BlockFieldMatrix.java

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package org.apache.commons.math4.legacy.linear;

import java.io.Serializable;

import org.apache.commons.math4.legacy.core.Field;
import org.apache.commons.math4.legacy.core.FieldElement;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.exception.NoDataException;
import org.apache.commons.math4.legacy.exception.NotStrictlyPositiveException;
import org.apache.commons.math4.legacy.exception.NullArgumentException;
import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
import org.apache.commons.math4.legacy.exception.OutOfRangeException;
import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
import org.apache.commons.math4.core.jdkmath.JdkMath;
import org.apache.commons.math4.legacy.core.MathArrays;

/**
 * Cache-friendly implementation of FieldMatrix using a flat arrays to store
 * square blocks of the matrix.
 * <p>
 * This implementation is specially designed to be cache-friendly. Square blocks are
 * stored as small arrays and allow efficient traversal of data both in row major direction
 * and columns major direction, one block at a time. This greatly increases performances
 * for algorithms that use crossed directions loops like multiplication or transposition.
 * </p>
 * <p>
 * The size of square blocks is a static parameter. It may be tuned according to the cache
 * size of the target computer processor. As a rule of thumbs, it should be the largest
 * value that allows three blocks to be simultaneously cached (this is necessary for example
 * for matrix multiplication). The default value is to use 36x36 blocks.
 * </p>
 * <p>
 * The regular blocks represent {@link #BLOCK_SIZE} x {@link #BLOCK_SIZE} squares. Blocks
 * at right hand side and bottom side which may be smaller to fit matrix dimensions. The square
 * blocks are flattened in row major order in single dimension arrays which are therefore
 * {@link #BLOCK_SIZE}<sup>2</sup> elements long for regular blocks. The blocks are themselves
 * organized in row major order.
 * </p>
 * <p>
 * As an example, for a block size of 36x36, a 100x60 matrix would be stored in 6 blocks.
 * Block 0 would be a Field[1296] array holding the upper left 36x36 square, block 1 would be
 * a Field[1296] array holding the upper center 36x36 square, block 2 would be a Field[1008]
 * array holding the upper right 36x28 rectangle, block 3 would be a Field[864] array holding
 * the lower left 24x36 rectangle, block 4 would be a Field[864] array holding the lower center
 * 24x36 rectangle and block 5 would be a Field[672] array holding the lower right 24x28
 * rectangle.
 * </p>
 * <p>
 * The layout complexity overhead versus simple mapping of matrices to java
 * arrays is negligible for small matrices (about 1%). The gain from cache efficiency leads
 * to up to 3-fold improvements for matrices of moderate to large size.
 * </p>
 * @param <T> the type of the field elements
 * @since 2.0
 */
public class BlockFieldMatrix<T extends FieldElement<T>> extends AbstractFieldMatrix<T> implements Serializable {
    /** Block size. */
    public static final int BLOCK_SIZE = 36;
    /** Serializable version identifier. */
    private static final long serialVersionUID = -4602336630143123183L;
    /** Blocks of matrix entries. */
    private final T[][] blocks;
    /** Number of rows of the matrix. */
    private final int rows;
    /** Number of columns of the matrix. */
    private final int columns;
    /** Number of block rows of the matrix. */
    private final int blockRows;
    /** Number of block columns of the matrix. */
    private final int blockColumns;

    /**
     * Create a new matrix with the supplied row and column dimensions.
     *
     * @param field Field to which the elements belong.
     * @param rows Number of rows in the new matrix.
     * @param columns Number of columns in the new matrix.
     * @throws NotStrictlyPositiveException if row or column dimension is not
     * positive.
     */
    public BlockFieldMatrix(final Field<T> field, final int rows,
                            final int columns)
        throws NotStrictlyPositiveException {
        super(field, rows, columns);
        this.rows    = rows;
        this.columns = columns;

        // number of blocks
        blockRows    = (rows    + BLOCK_SIZE - 1) / BLOCK_SIZE;
        blockColumns = (columns + BLOCK_SIZE - 1) / BLOCK_SIZE;

        // allocate storage blocks, taking care of smaller ones at right and bottom
        blocks = createBlocksLayout(field, rows, columns);
    }

    /**
     * Create a new dense matrix copying entries from raw layout data.
     * <p>The input array <em>must</em> already be in raw layout.</p>
     * <p>Calling this constructor is equivalent to call:
     * {@code matrix = new BlockFieldMatrix<T>(getField(), rawData.length, rawData[0].length,
     *                                   toBlocksLayout(rawData), false);}
     *
     * @param rawData Data for the new matrix, in raw layout.
     * @throws DimensionMismatchException if the {@code blockData} shape is
     * inconsistent with block layout.
     * @see #BlockFieldMatrix(int, int, FieldElement[][], boolean)
     */
    public BlockFieldMatrix(final T[][] rawData)
        throws DimensionMismatchException {
        this(rawData.length, rawData[0].length, toBlocksLayout(rawData), false);
    }

