text:
package org.apache.sysml.runtime.matrix.data;

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.concurrent.Callable;
import java.util.concurrent.ExecutionException;
import java.util.concurrent.ExecutorService;
import java.util.concurrent.Executors;
import java.util.concurrent.Future;

import org.apache.commons.math3.util.FastMath;
import org.apache.sysml.hops.OptimizerUtils;
import org.apache.sysml.lops.MapMultChain.ChainType;
import org.apache.sysml.lops.WeightedCrossEntropy.WCeMMType;
import org.apache.sysml.lops.WeightedDivMM.WDivMMType;
import org.apache.sysml.lops.WeightedSigmoid.WSigmoidType;
import org.apache.sysml.lops.WeightedSquaredLoss.WeightsType;
import org.apache.sysml.lops.WeightedUnaryMM.WUMMType;
import org.apache.sysml.runtime.DMLRuntimeException;
import org.apache.sysml.runtime.functionobjects.SwapIndex;
import org.apache.sysml.runtime.functionobjects.ValueFunction;
import org.apache.sysml.runtime.matrix.operators.ReorgOperator;
import org.apache.sysml.runtime.util.UtilFunctions;

/**
 * MB: Library for matrix multiplications including MM, MV, VV for all
 * combinations of dense, sparse, ultrasparse representations and special
 * operations such as transpose-self matrix multiplication.
 * <p>
 * In general all implementations use internally dense outputs
 * for direct access, but change the final result to sparse if necessary.
 * The only exceptions are ultra-sparse matrix mult, wsloss and wsigmoid.
 */
public class LibMatrixMult
{
	//internal configuration
	private static final boolean LOW_LEVEL_OPTIMIZATION = true;
	private static final long MEM_OVERHEAD_THRESHOLD = 2L*1024*1024; //MAX 2 MB
	private static final long PAR_MINFLOP_THRESHOLD = 2L*1024*1024; //MIN 2 MFLOP
	private static final int L2_CACHESIZE = 256 *1024; //256KB (common size)

	private LibMatrixMult() {
		//prevent instantiation via private constructor
	}

	////////////////////////////////
	// public matrix mult interface
	////////////////////////////////

	/**
	 * Performs a matrix multiplication and stores the result in the output matrix.
	 *
	 * All variants use a IKJ access pattern, and internally use dense output. After the
	 * actual computation, we recompute nnz and check for sparse/dense representation.
	 *
	 *
	 * @param m1 first matrix
	 * @param m2 second matrix
	 * @param ret result matrix
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	public static void matrixMult(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret)
		throws DMLRuntimeException
	{
		matrixMult(m1, m2, ret, 0, m1.rlen);
	}

	/**
	 * This method allows one to disabling exam sparsity. This feature is useful if matrixMult is used as an intermediate
	 * operation (for example: LibMatrixDNN). It makes sense for LibMatrixDNN because the output is internally
	 * consumed by another dense instruction, which makes repeated conversion to sparse wasteful.
	 * This should be used in rare cases and if you are unsure,
	 * use the method 'matrixMult(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret)' instead.
	 *
	 * @param m1 first matrix
	 * @param m2 second matrix
	 * @param ret result matrix
	 * @param fixedRet if true, output representation is fixed and nnzs not recomputed
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	public static void matrixMult(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, boolean fixedRet) throws DMLRuntimeException {
		matrixMult(m1, m2, ret, 0, m1.rlen, fixedRet);
	}

	public static void matrixMult(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, int rl, int ru) throws DMLRuntimeException {
		matrixMult(m1, m2, ret, rl, ru, false);
	}

	public static void matrixMult(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, int rl, int ru, boolean fixedRet)
		throws DMLRuntimeException
	{
		//check inputs / outputs
		if( m1.isEmptyBlock(false) || m2.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing: output allocation
		boolean ultraSparse = (fixedRet && ret.sparse)
			|| (!fixedRet && isUltraSparseMatrixMult(m1, m2));
		boolean tm2 = checkPrepMatrixMultRightInput(m1,m2);
		m2 = prepMatrixMultRightInput(m1, m2);
		ret.sparse = ultraSparse;
		ret.allocateBlock();

		//prepare row-upper for special cases of vector-matrix
		boolean pm2 = !ultraSparse &&
			checkParMatrixMultRightInputRows(m1, m2, Integer.MAX_VALUE);
		int ru2 = (pm2 && ru==m1.rlen) ? m2.rlen : ru;
		int cu = m2.clen;

		//core matrix mult computation
		if( ultraSparse )
			matrixMultUltraSparse(m1, m2, ret, 0, ru2);
		else if(!m1.sparse && !m2.sparse)
			matrixMultDenseDense(m1, m2, ret, tm2, pm2, 0, ru2, 0, cu);
		else if(m1.sparse && m2.sparse)
			matrixMultSparseSparse(m1, m2, ret, pm2, 0, ru2);
		else if(m1.sparse)
			matrixMultSparseDense(m1, m2, ret, pm2, 0, ru2);
		else
			matrixMultDenseSparse(m1, m2, ret, pm2, 0, ru2);

		//post-processing: nnz/representation
		if( !fixedRet ) {
			if( !ret.sparse )
				ret.recomputeNonZeros();
			ret.examSparsity();
		}

		//System.out.println("MM ("+m1.isInSparseFormat()+","+m1.getNumRows()+","+m1.getNumColumns()+","+m1.getNonZeros()+")x" +
		//		"("+m2.isInSparseFormat()+","+m2.getNumRows()+","+m2.getNumColumns()+","+m2.getNonZeros()+") in "+time.stop());
	}

	/**
	 * Performs a multi-threaded matrix multiplication and stores the result in the output matrix.
	 * The parameter k (k>=1) determines the max parallelism k' with k'=min(k, vcores, m1.rlen).
	 *
	 * @param m1 first matrix
	 * @param m2 second matrix
	 * @param ret result matrix
	 * @param k maximum parallelism
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	public static void matrixMult(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, int k)
		throws DMLRuntimeException
	{
		//check inputs / outputs
		if( m1.isEmptyBlock(false) || m2.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//check too high additional vector-matrix memory requirements (fallback to sequential)
		//check too small workload in terms of flops (fallback to sequential too)
		if( m1.rlen == 1 && (8L * m2.clen * k > MEM_OVERHEAD_THRESHOLD || !LOW_LEVEL_OPTIMIZATION || m2.clen==1 || m1.isUltraSparse() || m2.isUltraSparse())
			|| 2L * m1.rlen * m1.clen * m2.clen < PAR_MINFLOP_THRESHOLD )
		{
			matrixMult(m1, m2, ret);
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing: output allocation (in contrast to single-threaded,
		//we need to allocate sparse as well in order to prevent synchronization)
		boolean ultraSparse = isUltraSparseMatrixMult(m1, m2);
		boolean tm2 = checkPrepMatrixMultRightInput(m1,m2);
		m2 = prepMatrixMultRightInput(m1, m2);
		ret.sparse = ultraSparse;
		ret.allocateBlock();

		if (!ret.isThreadSafe()) {
			matrixMult(m1, m2, ret);
			return;
		}

		//prepare row-upper for special cases of vector-matrix / matrix-matrix
		boolean pm2r = !ultraSparse && checkParMatrixMultRightInputRows(m1, m2, k);
		boolean pm2c = !ultraSparse && checkParMatrixMultRightInputCols(m1, m2, k, pm2r);
		int num = pm2r ? m2.rlen : pm2c ? m2.clen : m1.rlen;

		//core multi-threaded matrix mult computation
		//(currently: always parallelization over number of rows)
		try {
			ExecutorService pool = Executors.newFixedThreadPool( k );
			ArrayList<MatrixMultTask> tasks = new ArrayList<>();
			ArrayList<Integer> blklens = getBalancedBlockSizesDefault(num, k, (pm2r||pm2c));
			for( int i=0, lb=0; i<blklens.size(); lb+=blklens.get(i), i++ )
				tasks.add(new MatrixMultTask(m1, m2, ret, tm2, pm2r, pm2c, lb, lb+blklens.get(i)));
			//execute tasks
			List<Future<Object>> taskret = pool.invokeAll(tasks);
			pool.shutdown();
			//aggregate partial results (nnz, ret for vector/matrix)
			ret.nonZeros = 0; //reset after execute
			for( Future<Object> task : taskret ) {
				if( pm2r ) //guaranteed single block
					vectAdd((double[])task.get(), ret.getDenseBlockValues(), 0, 0, ret.rlen*ret.clen);
				else
					ret.nonZeros += (Long)task.get();
			}
			if( pm2r )
				ret.recomputeNonZeros();
		}
		catch(Exception ex) {
			throw new DMLRuntimeException(ex);
		}

		//post-processing (nnz maintained in parallel)
		ret.examSparsity();

		//System.out.println("MM k="+k+" ("+m1.isInSparseFormat()+","+m1.getNumRows()+","+m1.getNumColumns()+","+m1.getNonZeros()+")x" +
		//		"("+m2.isInSparseFormat()+","+m2.getNumRows()+","+m2.getNumColumns()+","+m2.getNonZeros()+") in "+time.stop());
	}

	/**
	 * Performs a matrix multiplication chain operation of type t(X)%*%(X%*%v) or t(X)%*%(w*(X%*%v)).
	 *
	 * All variants use a IKJ access pattern, and internally use dense output. After the
	 * actual computation, we recompute nnz and check for sparse/dense representation.
	 *
	 * @param mX X matrix
	 * @param mV v matrix
	 * @param mW w matrix
	 * @param ret result matrix
	 * @param ct chain type
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	public static void matrixMultChain(MatrixBlock mX, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, ChainType ct)
		throws DMLRuntimeException
	{
		//check inputs / outputs (after that mV and mW guaranteed to be dense)
		if( mX.isEmptyBlock(false) || mV.isEmptyBlock(false) || (mW !=null && mW.isEmptyBlock(false)) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing: output allocation
		ret.sparse = false;
		ret.allocateDenseBlock();

		//core matrix mult chain computation
		if( mX.sparse )
			matrixMultChainSparse(mX, mV, mW, ret, ct, 0, mX.rlen);
		else
			matrixMultChainDense(mX, mV, mW, ret, ct, 0, mX.rlen);

		//post-processing
		ret.recomputeNonZeros();
		ret.examSparsity();

		//System.out.println("MMChain "+ct.toString()+" ("+mX.isInSparseFormat()+","+mX.getNumRows()+","+mX.getNumColumns()+","+mX.getNonZeros()+")x" +
		//		             "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	/**
	 * Performs a parallel matrix multiplication chain operation of type t(X)%*%(X%*%v) or t(X)%*%(w*(X%*%v)).
	 * The parameter k (k>=1) determines the max parallelism k' with k'=min(k, vcores, m1.rlen).
	 *
	 * NOTE: This multi-threaded mmchain operation has additional memory requirements of k*ncol(X)*8bytes
	 * for partial aggregation. Current max memory: 256KB; otherwise redirectly to sequential execution.
	 *
	 * @param mX X matrix
	 * @param mV v matrix
	 * @param mW w matrix
	 * @param ret result matrix
	 * @param ct chain type
	 * @param k maximum parallelism
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	public static void matrixMultChain(MatrixBlock mX, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, ChainType ct, int k)
		throws DMLRuntimeException
	{
		//check inputs / outputs (after that mV and mW guaranteed to be dense)
		if( mX.isEmptyBlock(false) || mV.isEmptyBlock(false) || (mW !=null && mW.isEmptyBlock(false)) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//check too high additional memory requirements (fallback to sequential)
		//check too small workload in terms of flops (fallback to sequential too)
		if( !checkParColumnAgg(mX, k, true) ) {
			matrixMultChain(mX, mV, mW, ret, ct);
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing (no need to check isThreadSafe)
		ret.sparse = false;
		ret.allocateDenseBlock();

		//core matrix mult chain computation
		//(currently: always parallelization over number of rows)
		try {
			ExecutorService pool = Executors.newFixedThreadPool( k );
			ArrayList<Integer> blklens = getBalancedBlockSizesDefault(mX.rlen, k, true);
			ArrayList<MatrixMultChainTask> tasks = new ArrayList<>();
			for( int i=0, lb=0; i<blklens.size(); lb+=blklens.get(i), i++ )
				tasks.add(new MatrixMultChainTask(mX, mV, mW, ct, lb, lb+blklens.get(i)));
			//execute tasks
			List<Future<double[]>> taskret = pool.invokeAll(tasks);
			pool.shutdown();
			//aggregate partial results
			for( Future<double[]> task : taskret )
				vectAdd(task.get(), ret.getDenseBlockValues(), 0, 0, mX.clen);
		}
		catch(Exception ex) {
			throw new DMLRuntimeException(ex);
		}

		//post-processing
		ret.recomputeNonZeros();
		ret.examSparsity();

		//System.out.println("MMChain "+ct.toString()+" k="+k+" ("+mX.isInSparseFormat()+","+mX.getNumRows()+","+mX.getNumColumns()+","+mX.getNonZeros()+")x" +
		//		              "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultTransposeSelf( MatrixBlock m1, MatrixBlock ret, boolean leftTranspose )
		throws DMLRuntimeException
	{
		//check inputs / outputs
		if( m1.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		m1 = prepMatrixMultTransposeSelfInput(m1, leftTranspose);
		ret.sparse = false;
		ret.allocateDenseBlock();

		if( m1.sparse )
			matrixMultTransposeSelfSparse(m1, ret, leftTranspose, 0, ret.rlen);
		else
			matrixMultTransposeSelfDense(m1, ret, leftTranspose, 0, ret.rlen );

		//post-processing
		long nnz = copyUpperToLowerTriangle(ret);
		ret.setNonZeros(nnz);
		ret.examSparsity();

		//System.out.println("TSMM ("+m1.isInSparseFormat()+","+m1.getNumRows()+","+m1.getNumColumns()+","+m1.getNonZeros()+","+leftTranspose+") in "+time.stop());
	}

	public static void matrixMultTransposeSelf( MatrixBlock m1, MatrixBlock ret, boolean leftTranspose, int k )
		throws DMLRuntimeException
	{
		//check inputs / outputs
		if( m1.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//check no parallelization benefit (fallback to sequential)
		//check too small workload in terms of flops (fallback to sequential too)
		if( ret.rlen == 1 || k <= 1
			|| leftTranspose && 1L * m1.rlen * m1.clen * m1.clen < PAR_MINFLOP_THRESHOLD
			|| !leftTranspose && 1L * m1.clen * m1.rlen * m1.rlen < PAR_MINFLOP_THRESHOLD)
		{
			matrixMultTransposeSelf(m1, ret, leftTranspose);
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing (no need to check isThreadSafe)
		m1 = prepMatrixMultTransposeSelfInput(m1, leftTranspose);
		ret.sparse = false;
		ret.allocateDenseBlock();

		//core multi-threaded matrix mult computation
		try {
			ExecutorService pool = Executors.newFixedThreadPool( k );
			ArrayList<MatrixMultTransposeTask> tasks = new ArrayList<>();
			//load balance via #tasks=2k due to triangular shape
			int blklen = (int)(Math.ceil((double)ret.rlen/(2*k)));
			for( int i=0; i<2*k & i*blklen<ret.rlen; i++ )
				tasks.add(new MatrixMultTransposeTask(m1, ret, leftTranspose, i*blklen, Math.min((i+1)*blklen, ret.rlen)));
			List<Future<Object>> rtasks = pool.invokeAll(tasks);
			pool.shutdown();
			for( Future<Object> rtask : rtasks )
				rtask.get(); //error handling
		}
		catch(Exception ex) {
			throw new DMLRuntimeException(ex);
		}

		//post-processing
		long nnz = copyUpperToLowerTriangle(ret);
		ret.setNonZeros(nnz);
		ret.examSparsity();

		//System.out.println("TSMM k="+k+" ("+m1.isInSparseFormat()+","+m1.getNumRows()+","+m1.getNumColumns()+","+m1.getNonZeros()+","+leftTranspose+") in "+time.stop());
	}

	public static void matrixMultPermute( MatrixBlock pm1, MatrixBlock m2, MatrixBlock ret1, MatrixBlock ret2 )
		throws DMLRuntimeException
	{
		//check inputs / outputs
		if( pm1.isEmptyBlock(false) || m2.isEmptyBlock(false) )
			return;

		//Timing time = new Timing(true);

		//pre-processing
		ret1.sparse = (m2.sparse || ret1.sparse);
		if( ret1.sparse )
			ret1.allocateSparseRowsBlock();
		else
			ret1.allocateDenseBlock();

		//core permutation mm computation
		if( m2.sparse )
			matrixMultPermuteSparse(pm1, m2, ret1, ret2, 0, pm1.rlen);
		else if( ret1.sparse )
			matrixMultPermuteDenseSparse(pm1, m2, ret1, ret2, 0, pm1.rlen);
		else
			matrixMultPermuteDense(pm1, m2, ret1, ret2, 0, pm1.rlen);

		//post-processing
		ret1.recomputeNonZeros();
		ret1.examSparsity();
		if( ret2 != null ) { //optional second output
			ret2.recomputeNonZeros();
			ret2.examSparsity();
		}

		//System.out.println("PMM Seq ("+pm1.isInSparseFormat()+","+pm1.getNumRows()+","+pm1.getNumColumns()+","+pm1.getNonZeros()+")x" +
		//                  "("+m2.isInSparseFormat()+","+m2.getNumRows()+","+m2.getNumColumns()+","+m2.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultPermute( MatrixBlock pm1, MatrixBlock m2, MatrixBlock ret1, MatrixBlock ret2, int k)
		throws DMLRuntimeException
	{
		//check inputs / outputs
		if( pm1.isEmptyBlock(false) || m2.isEmptyBlock(false) )
			return;

		//check no parallelization benefit (fallback to sequential)
		if (pm1.rlen == 1) {
			matrixMultPermute(pm1, m2, ret1, ret2);
			return;
		}

		//Timing time = new Timing(true);

		//allocate first output block (second allocated if needed)
		ret1.sparse = false;	  // no need to check isThreadSafe
		ret1.allocateDenseBlock();

		try
		{
			ExecutorService pool = Executors.newFixedThreadPool(k);
			ArrayList<MatrixMultPermuteTask> tasks = new ArrayList<>();
			int blklen = (int)(Math.ceil((double)pm1.rlen/k));
			for( int i=0; i<k & i*blklen<pm1.rlen; i++ )
				tasks.add(new MatrixMultPermuteTask(pm1, m2, ret1, ret2, i*blklen, Math.min((i+1)*blklen, pm1.rlen)));
			pool.invokeAll(tasks);
			pool.shutdown();
		}
		catch (InterruptedException e) {
			throw new DMLRuntimeException(e);
		}

		//post-processing
		ret1.recomputeNonZeros();
		ret1.examSparsity();
		if( ret2 != null ) { //optional second output
			ret2.recomputeNonZeros();
			ret2.examSparsity();
		}

		// System.out.println("PMM Par ("+pm1.isInSparseFormat()+","+pm1.getNumRows()+","+pm1.getNumColumns()+","+pm1.getNonZeros()+")x" +
		//                   "("+m2.isInSparseFormat()+","+m2.getNumRows()+","+m2.getNumColumns()+","+m2.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWSLoss(MatrixBlock mX, MatrixBlock mU, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, WeightsType wt)
		throws DMLRuntimeException
	{
		//check for empty result
		if( wt==WeightsType.POST && mW.isEmptyBlock(false)
			|| wt==WeightsType.POST_NZ && mX.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//core weighted square sum mm computation
		if( !mX.sparse && !mU.sparse && !mV.sparse && (mW==null || !mW.sparse)
			&& !mX.isEmptyBlock() && !mU.isEmptyBlock() && !mV.isEmptyBlock() && (mW==null || !mW.isEmptyBlock()))
			matrixMultWSLossDense(mX, mU, mV, mW, ret, wt, 0, mX.rlen);
		else if( mX.sparse && !mU.sparse && !mV.sparse && (mW==null || mW.sparse)
				&& !mX.isEmptyBlock() && !mU.isEmptyBlock() && !mV.isEmptyBlock() && (mW==null || !mW.isEmptyBlock()))
			matrixMultWSLossSparseDense(mX, mU, mV, mW, ret, wt, 0, mX.rlen);
		else
			matrixMultWSLossGeneric(mX, mU, mV, mW, ret, wt, 0, mX.rlen);

		//add correction for sparse wsloss w/o weight
		if( mX.sparse && wt==WeightsType.NONE )
			addMatrixMultWSLossNoWeightCorrection(mU, mV, ret, 1);

		//System.out.println("MMWSLoss " +wt.toString()+ " ("+mX.isInSparseFormat()+","+mX.getNumRows()+","+mX.getNumColumns()+","+mX.getNonZeros()+")x" +
		//                  "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWSLoss(MatrixBlock mX, MatrixBlock mU, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, WeightsType wt, int k)
		throws DMLRuntimeException
	{
		//check for empty result
		if( wt==WeightsType.POST && mW.isEmptyBlock(false)
			|| wt==WeightsType.POST_NZ && mX.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//check no parallelization benefit (fallback to sequential)
		//no need to check isThreadSafe (scalar output)
		if( mX.rlen == 1 ) {
			matrixMultWSLoss(mX, mU, mV, mW, ret, wt);
			return;
		}

