<Real> Q(comm), AHat(comm), G(comm);
    DistMultiVec<Real> c(comm), h(comm);

    // Q := | 0 0 0 |
    //      | 0 0 0 |
    //      | 0 0 I |
    // ==============
    Zeros( Q, 2*n+m, 2*n+m );
    {
        Int numLocalUpdates = 0;
        for( Int iLoc=0; iLoc<Q.LocalHeight(); ++iLoc )
            if( Q.GlobalRow(iLoc) >= 2*n )
                ++numLocalUpdates;
        Q.Reserve( numLocalUpdates );
        for( Int iLoc=0; iLoc<Q.LocalHeight(); ++iLoc )
            if( Q.GlobalRow(iLoc) >= 2*n )
                Q.QueueLocalUpdate( iLoc, Q.GlobalRow(iLoc), Real(1) );
        Q.ProcessLocalQueues();
    }

    // c := lambda*[1;1;0]
    // ===================
    Zeros( c, 2*n+m, 1 );
    auto& cLoc = c.Matrix();
    for( Int iLoc=0; iLoc<c.LocalHeight(); ++iLoc )
        if( c.GlobalRow(iLoc) < 2*n )
            cLoc(iLoc) = lambda;

    // \hat A := [A, -A, I]
    // ====================
    // NOTE: Since A and \hat A are the same height and each distributed within
    //       columns, it is possible to form \hat A from A without communication
    const Int numLocalEntriesA = A.NumLocalEntries();
    Zeros( AHat, m, 2*n+m );
    AHat.Reserve( 2*numLocalEntriesA+AHat.LocalHeight() );
    for( Int e=0; e<numLocalEntriesA; ++e )
    {
        AHat.QueueUpdate( A.Row(e), A.Col(e),    A.Value(e) );
        AHat.QueueUpdate( A.Row(e), A.Col(e)+n, -A.Value(e) );
    }
    for( Int iLoc=0; iLoc<AHat.LocalHeight(); ++iLoc )
    {
        const Int i = AHat.GlobalRow(iLoc);
        AHat.QueueLocalUpdate( iLoc, i+2*n, Real(1) );
    }
    AHat.ProcessLocalQueues();

    // G := | -I  0 0 |
    //      |  0 -I 0 |
    // ================
    Zeros( G, 2*n, 2*n+m );
    G.Reserve( G.LocalHeight() );
    for( Int iLoc=0; iLoc<G.LocalHeight(); ++iLoc )
    {
        const Int i = G.GlobalRow(iLoc);
        G.QueueLocalUpdate( iLoc, i, Real(-1) );
    }
    G.ProcessLocalQueues();

    // h := 0
    // ======
    Zeros( h, 2*n, 1 );

    // Solve the affine QP
    // ===================
    DistMultiVec<Real> xHat(comm), y(comm), z(comm), s(comm);
    QP( Q, AHat, G, b, c, h, xHat, y, z, s, ctrl );

    // x := u - v
    // ==========
    Zeros( x, n, 1 );
    Int numRemoteUpdates = 0;
    for( Int iLoc=0; iLoc<xHat.LocalHeight(); ++iLoc )
        if( xHat.GlobalRow(iLoc) < 2*n )
            ++numRemoteUpdates; 
        else
            break;
    x.Reserve( numRemoteUpdates );
    auto& xHatLoc = xHat.LockedMatrix();
    for( Int iLoc=0; iLoc<xHat.LocalHeight(); ++iLoc )
    {
        const Int i = xHat.GlobalRow(iLoc);
        if( i < n )
            x.QueueUpdate( i, 0, xHatLoc(iLoc) );
        else if( i < 2*n )
            x.QueueUpdate( i-n, 0, -xHatLoc(iLoc) );
        else
            break;
    }
//.........这里部分代码省略.........

void IPM
( const DistSparseMatrix<Real>& A,
  const DistMultiVec<Real>& d,
        Real lambda,
        DistMultiVec<Real>& x,
  const qp::affine::Ctrl<Real>& ctrl )
{
    EL_DEBUG_CSE
    const Int m = A.Height();
    const Int n = A.Width();
    mpi::Comm comm = A.Comm();

    DistSparseMatrix<Real> Q(comm), AHat(comm), G(comm);
    DistMultiVec<Real> c(comm), b(comm), h(comm);

    auto& dLoc = d.LockedMatrix();
    auto& cLoc = c.Matrix();
    auto& hLoc = h.Matrix();

    // Q := | I 0 0 |
    //      | 0 0 0 |
    //      | 0 0 0 |
    // ==============
    Zeros( Q, n+m+1, n+m+1 );
    {
        // Count the number of local entries in the top-left I
        // ---------------------------------------------------
        Int numLocalUpdates = 0;
        for( Int iLoc=0; iLoc<Q.LocalHeight(); ++iLoc )
            if( Q.GlobalRow(iLoc) < n )
                ++numLocalUpdates;
            else
                break;
        Q.Reserve( numLocalUpdates );
        for( Int iLoc=0; iLoc<Q.LocalHeight(); ++iLoc )
            if( Q.GlobalRow(iLoc) < n )
                Q.QueueLocalUpdate( iLoc, Q.GlobalRow(iLoc), Real(1) );
        Q.ProcessLocalQueues();
    }

    // c := [0;0;lambda]
    // =================
    Zeros( c, n+m+1, 1 );
    for( Int iLoc=0; iLoc<c.LocalHeight(); ++iLoc )
        if( c.GlobalRow(iLoc) > n )
            cLoc(iLoc) = lambda;

