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 RuizEquil ( DistSparseMatrix<Field>& A, DistMultiVec<Base<Field>>& dRow, DistMultiVec<Base<Field>>& dCol, bool progress ) { EL_DEBUG_CSE typedef Base<Field> Real; const Int m = A.Height(); const Int n = A.Width(); mpi::Comm comm = A.Comm(); dRow.SetComm( comm ); dCol.SetComm( comm ); Ones( dRow, m, 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); const Int indent = PushIndent(); for( Int iter=0; iter<maxIter; ++iter ) { // Rescale the columns // ------------------- ColumnMaxNorms( A, scales ); EntrywiseMap( scales, MakeFunction(DampScaling<Real>) ); DiagonalScale( LEFT, NORMAL, scales, dCol ); DiagonalSolve( RIGHT, NORMAL, scales, A ); // Rescale the rows // ---------------- RowMaxNorms( A, scales ); EntrywiseMap( scales, MakeFunction(DampScaling<Real>) ); DiagonalScale( LEFT, NORMAL, scales, dRow ); DiagonalSolve( LEFT, NORMAL, scales, A ); } SetIndent( indent ); }
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; } x.ProcessQueues(); }
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); Zeros( AHat, 0, n+2 ); DistMultiVec<Real> bHat(comm); Zeros( bHat, 0, 1 ); DistMultiVec<Real> xHat(comm), y(comm), z(comm), s(comm); SOCP( AHat, G, bHat, c, h, orders, firstInds, xHat, y, z, s, ctrl ); x = xHat( IR(2,END), ALL ); }