		//Timing time = new Timing(true);

		try
		{
			ExecutorService pool = Executors.newFixedThreadPool(k);
			ArrayList<MatrixMultWSLossTask> tasks = new ArrayList<>();
			int blklen = (int)(Math.ceil((double)mX.rlen/k));
			for( int i=0; i<k & i*blklen<mX.rlen; i++ )
				tasks.add(new MatrixMultWSLossTask(mX, mU, mV, mW, wt, i*blklen, Math.min((i+1)*blklen, mX.rlen)));
			List<Future<Double>> taskret = pool.invokeAll(tasks);
			pool.shutdown();
			//aggregate partial results
			sumScalarResults(taskret, ret);
		}
		catch( Exception e ) {
			throw new DMLRuntimeException(e);
		}

		//add correction for sparse wsloss w/o weight
		if( mX.sparse && wt==WeightsType.NONE )
			addMatrixMultWSLossNoWeightCorrection(mU, mV, ret, k);

		//System.out.println("MMWSLoss "+wt.toString()+" k="+k+" ("+mX.isInSparseFormat()+","+mX.getNumRows()+","+mX.getNumColumns()+","+mX.getNonZeros()+")x" +
		//                   "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWSigmoid(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WSigmoidType wt)
		throws DMLRuntimeException
	{
		//check for empty result
		if( mW.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		ret.sparse = mW.sparse;
		ret.allocateBlock();

		//core weighted square sum mm computation
		if( !mW.sparse && !mU.sparse && !mV.sparse && !mU.isEmptyBlock() && !mV.isEmptyBlock() )
			matrixMultWSigmoidDense(mW, mU, mV, ret, wt, 0, mW.rlen);
		else if( mW.sparse && !mU.sparse && !mV.sparse && !mU.isEmptyBlock() && !mV.isEmptyBlock())
			matrixMultWSigmoidSparseDense(mW, mU, mV, ret, wt, 0, mW.rlen);
		else
			matrixMultWSigmoidGeneric(mW, mU, mV, ret, wt, 0, mW.rlen);

		//post-processing
		ret.recomputeNonZeros();
		ret.examSparsity();

		//System.out.println("MMWSig "+wt.toString()+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                 "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWSigmoid(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WSigmoidType wt, int k)
		throws DMLRuntimeException
	{
		//check for empty result
		if( mW.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//check no parallelization benefit (fallback to sequential)
		if (mW.rlen == 1 || !MatrixBlock.isThreadSafe(mW.sparse)) {
			matrixMultWSigmoid(mW, mU, mV, ret, wt);
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		ret.sparse = mW.sparse;
		ret.allocateBlock();

		try
		{
			ExecutorService pool = Executors.newFixedThreadPool(k);
			ArrayList<MatrixMultWSigmoidTask> tasks = new ArrayList<>();
			int blklen = (int)(Math.ceil((double)mW.rlen/k));
			for( int i=0; i<k & i*blklen<mW.rlen; i++ )
				tasks.add(new MatrixMultWSigmoidTask(mW, mU, mV, ret, wt, i*blklen, Math.min((i+1)*blklen, mW.rlen)));
			//execute tasks
			List<Future<Long>> taskret = pool.invokeAll(tasks);
			pool.shutdown();
			//aggregate partial nnz and check for errors
			ret.nonZeros = 0; //reset after execute
			for( Future<Long> task : taskret )
				ret.nonZeros += task.get();
		}
		catch (Exception e) {
			throw new DMLRuntimeException(e);
		}

		//post-processing (nnz maintained in parallel)
		ret.examSparsity();

		//System.out.println("MMWSig "+wt.toString()+" k="+k+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                   "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop() + ".");
	}

	/**
	 * NOTE: This operation has limited NaN support, which is acceptable because all our sparse-safe operations
	 * have only limited NaN support. If this is not intended behavior, please disable the rewrite. In detail,
	 * this operator will produce for W/(U%*%t(V)) a zero intermediate for each zero in W (even if UVij is zero
	 * which would give 0/0=NaN) but INF/-INF for non-zero entries in V where the corresponding cell in (Y%*%X)
	 * is zero.
	 *
	 * @param mW matrix W
	 * @param mU matrix U
	 * @param mV matrix V
	 * @param mX matrix X
	 * @param ret result type
	 * @param wt weighted divide matrix multiplication type
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	public static void matrixMultWDivMM(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock mX, MatrixBlock ret, WDivMMType wt)
		throws DMLRuntimeException
	{
		//check for empty result
		if(   mW.isEmptyBlock(false)
		   || (wt.isLeft() && mU.isEmptyBlock(false))
		   || (wt.isRight() && mV.isEmptyBlock(false))
		   || (wt.isBasic() && mW.isEmptyBlock(false)))  {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		ret.sparse = wt.isBasic()?mW.sparse:false;
		ret.allocateBlock();

		//core weighted div mm computation
		boolean scalarX = wt.hasScalar();
		if( !mW.sparse && !mU.sparse && !mV.sparse && (mX==null || !mX.sparse || scalarX) && !mU.isEmptyBlock() && !mV.isEmptyBlock() )
			matrixMultWDivMMDense(mW, mU, mV, mX, ret, wt, 0, mW.rlen, 0, mW.clen);
		else if( mW.sparse && !mU.sparse && !mV.sparse && (mX==null || mX.sparse || scalarX) && !mU.isEmptyBlock() && !mV.isEmptyBlock())
			matrixMultWDivMMSparseDense(mW, mU, mV, mX, ret, wt, 0, mW.rlen, 0, mW.clen);
		else
			matrixMultWDivMMGeneric(mW, mU, mV, mX, ret, wt, 0, mW.rlen, 0, mW.clen);

		//post-processing
		ret.recomputeNonZeros();
		ret.examSparsity();

		//System.out.println("MMWDiv "+wt.toString()+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                 "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	/**
	 * NOTE: This operation has limited NaN support, which is acceptable because all our sparse-safe operations
	 * have only limited NaN support. If this is not intended behavior, please disable the rewrite. In detail,
	 * this operator will produce for W/(U%*%t(V)) a zero intermediate for each zero in W (even if UVij is zero
	 * which would give 0/0=NaN) but INF/-INF for non-zero entries in V where the corresponding cell in (Y%*%X)
	 * is zero.
	 *
	 * @param mW matrix W
	 * @param mU matrix U
	 * @param mV matrix V
	 * @param mX matrix X
	 * @param ret result matrix
	 * @param wt weighted divide matrix multiplication type
	 * @param k maximum parallelism
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	public static void matrixMultWDivMM(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock mX, MatrixBlock ret, WDivMMType wt, int k)
		throws DMLRuntimeException
	{
		//check for empty result
		if(   mW.isEmptyBlock(false)
		   || (wt.isLeft() && mU.isEmptyBlock(false))
		   || (wt.isRight() && mV.isEmptyBlock(false))
		   || (wt.isBasic() && mW.isEmptyBlock(false)))  {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		ret.sparse = wt.isBasic()?mW.sparse:false;
		ret.allocateBlock();

		if (!ret.isThreadSafe()){
			matrixMultWDivMM(mW, mU, mV, mX, ret, wt);
			return;
		}

		try
		{
			ExecutorService pool = Executors.newFixedThreadPool(k);
			ArrayList<MatrixMultWDivTask> tasks = new ArrayList<>();
			//create tasks (for wdivmm-left, parallelization over columns;
			//for wdivmm-right, parallelization over rows; both ensure disjoint results)
			if( wt.isLeft() ) {
				int blklen = (int)(Math.ceil((double)mW.clen/k));
				for( int j=0; j<k & j*blklen<mW.clen; j++ )
					tasks.add(new MatrixMultWDivTask(mW, mU, mV, mX, ret, wt, 0, mW.rlen, j*blklen, Math.min((j+1)*blklen, mW.clen)));
			}
			else { //basic/right
				int blklen = (int)(Math.ceil((double)mW.rlen/k));
				for( int i=0; i<k & i*blklen<mW.rlen; i++ )
					tasks.add(new MatrixMultWDivTask(mW, mU, mV, mX, ret, wt, i*blklen, Math.min((i+1)*blklen, mW.rlen), 0, mW.clen));
			}
			//execute tasks
			List<Future<Long>> taskret = pool.invokeAll(tasks);
			pool.shutdown();
			//aggregate partial nnz and check for errors
			ret.nonZeros = 0;  //reset after execute
			for( Future<Long> task : taskret )
				ret.nonZeros += task.get();
		}
		catch (Exception e) {
			throw new DMLRuntimeException(e);
		}

		//post-processing
		ret.examSparsity();

		//System.out.println("MMWDiv "+wt.toString()+" k="+k+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWCeMM(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, double eps, MatrixBlock ret, WCeMMType wt)
		throws DMLRuntimeException
	{
		//check for empty result
		if( mW.isEmptyBlock(false) )  {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		ret.sparse = false;
		ret.allocateDenseBlock();

		//core weighted cross entropy mm computation
		if( !mW.sparse && !mU.sparse && !mV.sparse && !mU.isEmptyBlock() && !mV.isEmptyBlock() )
			matrixMultWCeMMDense(mW, mU, mV, eps, ret, wt, 0, mW.rlen);
		else if( mW.sparse && !mU.sparse && !mV.sparse && !mU.isEmptyBlock() && !mV.isEmptyBlock())
			matrixMultWCeMMSparseDense(mW, mU, mV, eps, ret, wt, 0, mW.rlen);
		else
			matrixMultWCeMMGeneric(mW, mU, mV, eps, ret, wt, 0, mW.rlen);

		//System.out.println("MMWCe "+wt.toString()+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                 "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWCeMM(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, double eps, MatrixBlock ret, WCeMMType wt, int k)
		throws DMLRuntimeException
	{
		//check for empty result
		if( mW.isEmptyBlock(false) )  {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing (no need to check isThreadSafe)
		ret.sparse = false;
		ret.allocateDenseBlock();

		try
		{
			ExecutorService pool = Executors.newFixedThreadPool(k);
			ArrayList<MatrixMultWCeTask> tasks = new ArrayList<>();
			int blklen = (int)(Math.ceil((double)mW.rlen/k));
			for( int i=0; i<k & i*blklen<mW.rlen; i++ )
				tasks.add(new MatrixMultWCeTask(mW, mU, mV, eps, wt, i*blklen, Math.min((i+1)*blklen, mW.rlen)));
			List<Future<Double>> taskret = pool.invokeAll(tasks);
			pool.shutdown();
			//aggregate partial results
			sumScalarResults(taskret, ret);
		}
		catch( Exception e ) {
			throw new DMLRuntimeException(e);
		}

		//System.out.println("MMWCe "+wt.toString()+" k="+k+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                 "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWuMM(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WUMMType wt, ValueFunction fn)
		throws DMLRuntimeException
	{
		//check for empty result
		if( mW.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		ret.sparse = mW.sparse;
		ret.allocateBlock();

		//core weighted square sum mm computation
		if( !mW.sparse && !mU.sparse && !mV.sparse && !mU.isEmptyBlock() && !mV.isEmptyBlock() )
			matrixMultWuMMDense(mW, mU, mV, ret, wt, fn, 0, mW.rlen);
		else if( mW.sparse && !mU.sparse && !mV.sparse && !mU.isEmptyBlock() && !mV.isEmptyBlock())
			matrixMultWuMMSparseDense(mW, mU, mV, ret, wt, fn, 0, mW.rlen);
		else
			matrixMultWuMMGeneric(mW, mU, mV, ret, wt, fn, 0, mW.rlen);

		//post-processing
		ret.recomputeNonZeros();
		ret.examSparsity();

		//System.out.println("MMWu "+wt.toString()+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                 "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop());
	}

	public static void matrixMultWuMM(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WUMMType wt, ValueFunction fn, int k)
		throws DMLRuntimeException
	{
		//check for empty result
		if( mW.isEmptyBlock(false) ) {
			ret.examSparsity(); //turn empty dense into sparse
			return;
		}

		//check no parallelization benefit (fallback to sequential)
		if (mW.rlen == 1 || !MatrixBlock.isThreadSafe(mW.sparse)) {
			matrixMultWuMM(mW, mU, mV, ret, wt, fn);
			return;
		}

		//Timing time = new Timing(true);

		//pre-processing
		ret.sparse = mW.sparse;
		ret.allocateBlock();

		try
		{
			ExecutorService pool = Executors.newFixedThreadPool(k);
			ArrayList<MatrixMultWuTask> tasks = new ArrayList<>();
			int blklen = (int)(Math.ceil((double)mW.rlen/k));
			for( int i=0; i<k & i*blklen<mW.rlen; i++ )
				tasks.add(new MatrixMultWuTask(mW, mU, mV, ret, wt, fn, i*blklen, Math.min((i+1)*blklen, mW.rlen)));
			//execute tasks
			List<Future<Long>> taskret = pool.invokeAll(tasks);
			pool.shutdown();
			//aggregate partial nnz and check for errors
			ret.nonZeros = 0; //reset after execute
			for( Future<Long> task : taskret )
				ret.nonZeros += task.get();
		}
		catch (Exception e) {
			throw new DMLRuntimeException(e);
		}

		//post-processing (nnz maintained in parallel)
		ret.examSparsity();

		//System.out.println("MMWu "+wt.toString()+" k="+k+" ("+mW.isInSparseFormat()+","+mW.getNumRows()+","+mW.getNumColumns()+","+mW.getNonZeros()+")x" +
		//                   "("+mV.isInSparseFormat()+","+mV.getNumRows()+","+mV.getNumColumns()+","+mV.getNonZeros()+") in "+time.stop() + ".");
	}

	//////////////////////////////////////////
	// optimized matrix mult implementation //
	//////////////////////////////////////////

	private static void matrixMultDenseDense(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, boolean tm2, boolean pm2, int rl, int ru, int cl, int cu)
		throws DMLRuntimeException
	{
		DenseBlock a = m1.getDenseBlock();
		DenseBlock b = m2.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();
		final int m = m1.rlen;
		final int n = m2.clen;
		final int cd = m1.clen;

		if( LOW_LEVEL_OPTIMIZATION ) {
			if( m==1 && n==1 ) {            //DOT PRODUCT
				double[] avals = a.valuesAt(0);
				double[] bvals = b.valuesAt(0);
				c.set(0, 0, dotProduct(avals, bvals, cd));
			}
			else if( n>1 && cd == 1 ) {     //OUTER PRODUCT
				double[] avals = a.valuesAt(0);
				double[] bvals = b.valuesAt(0);
				for( int i=rl; i < ru; i++) {
					double[] cvals = c.values(i);
					int cix = c.pos(i);
					if( avals[i] == 1 )
						System.arraycopy(bvals, 0, cvals, cix, n);
					else if( avals[i] != 0 )
						vectMultiplyWrite(avals[i], bvals, cvals, 0, cix, n);
					else
						Arrays.fill(cvals, cix, cix+n, 0);
				}
			}
			else if( n==1 && cd == 1 ) {    //VECTOR-SCALAR
				double[] avals = a.valuesAt(0);
				double[] cvals = c.valuesAt(0);
				vectMultiplyWrite(b.get(0,0), avals, cvals, rl, rl, ru-rl);
			}
			else if( n==1 && cd<=2*1024 ) { //MATRIX-VECTOR (short rhs)
				matrixMultDenseDenseMVShortRHS(a, b, c, cd, rl, ru);
			}
			else if( n==1 ) {               //MATRIX-VECTOR (tall rhs)
				matrixMultDenseDenseMVTallRHS(a, b, c, cd, rl, ru);
			}
			else if( pm2 && m==1 ) {        //VECTOR-MATRIX
				matrixMultDenseDenseVM(a, b, c, n, cd, rl, ru);
			}
			else if( pm2 && m<=16 ) {       //MATRIX-MATRIX (short lhs)
				matrixMultDenseDenseMMShortLHS(a, b, c, m, n, cd, rl, ru);
			}
			else if( tm2 ) {                //MATRIX-MATRIX (skinny rhs)
				matrixMultDenseDenseMMSkinnyRHS(a, b, c, m2.rlen, cd, rl, ru);
			}
			else {                          //MATRIX-MATRIX
				matrixMultDenseDenseMM(a, b, c, n, cd, rl, ru, cl, cu);
			}
		}
		else {
			for( int i = rl; i < ru; i++) {
				double[] avals = a.values(i);
				double[] cvals = c.values(i);
				int aix = a.pos(i), cix = c.pos(i);
				for( int k = 0; k < cd; k++) {
					double val = avals[aix + k];
					if( val != 0 ) {
						double[] bvals = b.values(k);
						int bix = b.pos(k);
						for( int j = 0; j < n; j++)
							cvals[cix+j] += val * bvals[bix+j];
					}
				}
			}
		}
	}

	private static void matrixMultDenseDenseMVShortRHS(DenseBlock a, DenseBlock b, DenseBlock c, int cd, int rl, int ru)
		throws DMLRuntimeException
	{
		double[] bvals = b.valuesAt(0);
		double[] cvals = c.valuesAt(0);
		for( int i=rl; i < ru; i++ )
			cvals[i] = dotProduct(a.values(i), bvals, a.pos(i), 0, cd);
	}

	private static void matrixMultDenseDenseMVTallRHS(DenseBlock a, DenseBlock b, DenseBlock c, int cd, int rl, int ru)
		throws DMLRuntimeException
	{
		final int blocksizeI = 32;
		final int blocksizeK = 2*1024; //16KB vector blocks (L1)
		double[] bvals = b.valuesAt(0);
		double[] cvals = c.valuesAt(0);
		for( int bi=rl; bi<ru; bi+=blocksizeI ) {
			int bimin = Math.min(bi+blocksizeI, ru);
			for( int bk=0; bk<cd; bk+=blocksizeK ) {
				int bkmin = Math.min(bk+blocksizeK, cd);
				for( int i=bi; i<bimin; i++)
					cvals[i] += dotProduct(a.values(i), bvals, a.pos(i,bk), bk, bkmin-bk);
			}
		}
	}

	private static void matrixMultDenseDenseVM(DenseBlock a, DenseBlock b, DenseBlock c, int n, int cd, int rl, int ru)
		throws DMLRuntimeException
	{
		double[] avals = a.valuesAt(0); //vector
		double[] cvals = c.valuesAt(0); //vector

		//parallelization over rows in rhs matrix
		//rest not aligned to blocks of 2 rows
		final int kn = b.isContiguous() ? rl+(ru-rl)%2 : ru;
		for( int k = rl; k < kn; k++ )
			if( avals[k] != 0 )
				vectMultiplyAdd(avals[k], b.values(k), cvals, b.pos(k), 0, n);

		//compute blocks of 2 rows (2 instead of 4 for small n<64)
		double[] bvals = b.valuesAt(0); //only for special case
		for( int k=kn, bix=kn*n; k<ru; k+=2, bix+=2*n ){
			if( avals[k] != 0 && avals[k+1] != 0  )
				vectMultiplyAdd2(avals[k], avals[k+1], bvals, cvals, bix, bix+n, 0, n);
			else if( avals[k] != 0 )
				vectMultiplyAdd(avals[k], bvals, cvals, bix, 0, n);
			else if( avals[k+1] != 0 )
				vectMultiplyAdd(avals[k+1], bvals, cvals, bix+n, 0, n);
		}
	}

	private static void matrixMultDenseDenseMMShortLHS(DenseBlock a, DenseBlock b, DenseBlock c, int m, int n, int cd, int rl, int ru)
		throws DMLRuntimeException
	{
		//cache-conscious parallelization over rows in rhs matrix
		final int kn = (ru-rl)%4;

		//rest not aligned to blocks of 2 rows
		for( int i=0; i<m; i++ ) {
			double[] avals = a.values(i), cvals = c.values(i);
			int aix = a.pos(i), cix = c.pos(i);
			for( int k=rl; k<rl+kn; k++ )
				if( avals[aix+k] != 0 )
					vectMultiplyAdd(avals[aix+k], b.values(k), cvals, b.pos(k), cix, n);
		}

		final int blocksizeK = 48;
		final int blocksizeJ = 1024;

		//blocked execution
		for( int bk = rl+kn; bk < ru; bk+=blocksizeK ) {
			int bkmin = Math.min(ru, bk+blocksizeK);
			for( int bj = 0; bj < n; bj+=blocksizeJ ) {
				//compute blocks of 4 rows in rhs w/ IKJ
				int bjlen = Math.min(n, bj+blocksizeJ)-bj;
				for( int i=0; i<m; i++ ) {
					double[] avals = a.values(i), cvals = c.values(i);
					int aix = a.pos(i), cix = c.pos(i, bj);
					if( b.isContiguous(bk, bkmin-1) ) {
						double[] bvals = b.values(bk);
						for( int k=bk, bix=b.pos(bk, bj); k<bkmin; k+=4, bix+=4*n )
							vectMultiplyAdd4(avals[aix+k], avals[aix+k+1], avals[aix+k+2], avals[aix+k+3],
								bvals, cvals, bix, bix+n, bix+2*n, bix+3*n, cix, bjlen);
					}
					else {
						for( int k=rl; k<rl+kn; k++ )
							if( avals[aix+k] != 0 )
								vectMultiplyAdd(avals[aix+k], b.values(k), cvals, b.pos(k), cix, n);
					}
				}
			}
		}
	}

	private static void matrixMultDenseDenseMMSkinnyRHS(DenseBlock a, DenseBlock b, DenseBlock c, int n2, int cd, int rl, int ru)
		throws DMLRuntimeException
	{
		//note: prepared rhs input via transpose for: m > n && cd > 64 && n < 64
		//however, explicit flag required since dimension change m2
		for( int i=rl; i < ru; i++ ) {
			double[] avals = a.values(i), cvals = c.values(i);
			int aix = a.pos(i), cix = c.pos(i);
			for( int j=0; j<n2; j++ )
				cvals[cix+j] = dotProduct(avals, b.values(j), aix, b.pos(j), cd);
		}
	}

	private static void matrixMultDenseDenseMM(DenseBlock a, DenseBlock b, DenseBlock c, int n, int cd, int rl, int ru, int cl, int cu)
		throws DMLRuntimeException
	{
		//1) Unrolled inner loop (for better instruction-level parallelism)
		//2) Blocked execution (for less cache trashing in parallel exec)
		//3) Asymmetric block sizes (for less misses in inner loop, yet blocks in L1/L2)

		final int blocksizeI = 32; //64//256KB c block (typical L2 size per core), 32KB a block
		final int blocksizeK = 24; //64//256KB b block (typical L2 size per core), used while read 512B of a / read/write 4KB of c
		final int blocksizeJ = 1024; //512//4KB (typical main-memory page size), for scan