    // AHat = []
    // =========
    Zeros( AHat, 0, n+m+1 );

    // b = []
    // ======
    Zeros( b, 0, 1 );

    // G := |-diag(d) A, -d, -I|
    //      |      0,     0, -I|
    // =========================
    Zeros( G, 2*m, n+m+1 );
    G.Reserve
    ( A.NumLocalEntries()+d.LocalHeight()+G.LocalHeight(),
      A.NumLocalEntries()+d.LocalHeight() );
    for( Int e=0; e<A.NumLocalEntries(); ++e )
    {
        const Int i = A.Row(e);
        const Int j = A.Col(e);
        const Int iLoc = A.LocalRow(i);
        const Real value = -dLoc(iLoc)*A.Value(e);
        G.QueueUpdate( i, j, value );
    }
    for( Int iLoc=0; iLoc<d.LocalHeight(); ++iLoc )
    {
        const Int i = d.GlobalRow(iLoc);
        G.QueueUpdate( i, n, -dLoc(iLoc) );
    }
    for( Int iLoc=0; iLoc<G.LocalHeight(); ++iLoc )
    {
        const Int i = G.GlobalRow(iLoc);
        if( i < m )
            G.QueueLocalUpdate( iLoc, i+n+1, Real(-1) );
        else
            G.QueueLocalUpdate( iLoc, (i-m)+n+1, Real(-1) );
    }
    G.ProcessQueues();

    // h := [-ones(m,1); zeros(m,1)]
    // =============================
    Zeros( h, 2*m, 1 );
    for( Int iLoc=0; iLoc<h.LocalHeight(); ++iLoc )
        if( h.GlobalRow(iLoc) < m )
            hLoc(iLoc) = Real(-1);
        else
            break;

    // Solve the affine QP
    // ===================
    DistMultiVec<Real> y(comm), z(comm), s(comm);
    QP( Q, AHat, G, b, c, h, x, y, z, s, ctrl );
}

void RLS
( const DistSparseMatrix<Real>& A, 
  const DistMultiVec<Real>& b, 
        Real rho,
        DistMultiVec<Real>& x, 
  const socp::affine::Ctrl<Real>& ctrl )
{
    DEBUG_CSE
    const Int m = A.Height();
    const Int n = A.Width();
    mpi::Comm comm = A.Comm();

    DistMultiVec<Int> orders(comm), firstInds(comm);
    Zeros( orders, m+n+3, 1 );
    Zeros( firstInds, m+n+3, 1 );
    {
        const Int localHeight = orders.LocalHeight();
        for( Int iLoc=0; iLoc<localHeight; ++iLoc )
        {
            const Int i = orders.GlobalRow(iLoc);
            if( i < m+1 )
            {
                orders.SetLocal( iLoc, 0, m+1 );
                firstInds.SetLocal( iLoc, 0, 0 );
            }
            else
            {
                orders.SetLocal( iLoc, 0, n+2 ); 
                firstInds.SetLocal( iLoc, 0, m+1 );
            }
        }
    }

    // G := | -1  0  0 |
    //      |  0  0  A |
    //      |  0 -1  0 |
    //      |  0  0 -I |
    //      |  0  0  0 |
    DistSparseMatrix<Real> G(comm);
    {
        Zeros( G, m+n+3, n+2 );

        // Count the number of entries of G to reserve
        // -------------------------------------------
        Int numLocalUpdates = 0;
        const Int localHeight = G.LocalHeight();
        for( Int iLoc=0; iLoc<localHeight; ++iLoc )
        {
            const Int i = G.GlobalRow(iLoc);
            if( i == 0 || i == m+1 || (i>m+1 && i<m+n+2) )
                ++numLocalUpdates;
        }
        const Int numEntriesA = A.NumLocalEntries();
        G.Reserve( numLocalUpdates+numEntriesA, numEntriesA );

        // Queue the local updates
        // ----------------------- 
        for( Int iLoc=0; iLoc<localHeight; ++iLoc )
        {
            const Int i = G.GlobalRow(iLoc);
            if( i == 0 )
                G.QueueLocalUpdate( iLoc, 0, -1 );
            else if( i == m+1 )
                G.QueueLocalUpdate( iLoc, 1, -1 );
            else if( i > m+1 && i < m+n+2 )
                G.QueueLocalUpdate( iLoc, i-m, -1 );
        }

        // Queue the remote updates
        // ------------------------
        for( Int e=0; e<numEntriesA; ++e )
            G.QueueUpdate( A.Row(e)+1, A.Col(e)+2, A.Value(e) );

        G.ProcessQueues();
    }

    // h := | 0 |
    //      | b |
    //      | 0 |
    //      | 0 |
    //      | 1 |
    DistMultiVec<Real> h(comm);
    Zeros( h, m+n+3, 1 ); 
    auto& bLoc = b.LockedMatrix();
    {
        const Int bLocalHeight = b.LocalHeight();
        h.Reserve( bLocalHeight );
        for( Int iLoc=0; iLoc<bLocalHeight; ++iLoc )
            h.QueueUpdate( b.GlobalRow(iLoc)+1, 0, bLoc(iLoc) );
        h.ProcessQueues();
    }
    h.Set( END, 0, 1 );

    // c := [1; rho; 0]
    DistMultiVec<Real> c(comm);
    Zeros( c, n+2, 1 );
    c.Set( 0, 0, 1 );
    c.Set( 1, 0, rho );

    DistSparseMatrix<Real> AHat(comm);
//.........这里部分代码省略.........