		//temporary arrays (nnz a, b index)
		double[] ta = new double[ blocksizeK ];
		int[]  tbi  = new int[ blocksizeK ];

		//blocked execution
		for( int bi = rl; bi < ru; bi+=blocksizeI )
			for( int bk = 0, bimin = Math.min(ru, bi+blocksizeI); bk < cd; bk+=blocksizeK )
				for( int bj = cl, bkmin = Math.min(cd, bk+blocksizeK); bj < cu; bj+=blocksizeJ )
				{
					int bklen = bkmin-bk;
					int bjlen = Math.min(cu, bj+blocksizeJ)-bj;

					//core sub block matrix multiplication
					for( int i = bi; i < bimin; i++) {
						double[] avals = a.values(i), cvals = c.values(i);
						int aixi = a.pos(i, bk), cixj = c.pos(i, bj);

						if( b.isContiguous(bk, bkmin-1) ) {
							double[] bvals = b.values(bk);
							int bkpos = b.pos(bk, bj);

							//determine nnz of a (for sparsity-aware skipping of rows)
							int knnz = copyNonZeroElements(avals, aixi, bkpos, bj, n, ta, tbi, bklen);

							//rest not aligned to blocks of 4 rows
							final int bn = knnz % 4;
							switch( bn ){
								case 1: vectMultiplyAdd(ta[0], bvals, cvals, tbi[0], cixj, bjlen); break;
								case 2: vectMultiplyAdd2(ta[0],ta[1], bvals, cvals, tbi[0], tbi[1], cixj, bjlen); break;
								case 3: vectMultiplyAdd3(ta[0],ta[1],ta[2], bvals, cvals, tbi[0], tbi[1],tbi[2], cixj, bjlen); break;
							}

							//compute blocks of 4 rows (core inner loop)
							for( int k = bn; k<knnz; k+=4 ){
								vectMultiplyAdd4( ta[k], ta[k+1], ta[k+2], ta[k+3], bvals, cvals,
									tbi[k], tbi[k+1], tbi[k+2], tbi[k+3], cixj, bjlen );
							}
						}
						else {
							for( int k = bk; k<bkmin; k++ ) {
								if( avals[k] != 0 )
									vectMultiplyAdd( avals[k], b.values(k),
										cvals, b.pos(k, bj), cixj, bjlen );
							}
						}
					}
				}
	}

	private static void matrixMultDenseSparse(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, boolean pm2, int rl, int ru)
		throws DMLRuntimeException
	{
		DenseBlock a = m1.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();
		int m = m1.rlen;
		int cd = m1.clen;

		// MATRIX-MATRIX (VV, MV not applicable here because V always dense)
		if( LOW_LEVEL_OPTIMIZATION )
		{
			SparseBlock b = m2.sparseBlock;

			if( pm2 && m==1 ) {        //VECTOR-MATRIX
				//parallelization over rows in rhs matrix
				double[] avals = a.valuesAt(0); //vector
				double[] cvals = c.valuesAt(0); //vector
				for( int k=rl; k<ru; k++ )
					if( avals[k] != 0 && !b.isEmpty(k) ) {
						vectMultiplyAdd(avals[k], b.values(k), cvals,
							b.indexes(k), b.pos(k), 0, b.size(k));
					}
			}
			else {                     //MATRIX-MATRIX
				//best effort blocking, without blocking over J because it is
				//counter-productive, even with front of current indexes
				final int blocksizeK = 32;
				final int blocksizeI = 32;

				//blocked execution
				for( int bi = rl; bi < ru; bi+=blocksizeI )
					for( int bk = 0, bimin = Math.min(ru, bi+blocksizeI); bk < cd; bk+=blocksizeK ) {
						int bkmin = Math.min(cd, bk+blocksizeK);
						//core sub block matrix multiplication
						for(int i = bi; i < bimin; i++) {
							double[] avals = a.values(i), cvals = c.values(i);
							int aix = a.pos(i), cix = c.pos(i);
							for( int k = bk; k < bkmin; k++ ) {
								double aval = avals[aix+k];
								if( aval == 0 || b.isEmpty(k) )
									continue;
								vectMultiplyAdd(aval, b.values(k), cvals,
									b.indexes(k), b.pos(k), cix, b.size(k));
							}
						}
					}
			}
		}
		else {
			SparseBlock b = m2.sparseBlock;
			for( int i=rl; i < ru; i++ ) {
				double[] avals = a.values(i), cvals = c.values(i);
				int aix = a.pos(i), cix = c.pos(i);
				for(int k = 0; k < cd; k++ ) {
					double val = avals [aix];
					if( val == 0 || b.isEmpty(k) ) continue;
					int bpos = b.pos(k);
					int blen = b.size(k);
					int[] bix = b.indexes(k);
					double[] bvals = b.values(k);
					for(int j = bpos; j < bpos+blen; j++)
						cvals[cix+bix[j]] += val * bvals[j];
				}
			}
		}
	}

	private static void matrixMultSparseDense(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, boolean pm2, int rl, int ru)
		throws DMLRuntimeException
	{
		SparseBlock a = m1.sparseBlock;
		DenseBlock b = m2.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();
		final int m = m1.rlen;
		final int n = m2.clen;
		final int cd = m2.rlen;
		final long xsp = (long)m*cd/m1.nonZeros;

		if( LOW_LEVEL_OPTIMIZATION ) {
			if( m==1 && n==1 ) {            //DOT PRODUCT
				if( !a.isEmpty(0) )
					c.set(0, 0, dotProduct(a.values(0), b.values(0), a.indexes(0), a.pos(0), 0, a.size(0)));
			}
			else if( n==1 && cd<=2*1024 ) { //MATRIX-VECTOR (short rhs)
				matrixMultSparseDenseMVShortRHS(a, b, c, rl, ru);
			}
			else if( n==1 ) {               //MATRIX-VECTOR (tall rhs)
				matrixMultSparseDenseMVTallRHS(a, b, c, cd, xsp, rl, ru);
			}
			else if( pm2 && m==1 ) {        //VECTOR-MATRIX
				matrixMultSparseDenseVM(a, b, c, n, rl, ru);
			}
			else if( pm2 && m<=16 ) {       //MATRIX-MATRIX (short lhs)
				matrixMultSparseDenseMMShortLHS(a, b, c, n, cd, rl, ru);
			}
			else if( n<=64 ) {              //MATRIX-MATRIX (skinny rhs)
				matrixMultSparseDenseMMSkinnyRHS(a, b, c, n, rl, ru);
			}
			else {                          //MATRIX-MATRIX
				matrixMultSparseDenseMM(a, b, c, n, cd, xsp, rl, ru);
			}
		}
		else {
			for( int i=rl; i<ru; i++ ) {
				if( a.isEmpty(i) ) continue;
				int apos = a.pos(i);
				int alen = a.size(i);
				int[] aix = a.indexes(i);
				double[] avals = a.values(i);
				double[] cvals = c.values(i);
				int cix = c.pos(i);
				for(int k = apos; k < apos+alen; k++) {
					double val = avals[k];
					double[] bvals = b.values(k);
					int bix = b.pos(k);
					for(int j = 0; j < n; j++)
						cvals[cix+j] += val * bvals[bix+j];
				}
			}
		}
	}

	private static void matrixMultSparseDenseMVShortRHS(SparseBlock a, DenseBlock b, DenseBlock c, int rl, int ru)
		throws DMLRuntimeException
	{
		double[] bvals = b.valuesAt(0);
		double[] cvals = c.valuesAt(0);
		for( int i=rl; i<ru; i++ )
			if( !a.isEmpty(i) )
				cvals[i] = dotProduct(a.values(i), bvals,
					a.indexes(i), a.pos(i), 0, a.size(i));
	}

	private static void matrixMultSparseDenseMVTallRHS(SparseBlock a, DenseBlock b, DenseBlock c, int cd, long xsp, int rl, int ru)
		throws DMLRuntimeException
	{
		double[] bvals = b.valuesAt(0);
		double[] cvals = c.valuesAt(0);

		final int blocksizeI = 32;
		final int blocksizeK = (int)Math.max(2*1024,2*1024*xsp/32); //~ 16KB L1
		int[] curk = new int[blocksizeI];

		for( int bi = rl; bi < ru; bi+=blocksizeI ) {
			Arrays.fill(curk, 0); //reset positions
			for( int bk=0, bimin = Math.min(ru, bi+blocksizeI); bk<cd; bk+=blocksizeK ) {
				for( int i=bi, bkmin = Math.min(bk+blocksizeK, cd); i<bimin; i++) {
					if( a.isEmpty(i) ) continue;
					int apos = a.pos(i);
					int alen = a.size(i);
					int[] aix = a.indexes(i);
					double[] avals = a.values(i);
					int k = curk[i-bi] + apos;
					for( ; k<apos+alen && aix[k]<bkmin; k++ )
						cvals[i] += avals[k] * bvals[aix[k]];
					curk[i-bi] = k - apos;
				}
			}
		}
	}

	private static void matrixMultSparseDenseVM(SparseBlock a, DenseBlock b, DenseBlock c, int n, int rl, int ru)
		throws DMLRuntimeException
	{
		if( a.isEmpty(0) )
			return;

		//parallelization over rows in rhs matrix
		int alen = a.size(0);
		int[] aix = a.indexes(0);
		double[] avals = a.values(0);
		double[] cvals = c.valuesAt(0);
		int rlix = (rl==0) ? 0 : a.posFIndexGTE(0,rl);
		rlix = (rlix>=0) ? rlix : alen;

		if( b.isContiguous() ) {
			double[] bvals = b.valuesAt(0);
			for( int k=rlix; k<alen && aix[k]<ru; k++ )
				if( k+1<alen && aix[k+1]<ru )
					vectMultiplyAdd2(avals[k], avals[k+1], bvals, cvals, aix[k]*n, aix[++k]*n, 0, n);
				else
					vectMultiplyAdd(avals[k], bvals, cvals, aix[k]*n, 0, n);
		}
		else {
			for( int k=rlix; k<alen && aix[k]<ru; k++ )
				vectMultiplyAdd(avals[k], b.values(aix[k]), cvals, b.pos(aix[k]), 0, n);
		}
	}

	private static void matrixMultSparseDenseMMShortLHS(SparseBlock a, DenseBlock b, DenseBlock c, int n, int cd, int rl, int ru)
		throws DMLRuntimeException
	{
		int arlen = a.numRows();
		for( int i=0; i<arlen; i++ ) {
			if( a.isEmpty(i) ) continue;
			int apos = a.pos(i);
			int alen = a.size(i);
			int[] aix = a.indexes(i);
			double[] avals = a.values(i);
			double[] cvals = c.values(i);
			int cix = c.pos(i);

			int k1 = (rl==0) ? 0 : a.posFIndexGTE(i, rl);
			k1 = (k1>=0) ? apos+k1 : apos+alen;
			int k2 = (ru==cd) ? alen : a.posFIndexGTE(i, ru);
			k2 = (k2>=0) ? apos+k2 : apos+alen;

			if( k1==k2 || b.isContiguous(aix[k1], aix[k2-1]) ) {
				double[] bvals = b.values(aix[k1]);
				int base = aix[k1]*n - b.pos(aix[k1]);
				//rest not aligned to blocks of 4 rows
				final int bn = (k2-k1) % 4;
				switch( bn ){
					case 1: vectMultiplyAdd(avals[k1], bvals, cvals, aix[k1]*n-base, cix, n); break;
					case 2: vectMultiplyAdd2(avals[k1],avals[k1+1], bvals, cvals, aix[k1]*n-base, aix[k1+1]*n-base, cix, n); break;
					case 3: vectMultiplyAdd3(avals[k1],avals[k1+1],avals[k1+2], bvals, cvals, aix[k1]*n-base, aix[k1+1]*n-base, aix[k1+2]*n-base, cix, n); break;
				}

				//compute blocks of 4 rows (core inner loop)
				for( int k = k1+bn; k<k2; k+=4 ) {
					vectMultiplyAdd4( avals[k], avals[k+1], avals[k+2], avals[k+3], bvals, cvals,
						aix[k]*n-base, aix[k+1]*n-base, aix[k+2]*n-base, aix[k+3]*n-base, cix, n );
				}
			}
			else {
				for( int k = k1; k<k2; k++ )
					vectMultiplyAdd( avals[k], b.values(aix[k]), cvals, b.pos(aix[k]), cix, n );
			}
		}
	}

	private static void matrixMultSparseDenseMMSkinnyRHS(SparseBlock a, DenseBlock b, DenseBlock c, int n, int rl, int ru)
		throws DMLRuntimeException
	{
		//no blocking since b and c fit into cache anyway
		for( int i=rl, cix=rl*n; i<ru; i++, cix+=n ) {
			if( a.isEmpty(i) ) continue;
			int apos = a.pos(i);
			int alen = a.size(i);
			int[] aix = a.indexes(i);
			double[] avals = a.values(i);
			double[] cvals = c.values(i);
			//rest not aligned to blocks of 4 rows
			int bn = b.isContiguous() ? alen%4 : alen;
			for( int k=apos; k<apos+bn; k++ )
				vectMultiplyAdd(avals[k], b.values(aix[k]), cvals, b.pos(aix[k]), cix, n);
			//compute blocks of 4 rows (core inner loop)
			double[] bvals = b.valuesAt(0); //only for contiguous
			for( int k=apos+bn; k<apos+alen; k+=4 )
				vectMultiplyAdd4( avals[k], avals[k+1], avals[k+2], avals[k+3], bvals, cvals,
					aix[k]*n, aix[k+1]*n, aix[k+2]*n, aix[k+3]*n, cix, n );
		}
	}

	private static void matrixMultSparseDenseMM(SparseBlock a, DenseBlock b, DenseBlock c, int n, int cd, long xsp, int rl, int ru)
		throws DMLRuntimeException
	{
		//blocksizes to fit blocks of B (dense) and several rows of A/C in common L2 cache size,
		//while blocking A/C for L1/L2 yet allowing long scans (2 pages) in the inner loop over j
		//in case of almost ultra-sparse matrices, we cannot ensure the blocking for the rhs and
		//output - however, in this case it's unlikely that we consume every cache line in the rhs
		final int blocksizeI = (int) (8L*xsp);
		final int blocksizeK = (int) (8L*xsp);
		final int blocksizeJ = 1024;

		//temporary array of current sparse positions
		int[] curk = new int[Math.min(blocksizeI, ru-rl)];

		//blocked execution over IKJ
		for( int bi = rl; bi < ru; bi+=blocksizeI ) {
			Arrays.fill(curk, 0); //reset positions
			for( int bk = 0, bimin = Math.min(ru, bi+blocksizeI); bk < cd; bk+=blocksizeK ) {
				for( int bj = 0, bkmin = Math.min(cd, bk+blocksizeK); bj < n; bj+=blocksizeJ ) {
					int bjlen = Math.min(n, bj+blocksizeJ)-bj;

					//core sub block matrix multiplication
					for( int i=bi; i<bimin; i++ ) {
						if( a.isEmpty(i) ) continue;
						int apos = a.pos(i);
						int alen = a.size(i);
						int[] aix = a.indexes(i);
						double[] avals = a.values(i);
						double[] cvals = c.values(i);
						int cix = c.pos(i, bj);

						int k = curk[i-bi] + apos;
						//rest not aligned to blocks of 4 rows
						int bn = b.isContiguous() ? alen%4 : alen;
						for( ; k<apos+bn && aix[k]<bkmin; k++ )
							vectMultiplyAdd(avals[k], b.values(aix[k]), cvals, b.pos(aix[k],bj), cix, bjlen);
						//compute blocks of 4 rows (core inner loop), allowed to exceed bkmin
						double[] bvals = b.valuesAt(0); //only for contiguous
						for( ; k<apos+alen && aix[k]<bkmin; k+=4 )
							vectMultiplyAdd4( avals[k], avals[k+1], avals[k+2], avals[k+3], bvals, cvals,
								aix[k]*n+bj, aix[k+1]*n+bj, aix[k+2]*n+bj, aix[k+3]*n+bj, cix, bjlen );
						//update positions on last bj block
						if( bj+bjlen==n )
							curk[i-bi] = k - apos;
					}
				}
			}
		}
	}

	private static void matrixMultSparseSparse(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, boolean pm2, int rl, int ru)
		throws DMLRuntimeException
	{
		SparseBlock a = m1.sparseBlock;
		SparseBlock b = m2.sparseBlock;
		DenseBlock c = ret.getDenseBlock();
		int m = m1.rlen;
		int cd = m1.clen;

		// MATRIX-MATRIX (VV, MV not applicable here because V always dense)
		if(LOW_LEVEL_OPTIMIZATION)
		{
			if( pm2 && m==1 )          //VECTOR-MATRIX
			{
				//parallelization over rows in rhs matrix
				if( !a.isEmpty(0) )
				{
					int alen = a.size(0);
					int[] aix = a.indexes(0);
					double[] avals = a.values(0);
					double[] cvals = c.valuesAt(0);
					int rlix = (rl==0) ? 0 : a.posFIndexGTE(0,rl);
					rlix = (rlix>=0) ? rlix : alen;

					for( int k=rlix; k<alen && aix[k]<ru; k++ )
						if( !b.isEmpty(aix[k]) ) {
							int bpos = b.pos(aix[k]);
							int blen = b.size(aix[k]);
							int[] bix = b.indexes(aix[k]);
							double[] bvals = b.values(aix[k]);
							vectMultiplyAdd(avals[k], bvals, cvals, bix, bpos, 0, blen);
						}
				}
			}
			else                       //MATRIX-MATRIX
			{
				//block sizes for best-effort blocking w/ sufficient row reuse in B yet small overhead
				final int blocksizeI = 32;
				final int blocksizeK = (int)Math.max(32, UtilFunctions.nextIntPow2(
						(int)Math.pow((double)m*cd/m1.nonZeros,2)));

				//temporary array of current sparse positions
				int[] curk = new int[Math.min(blocksizeI, ru-rl)];

				//blocked execution over IK
				for( int bi = rl; bi < ru; bi+=blocksizeI ) {
					Arrays.fill(curk, 0); //reset positions
					for( int bk = 0, bimin = Math.min(ru, bi+blocksizeI); bk < cd; bk+=blocksizeK ) {
						final int bkmin = Math.min(cd, bk+blocksizeK);
						//core sub block matrix multiplication
						for( int i=bi; i<bimin; i++ ) {
							if( a.isEmpty(i) ) continue;
							final int apos = a.pos(i);
							final int alen = a.size(i);
							int[] aix = a.indexes(i);
							double[] avals = a.values(i);
							double[] cvals = c.values(i);
							int cix = c.pos(i);
							int k = curk[i-bi] + apos;
							for(; k < apos+alen && aix[k]<bkmin; k++) {
								if( b.isEmpty(aix[k]) ) continue;
								vectMultiplyAdd(avals[k], b.values(aix[k]), cvals,
									b.indexes(aix[k]), b.pos(aix[k]), cix, b.size(aix[k]));
							}
							curk[i-bi] = k - apos;
						}
					}
				}
			}
		}
		else {
			for( int i=rl; i<Math.min(ru, a.numRows()); i++ ) {
				if( a.isEmpty(i) ) continue;
				int apos = a.pos(i);
				int alen = a.size(i);
				int[] aix = a.indexes(i);
				double[] avals = a.values(i);
				double[] cvals = c.values(i);
				int cix = c.pos(i);
				for(int k = apos; k < apos+alen; k++) {
					if( b.isEmpty(aix[k]) ) continue;
					double val = avals[k];
					int bpos = b.pos(aix[k]);
					int blen = b.size(aix[k]);
					int[] bix = b.indexes(aix[k]);
					double[] bvals = b.values(aix[k]);
					for(int j = bpos; j < bpos+blen; j++)
						cvals[cix+bix[j]] += val * bvals[j];
				}
			}
		}
	}

	/**
	 * This implementation applies to any combination of dense/sparse if at least one
	 * input is ultrasparse (sparse and very few nnz). In that case, most importantly,
	 * we want to create a sparse output and only iterate over the few nnz as the major
	 * dimension. Low-level optimization have less importance in that case and having
	 * this generic implementation helps to reduce the implementations from (2+1)^2
	 * to 2^2+1.
	 *
	 * @param m1 first matrix
	 * @param m2 second matrix
	 * @param ret result matrix
	 * @param rl row lower bound
	 * @param ru row upper bound
	 * @throws DMLRuntimeException if DMLRuntimeException occurs
	 */
	private static void matrixMultUltraSparse(MatrixBlock m1, MatrixBlock m2, MatrixBlock ret, int rl, int ru)
		throws DMLRuntimeException
	{
		final boolean leftUS = m1.isUltraSparse()
			|| (m1.isUltraSparse(false) && !m2.isUltraSparse());
		final int m  = m1.rlen;
		final int cd = m1.clen;
		final int n  = m2.clen;

		if( leftUS ) //left is ultra-sparse (IKJ)
		{
			SparseBlock a = m1.sparseBlock;
			SparseBlock c = ret.sparseBlock;
			boolean rightSparse = m2.sparse;

			for( int i=rl; i<ru; i++ )
			{
				if( !a.isEmpty(i) )
				{
					int apos = a.pos(i);
					int alen = a.size(i);
					int[] aixs = a.indexes(i);
					double[] avals = a.values(i);

					if( alen==1 && avals[apos]==1 ) //ROW SELECTION (no aggregation)
					{
						int aix = aixs[apos];
						if( rightSparse ) { //sparse right matrix (full row copy)
							if( !m2.sparseBlock.isEmpty(aix) ) {
								ret.rlen=m;
								ret.allocateSparseRowsBlock(false); //allocation on demand
								boolean ldeep = (m2.sparseBlock instanceof SparseBlockMCSR);
								ret.sparseBlock.set(i, m2.sparseBlock.get(aix), ldeep);
								ret.nonZeros += ret.sparseBlock.size(i);
							}
						}
						else { //dense right matrix (append all values)
							int lnnz = (int)m2.recomputeNonZeros(aix, aix, 0, n-1);
							if( lnnz > 0 ) {
								c.allocate(i, lnnz); //allocate once
								for( int j=0; j<n; j++ )
									c.append(i, j, m2.quickGetValue(aix, j));
								ret.nonZeros += lnnz;
							}
						}
					}
					else //GENERAL CASE
					{
						for( int k=apos; k<apos+alen; k++ )
						{
							double aval = avals[k];
							int aix = aixs[k];
							for( int j=0; j<n; j++ )
							{
								double cval = ret.quickGetValue(i, j);
								double cvald = aval*m2.quickGetValue(aix, j);
								if( cvald != 0 )
									ret.quickSetValue(i, j, cval+cvald);
							}
						}
					}
				}
			}
		}
		else //right is ultra-sparse (KJI)
		{
			SparseBlock b = m2.sparseBlock;

			for(int k = 0; k < cd; k++ ) {
				if( b.isEmpty(k) ) continue;
				int bpos = b.pos(k);
				int blen = b.size(k);
				int[] bixs = b.indexes(k);
				double[] bvals = b.values(k);
				for( int j=bpos; j<bpos+blen; j++ ) {
					double bval = bvals[j];
					int bix = bixs[j];
					for( int i=rl; i<ru; i++ ) {
						double cvald = bval*m1.quickGetValue(i, k);
						if( cvald != 0 ){
							double cval = ret.quickGetValue(i, bix);
							ret.quickSetValue(i, bix, cval+cvald);
						}
					}
				}
			}
		}
		//no need to recompute nonzeros because maintained internally
	}

	private static void matrixMultChainDense(MatrixBlock mX, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, ChainType ct, int rl, int ru)
	{
		DenseBlock a = mX.getDenseBlock();
		double[] b = mV.getDenseBlockValues();
		double[] w = (mW!=null) ? mW.getDenseBlockValues() : null;
		double[] c = ret.getDenseBlockValues();
		final int cd = mX.clen; //features in X
		boolean weights = (ct == ChainType.XtwXv);
		boolean weights2 = (ct == ChainType.XtXvy);

		//temporary array for cache blocking
		//(blocksize chosen to fit b+v in L2 (256KB) for default 1k blocks)
		final int blocksizeI = 24; // constraint: factor of 4
		final int blocksizeJ = 1024;
		double[] tmp = new double[blocksizeI];
		final int bn = (ru-rl) % blocksizeI;

		//compute rest (not aligned to blocksize)
		for( int i=rl; i < rl+bn; i++ ) {
			double[] avals = a.values(i);
			int aix = a.pos(i);
			double val = dotProduct(avals, b, aix, 0, cd);
			val *= (weights) ? w[i] : 1;
			val -= (weights2) ? w[i] : 0;
			vectMultiplyAdd(val, avals, c, aix, 0, cd);
		}

		//blockwise mmchain computation
		for( int bi=rl+bn; bi < ru; bi+=blocksizeI )
		{
			//compute 1st matrix-vector for row block
			Arrays.fill(tmp, 0);
			for( int bj=0; bj<cd; bj+=blocksizeJ ) {
				int bjmin = Math.min(cd-bj, blocksizeJ);
				for( int i=0; i < blocksizeI; i++ )
					tmp[i] += dotProduct(a.values(bi+i), b, a.pos(bi+i,bj), bj, bjmin);
			}

			//multiply/subtract weights (in-place), if required
			if( weights )
				vectMultiply(w, tmp, bi, 0, blocksizeI);
			else if( weights2 )
				vectSubtract(w, tmp, bi, 0, blocksizeI);

			//compute 2nd matrix vector for row block and aggregate
			for( int bj = 0; bj<cd; bj+=blocksizeJ ) {
				int bjmin = Math.min(cd-bj, blocksizeJ);
				if( a.isContiguous() ) {
					double[] avals = a.values(0);
					for( int i=0, aix=bi*cd+bj; i<blocksizeI; i+=4, aix+=4*cd )
						vectMultiplyAdd4(tmp[i], tmp[i+1], tmp[i+2], tmp[i+3],
							avals, c, aix, aix+cd, aix+2*cd, aix+3*cd, bj, bjmin);
				}
				else {
					for( int i=0; i<blocksizeI; i++ )
						vectMultiplyAdd(tmp[i], a.values(bi+i), c, a.pos(bi+i,bj), bj, bjmin);
				}
			}
		}
	}

	private static void matrixMultChainSparse(MatrixBlock mX, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, ChainType ct, int rl, int ru)
	{
		SparseBlock a = mX.sparseBlock;
		double[] b = mV.getDenseBlockValues();
		double[] w = (mW!=null) ? mW.getDenseBlockValues() : null;
		double[] c = ret.getDenseBlockValues();
		boolean weights = (ct == ChainType.XtwXv);
		boolean weights2 = (ct == ChainType.XtXvy);

		//row-wise mmchain computation
		for( int i=rl; i < ru; i++ ) {
			if( a.isEmpty(i) || (weights && w[i]==0) )
				continue;
			int apos = a.pos(i);
			int alen = a.size(i);
			int[] aix = a.indexes(i);
			double[] avals = a.values(i);

			//compute 1st matrix-vector dot product
			double val = dotProduct(avals, b, aix, apos, 0, alen);

			//multiply/subtract weights, if required
			val *= (weights) ? w[i] : 1;
			val -= (weights2) ? w[i] : 0;

			//compute 2nd matrix vector and aggregate
			if( val != 0 )
				vectMultiplyAdd(val, avals, c, aix, apos, 0, alen);
		}
	}

	private static void matrixMultTransposeSelfDense( MatrixBlock m1, MatrixBlock ret, boolean leftTranspose, int rl, int ru )
		throws DMLRuntimeException
	{
		//2) transpose self matrix multiply dense
		// (compute only upper-triangular matrix due to symmetry)
		DenseBlock a = m1.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();
		int m = m1.rlen;
		int n = m1.clen;

		if( leftTranspose ) // t(X)%*%X
		{
			if( LOW_LEVEL_OPTIMIZATION )
			{
				if( n==1 ) //VECTOR (col)
				{
					double[] avals = a.valuesAt(0);
					c.set(0, 0, dotProduct(avals, avals, m));
				}
				else //MATRIX
				{
					//1) Unrolled inner loop (for better instruction-level parallelism)
					//2) Blocked execution (for less cache trashing in parallel exec)
					//3) Asymmetric block sizes (for less misses in inner loop, yet blocks in L1/L2)

					final int blocksizeI = 32; //64//256KB c block (typical L2 size per core), 32KB a block
					final int blocksizeK = 24; //64//256KB b block (typical L2 size per core), used while read 512B of a / read/write 4KB of c
					final int blocksizeJ = 1024; //512//4KB (typical main-memory page size), for scan

					//temporary arrays (nnz a, b index)
					double[] ta = new double[ blocksizeK ];
					int[]  tbi  = new int[ blocksizeK ];

					final int mx = ru;
					final int cdx = m;
					final int nx = n;

					//blocked execution
					for( int bi = rl; bi < mx; bi+=blocksizeI ) //from bi due to symmetry
						for( int bk = 0, bimin = Math.min(mx, bi+blocksizeI); bk < cdx; bk+=blocksizeK )
							for( int bj = bi, bkmin = Math.min(cdx, bk+blocksizeK); bj < nx; bj+=blocksizeJ )
							{
								int bklen = bkmin-bk;
								int bjlen = Math.min(nx, bj+blocksizeJ)-bj;

								//core sub block matrix multiplication
								for( int i = bi; i < bimin; i++)
								{
									double[] cvals = c.values(i);
									int cixj = c.pos(i, bj);

									if( a.isContiguous(bk, bkmin-1) ) {
										double[] avals = a.values(bk);
										int aixi = a.pos(bk, i);

										//determine nnz of a (for sparsity-aware skipping of rows)
										int knnz = copyNonZeroElements(avals, aixi, bk, bj, n, nx, ta, tbi, bklen);

										//rest not aligned to blocks of 4 rows
										final int bn = knnz % 4;
										switch( bn ){
											case 1: vectMultiplyAdd(ta[0], avals, cvals, tbi[0], cixj, bjlen); break;
											case 2: vectMultiplyAdd2(ta[0],ta[1], avals, cvals, tbi[0], tbi[1], cixj, bjlen); break;
											case 3: vectMultiplyAdd3(ta[0],ta[1],ta[2], avals, cvals, tbi[0], tbi[1],tbi[2], cixj, bjlen); break;
										}

										//compute blocks of 4 rows (core inner loop)
										for( int k = bn; k<knnz; k+=4 ){
											vectMultiplyAdd4( ta[k], ta[k+1], ta[k+2], ta[k+3], avals, cvals,
												tbi[k], tbi[k+1], tbi[k+2], tbi[k+3], cixj, bjlen );
										}
									}
									else {
										for( int k = bk; k<bkmin; k++ ) {
											double[] avals = a.values(bk);
											int aix = a.pos(bk, i);
											if( avals[aix] != 0 )
												vectMultiplyAdd( avals[aix], a.values(k),
													cvals, a.pos(k, bj), cixj, bjlen );
										}
									}
								}
							}
				}
			}
			else {
				for( int k = 0; k < m; k++ ) {
					double[] avals = a.values(k);
					int aix = a.pos(k);
					for( int i = rl; i < ru; i++ ) {
						double[] cvals = c.values(i);
						int cix = c.pos(i);
						double val = avals[ aix+i ];
						if( val != 0 ) {
							for(int j = i; j < n; j++) //from i due to symmetry
								cvals[ cix+j ] += val * avals[ aix+j ];
						}
					}
				}
			}
		}
		else // X%*%t(X)
		{
			if(LOW_LEVEL_OPTIMIZATION)
			{
				if( m==1 ) //VECTOR
				{
					double[] avals = a.valuesAt(0);
					c.set(0, 0, dotProduct(avals, avals, n));
				}
				else //MATRIX
				{
					//algorithm: scan c, foreach ci,j: scan row of a and t(a) (IJK)

					//1) Unrolled inner loop, for better ILP
					//2) Blocked execution, for less cache trashing in parallel exec
					//   (we block such that lhs, rhs, and output roughly fit into L2, output in L1)
					//3) Asymmetric block sizes and exploitation of result symmetry
					int blocksizeK = 1024; //two memory pages for sufficiently long scans
					int blocksizeIJ = L2_CACHESIZE / 8 / blocksizeK / 2 - 1; //15

					//blocked execution over IKJ (lhs/rhs in L2, output in L1)
					for( int bi = rl; bi<ru; bi+=blocksizeIJ )
						for( int bk = 0, bimin = Math.min(ru, bi+blocksizeIJ); bk<n; bk+=blocksizeK )
							for( int bj = bi, bklen = Math.min(blocksizeK, n-bk); bj<m; bj+=blocksizeIJ ) {
								//core tsmm block operation (15x15 vectors of length 1K elements)
								int bjmin = Math.min(m, bj+blocksizeIJ);
								for( int i=bi; i<bimin; i++ ) {
									final int bjmax = Math.max(i,bj); //from i due to symmetry
									double[] avals = a.values(i), cvals = c.values(i);
									int aix = a.pos(i, bk), cix = c.pos(i);
									for(int j=bjmax; j <bjmin; j++)
										cvals[ cix+j ] += dotProduct(avals, a.values(j), aix, a.pos(j, bk), bklen);
								}
							}
				}
			}
			else
			{
				for( int i = rl; i < ru; i++ ) {
					double[] avals1 = a.values(i), cvals = c.values(i);
					int aix1 = a.pos(i), cix = c.pos(i);
					for(int j = i; j < m; j++) { //from i due to symmetry
						double[] avals2 = a.values(j);
						int aix2 = a.pos(j);
						double val = 0;
						for(int k = 0; k < n; k++)
							val += avals1[aix1+k] * avals2[aix2+k];
						cvals[cix+j] = val;
					}
				}
			}
		}
	}

	private static void matrixMultTransposeSelfSparse( MatrixBlock m1, MatrixBlock ret, boolean leftTranspose, int rl, int ru )
		throws DMLRuntimeException
	{
		//2) transpose self matrix multiply sparse
		// (compute only upper-triangular matrix due to symmetry)
		SparseBlock a = m1.sparseBlock;
		DenseBlock c = ret.getDenseBlock();
		int m = m1.rlen;

		if( leftTranspose ) // t(X)%*%X
		{
			//only general case (because vectors always dense)
			//algorithm: scan rows, foreach row self join (KIJ)
			if( LOW_LEVEL_OPTIMIZATION )
			{
				int arlen = a.numRows();
				for( int r=0; r<arlen; r++ ) {
					if( a.isEmpty(r) ) continue;
					int apos = a.pos(r);
					int alen = a.size(r);
					int[] aix = a.indexes(r);
					double[] avals = a.values(r);
					int rlix = (rl==0) ? 0 : a.posFIndexGTE(r, rl);
					rlix = (rlix>=0) ? apos+rlix : apos+alen;
					int len = apos + alen;
					for(int i = rlix; i < len && aix[i]<ru; i++) {
						double val = avals[i];
						if( val != 0 )
							vectMultiplyAdd(val, avals, c.values(aix[i]),
								aix, i, c.pos(aix[i]), len-i);
					}
				}
			}
			else
			{
				for( int r=0; r<a.numRows(); r++ ) {
					if( a.isEmpty(r) ) continue;
					int apos = a.pos(r);
					int alen = a.size(r);
					int[] aix = a.indexes(r);
					double[] avals = a.values(r);
					int rlix = (rl==0) ? 0 : a.posFIndexGTE(r, rl);
					rlix = (rlix>=0) ? apos+rlix : apos+alen;
					for(int i = rlix; i < apos+alen && aix[i]<ru; i++) {
						double val = avals[i];
						if( val != 0 ) {
							double[] cvals = c.values(aix[i]);
							int cix = c.pos(aix[i]);
							for(int j = i; j < apos+alen; j++)
								cvals[cix+aix[j]] += val * avals[j];
						}
					}
				}
			}
		}
		else // X%*%t(X)
		{
			if( m==1 ) //VECTOR
			{
				if( !m1.sparseBlock.isEmpty(0) ) {
					int alen = m1.sparseBlock.size(0); //pos always 0
					double[] avals = a.values(0);
					c.set(0, 0, dotProduct(avals, avals, alen));
				}
			}
			else //MATRIX
			{
				//note: reorg to similar layout as t(X)%*%X because faster than
				//direct computation with IJK (no dependencies/branches in inner loop)
				//see preprocessMatrixMultTransposeSelf m1<-tmpBlock
				m = m1.clen;

				//algorithm: scan rows, foreach row self join (KIJ)
				if( LOW_LEVEL_OPTIMIZATION )
				{
					int arlen = a.numRows();
					for( int r=0; r<arlen; r++ ) {
						if( a.isEmpty(r) ) continue;
						int apos = a.pos(r);
						int alen = a.size(r);
						int[] aix = a.indexes(r);
						double[] avals = a.values(r);
						int rlix = (rl==0) ? 0 : a.posFIndexGTE(r, rl);
						rlix = (rlix>=0) ? apos+rlix : apos+alen;
						for(int i = rlix; i < apos+alen && aix[i]<ru; i++) {
							double val = avals[i];
							if( val != 0 )
								vectMultiplyAdd(val, avals, c.values(aix[i]),
									aix, i, c.pos(aix[i]), alen-i);
						}
					}
				}
				else
				{
					for( int r=0; r<a.numRows(); r++ ) {
						if( a.isEmpty(r) ) continue;
						int apos = a.pos(r);
						int alen = a.size(r);
						int[] aix = a.indexes(r);
						double[] avals = a.values(r);
						int rlix = (rl==0) ? 0 : a.posFIndexGTE(r,rl);
						rlix = (rlix>=0) ? apos+rlix : apos+alen;
						for(int i = rlix; i < apos+alen && aix[i]<ru; i++) {
							double val = avals[i];
							if( val != 0 ) {
								double[] cvals = c.values(aix[i]);
								int cix = c.pos(aix[i]);
								for( int j = i; j < alen; j++ )
									cvals[cix+aix[j]] += val * avals[j];
							}
						}
					}
				}
			}
		}
	}

	private static void matrixMultPermuteDense(MatrixBlock pm1, MatrixBlock m2, MatrixBlock ret1, MatrixBlock ret2, int rl, int ru)
		throws DMLRuntimeException
	{
		double[] a = pm1.getDenseBlockValues();
		DenseBlock b = m2.getDenseBlock();
		DenseBlock c = ret1.getDenseBlock();

		final int n = m2.clen;
		final int brlen = ret1.getNumRows();
		int lastblk = -1;

		for( int i=rl; i<ru; i++ ) {
			//compute block index and in-block indexes
			int pos = UtilFunctions.toInt( a[ i ]); //safe cast
			if( pos > 0 ) { //selected row
				int bpos = (pos-1) % brlen;
				int blk = (pos-1) / brlen;
				//allocate and switch to second output block
				//(never happens in cp, correct for multi-threaded usage)
				if( lastblk!=-1 && lastblk<blk ) {
					ret2.sparse = false;
					ret2.allocateDenseBlock();
					c = ret2.getDenseBlock();
				}
				//memcopy entire dense row into target position
				System.arraycopy(b.values(i), b.pos(i), c.values(bpos), c.pos(bpos), n);
				lastblk = blk;
			}
		}
	}

	private static void matrixMultPermuteDenseSparse( MatrixBlock pm1, MatrixBlock m2, MatrixBlock ret1, MatrixBlock ret2, int rl, int ru)
	{
		double[] a = pm1.getDenseBlockValues(); //vector
		DenseBlock b = m2.getDenseBlock();
		SparseBlock c = ret1.sparseBlock;

		final int n = m2.clen;
		final int brlen = ret1.getNumRows();

		int lastblk = -1;
		for( int i=rl; i<ru; i++ ) {
			//compute block index and in-block indexes
			int pos = UtilFunctions.toInt( a[ i ]); //safe cast
			if( pos > 0 ) { //selected row
				double[] bvals = b.values(i);
				int bix = b.pos(i);
				int bpos = (pos-1) % brlen;
				int blk = (pos-1) / brlen;
				//allocate and switch to second output block
				//(never happens in cp, correct for multi-threaded usage)
				if( lastblk!=-1 && lastblk<blk ){
					ret2.sparse = true;
					ret2.rlen=ret1.rlen;
					ret2.allocateSparseRowsBlock();
					c = ret2.sparseBlock;
				}
				//append entire dense row into sparse target position
				for( int j=0; j<n; j++ )
					c.append(bpos, j, bvals[bix+j]);
				lastblk = blk;
			}
		}
	}

	private static void matrixMultPermuteSparse( MatrixBlock pm1, MatrixBlock m2, MatrixBlock ret1, MatrixBlock ret2, int rl, int ru)
	{
		double[] a = pm1.getDenseBlockValues(); //vector
		SparseBlock b = m2.sparseBlock;
		SparseBlock c = ret1.sparseBlock;

		final int brlen = ret1.getNumRows();

		int lastblk = -1;
		for( int i=rl; i<ru; i++ )  {
			//compute block index and in-block indexes
			int pos = UtilFunctions.toInt( a[ i ]); //safe cast
			if( pos > 0 ) { //selected row
				int bpos = (pos-1) % brlen;
				int blk = (pos-1) / brlen;
				//allocate and switch to second output block
				//(never happens in cp, correct for multi-threaded usage)
				if( lastblk!=-1 && lastblk<blk ){
					ret2.sparse = true;
					ret2.allocateSparseRowsBlock();
					c = ret2.sparseBlock;
				}
				//memcopy entire sparse row into target position
				c.set(bpos, b.get(i), true);
				lastblk = blk;
			}
		}

	}

	private static void matrixMultWSLossDense(MatrixBlock mX, MatrixBlock mU, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, WeightsType wt, int rl, int ru)
	{
		DenseBlock x = mX.getDenseBlock();
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		DenseBlock w = (mW!=null)? mW.getDenseBlock() : null;
		final int n = mX.clen;
		final int cd = mU.clen;
		double wsloss = 0;

		// approach: iterate over all cells of X
		//cache-conscious blocking: due to blocksize constraint (default 1000),
		//a blocksize of 16 allows to fit blocks of UV into L2 cache (256KB)

		final int blocksizeIJ = 16; //u/v block (max at typical L2 size)

		//blocked execution
		for( int bi = rl; bi < ru; bi+=blocksizeIJ ) {
			int bimin = Math.min(ru, bi+blocksizeIJ);
			for( int bj = 0; bj < n; bj+=blocksizeIJ ){
				int bjmin = Math.min(n, bj+blocksizeIJ);

				// Pattern 1) sum (W * (X - U %*% t(V)) ^ 2) (post weighting)
				if( wt==WeightsType.POST ) {
					for( int i=bi; i<bimin; i++ ) {
						double[] wvals = w.values(i), xvals = x.values(i), uvals = u.values(i);
						int xix = x.pos(i), uix = u.pos(i);
						for( int j=bj; j<bjmin; j++ ) {
							double wij = wvals[xix+j];
							if( wij != 0 ) {
								double uvij = dotProduct(uvals, v.values(j), uix, v.pos(j), cd);
								wsloss += wij*(xvals[xix+j]-uvij)*(xvals[xix+j]-uvij); //^2
							}
						}
					}
				}
				// Pattern 1b) sum ((X!=0) * (X - U %*% t(V)) ^ 2) (post_nz weighting)
				else if( wt==WeightsType.POST_NZ ) {
					for( int i=bi; i<bimin; i++ ) {
						double[] xvals = x.values(i), uvals = u.values(i);
						int xix = x.pos(i), uix = u.pos(i);
						for( int j=bj; j<bjmin; j++ ) {
							double xij = xvals[xix+j];
							if( xij != 0 ) {
								double uvij = dotProduct(uvals, v.values(j), uix, v.pos(j), cd);
								wsloss += (xij-uvij)*(xij-uvij); //^2
							}
						}
					}
				}
				// Pattern 2) sum ((X - W * (U %*% t(V))) ^ 2) (pre weighting)
				else if( wt==WeightsType.PRE ) {
					for( int i=bi; i<bimin; i++ ) {
						double[] wvals = w.values(i), xvals = x.values(i), uvals = u.values(i);
						int xix = x.pos(i), uix = u.pos(i);
						for( int j=bj; j<bjmin; j++ ) {
							double wij = wvals[xix+j];
							double uvij = 0;
							if( wij != 0 )
								uvij = dotProduct(uvals, v.values(j), uix, v.pos(j), cd);
							wsloss += (xvals[xix+j]-wij*uvij)*(xvals[xix+j]-wij*uvij); //^2
						}
					}
				}
				// Pattern 3) sum ((X - (U %*% t(V))) ^ 2) (no weighting)
				else if( wt==WeightsType.NONE ) {
					for( int i=bi; i<bimin; i++ ) {
						double[] xvals = x.values(i), uvals = u.values(i);
						int xix = x.pos(i), uix = u.pos(i);
						for( int j=bj; j<bjmin; j++) {
							double uvij = dotProduct(uvals, v.values(j), uix, v.pos(j), cd);
							wsloss += (xvals[xix+j]-uvij)*(xvals[xix+j]-uvij); //^2
						}
					}
				}
			}
		}
		ret.quickSetValue(0, 0, wsloss);
	}

	private static void matrixMultWSLossSparseDense(MatrixBlock mX, MatrixBlock mU, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, WeightsType wt, int rl, int ru)
	{
		SparseBlock x = mX.sparseBlock;
		SparseBlock w = (mW!=null)? mW.sparseBlock : null;
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		final int n = mX.clen;
		final int cd = mU.clen;
		double wsloss = 0;

		// Pattern 1) sum (W * (X - U %*% t(V)) ^ 2) (post weighting)
		if( wt==WeightsType.POST ) {
			// approach: iterate over W, point-wise in order to exploit sparsity
			for( int i=rl; i<ru; i++ ) {
				if( w.isEmpty(i) ) continue;
				int wpos = w.pos(i);
				int wlen = w.size(i);
				int[] wix = w.indexes(i);
				double[] wval = w.values(i);
				double[] uvals = u.values(i);
				int uix = u.pos(i);
				if( w.isAligned(i, x) ) {
					//O(n) where n is nnz in w/x
					double[] xval = x.values(i);
					for( int k=wpos; k<wpos+wlen; k++ ) {
						double uvij = dotProduct(uvals, v.values(wix[k]), uix, v.pos(wix[k]), cd);
						wsloss += wval[k]*(xval[k]-uvij)*(xval[k]-uvij);
					}
				}
				else {
					//O(n log m) where n/m is nnz in w/x
					for( int k=wpos; k<wpos+wlen; k++ ) {
						double xi = mX.quickGetValue(i, wix[k]);
						double uvij = dotProduct(uvals, v.values(wix[k]), uix, v.pos(wix[k]), cd);
						wsloss += wval[k]*(xi-uvij)*(xi-uvij);
					}
				}
			}
		}
		// Pattern 1b) sum ((X!=0) * (X - U %*% t(V)) ^ 2) (post weighting)
		else if( wt==WeightsType.POST_NZ ) {
			// approach: iterate over W, point-wise in order to exploit sparsity
			// blocked over ij, while maintaining front of column indexes, where the
			// blocksize is chosen such that we reuse each vector on average 8 times.
			final int blocksizeIJ = (int) (8L*mX.rlen*mX.clen/mX.nonZeros);
			int[] curk = new int[blocksizeIJ];

			for( int bi=rl; bi<ru; bi+=blocksizeIJ ) {
				int bimin = Math.min(ru, bi+blocksizeIJ);
				//prepare starting indexes for block row
				Arrays.fill(curk, 0);
				//blocked execution over column blocks
				for( int bj=0; bj<n; bj+=blocksizeIJ ) {
					int bjmin = Math.min(n, bj+blocksizeIJ);
					for( int i=bi; i<bimin; i++ ) {
						if( x.isEmpty(i) ) continue;
						int xpos = x.pos(i);
						int xlen = x.size(i);
						int[] xix = x.indexes(i);
						double[] xval = x.values(i), uvals = u.values(i);
						int uix = u.pos(i);
						int k = xpos + curk[i-bi];
						for( ; k<xpos+xlen && xix[k]<bjmin; k++ ) {
							double uvij = dotProduct(uvals, v.values(xix[k]), uix, v.pos(xix[k]), cd);
							wsloss += (xval[k]-uvij)*(xval[k]-uvij);
						}
						curk[i-bi] = k - xpos;
					}
				}
			}
		}
		// Pattern 2) sum ((X - W * (U %*% t(V))) ^ 2) (pre weighting)
		else if( wt==WeightsType.PRE ) {
			// approach: iterate over all cells of X maybe sparse and dense
			// (note: tuning similar to pattern 3 possible but more complex)
			for( int i=rl; i<ru; i++ ) {
				double[] uvals = u.values(i);
				int uix = u.pos(i);
				for( int j=0; j<n; j++ ) {
					double xij = mX.quickGetValue(i, j);
					double wij = mW.quickGetValue(i, j);
					double uvij = 0;
					if( wij != 0 )
						uvij = dotProduct(uvals, v.values(j), uix, v.pos(j), cd);
					wsloss += (xij-wij*uvij)*(xij-wij*uvij);
				}
			}
		}
		// Pattern 3) sum ((X - (U %*% t(V))) ^ 2) (no weighting)
		else if( wt==WeightsType.NONE ) {
			//approach: use sparsity-exploiting pattern rewrite sum((X-(U%*%t(V)))^2)
			//-> sum(X^2)-sum(2*X*(U%*%t(V))))+sum((t(U)%*%U)*(t(V)%*%V)), where each
			//parallel task computes sum(X^2)-sum(2*X*(U%*%t(V)))) and the last term
			//sum((t(U)%*%U)*(t(V)%*%V)) is computed once via two tsmm operations.

			final int blocksizeIJ = (int) (8L*mX.rlen*mX.clen/mX.nonZeros);
			int[] curk = new int[blocksizeIJ];

			for( int bi=rl; bi<ru; bi+=blocksizeIJ ) {
				int bimin = Math.min(ru, bi+blocksizeIJ);
				//prepare starting indexes for block row
				Arrays.fill(curk, 0);
				//blocked execution over column blocks
				for( int bj=0; bj<n; bj+=blocksizeIJ ) {
					int bjmin = Math.min(n, bj+blocksizeIJ);
					for( int i=bi; i<bimin; i++ ) {
						if( x.isEmpty(i) ) continue;
						int xpos = x.pos(i);
						int xlen = x.size(i);
						int[] xix = x.indexes(i);
						double[] xval = x.values(i);
						double[] uvals = u.values(i);
						int uix = u.pos(i);
						int k = xpos + curk[i-bi];
						for( ; k<xpos+xlen && xix[k]<bjmin; k++ ) {
							double xij = xval[k];
							double uvij = dotProduct(uvals, v.values(xix[k]), uix, v.pos(xix[k]), cd);
							wsloss += xij * xij - 2 * xij * uvij;
						}
						curk[i-bi] = k - xpos;
					}
				}
			}
		}

		ret.quickSetValue(0, 0, wsloss);
	}

	private static void matrixMultWSLossGeneric (MatrixBlock mX, MatrixBlock mU, MatrixBlock mV, MatrixBlock mW, MatrixBlock ret, WeightsType wt, int rl, int ru)
	{
		final int n = mX.clen;
		final int cd = mU.clen;
		double wsloss = 0;

		// Pattern 1) sum (W * (X - U %*% t(V)) ^ 2) (post weighting)
		if( wt==WeightsType.POST )
		{
			// approach: iterate over W, point-wise in order to exploit sparsity
			if( mW.sparse ) //SPARSE
			{
				SparseBlock w = mW.sparseBlock;
				for( int i=rl; i<ru; i++ ) {
					if( w.isEmpty(i) ) continue;
					int wpos = w.pos(i);
					int wlen = w.size(i);
					int[] wix = w.indexes(i);
					double[] wval = w.values(i);
					for( int k=wpos; k<wpos+wlen; k++ ) {
						double uvij = dotProductGeneric(mU, mV, i, wix[k], cd);
						double xi = mX.quickGetValue(i, wix[k]);
						wsloss += wval[k]*(xi-uvij)*(xi-uvij);
					}
				}
			}
			else //DENSE
			{
				DenseBlock w = mW.getDenseBlock();
				for( int i=rl; i<ru; i++ ) {
					double[] wvals = w.values(i);
					int wix = w.pos(i);
					for( int j=0; j<n; j++)
						if( wvals[wix+j] != 0 ) {
							double uvij = dotProductGeneric(mU, mV, i, j, cd);
							double xij = mX.quickGetValue(i, j);
							wsloss += wvals[wix+j]*(xij-uvij)*(xij-uvij);
						}
				}
			}
		}
		// Pattern 1b) sum ((X!=0) * (X - U %*% t(V)) ^ 2) (post weighting)
		else if( wt==WeightsType.POST_NZ )
		{
			// approach: iterate over W, point-wise in order to exploit sparsity
			if( mX.sparse ) //SPARSE
			{
				SparseBlock x = mX.sparseBlock;
				for( int i=rl; i<ru; i++ ) {
					if( x.isEmpty(i) ) continue;
					int xpos = x.pos(i);
					int xlen = x.size(i);
					int[] xix = x.indexes(i);
					double[] xval = x.values(i);
					for( int k=xpos; k<xpos+xlen; k++ ) {
						double uvij = dotProductGeneric(mU, mV, i, xix[k], cd);
						wsloss += (xval[k]-uvij)*(xval[k]-uvij);
					}
				}
			}
			else //DENSE
			{
				DenseBlock x = mX.getDenseBlock();
				for( int i=rl; i<ru; i++ ) {
					double[] xvals = x.values(i);
					int xix = x.pos(i);
					for( int j=0; j<n; j++) {
						double xij = xvals[xix+j];
						if( xij != 0 ) {
							double uvij = dotProductGeneric(mU, mV, i, j, cd);
							wsloss += (xij-uvij)*(xij-uvij);
						}
					}
				}
			}
		}
		// Pattern 2) sum ((X - W * (U %*% t(V))) ^ 2) (pre weighting)
		else if( wt==WeightsType.PRE )
		{
			// approach: iterate over all cells of X maybe sparse and dense
			for( int i=rl; i<ru; i++ )
				for( int j=0; j<n; j++) {
					double xij = mX.quickGetValue(i, j);
					double wij = mW.quickGetValue(i, j);
					double uvij = 0;
					if( wij != 0 )
						uvij = dotProductGeneric(mU, mV, i, j, cd);
					wsloss += (xij-wij*uvij)*(xij-wij*uvij);
				}
		}
		// Pattern 3) sum ((X - (U %*% t(V))) ^ 2) (no weighting)
		else if( wt==WeightsType.NONE )
		{
			//approach: use sparsity-exploiting pattern rewrite sum((X-(U%*%t(V)))^2)
			//-> sum(X^2)-sum(2*X*(U%*%t(V))))+sum((t(U)%*%U)*(t(V)%*%V)), where each
			//parallel task computes sum(X^2)-sum(2*X*(U%*%t(V)))) and the last term
			//sum((t(U)%*%U)*(t(V)%*%V)) is computed once via two tsmm operations.

			if( mX.sparse ) { //SPARSE
				SparseBlock x = mX.sparseBlock;
				for( int i=rl; i<ru; i++ ) {
					if( x.isEmpty(i) ) continue;
					int xpos = x.pos(i);
					int xlen = x.size(i);
					int[] xix = x.indexes(i);
					double[] xval = x.values(i);
					for( int k=xpos; k<xpos+xlen; k++ ) {
						double xij = xval[k];
						double uvij = dotProductGeneric(mU, mV, i, xix[k], cd);
						wsloss += xij * xij - 2 * xij * uvij;
					}
				}
			}
			else { //DENSE
				DenseBlock x = mX.getDenseBlock();
				for( int i=rl; i<ru; i++ ) {
					double[] xvals = x.values(i);
					int xix = x.pos(i);
					for( int j=0; j<n; j++)
						if( xvals[xix+j] != 0 ) {
							double xij = xvals[xix+j];
							double uvij = dotProductGeneric(mU, mV, i, j, cd);
							wsloss += xij * xij - 2 * xij * uvij;
						}
				}
			}
		}

		ret.quickSetValue(0, 0, wsloss);
	}

	private static void addMatrixMultWSLossNoWeightCorrection(MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, int k)
		throws DMLRuntimeException
	{
		MatrixBlock tmp1 = new MatrixBlock(mU.clen, mU.clen, false);
		MatrixBlock tmp2 = new MatrixBlock(mU.clen, mU.clen, false);
		matrixMultTransposeSelf(mU, tmp1, true, k);
		matrixMultTransposeSelf(mV, tmp2, true, k);
		ret.quickSetValue(0, 0, ret.quickGetValue(0, 0) +
			((tmp1.sparse || tmp2.sparse) ? dotProductGeneric(tmp1, tmp2) :
			dotProduct(tmp1.getDenseBlockValues(), tmp2.getDenseBlockValues(), mU.clen*mU.clen)));
	}

	private static void matrixMultWSigmoidDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WSigmoidType wt, int rl, int ru)
		throws DMLRuntimeException
	{
		DenseBlock w = mW.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		final int n = mW.clen;
		final int cd = mU.clen;

		//note: cannot compute U %*% t(V) in-place of result w/ regular mm because
		//t(V) comes in transformed form and hence would require additional memory

		boolean flagminus = (wt==WSigmoidType.MINUS || wt==WSigmoidType.LOG_MINUS);
		boolean flaglog = (wt==WSigmoidType.LOG || wt==WSigmoidType.LOG_MINUS);

		//approach: iterate over non-zeros of w, selective mm computation
		//cache-conscious blocking: due to blocksize constraint (default 1000),
		//a blocksize of 16 allows to fit blocks of UV into L2 cache (256KB)

		final int blocksizeIJ = 16; //u/v block (max at typical L2 size)

		//blocked execution
		for( int bi = rl; bi < ru; bi+=blocksizeIJ ) {
			int bimin = Math.min(ru, bi+blocksizeIJ);
			for( int bj = 0; bj < n; bj+=blocksizeIJ ) {
				int bjmin = Math.min(n, bj+blocksizeIJ);
				//core wsigmoid computation
				for( int i=bi; i<bimin; i++ ) {
					double[] wvals = w.values(i), uvals = u.values(i), cvals = c.values(i);
					int wix = w.pos(i), uix = u.pos(i);
					for( int j=bj; j<bjmin; j++) {
						double wij = wvals[wix+j];
						if( wij != 0 )
							cvals[wix+j] = wsigmoid(wij, uvals, v.values(j),
								uix, v.pos(j), flagminus, flaglog, cd);
					}
				}
			}
		}
	}

	private static void matrixMultWSigmoidSparseDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WSigmoidType wt, int rl, int ru)
		throws DMLRuntimeException
	{
		SparseBlock w = mW.sparseBlock;
		SparseBlock c = ret.sparseBlock;
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		final int cd = mU.clen;

		boolean flagminus = (wt==WSigmoidType.MINUS || wt==WSigmoidType.LOG_MINUS);
		boolean flaglog = (wt==WSigmoidType.LOG || wt==WSigmoidType.LOG_MINUS);

		//approach: iterate over non-zeros of w, selective mm computation
		for( int i=rl; i<ru; i++ ) {
			if( w.isEmpty(i) ) continue;
			int wpos = w.pos(i);
			int wlen = w.size(i);
			int[] wix = w.indexes(i);
			double[] wval = w.values(i);
			double[] uvals = u.values(i);
			int uix = u.pos(i);
			c.allocate(i, wlen);
			for( int k=wpos; k<wpos+wlen; k++ ) {
				double cval = wsigmoid(wval[k], uvals, v.values(wix[k]),
					uix, v.pos(wix[k]), flagminus, flaglog, cd);
				c.append(i, wix[k], cval);
			}
		}
	}

	private static void matrixMultWSigmoidGeneric (MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WSigmoidType wt, int rl, int ru)
		throws DMLRuntimeException
	{
		final int n = mW.clen;
		final int cd = mU.clen;

		boolean flagminus = (wt==WSigmoidType.MINUS || wt==WSigmoidType.LOG_MINUS);
		boolean flaglog = (wt==WSigmoidType.LOG || wt==WSigmoidType.LOG_MINUS);

		//approach: iterate over non-zeros of w, selective mm computation
		if( mW.sparse ) //SPARSE
		{
			//w and c always in same representation
			SparseBlock w = mW.sparseBlock;
			SparseBlock c = ret.sparseBlock;
			for( int i=rl; i<ru; i++ ) {
				if( w.isEmpty(i) ) continue;
				int wpos = w.pos(i);
				int wlen = w.size(i);
				int[] wix = w.indexes(i);
				double[] wval = w.values(i);
				c.allocate(i, wlen);
				for( int k=wpos; k<wpos+wlen; k++ ) {
					double cval = wsigmoid(wval[k], mU, mV, i, wix[k], flagminus, flaglog, cd);
					c.append(i, wix[k], cval);
				}
			}
		}
		else //DENSE
		{
			//w and c always in same representation
			DenseBlock w = mW.getDenseBlock();
			DenseBlock c = ret.getDenseBlock();
			for( int i=rl; i<ru; i++ ) {
				double[] wvals = w.values(i), cvals = c.values(i);
				int ix = w.pos(i);
				for( int j=0; j<n; j++ ) {
					double wij = wvals[ix+j];
					if( wij != 0 )
						cvals[ix+j] = wsigmoid(wij, mU, mV, i, j, flagminus, flaglog, cd);
				}
			}
		}
	}

	private static void matrixMultWDivMMDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock mX, MatrixBlock ret, WDivMMType wt, int rl, int ru, int cl, int cu)
		throws DMLRuntimeException
	{
		final boolean basic = wt.isBasic();
		final boolean left = wt.isLeft();
		final boolean mult = wt.isMult();
		final boolean minus = wt.isMinus();
		final boolean four = wt.hasFourInputs();
		final boolean scalar = wt.hasScalar();
		final double eps = scalar ? mX.quickGetValue(0, 0) : 0;
		final int cd = mU.clen;

		DenseBlock w = mW.getDenseBlock();
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		DenseBlock x = (mX==null) ? null : mX.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();

		//approach: iterate over non-zeros of w, selective mm computation
		//cache-conscious blocking: due to blocksize constraint (default 1000),
		//a blocksize of 16 allows to fit blocks of UV into L2 cache (256KB)

		final int blocksizeIJ = 16; //u/v block (max at typical L2 size)

		//blocked execution
		for( int bi = rl; bi < ru; bi+=blocksizeIJ ) {
			int bimin = Math.min(ru, bi+blocksizeIJ);
			for( int bj = cl; bj < cu; bj+=blocksizeIJ ) {
				int bjmin = Math.min(cu, bj+blocksizeIJ);
				//core wsigmoid computation
				for( int i=bi; i<bimin; i++ ) {
					double[] wvals = w.values(i), uvals = u.values(i);
					double[] xvals = four ? x.values(i) : null;
					int wix = w.pos(i), uix = u.pos(i);
					for( int j=bj; j<bjmin; j++ )
						if( wvals[wix+j] != 0 ) {
							double[] cvals = c.values((basic||!left) ? i : j);
							if( basic )
								cvals[wix+j] = wvals[wix+j] * dotProduct(uvals, v.values(j), uix, v.pos(j), cd);
							else if( four ) { //left/right
								if (scalar)
									wdivmm(wvals[wix+j], eps, uvals, v.values(j), cvals, uix, v.pos(j), left, scalar, cd);
								else
									wdivmm(wvals[wix+j], xvals[wix+j], uvals, v.values(j), cvals, uix, v.pos(j), left, scalar, cd);
							}
							else //left/right minus/default
								wdivmm(wvals[wix+j], uvals, v.values(j), cvals, uix, v.pos(j), left, mult, minus, cd);
						}
				}
			}
		}
	}

	private static void matrixMultWDivMMSparseDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock mX, MatrixBlock ret, WDivMMType wt, int rl, int ru, int cl, int cu)
		throws DMLRuntimeException
	{
		final boolean basic = wt.isBasic();
		final boolean left = wt.isLeft();
		final boolean mult = wt.isMult();
		final boolean minus = wt.isMinus();
		final boolean four = wt.hasFourInputs();
		final boolean scalar = wt.hasScalar();
		final double eps = scalar ? mX.quickGetValue(0, 0) : 0;
		final int cd = mU.clen;

		SparseBlock w = mW.sparseBlock;
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();
		SparseBlock x = (mX==null) ? null : mX.sparseBlock;

		//approach: iterate over non-zeros of w, selective mm computation
		//blocked over ij, while maintaining front of column indexes, where the
		//blocksize is chosen such that we reuse each  Ui/Vj vector on average 8 times,
		//with custom blocksizeJ for wdivmm_left to avoid LLC misses on output.
		final int blocksizeI = (int) (8L*mW.rlen*mW.clen/mW.nonZeros);
		final int blocksizeJ = left ? Math.max(8,Math.min(L2_CACHESIZE/(mU.clen*8), blocksizeI)) : blocksizeI;

		int[] curk = new int[blocksizeI];
		boolean[] aligned = (four&&!scalar) ? new boolean[blocksizeI] : null;

		//blocked execution over row blocks
		for( int bi=rl; bi<ru; bi+=blocksizeI )
		{
			int bimin = Math.min(ru, bi+blocksizeI);
			//prepare starting indexes for block row
			for( int i=bi; i<bimin; i++ ) {
				int k = (cl==0||w.isEmpty(i)) ? 0 : w.posFIndexGTE(i,cl);
				curk[i-bi] = (k>=0) ? k : mW.clen;
			}
			//prepare alignment info if necessary
			if( four && !scalar )
				for( int i=bi; i<bimin; i++ )
					aligned[i-bi] = w.isAligned(i-bi, x);

			//blocked execution over column blocks
			for( int bj=cl; bj<cu; bj+=blocksizeJ )
			{
				int bjmin = Math.min(cu, bj+blocksizeJ);
				//core wdivmm block matrix mult
				for( int i=bi; i<bimin; i++ ) {
					if( w.isEmpty(i) ) continue;

					int wpos = w.pos(i);
					int wlen = w.size(i);
					int[] wix = w.indexes(i);
					double[] wval = w.values(i);
					double[] uvals = u.values(i);
					int uix = u.pos(i);

					int k = wpos + curk[i-bi];
					if( basic ) {
						for( ; k<wpos+wlen && wix[k]<bjmin; k++ )
							ret.appendValue( i, wix[k], wval[k] *dotProduct(
								uvals, v.values(wix[k]), uix, v.pos(wix[k]), cd));
					}
					else if( four ) { //left/right
						//checking alignment per row is ok because early abort if false,
						//row nnz likely fit in L1/L2 cache, and asymptotically better if aligned
						if( !scalar && w.isAligned(i, x) ) {
							//O(n) where n is nnz in w/x
							double[] xvals = x.values(i);
							for( ; k<wpos+wlen && wix[k]<bjmin; k++ ) {
								double[] cvals = c.values(left ? wix[k] : i);
								wdivmm(wval[k], xvals[k], uvals, v.values(wix[k]),
									cvals, uix, v.pos(wix[k]), left, scalar, cd);
							}
						}
						else {
							//scalar or O(n log m) where n/m are nnz in w/x
							for( ; k<wpos+wlen && wix[k]<bjmin; k++ ) {
								double[] cvals = c.values(left ? wix[k] : i);
								if (scalar)
									wdivmm(wval[k], eps, uvals, v.values(wix[k]),
										cvals, uix, v.pos(wix[k]), left, scalar, cd);
								else
									wdivmm(wval[k], x.get(i, wix[k]), uvals,
										v.values(wix[k]), cvals, uix, v.pos(wix[k]), left, scalar, cd);
							}
						}
					}
					else { //left/right minus default
						for( ; k<wpos+wlen && wix[k]<bjmin; k++ ) {
							double[] cvals = c.values(left ? wix[k] : i);
							wdivmm(wval[k], uvals, v.values(wix[k]), cvals,
								uix, v.pos(wix[k]), left, mult, minus, cd);
						}
					}
					curk[i-bi] = k - wpos;
				}
			}
		}
	}

	private static void matrixMultWDivMMGeneric(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock mX, MatrixBlock ret, WDivMMType wt, int rl, int ru, int cl, int cu)
		throws DMLRuntimeException
	{
		final boolean basic = wt.isBasic();
		final boolean left = wt.isLeft();
		final boolean mult = wt.isMult();
		final boolean minus = wt.isMinus();
		final boolean four = wt.hasFourInputs();
		final boolean scalar = wt.hasScalar();
		final double eps = scalar ? mX.quickGetValue(0, 0) : 0;
		final int cd = mU.clen;

		//output always in dense representation
		DenseBlock c = ret.getDenseBlock();

		//approach: iterate over non-zeros of w, selective mm computation
		if( mW.sparse ) //SPARSE
		{
			SparseBlock w = mW.sparseBlock;

			for( int i=rl; i<ru; i++ ) {
				if( w.isEmpty(i) ) continue;
				int wpos = w.pos(i);
				int wlen = w.size(i);
				int[] wix = w.indexes(i);
				double[] wval = w.values(i);
				int k = (cl==0) ? 0 : w.posFIndexGTE(i,cl);
				k = (k>=0) ? wpos+k : wpos+wlen;
				for( ; k<wpos+wlen && wix[k]<cu; k++ ) {
					double[] cvals = c.values((basic||!left) ? i : wix[k]);
					if( basic ) {
						double uvij = dotProductGeneric(mU,mV, i, wix[k], cd);
						ret.appendValue(i, wix[k], uvij);
					}
					else if( four ) { //left/right
						double xij = scalar ? eps : mX.quickGetValue(i, wix[k]);
						wdivmm(wval[k], xij, mU, mV, cvals, i, wix[k], left, scalar, cd);
					}
					else { //left/right minus/default
						wdivmm(wval[k], mU, mV, cvals, i, wix[k], left, mult, minus, cd);
					}
				}
			}
		}
		else //DENSE
		{
			DenseBlock w = mW.getDenseBlock();
			for( int i=rl; i<ru; i++ ) {
				double[] wvals = w.values(i);
				int ix = w.pos(i);
				for( int j=cl; j<cu; j++)
					if( wvals[ix+j] != 0 ) {
						double[] cvals = c.values((basic||!left) ? i : j);
						if( basic ) {
							cvals[ix+j] = dotProductGeneric(mU,mV, i, j, cd);
						}
						else if( four ) { //left/right
							double xij = scalar ? eps : mX.quickGetValue(i, j);
							wdivmm(wvals[ix+j], xij, mU, mV, cvals, i, j, left, scalar, cd);
						}
						else { //left/right minus/default
							wdivmm(wvals[ix+j], mU, mV, cvals, i, j, left, mult, minus, cd);
						}
					}
			}
		}
	}

	private static void matrixMultWCeMMDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, double eps, MatrixBlock ret, WCeMMType wt, int rl, int ru)
	{
		DenseBlock w = mW.getDenseBlock();
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		final int n = mW.clen;
		final int cd = mU.clen;
		double wceval = 0;

		// approach: iterate over all cells of X
		//cache-conscious blocking: due to blocksize constraint (default 1000),
		//a blocksize of 16 allows to fit blocks of UV into L2 cache (256KB)
		final int blocksizeIJ = 16; //u/v block (max at typical L2 size)

		//blocked execution
		for( int bi = rl; bi < ru; bi+=blocksizeIJ ) {
			int bimin = Math.min(ru, bi+blocksizeIJ);
			for( int bj = 0; bj < n; bj+=blocksizeIJ ) {
				int bjmin = Math.min(n, bj+blocksizeIJ);
				for( int i=bi; i<bimin; i++ ) {
					double[] wvals = w.values(i), uvals = u.values(i);
					int wix = w.pos(i), uix = u.pos(i);
					for( int j=bj; j<bjmin; j++ ) {
						double wij = wvals[wix+j];
						if( wij != 0 ) {
							double uvij = dotProduct(uvals, v.values(j), uix, v.pos(j), cd);
							wceval += wij * Math.log(uvij + eps);
						}
					}
				}
			}
		}
		ret.quickSetValue(0, 0, wceval);
	}

	private static void matrixMultWCeMMSparseDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, double eps, MatrixBlock ret, WCeMMType wt, int rl, int ru)
	{
		SparseBlock w = mW.sparseBlock;
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		final int n = mW.clen;
		final int cd = mU.clen;
		double wceval = 0;

		// approach: iterate over W, point-wise in order to exploit sparsity
		// blocked over ij, while maintaining front of column indexes, where the
		// blocksize is chosen such that we reuse each vector on average 8 times.
		final int blocksizeIJ = (int) (8L*mW.rlen*mW.clen/mW.nonZeros);
		int[] curk = new int[blocksizeIJ];

		for( int bi=rl; bi<ru; bi+=blocksizeIJ ) {
			int bimin = Math.min(ru, bi+blocksizeIJ);
			//prepare starting indexes for block row
			Arrays.fill(curk, 0);
			//blocked execution over column blocks
			for( int bj=0; bj<n; bj+=blocksizeIJ ) {
				int bjmin = Math.min(n, bj+blocksizeIJ);
				for( int i=bi; i<bimin; i++ ) {
					if( w.isEmpty(i) ) continue;
					int wpos = w.pos(i);
					int wlen = w.size(i);
					int[] wix = w.indexes(i);
					double[] wvals = w.values(i);
					double[] uvals = u.values(i);
					int uix = u.pos(i);
					int k = wpos + curk[i-bi];
					for( ; k<wpos+wlen && wix[k]<bjmin; k++ ) {
						double uvij = dotProduct(uvals, v.values(wix[k]), uix, v.pos(wix[k]), cd);
						wceval += wvals[k] * Math.log(uvij + eps);
					}
					curk[i-bi] = k - wpos;
				}
			}
		}
		ret.quickSetValue(0, 0, wceval);
	}

	private static void matrixMultWCeMMGeneric(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, double eps, MatrixBlock ret, WCeMMType wt, int rl, int ru)
	{
		final int n = mW.clen;
		final int cd = mU.clen;
		double wceval = 0;

		//approach: iterate over non-zeros of w, selective mm computation
		if( mW.sparse ) //SPARSE
		{
			SparseBlock w = mW.sparseBlock;
			for( int i=rl; i<ru; i++ ) {
				if( w.isEmpty(i) ) continue;
				int wpos = w.pos(i);
				int wlen = w.size(i);
				int[] wix = w.indexes(i);
				double[] wval = w.values(i);
				for( int k=wpos; k<wpos+wlen; k++ ) {
					double uvij = dotProductGeneric(mU, mV, i, wix[k], cd);
					wceval += wval[k] * Math.log(uvij + eps);
				}
			}
		}
		else //DENSE
		{
			DenseBlock w = mW.getDenseBlock();
			for( int i=rl; i<ru; i++ ) {
				double[] wvals = w.values(i);
				int wix = w.pos(i);
				for( int j=0; j<n; j++ ) {
					double wij = wvals[wix+j];
					if( wij != 0 ) {
						double uvij = dotProductGeneric(mU, mV, i, j, cd);
						wceval += wij * Math.log(uvij + eps);
					}
				}
			}
		}

		ret.quickSetValue(0, 0, wceval);
	}

	private static void matrixMultWuMMDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WUMMType wt, ValueFunction fn, int rl, int ru)
		throws DMLRuntimeException
	{
		DenseBlock w = mW.getDenseBlock();
		DenseBlock c = ret.getDenseBlock();
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		final int n = mW.clen;
		final int cd = mU.clen;

		//note: cannot compute U %*% t(V) in-place of result w/ regular mm because
		//t(V) comes in transformed form and hence would require additional memory

		boolean flagmult = (wt==WUMMType.MULT);

		//approach: iterate over non-zeros of w, selective mm computation
		//cache-conscious blocking: due to blocksize constraint (default 1000),
		//a blocksize of 16 allows to fit blocks of UV into L2 cache (256KB)

		final int blocksizeIJ = 16; //u/v block (max at typical L2 size)

		//blocked execution
		for( int bi = rl; bi < ru; bi+=blocksizeIJ ) {
			int bimin = Math.min(ru, bi+blocksizeIJ);
			for( int bj = 0; bj < n; bj+=blocksizeIJ ) {
				int bjmin = Math.min(n, bj+blocksizeIJ);
				//core wsigmoid computation
				for( int i=bi; i<bimin; i++ ) {
					double[] wvals = w.values(i), uvals = u.values(i), cvals = c.values(i);
					int wix = w.pos(i), uix = u.pos(i);
					for( int j=bj; j<bjmin; j++ ) {
						double wij = wvals[wix+j];
						if( wij != 0 )
							cvals[wix+j] = wumm(wij, uvals, v.values(j),
								uix, v.pos(j), flagmult, fn, cd);
					}
				}
			}
		}
	}

	private static void matrixMultWuMMSparseDense(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WUMMType wt, ValueFunction fn, int rl, int ru)
		throws DMLRuntimeException
	{
		SparseBlock w = mW.sparseBlock;
		SparseBlock c = ret.sparseBlock;
		DenseBlock u = mU.getDenseBlock();
		DenseBlock v = mV.getDenseBlock();
		final int cd = mU.clen;
		boolean flagmult = (wt==WUMMType.MULT);

		//approach: iterate over non-zeros of w, selective mm computation
		for( int i=rl; i<ru; i++ ) {
			if( w.isEmpty(i) ) continue;
			int wpos = w.pos(i);
			int wlen = w.size(i);
			int[] wix = w.indexes(i);
			double[] wvals = w.values(i);
			double[] uvals = u.values(i);
			int uix = u.pos(i);
			c.allocate(i, wlen);
			for( int k=wpos; k<wpos+wlen; k++ ) {
				double cval = wumm(wvals[k], uvals, v.values(wix[k]),
					uix, v.pos(wix[k]), flagmult, fn, cd);
				c.append(i, wix[k], cval);
			}
		}
	}

	private static void matrixMultWuMMGeneric (MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WUMMType wt, ValueFunction fn, int rl, int ru)
		throws DMLRuntimeException
	{
		final int n = mW.clen;
		final int cd = mU.clen;
		boolean flagmult = (wt==WUMMType.MULT);

		//approach: iterate over non-zeros of w, selective mm computation
		if( mW.sparse ) { //SPARSE
			//w and c always in same representation
			SparseBlock w = mW.sparseBlock;
			SparseBlock c = ret.sparseBlock;
			for( int i=rl; i<ru; i++ ) {
				if( w.isEmpty(i) ) continue;
				int wpos = w.pos(i);
				int wlen = w.size(i);
				int[] wix = w.indexes(i);
				double[] wval = w.values(i);
				c.allocate(i, wlen);
				for( int k=wpos; k<wpos+wlen; k++ ) {
					double cval = wumm(wval[k], mU, mV, i, wix[k], flagmult, fn, cd);
					c.append(i, wix[k], cval);
				}
			}
		}
		else { //DENSE
			//w and c always in same representation
			DenseBlock w = mW.getDenseBlock();
			DenseBlock c = ret.getDenseBlock();
			for( int i=rl; i<ru; i++ ) {
				double[] wvals = w.values(i), cvals = c.values(i);
				int ix = w.pos(i);
				for( int j=0; j<n; j++) {
					double wij = wvals[ix+j];
					if( wij != 0 )
						cvals[ix+j] = wumm(wij, mU, mV, i, j, flagmult, fn, cd);
				}
			}
		}
	}

	////////////////////////////////////////////
	// performance-relevant utility functions //
	////////////////////////////////////////////

	/**
	 * Computes the dot-product of two vectors. Experiments (on long vectors of
	 * 10^7 values) showed that this generic function provides equivalent performance
	 * even for the specific case of dotProduct(a,a,len) as used for TSMM.
	 *
	 * @param a first vector
	 * @param b second vector
	 * @param len length
	 * @return dot product of the two input vectors
	 */
	private static double dotProduct( double[] a, double[] b, final int len )
	{
		double val = 0;
		final int bn = len%8;

		//compute rest
		for( int i = 0; i < bn; i++ )
			val += a[ i ] * b[ i ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int i = bn; i < len; i+=8 )
		{
			//read 64B cachelines of a and b
			//compute cval' = sum(a * b) + cval
			val += a[ i+0 ] * b[ i+0 ]
			     + a[ i+1 ] * b[ i+1 ]
			     + a[ i+2 ] * b[ i+2 ]
			     + a[ i+3 ] * b[ i+3 ]
			     + a[ i+4 ] * b[ i+4 ]
			     + a[ i+5 ] * b[ i+5 ]
			     + a[ i+6 ] * b[ i+6 ]
			     + a[ i+7 ] * b[ i+7 ];
		}

		//scalar result
		return val;
	}

	//note: public for use by codegen for consistency
	public static double dotProduct( double[] a, double[] b, int ai, int bi, final int len )
	{
		double val = 0;
		final int bn = len%8;

		//compute rest
		for( int i = 0; i < bn; i++, ai++, bi++ )
			val += a[ ai ] * b[ bi ];

		//unrolled 8-block (for better instruction-level parallelism)
		for( int i = bn; i < len; i+=8, ai+=8, bi+=8 )
		{
			//read 64B cachelines of a and b
			//compute cval' = sum(a * b) + cval
			val += a[ ai+0 ] * b[ bi+0 ]
			     + a[ ai+1 ] * b[ bi+1 ]
			     + a[ ai+2 ] * b[ bi+2 ]
			     + a[ ai+3 ] * b[ bi+3 ]
			     + a[ ai+4 ] * b[ bi+4 ]
			     + a[ ai+5 ] * b[ bi+5 ]
			     + a[ ai+6 ] * b[ bi+6 ]
			     + a[ ai+7 ] * b[ bi+7 ];
		}

		//scalar result
		return val;
	}

	//note: public for use by codegen for consistency
	public static double dotProduct( double[] a, double[] b, int[] aix, int ai, final int bi, final int len )
	{
		double val = 0;
		final int bn = len%8;

		//compute rest
		for( int i = ai; i < ai+bn; i++ )
			val += a[ i ] * b[ bi+aix[i] ];

		//unrolled 8-block (for better instruction-level parallelism)
		for( int i = ai+bn; i < ai+len; i+=8 )
		{
			//read 64B cacheline of a
			//read 64B of b via 'gather'
			//compute cval' = sum(a * b) + cval
			val += a[ i+0 ] * b[ bi+aix[i+0] ]
			     + a[ i+1 ] * b[ bi+aix[i+1] ]
			     + a[ i+2 ] * b[ bi+aix[i+2] ]
			     + a[ i+3 ] * b[ bi+aix[i+3] ]
			     + a[ i+4 ] * b[ bi+aix[i+4] ]
			     + a[ i+5 ] * b[ bi+aix[i+5] ]
			     + a[ i+6 ] * b[ bi+aix[i+6] ]
			     + a[ i+7 ] * b[ bi+aix[i+7] ];
		}

		//scalar result
		return val;
	}

	//note: public for use by codegen for consistency
	public static void vectMultiplyAdd( final double aval, double[] b, double[] c, int bi, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, bi++, ci++)
			c[ ci ] += aval * b[ bi ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, bi+=8, ci+=8)
		{
			//read 64B cachelines of b and c
			//compute c' = aval * b + c
			//write back 64B cacheline of c = c'
			c[ ci+0 ] += aval * b[ bi+0 ];
			c[ ci+1 ] += aval * b[ bi+1 ];
			c[ ci+2 ] += aval * b[ bi+2 ];
			c[ ci+3 ] += aval * b[ bi+3 ];
			c[ ci+4 ] += aval * b[ bi+4 ];
			c[ ci+5 ] += aval * b[ bi+5 ];
			c[ ci+6 ] += aval * b[ bi+6 ];
			c[ ci+7 ] += aval * b[ bi+7 ];
		}
	}

    private static void vectMultiplyAdd2( final double aval1, final double aval2, double[] b, double[] c, int bi1, int bi2, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, bi1++, bi2++, ci++ )
			c[ ci ] += aval1 * b[ bi1 ] + aval2 * b[ bi2 ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, bi1+=8, bi2+=8, ci+=8 )
		{
			//read 64B cachelines of b (2x) and c
			//compute c' = aval_1 * b_1 + aval_2 * b_2 + c
			//write back 64B cacheline of c = c'
			c[ ci+0 ] += aval1 * b[ bi1+0 ] + aval2 * b[ bi2+0 ];
			c[ ci+1 ] += aval1 * b[ bi1+1 ] + aval2 * b[ bi2+1 ];
			c[ ci+2 ] += aval1 * b[ bi1+2 ] + aval2 * b[ bi2+2 ];
			c[ ci+3 ] += aval1 * b[ bi1+3 ] + aval2 * b[ bi2+3 ];
			c[ ci+4 ] += aval1 * b[ bi1+4 ] + aval2 * b[ bi2+4 ];
			c[ ci+5 ] += aval1 * b[ bi1+5 ] + aval2 * b[ bi2+5 ];
			c[ ci+6 ] += aval1 * b[ bi1+6 ] + aval2 * b[ bi2+6 ];
			c[ ci+7 ] += aval1 * b[ bi1+7 ] + aval2 * b[ bi2+7 ];
		}
	}

	private static void vectMultiplyAdd3( final double aval1, final double aval2, final double aval3, double[] b, double[] c, int bi1, int bi2, int bi3, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, bi1++, bi2++, bi3++, ci++ )
			c[ ci ] += aval1 * b[ bi1 ] + aval2 * b[ bi2 ] + aval3 * b[ bi3 ];

		//unrolled 8-block (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, bi1+=8, bi2+=8, bi3+=8, ci+=8 )
		{
			//read 64B cachelines of b (3x) and c
			//compute c' = aval_1 * b_1 + aval_2 * b_2 + c
			//write back 64B cacheline of c = c'
			c[ ci+0 ] += aval1 * b[ bi1+0 ] + aval2 * b[ bi2+0 ] + aval3 * b[ bi3+0 ];
			c[ ci+1 ] += aval1 * b[ bi1+1 ] + aval2 * b[ bi2+1 ] + aval3 * b[ bi3+1 ];
			c[ ci+2 ] += aval1 * b[ bi1+2 ] + aval2 * b[ bi2+2 ] + aval3 * b[ bi3+2 ];
			c[ ci+3 ] += aval1 * b[ bi1+3 ] + aval2 * b[ bi2+3 ] + aval3 * b[ bi3+3 ];
			c[ ci+4 ] += aval1 * b[ bi1+4 ] + aval2 * b[ bi2+4 ] + aval3 * b[ bi3+4 ];
			c[ ci+5 ] += aval1 * b[ bi1+5 ] + aval2 * b[ bi2+5 ] + aval3 * b[ bi3+5 ];
			c[ ci+6 ] += aval1 * b[ bi1+6 ] + aval2 * b[ bi2+6 ] + aval3 * b[ bi3+6 ];
			c[ ci+7 ] += aval1 * b[ bi1+7 ] + aval2 * b[ bi2+7 ] + aval3 * b[ bi3+7 ];
		}
	}

	private static void vectMultiplyAdd4( final double aval1, final double aval2, final double aval3, final double aval4, double[] b, double[] c, int bi1, int bi2, int bi3, int bi4, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, bi1++, bi2++, bi3++, bi4++, ci++ )
			c[ ci ] += aval1 * b[ bi1 ] + aval2 * b[ bi2 ] + aval3 * b[ bi3 ] + aval4 * b[ bi4 ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, bi1+=8, bi2+=8, bi3+=8, bi4+=8, ci+=8)
		{
			//read 64B cachelines of b (4x) and c
			//compute c' = aval_1 * b_1 + aval_2 * b_2 + c
			//write back 64B cacheline of c = c'
			c[ ci+0 ] += aval1 * b[ bi1+0 ] + aval2 * b[ bi2+0 ] + aval3 * b[ bi3+0 ] + aval4 * b[ bi4+0 ];
			c[ ci+1 ] += aval1 * b[ bi1+1 ] + aval2 * b[ bi2+1 ] + aval3 * b[ bi3+1 ] + aval4 * b[ bi4+1 ];
			c[ ci+2 ] += aval1 * b[ bi1+2 ] + aval2 * b[ bi2+2 ] + aval3 * b[ bi3+2 ] + aval4 * b[ bi4+2 ];
			c[ ci+3 ] += aval1 * b[ bi1+3 ] + aval2 * b[ bi2+3 ] + aval3 * b[ bi3+3 ] + aval4 * b[ bi4+3 ];
			c[ ci+4 ] += aval1 * b[ bi1+4 ] + aval2 * b[ bi2+4 ] + aval3 * b[ bi3+4 ] + aval4 * b[ bi4+4 ];
			c[ ci+5 ] += aval1 * b[ bi1+5 ] + aval2 * b[ bi2+5 ] + aval3 * b[ bi3+5 ] + aval4 * b[ bi4+5 ];
			c[ ci+6 ] += aval1 * b[ bi1+6 ] + aval2 * b[ bi2+6 ] + aval3 * b[ bi3+6 ] + aval4 * b[ bi4+6 ];
			c[ ci+7 ] += aval1 * b[ bi1+7 ] + aval2 * b[ bi2+7 ] + aval3 * b[ bi3+7 ] + aval4 * b[ bi4+7 ];
		}
	}

	@SuppressWarnings("unused")
	private static void vectMultiplyAdd( final double aval, double[] b, double[] c, int[] bix, final int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++ )
			c[ ci + bix[j] ] += aval * b[ j ];

		//unrolled 8-block (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8 )
		{
			//read 64B cacheline of b
			//read 64B of c via 'gather'
			//compute c' = aval * b + c
			//write back 64B of c = c' via 'scatter'
			c[ ci+bix[j+0] ] += aval * b[ j+0 ];
			c[ ci+bix[j+1] ] += aval * b[ j+1 ];
			c[ ci+bix[j+2] ] += aval * b[ j+2 ];
			c[ ci+bix[j+3] ] += aval * b[ j+3 ];
			c[ ci+bix[j+4] ] += aval * b[ j+4 ];
			c[ ci+bix[j+5] ] += aval * b[ j+5 ];
			c[ ci+bix[j+6] ] += aval * b[ j+6 ];
			c[ ci+bix[j+7] ] += aval * b[ j+7 ];
		}
	}

	//note: public for use by codegen for consistency
	public static void vectMultiplyAdd( final double aval, double[] b, double[] c, int[] bix, final int bi, final int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = bi; j < bi+bn; j++ )
			c[ ci + bix[j] ] += aval * b[ j ];

		//unrolled 8-block (for better instruction-level parallelism)
		for( int j = bi+bn; j < bi+len; j+=8 )
		{
			//read 64B cacheline of b
			//read 64B of c via 'gather'
			//compute c' = aval * b + c
			//write back 64B of c = c' via 'scatter'
			c[ ci+bix[j+0] ] += aval * b[ j+0 ];
			c[ ci+bix[j+1] ] += aval * b[ j+1 ];
			c[ ci+bix[j+2] ] += aval * b[ j+2 ];
			c[ ci+bix[j+3] ] += aval * b[ j+3 ];
			c[ ci+bix[j+4] ] += aval * b[ j+4 ];
			c[ ci+bix[j+5] ] += aval * b[ j+5 ];
			c[ ci+bix[j+6] ] += aval * b[ j+6 ];
			c[ ci+bix[j+7] ] += aval * b[ j+7 ];
		}
	}

	//note: public for use by codegen for consistency
	public static void vectMultiplyWrite( final double aval, double[] b, double[] c, int bi, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, bi++, ci++)
			c[ ci ] = aval * b[ bi ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, bi+=8, ci+=8)
		{
			//read 64B cachelines of b and c
			//compute c' = aval * b + c
			//write back 64B cacheline of c = c'
			c[ ci+0 ] = aval * b[ bi+0 ];
			c[ ci+1 ] = aval * b[ bi+1 ];
			c[ ci+2 ] = aval * b[ bi+2 ];
			c[ ci+3 ] = aval * b[ bi+3 ];
			c[ ci+4 ] = aval * b[ bi+4 ];
			c[ ci+5 ] = aval * b[ bi+5 ];
			c[ ci+6 ] = aval * b[ bi+6 ];
			c[ ci+7 ] = aval * b[ bi+7 ];
		}
	}

	//note: public for use by codegen for consistency
	public static void vectMultiplyWrite( double[] a, double[] b, double[] c, int ai, int bi, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, ai++, bi++, ci++)
			c[ ci ] = a[ ai ] * b[ bi ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, ai+=8, bi+=8, ci+=8)
		{
			//read 64B cachelines of a and b
			//compute c' = a * b
			//write back 64B cacheline of c = c'
			c[ ci+0 ] = a[ ai+0 ] * b[ bi+0 ];
			c[ ci+1 ] = a[ ai+1 ] * b[ bi+1 ];
			c[ ci+2 ] = a[ ai+2 ] * b[ bi+2 ];
			c[ ci+3 ] = a[ ai+3 ] * b[ bi+3 ];
			c[ ci+4 ] = a[ ai+4 ] * b[ bi+4 ];
			c[ ci+5 ] = a[ ai+5 ] * b[ bi+5 ];
			c[ ci+6 ] = a[ ai+6 ] * b[ bi+6 ];
			c[ ci+7 ] = a[ ai+7 ] * b[ bi+7 ];
		}
	}

	private static void vectMultiply( double[] a, double[] c, int ai, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, ai++, ci++)
			c[ ci ] *= a[ ai ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, ai+=8, ci+=8)
		{
			//read 64B cachelines of a and c
			//compute c' = c * a
			//write back 64B cacheline of c = c'
			c[ ci+0 ] *= a[ ai+0 ];
			c[ ci+1 ] *= a[ ai+1 ];
			c[ ci+2 ] *= a[ ai+2 ];
			c[ ci+3 ] *= a[ ai+3 ];
			c[ ci+4 ] *= a[ ai+4 ];
			c[ ci+5 ] *= a[ ai+5 ];
			c[ ci+6 ] *= a[ ai+6 ];
			c[ ci+7 ] *= a[ ai+7 ];
		}
	}

	//note: public for use by codegen for consistency
	public static void vectAdd( double[] a, double bval, double[] c, int ai, int ci, final int len ) {
		final int bn = len%8;
		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, ai++, ci++)
			c[ ci ] += a[ ai ];
		//unrolled 8-block  (for better ILP)
		for( int j = bn; j < len; j+=8, ai+=8, ci+=8) {
			c[ ci+0 ] += a[ ai+0 ] + bval;
			c[ ci+1 ] += a[ ai+1 ] + bval;
			c[ ci+2 ] += a[ ai+2 ] + bval;
			c[ ci+3 ] += a[ ai+3 ] + bval;
			c[ ci+4 ] += a[ ai+4 ] + bval;
			c[ ci+5 ] += a[ ai+5 ] + bval;
			c[ ci+6 ] += a[ ai+6 ] + bval;
			c[ ci+7 ] += a[ ai+7 ] + bval;
		}
	}

	//note: public for use by codegen for consistency
	public static void vectAdd( double[] a, double[] c, int ai, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, ai++, ci++)
			c[ ci ] += a[ ai ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, ai+=8, ci+=8)
		{
			//read 64B cachelines of a and c
			//compute c' = c * a
			//write back 64B cacheline of c = c'
			c[ ci+0 ] += a[ ai+0 ];
			c[ ci+1 ] += a[ ai+1 ];
			c[ ci+2 ] += a[ ai+2 ];
			c[ ci+3 ] += a[ ai+3 ];
			c[ ci+4 ] += a[ ai+4 ];
			c[ ci+5 ] += a[ ai+5 ];
			c[ ci+6 ] += a[ ai+6 ];
			c[ ci+7 ] += a[ ai+7 ];
		}
	}

	public static void vectAdd( double[] a, double[] c, int[] aix, int ai, int ci, final int alen ) {
		final int bn = alen%8;
		//rest, not aligned to 8-blocks
		for( int j = ai; j < ai+bn; j++ )
			c[ ci+aix[j] ] += a[ j ];
		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = ai+bn; j < ai+alen; j+=8 ) {
			c[ ci+aix[j+0] ] += a[ j+0 ];
			c[ ci+aix[j+1] ] += a[ j+1 ];
			c[ ci+aix[j+2] ] += a[ j+2 ];
			c[ ci+aix[j+3] ] += a[ j+3 ];
			c[ ci+aix[j+4] ] += a[ j+4 ];
			c[ ci+aix[j+5] ] += a[ j+5 ];
			c[ ci+aix[j+6] ] += a[ j+6 ];
			c[ ci+aix[j+7] ] += a[ j+7 ];
		}
	}

	private static void vectAdd4( double[] a1, double[] a2, double[] a3, double[] a4, double[] c, int ai, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, ai++, ci++)
			c[ ci ] += a1[ ai ] + a2[ ai ] + a3[ ai ] + a4[ ai ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, ai+=8, ci+=8)
		{
			//read 64B cachelines of a (4x) and c
			//compute c' = c + a1 + a2 + a3 + a4
			//write back 64B cacheline of c = c'
			c[ ci+0 ] += a1[ ai+0 ] + a2[ ai+0 ] + a3[ ai+0 ] + a4[ ai+0 ];
			c[ ci+1 ] += a1[ ai+1 ] + a2[ ai+1 ] + a3[ ai+1 ] + a4[ ai+1 ];
			c[ ci+2 ] += a1[ ai+2 ] + a2[ ai+2 ] + a3[ ai+2 ] + a4[ ai+2 ];
			c[ ci+3 ] += a1[ ai+3 ] + a2[ ai+3 ] + a3[ ai+3 ] + a4[ ai+3 ];
			c[ ci+4 ] += a1[ ai+4 ] + a2[ ai+4 ] + a3[ ai+4 ] + a4[ ai+4 ];
			c[ ci+5 ] += a1[ ai+5 ] + a2[ ai+5 ] + a3[ ai+5 ] + a4[ ai+5 ];
			c[ ci+6 ] += a1[ ai+6 ] + a2[ ai+6 ] + a3[ ai+6 ] + a4[ ai+6 ];
			c[ ci+7 ] += a1[ ai+7 ] + a2[ ai+7 ] + a3[ ai+7 ] + a4[ ai+7 ];
		}
	}

	public static void vectAddInPlace(double aval, double[] c, final int ci, final int len) {
		final int bn = len%8;
		//rest, not aligned to 8-blocks
		for( int j = ci; j < ci+bn; j++)
			c[ j ] += aval;
		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = ci+bn; j < ci+len; j+=8) {
			c[ j+0 ] += aval; c[ j+1 ] += aval;
			c[ j+2 ] += aval; c[ j+3 ] += aval;
			c[ j+4 ] += aval; c[ j+5 ] += aval;
			c[ j+6 ] += aval; c[ j+7 ] += aval;
		}
	}

	private static void vectSubtract( double[] a, double[] c, int ai, int ci, final int len )
	{
		final int bn = len%8;

		//rest, not aligned to 8-blocks
		for( int j = 0; j < bn; j++, ai++, ci++)
			c[ ci ] -= a[ ai ];

		//unrolled 8-block  (for better instruction-level parallelism)
		for( int j = bn; j < len; j+=8, ai+=8, ci+=8)
		{
			//read 64B cachelines of a and c
			//compute c' = c * a
			//write back 64B cacheline of c = c'
			c[ ci+0 ] -= a[ ai+0 ];
			c[ ci+1 ] -= a[ ai+1 ];
			c[ ci+2 ] -= a[ ai+2 ];
			c[ ci+3 ] -= a[ ai+3 ];
			c[ ci+4 ] -= a[ ai+4 ];
			c[ ci+5 ] -= a[ ai+5 ];
			c[ ci+6 ] -= a[ ai+6 ];
			c[ ci+7 ] -= a[ ai+7 ];
		}
	}

	private static double wsigmoid( final double wij, double[] u, double[] v, final int uix, final int vix, final boolean flagminus, final boolean flaglog, final int len )
	{
		//compute dot product over ui vj
		double uvij = dotProduct(u, v, uix, vix, len);

		//compute core sigmoid function
		double cval = flagminus ?
				1 / (1 + FastMath.exp(uvij)) :
				1 / (1 + FastMath.exp(-uvij));

		//compute weighted output
		return wij * ((flaglog) ? Math.log(cval) : cval);
	}

	private static double wsigmoid( final double wij, MatrixBlock u, MatrixBlock v, final int uix, final int vix, final boolean flagminus, final boolean flaglog, final int len )
	{
		//compute dot product over ui vj
		double uvij = dotProductGeneric(u, v, uix, vix, len);

		//compute core sigmoid function
		double cval = flagminus ?
				1 / (1 + FastMath.exp(uvij)) :
				1 / (1 + FastMath.exp(-uvij));

		//compute weighted output
		return wij * ((flaglog) ? Math.log(cval) : cval);
	}

	private static void wdivmm( final double wij, double[] u, double[] v, double[] c, final int uix, final int vix, final boolean left, final boolean mult, final boolean minus, final int len )
	{
		//compute dot product over ui vj
		double uvij = dotProduct(u, v, uix, vix, len);

		//compute core wdivmm
		double tmpval = minus ? uvij - wij :
			mult ? wij * uvij : wij / uvij;

		//prepare inputs for final mm
		int bix = left ? uix : vix;
		int cix = left ? vix : uix;
		double[] b = left ? u : v;

		//compute final mm output
		vectMultiplyAdd(tmpval, b, c, bix, cix, len);
	}

	private static void wdivmm( final double wij, final double xij, double[] u, double[] v, double[] c, final int uix, final int vix, final boolean left, final boolean scalar, final int len )
	{
		//compute dot product over ui vj
		double uvij = dotProduct(u, v, uix, vix, len);

		//compute core wdivmm
		double tmpval = scalar ? wij / (uvij + xij) : wij * (uvij - xij);

		//prepare inputs for final mm
		int bix = left ? uix : vix;
		int cix = left ? vix : uix;
		double[] b = left ? u : v;

		//compute final mm output
		vectMultiplyAdd(tmpval, b, c, bix, cix, len);
	}

	private static void wdivmm( final double wij, MatrixBlock u, MatrixBlock v, double[] c, final int uix, final int vix, final boolean left, boolean mult, final boolean minus, final int len )
	{
		//compute dot product over ui vj
		double uvij = dotProductGeneric(u, v, uix, vix, len);

		//compute core wdivmm
		double wtmp = minus ? uvij - wij :
			mult ? wij * uvij : wij / uvij;

		//prepare inputs for final mm
		int bix = left ? uix : vix;
		int cix = left ? vix*len : uix*len;
		MatrixBlock b = left ? u : v;

		//compute final mm
		for( int k2=0; k2<len; k2++ )
			c[cix+k2] += b.quickGetValue(bix, k2) * wtmp;
	}

	private static void wdivmm( final double wij, final double xij, MatrixBlock u, MatrixBlock v, double[] c, final int uix, final int vix, final boolean left, final boolean scalar, final int len )
	{
		//compute dot product over ui vj
		double uvij = dotProductGeneric(u, v, uix, vix, len);

		//compute core wdivmm
		double wtmp = scalar ? wij / (uvij + xij) : wij * (uvij - xij);

		//prepare inputs for final mm
		int bix = left ? uix : vix;
		int cix = left ? vix*len : uix*len;
		MatrixBlock b = left ? u : v;

		//compute final mm
		for( int k2=0; k2<len; k2++ )
			c[cix+k2] += b.quickGetValue(bix, k2) * wtmp;
	}

	private static double wumm( final double wij, double[] u, double[] v, final int uix, final int vix, final boolean flagmult, ValueFunction fn, final int len )
		throws DMLRuntimeException
	{
		//compute dot product over ui vj
		double uvij = dotProduct(u, v, uix, vix, len);

		//compute unary operations
		double cval = fn.execute(uvij);

		//compute weighted output
		return flagmult ? wij * cval : wij / cval;
	}

	private static double wumm( final double wij, MatrixBlock u, MatrixBlock v, final int uix, final int vix, final boolean flagmult, ValueFunction fn, final int len )
		throws DMLRuntimeException
	{
		//compute dot product over ui vj
		double uvij = dotProductGeneric(u, v, uix, vix, len);

		//compute unary operations
		double cval = fn.execute(uvij);

		//compute weighted output
		return flagmult ? wij * cval : wij / cval;
	}

	private static double dotProductGeneric(MatrixBlock a, MatrixBlock b, final int ai, final int bi, int len)
	{
		double val = 0;
		for( int k2=0; k2<len; k2++ )
			val += a.quickGetValue(ai, k2) * b.quickGetValue(bi, k2);

		return val;
	}

	private static double dotProductGeneric(MatrixBlock a, MatrixBlock b)
	{
		double val = 0;
		for( int i=0; i<a.getNumRows(); i++ )
			for( int j=0; j<a.getNumColumns(); j++ )
				val += a.quickGetValue(i, j) * b.quickGetValue(i, j);

		return val;
	}

	/**
	 * Used for all version of TSMM where the result is known to be symmetric.
	 * Hence, we compute only the upper triangular matrix and copy this partial
	 * result down to lower triangular matrix once.
	 *
	 * @param ret matrix
	 * @return number of non zeros
	 */
	public static long copyUpperToLowerTriangle( MatrixBlock ret )
	{
		//ret is guaranteed to be a squared, symmetric matrix
		if( ret.rlen != ret.clen )
			throw new RuntimeException("Invalid non-squared input matrix.");

		final double[] c = ret.getDenseBlockValues();
		final int n = ret.rlen;
		long nnz = 0;

		//blocked execution (2x128KB for L2 blocking)
		final int blocksizeIJ = 128;

		//handle blocks on diagonal
		for( int bi = 0; bi<n; bi+=blocksizeIJ ) {
			int bimin = Math.min(bi+blocksizeIJ, n);
			for( int i=bi, rix=bi*n; i<bimin; i++, rix+=n ) {
				LibMatrixReorg.transposeRow(c, c, rix+bi, bi*n+i, n, bimin-bi);
				nnz += (c[rix+i] != 0) ? 1 : 0; //for diagonal element
				for( int j=rix+i+1; j<rix+bimin; j++ )
					nnz += (c[j] != 0) ? 2 : 0;
			}
		}

		//handle non-diagonal blocks (full block copies)
		for( int bi = 0; bi<n; bi+=blocksizeIJ ) {
			int bimin = Math.min(bi+blocksizeIJ, n);
			for( int bj = bi; bj<n; bj+=blocksizeIJ )
				if( bi != bj ) { //not on diagonal
					int bjmin = Math.min(bj+blocksizeIJ, n);
					for( int i=bi, rix=bi*n; i<bimin; i++, rix+=n ) {
						LibMatrixReorg.transposeRow(c, c, rix+bj, bj*n+i, n, bjmin-bj);
						for( int j=rix+bj; j<rix+bjmin; j++ )
							nnz += (c[j] != 0) ? 2 : 0;
					}
				}
		}

		return nnz;
	}

	private static MatrixBlock prepMatrixMultTransposeSelfInput( MatrixBlock m1, boolean leftTranspose )
		throws DMLRuntimeException
	{
		MatrixBlock ret = m1;

		if( !leftTranspose && m1.sparse && m1.rlen > 1) //X%*%t(X) SPARSE MATRIX
		{
			//directly via LibMatrixReorg in order to prevent sparsity change
			MatrixBlock tmpBlock = new MatrixBlock(m1.clen, m1.rlen, m1.sparse);
			LibMatrixReorg.reorg(m1, tmpBlock, new ReorgOperator(SwapIndex.getSwapIndexFnObject()));
			ret = tmpBlock;
		}

		return ret;
	}

	private static boolean checkPrepMatrixMultRightInput( MatrixBlock m1, MatrixBlock m2 ) {
		//transpose if dense-dense, skinny rhs matrix (not vector), and memory guarded by output
		return (LOW_LEVEL_OPTIMIZATION && !m1.sparse && !m2.sparse
			&& isSkinnyRightHandSide(m1.rlen, m1.clen, m2.rlen, m2.clen, true));
	}

	//note: public for use by codegen for consistency
	public static boolean isSkinnyRightHandSide(long m1rlen, long m1clen, long m2rlen, long m2clen, boolean inclCacheSize) {
		return m1rlen > m2clen && m2rlen > m2clen && m2clen > 1
			&& m2clen < 64 && (!inclCacheSize || 8*m2rlen*m2clen < L2_CACHESIZE);
	}

	public static boolean checkParColumnAgg(MatrixBlock m1, int k, boolean inclFLOPs) {
		return (8L * m1.clen * k <= MEM_OVERHEAD_THRESHOLD
			&& (!inclFLOPs || 4L * m1.rlen * m1.clen / (m1.sparse?2:1) >= PAR_MINFLOP_THRESHOLD));
	}

	private static boolean checkParMatrixMultRightInputRows( MatrixBlock m1, MatrixBlock m2, int k ) {
		//parallelize over rows in rhs matrix if number of rows in lhs/output is very small
		return (m1.rlen==1 && LOW_LEVEL_OPTIMIZATION && m2.clen>1 && !(m1.isUltraSparse()||m2.isUltraSparse()))
			|| (m1.rlen<=16 && LOW_LEVEL_OPTIMIZATION && m2.clen>1 && m2.rlen > m1.rlen
			   && ( !m1.isUltraSparse() && !m2.sparse ) //dense-dense / sparse/dense
			   && (long)k * 8 * m1.rlen * m2.clen < MEM_OVERHEAD_THRESHOLD );
	}

	private static boolean checkParMatrixMultRightInputCols( MatrixBlock m1, MatrixBlock m2, int k, boolean pm2r ) {
		//parallelize over cols in rhs matrix if dense, number of cols in rhs is large, and lhs fits in l2
		return (LOW_LEVEL_OPTIMIZATION && !m1.sparse && !m2.sparse
				&& m2.clen > k * 1024 && m1.rlen < k * 32 && !pm2r
				&& 8*m1.rlen*m1.clen < 256*1024 ); //lhs fits in L2 cache
	}

	public static boolean isUltraSparseMatrixMult(MatrixBlock m1, MatrixBlock m2) {
		//note: ultra-sparse matrix mult implies also sparse outputs, hence we need
		//to be conservative an cannot use this for all ultra-sparse matrices.
		return (m1.isUltraSparse() || m2.isUltraSparse()) //base case
			|| (m1.isUltraSparsePermutationMatrix()
				&& OptimizerUtils.getSparsity(m2.rlen, m2.clen, m2.nonZeros)<1.0);
	}

	private static MatrixBlock prepMatrixMultRightInput( MatrixBlock m1, MatrixBlock m2 )
		throws DMLRuntimeException
	{
		MatrixBlock ret = m2;

		//transpose if dense-dense, skinny rhs matrix (not vector), and memory guarded by output
		if( checkPrepMatrixMultRightInput(m1, m2)  ) {
			MatrixBlock tmpBlock = new MatrixBlock(m2.clen, m2.rlen, m2.sparse);
			LibMatrixReorg.reorg(m2, tmpBlock, new ReorgOperator(SwapIndex.getSwapIndexFnObject()));
			ret = tmpBlock;
		}

		return ret;
	}

	private static int copyNonZeroElements( double[] a, final int aixi, final int bixk, final int bj, final int n, double[] tmpa, int[] tmpbi, final int bklen )
	{
		int knnz = 0;
		for( int k = 0; k < bklen; k++ )
			if( a[ aixi+k ] != 0 ) {
				tmpa[ knnz ] = a[ aixi+k ];
				tmpbi[ knnz ] = bixk + k*n;
				knnz ++;
			}

		return knnz;
	}

	private static int copyNonZeroElements( double[] a, int aixi, final int bk, final int bj, final int n, final int nx, double[] tmpa, int[] tmpbi, final int bklen )
	{
		int knnz = 0;
		for( int k = 0; k < bklen; k++, aixi+=n )
			if( a[ aixi ] != 0 ) {
				tmpa[ knnz ] = a[ aixi ];
				tmpbi[ knnz ] = (bk+k) * nx + bj; //scan index on b
				knnz ++;
			}

		return knnz;
	}

	private static void sumScalarResults(List<Future<Double>> tasks, MatrixBlock ret)
		throws InterruptedException, ExecutionException
	{
		//aggregate partial results and check for errors
		double val = 0;
		for(Future<Double> task : tasks)
			val += task.get();
		ret.quickSetValue(0, 0, val);
	}

	@SuppressWarnings("unused")
	private static void sumDenseResults( double[][] partret, double[] ret )
	{
		final int len = ret.length;
		final int k = partret.length;
		final int bk = k % 4;
		final int blocksize = 2 * 1024; //16KB (half of common L1 data)

		//cache-conscious aggregation to prevent repreated scans/writes of ret
		for( int bi=0; bi<len; bi+=blocksize ) {
			int llen = Math.min(len-bi, blocksize);

			//aggregate next block from all partial results
			for( int j=0; j<bk; j++ ) //rest (not aligned to 4)
				vectAdd(partret[j], ret, bi, bi, llen);
			for( int j=bk; j<k; j+=4 ) //4 partial results at a time
				vectAdd4(partret[j], partret[j+1], partret[j+2], partret[j+3], ret, bi, bi, llen);
		}

	}

	public static ArrayList<Integer> getBalancedBlockSizesDefault(int len, int k, boolean constK) {
		int nk = constK ? k : UtilFunctions.roundToNext(Math.min(8*k,len/32), k);
		return getBalancedBlockSizes(len, nk);
	}

	public static ArrayList<Integer> getAlignedBlockSizes(int len, int k, int align) {
		int blklen = (int)(Math.ceil((double)len/k));
		blklen += ((blklen%align != 0) ? align-blklen%align : 0);
		ArrayList<Integer> ret = new ArrayList<>();
		for(int i=0; i<len; i+=blklen)
			ret.add(Math.min(blklen, len-i));
		return ret;
	}

	private static ArrayList<Integer> getBalancedBlockSizes(int len, int k) {
		ArrayList<Integer> ret = new ArrayList<>();
		int base = len / k;
		int rest = len % k;
		for( int i=0; i<k; i++ ) {
			int val = base + (i<rest?1:0);
			if( val > 0 )
				ret.add(val);
		}
		return ret;
	}

	/////////////////////////////////////////////////////////
	// Task Implementations for Multi-Threaded Operations  //
	/////////////////////////////////////////////////////////

	private static class MatrixMultTask implements Callable<Object>
	{
		private final MatrixBlock _m1;
		private final MatrixBlock _m2;
		private MatrixBlock _ret = null;
		private final boolean _tm2; //transposed m2
		private final boolean _pm2r; //par over m2 rows
		private final boolean _pm2c; //par over m2 rows
		private final int _rl;
		private final int _ru;

		protected MatrixMultTask( MatrixBlock m1, MatrixBlock m2, MatrixBlock ret,
				boolean tm2, boolean pm2r, boolean pm2c, int rl, int ru )
		{
			_m1 = m1;
			_m2 = m2;
			_tm2 = tm2;
			_pm2r = pm2r;
			_pm2c = pm2c;
			_rl = rl;
			_ru = ru;

			if( pm2r ) { //vector-matrix / matrix-matrix
				//allocate local result for partial aggregation
				_ret = new MatrixBlock(ret.rlen, ret.clen, false);
			}
			else { //default case
				_ret = ret;
			}
		}

		@Override
		public Object call() throws DMLRuntimeException
		{
			//setup target index ranges
			int rl = _pm2c ? 0 : _rl;
			int ru = _pm2c ? _m1.rlen : _ru;
			int cl = _pm2c ? _rl : 0;
			int cu = _pm2c ? _ru : _ret.clen;

			//thread-local allocation
			if( _pm2r )
				_ret.allocateDenseBlock();

			//compute block matrix multiplication
			if( _ret.sparse ) //ultra-sparse
				matrixMultUltraSparse(_m1, _m2, _ret, rl, ru);
			else if(!_m1.sparse && !_m2.sparse)
				matrixMultDenseDense(_m1, _m2, _ret, _tm2, _pm2r, rl, ru, cl, cu);
			else if(_m1.sparse && _m2.sparse)
				matrixMultSparseSparse(_m1, _m2, _ret, _pm2r, rl, ru);
			else if(_m1.sparse)
				matrixMultSparseDense(_m1, _m2, _ret, _pm2r, rl, ru);
			else
				matrixMultDenseSparse(_m1, _m2, _ret, _pm2r, rl, ru);

			//maintain block nnz (upper bounds inclusive)
			if( !_pm2r )
				return _ret.recomputeNonZeros(rl, ru-1, cl, cu-1);
			else
				return _ret.getDenseBlockValues();
		}
	}

	private static class MatrixMultChainTask implements Callable<double[]>
	{
		private MatrixBlock _m1  = null;
		private MatrixBlock _m2  = null;
		private MatrixBlock _m3  = null;
		private ChainType _ct = null;
		private int _rl = -1;
		private int _ru = -1;

		protected MatrixMultChainTask( MatrixBlock mX, MatrixBlock mV, MatrixBlock mW, ChainType ct, int rl, int ru )
			throws DMLRuntimeException
		{
			_m1 = mX;
			_m2 = mV;
			_m3 = mW;
			_ct = ct;
			_rl = rl;
			_ru = ru;
		}

		@Override
		public double[] call() throws DMLRuntimeException
		{
			//thread-local allocation for partial aggregation
			MatrixBlock ret = new MatrixBlock(1, _m1.clen, false);
			ret.allocateDenseBlock();

			if( _m1.sparse )
				matrixMultChainSparse(_m1, _m2, _m3, ret, _ct, _rl, _ru);
			else
				matrixMultChainDense(_m1, _m2, _m3, ret, _ct, _rl, _ru);

			//NOTE: we dont do global aggregation from concurrent tasks in order
			//to prevent synchronization (sequential aggregation led to better
			//performance after JIT)
			return ret.getDenseBlockValues();
		}
	}

	private static class MatrixMultTransposeTask implements Callable<Object>
	{
		private final MatrixBlock _m1;
		private final MatrixBlock _ret;
		private final boolean _left;
		private final int _rl;
		private final int _ru;

		protected MatrixMultTransposeTask( MatrixBlock m1, MatrixBlock ret, boolean left, int rl, int ru )
		{
			_m1 = m1;
			_ret = ret;
			_left = left;
			_rl = rl;
			_ru = ru;
		}

		@Override
		public Object call() throws DMLRuntimeException {
			if( _m1.sparse )
				matrixMultTransposeSelfSparse(_m1, _ret, _left, _rl, _ru);
			else
				matrixMultTransposeSelfDense(_m1, _ret, _left, _rl, _ru);
			return null;
		}
	}

	private static class MatrixMultPermuteTask implements Callable<Object>
	{
		private MatrixBlock _pm1  = null;
		private MatrixBlock _m2 = null;
		private MatrixBlock _ret1 = null;
		private MatrixBlock _ret2 = null;
		private int _rl = -1;
		private int _ru = -1;

		protected MatrixMultPermuteTask( MatrixBlock pm1, MatrixBlock m2, MatrixBlock ret1, MatrixBlock ret2, int rl, int ru)
		{
			_pm1 = pm1;
			_m2 = m2;
			_ret1 = ret1;
			_ret2 = ret2;
			_rl = rl;
			_ru = ru;
		}

		@Override
		public Object call() throws DMLRuntimeException
		{
			if( _m2.sparse )
				matrixMultPermuteSparse(_pm1, _m2, _ret1, _ret2, _rl, _ru);
			else if( _ret1.sparse )
				matrixMultPermuteDenseSparse(_pm1, _m2, _ret1, _ret2, _rl, _ru);
			else
				matrixMultPermuteDense(_pm1, _m2, _ret1, _ret2, _rl, _ru);

			return null;
		}
	}

	private static class MatrixMultWSLossTask implements Callable<Double>
	{
		private MatrixBlock _mX = null;
		private MatrixBlock _mU = null;
		private MatrixBlock _mV = null;
		private MatrixBlock _mW = null;
		private MatrixBlock _ret = null;
		private WeightsType _wt = null;
		private int _rl = -1;
		private int _ru = -1;

		protected MatrixMultWSLossTask(MatrixBlock mX, MatrixBlock mU, MatrixBlock mV, MatrixBlock mW, WeightsType wt, int rl, int ru)
			throws DMLRuntimeException
		{
			_mX = mX;
			_mU = mU;
			_mV = mV;
			_mW = mW;
			_wt = wt;
			_rl = rl;
			_ru = ru;

			//allocate local result for partial aggregation
			_ret = new MatrixBlock(1, 1, false);
			_ret.allocateDenseBlock();
		}

		@Override
		public Double call() throws DMLRuntimeException
		{
			if( !_mX.sparse && !_mU.sparse && !_mV.sparse && (_mW==null || !_mW.sparse)
				&& !_mX.isEmptyBlock() && !_mU.isEmptyBlock() && !_mV.isEmptyBlock()
				&& (_mW==null || !_mW.isEmptyBlock()))
				matrixMultWSLossDense(_mX, _mU, _mV, _mW, _ret, _wt, _rl, _ru);
			else if( _mX.sparse && !_mU.sparse && !_mV.sparse && (_mW==null || _mW.sparse)
				    && !_mX.isEmptyBlock() && !_mU.isEmptyBlock() && !_mV.isEmptyBlock()
				    && (_mW==null || !_mW.isEmptyBlock()))
				matrixMultWSLossSparseDense(_mX, _mU, _mV, _mW, _ret, _wt, _rl, _ru);
			else
				matrixMultWSLossGeneric(_mX, _mU, _mV, _mW, _ret, _wt, _rl, _ru);

			return _ret.quickGetValue(0, 0);
		}
	}

	private static class MatrixMultWSigmoidTask implements Callable<Long>
	{
		private MatrixBlock _mW = null;
		private MatrixBlock _mU = null;
		private MatrixBlock _mV = null;
		private MatrixBlock _ret = null;
		private WSigmoidType _wt = null;
		private int _rl = -1;
		private int _ru = -1;

		protected MatrixMultWSigmoidTask(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WSigmoidType wt, int rl, int ru)
			throws DMLRuntimeException
		{
			_mW = mW;
			_mU = mU;
			_mV = mV;
			_ret = ret;
			_wt = wt;
			_rl = rl;
			_ru = ru;
		}

		@Override
		public Long call() throws DMLRuntimeException
		{
			//core weighted square sum mm computation
			if( !_mW.sparse && !_mU.sparse && !_mV.sparse && !_mU.isEmptyBlock() && !_mV.isEmptyBlock() )
				matrixMultWSigmoidDense(_mW, _mU, _mV, _ret, _wt, _rl, _ru);
			else if( _mW.sparse && !_mU.sparse && !_mV.sparse && !_mU.isEmptyBlock() && !_mV.isEmptyBlock())
				matrixMultWSigmoidSparseDense(_mW, _mU, _mV, _ret, _wt, _rl, _ru);
			else
				matrixMultWSigmoidGeneric(_mW, _mU, _mV, _ret, _wt, _rl, _ru);

			//maintain block nnz (upper bounds inclusive)
			return _ret.recomputeNonZeros(_rl, _ru-1, 0, _ret.getNumColumns()-1);
		}
	}

	private static class MatrixMultWDivTask implements Callable<Long>
	{
		private MatrixBlock _mW = null;
		private MatrixBlock _mU = null;
		private MatrixBlock _mV = null;
		private MatrixBlock _mX = null;
		private MatrixBlock _ret = null;
		private WDivMMType _wt = null;
		private int _rl = -1;
		private int _ru = -1;
		private int _cl = -1;
		private int _cu = -1;

		protected MatrixMultWDivTask(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock mX, MatrixBlock ret, WDivMMType wt, int rl, int ru, int cl, int cu)
			throws DMLRuntimeException
		{
			_mW = mW;
			_mU = mU;
			_mV = mV;
			_mX = mX;
			_wt = wt;
			_rl = rl;
			_ru = ru;
			_cl = cl;
			_cu = cu;
			_ret = ret;
		}

		@Override
		public Long call() throws DMLRuntimeException
		{
			//core weighted div mm computation
			boolean scalarX = _wt.hasScalar();
			if( !_mW.sparse && !_mU.sparse && !_mV.sparse && (_mX==null || !_mX.sparse || scalarX) && !_mU.isEmptyBlock() && !_mV.isEmptyBlock() )
				matrixMultWDivMMDense(_mW, _mU, _mV, _mX, _ret, _wt, _rl, _ru, _cl, _cu);
			else if( _mW.sparse && !_mU.sparse && !_mV.sparse && (_mX==null || _mX.sparse || scalarX) && !_mU.isEmptyBlock() && !_mV.isEmptyBlock())
				matrixMultWDivMMSparseDense(_mW, _mU, _mV, _mX, _ret, _wt, _rl, _ru, _cl, _cu);
			else
				matrixMultWDivMMGeneric(_mW, _mU, _mV, _mX, _ret, _wt, _rl, _ru, _cl, _cu);

			//maintain partial nnz for right (upper bounds inclusive)
			int rl = _wt.isLeft() ? _cl : _rl;
			int ru = _wt.isLeft() ? _cu : _ru;
			return _ret.recomputeNonZeros(rl, ru-1, 0, _ret.getNumColumns()-1);
		}
	}

	private static class MatrixMultWCeTask implements Callable<Double>
	{
		private MatrixBlock _mW = null;
		private MatrixBlock _mU = null;
		private MatrixBlock _mV = null;
		private double _eps = 0.0;
		private MatrixBlock _ret = null;
		private WCeMMType _wt = null;
		private int _rl = -1;
		private int _ru = -1;

		protected MatrixMultWCeTask(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, double eps, WCeMMType wt, int rl, int ru)
			throws DMLRuntimeException
		{
			_mW = mW;
			_mU = mU;
			_mV = mV;
			_eps = eps;
			_wt = wt;
			_rl = rl;
			_ru = ru;

			//allocate local result for partial aggregation
			_ret = new MatrixBlock(1, 1, false);
			_ret.allocateDenseBlock();
		}

		@Override
		public Double call() throws DMLRuntimeException
		{
			//core weighted cross entropy mm computation
			if( !_mW.sparse && !_mU.sparse && !_mV.sparse && !_mU.isEmptyBlock() && !_mV.isEmptyBlock() )
				matrixMultWCeMMDense(_mW, _mU, _mV, _eps, _ret, _wt, _rl, _ru);
			else if( _mW.sparse && !_mU.sparse && !_mV.sparse && !_mU.isEmptyBlock() && !_mV.isEmptyBlock())
				matrixMultWCeMMSparseDense(_mW, _mU, _mV, _eps, _ret, _wt, _rl, _ru);
			else
				matrixMultWCeMMGeneric(_mW, _mU, _mV, _eps, _ret, _wt, _rl, _ru);

			return _ret.quickGetValue(0, 0);
		}
	}

	private static class MatrixMultWuTask implements Callable<Long>
	{
		private MatrixBlock _mW = null;
		private MatrixBlock _mU = null;
		private MatrixBlock _mV = null;
		private MatrixBlock _ret = null;
		private WUMMType _wt = null;
		private ValueFunction _fn = null;
		private int _rl = -1;
		private int _ru = -1;

		protected MatrixMultWuTask(MatrixBlock mW, MatrixBlock mU, MatrixBlock mV, MatrixBlock ret, WUMMType wt, ValueFunction fn, int rl, int ru)
			throws DMLRuntimeException
		{
			_mW = mW;
			_mU = mU;
			_mV = mV;
			_ret = ret;
			_wt = wt;
			_fn = fn;
			_rl = rl;
			_ru = ru;
		}

		@Override
		public Long call() throws DMLRuntimeException
		{
			//core weighted square sum mm computation
			if( !_mW.sparse && !_mU.sparse && !_mV.sparse && !_mU.isEmptyBlock() && !_mV.isEmptyBlock() )
				matrixMultWuMMDense(_mW, _mU, _mV, _ret, _wt, _fn, _rl, _ru);
			else if( _mW.sparse && !_mU.sparse && !_mV.sparse && !_mU.isEmptyBlock() && !_mV.isEmptyBlock())
				matrixMultWuMMSparseDense(_mW, _mU, _mV, _ret, _wt, _fn, _rl, _ru);
			else
				matrixMultWuMMGeneric(_mW, _mU, _mV, _ret, _wt, _fn, _rl, _ru);

			//maintain block nnz (upper bounds inclusive)
			return _ret.recomputeNonZeros(_rl, _ru-1, 0, _ret.getNumColumns()-1);
		}
	}
}