    /**
     * Create a new dense matrix copying entries from block layout data.
     * <p>The input array <em>must</em> already be in blocks layout.</p>
     * @param rows  the number of rows in the new matrix
     * @param columns  the number of columns in the new matrix
     * @param blockData data for new matrix
     * @param copyArray if true, the input array will be copied, otherwise
     * it will be referenced
     *
     * @throws DimensionMismatchException if the {@code blockData} shape is
     * inconsistent with block layout.
     * @throws NotStrictlyPositiveException if row or column dimension is not
     * positive.
     * @see #createBlocksLayout(Field, int, int)
     * @see #toBlocksLayout(FieldElement[][])
     * @see #BlockFieldMatrix(FieldElement[][])
     */
    public BlockFieldMatrix(final int rows, final int columns,
                            final T[][] blockData, final boolean copyArray)
        throws DimensionMismatchException, NotStrictlyPositiveException {
        super(extractField(blockData), rows, columns);
        this.rows    = rows;
        this.columns = columns;

        // number of blocks
        blockRows    = (rows    + BLOCK_SIZE - 1) / BLOCK_SIZE;
        blockColumns = (columns + BLOCK_SIZE - 1) / BLOCK_SIZE;

        if (copyArray) {
            // allocate storage blocks, taking care of smaller ones at right and bottom
            blocks = MathArrays.buildArray(getField(), blockRows * blockColumns, -1);
        } else {
            // reference existing array
            blocks = blockData;
        }

        int index = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int iHeight = blockHeight(iBlock);
            for (int jBlock = 0; jBlock < blockColumns; ++jBlock, ++index) {
                if (blockData[index].length != iHeight * blockWidth(jBlock)) {
                    throw new DimensionMismatchException(blockData[index].length,
                                                         iHeight * blockWidth(jBlock));
                }
                if (copyArray) {
                    blocks[index] = blockData[index].clone();
                }
            }
        }
    }

    /**
     * Convert a data array from raw layout to blocks layout.
     * <p>
     * Raw layout is the straightforward layout where element at row i and
     * column j is in array element <code>rawData[i][j]</code>. Blocks layout
     * is the layout used in {@link BlockFieldMatrix} instances, where the matrix
     * is split in square blocks (except at right and bottom side where blocks may
     * be rectangular to fit matrix size) and each block is stored in a flattened
     * one-dimensional array.
     * </p>
     * <p>
     * This method creates an array in blocks layout from an input array in raw layout.
     * It can be used to provide the array argument of the {@link
     * #BlockFieldMatrix(int, int, FieldElement[][], boolean)}
     * constructor.
     * </p>
     * @param <T> Type of the field elements.
     * @param rawData Data array in raw layout.
     * @return a new data array containing the same entries but in blocks layout
     * @throws DimensionMismatchException if {@code rawData} is not rectangular
     *  (not all rows have the same length).
     * @see #createBlocksLayout(Field, int, int)
     * @see #BlockFieldMatrix(int, int, FieldElement[][], boolean)
     */
    public static <T extends FieldElement<T>> T[][] toBlocksLayout(final T[][] rawData)
        throws DimensionMismatchException {

        final int rows         = rawData.length;
        final int columns      = rawData[0].length;
        final int blockRows    = (rows    + BLOCK_SIZE - 1) / BLOCK_SIZE;
        final int blockColumns = (columns + BLOCK_SIZE - 1) / BLOCK_SIZE;

        // safety checks
        for (int i = 0; i < rawData.length; ++i) {
            final int length = rawData[i].length;
            if (length != columns) {
                throw new DimensionMismatchException(columns, length);
            }
        }

        // convert array
        final Field<T> field = extractField(rawData);
        final T[][] blocks = MathArrays.buildArray(field, blockRows * blockColumns, -1);
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart  = iBlock * BLOCK_SIZE;
            final int pEnd    = JdkMath.min(pStart + BLOCK_SIZE, rows);
            final int iHeight = pEnd - pStart;
            for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
                final int qStart = jBlock * BLOCK_SIZE;
                final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
                final int jWidth = qEnd - qStart;

                // allocate new block
                final T[] block = MathArrays.buildArray(field, iHeight * jWidth);
                blocks[blockIndex] = block;

                // copy data
                int index = 0;
                for (int p = pStart; p < pEnd; ++p) {
                    System.arraycopy(rawData[p], qStart, block, index, jWidth);
                    index += jWidth;
                }

                ++blockIndex;
            }
        }

        return blocks;
    }

    /**
     * Create a data array in blocks layout.
     * <p>
     * This method can be used to create the array argument of the {@link
     * #BlockFieldMatrix(int, int, FieldElement[][], boolean)}
     * constructor.
     * </p>
     * @param <T> Type of the field elements.
     * @param field Field to which the elements belong.
     * @param rows Number of rows in the new matrix.
     * @param columns Number of columns in the new matrix.
     * @return a new data array in blocks layout.
     * @see #toBlocksLayout(FieldElement[][])
     * @see #BlockFieldMatrix(int, int, FieldElement[][], boolean)
     */
    public static <T extends FieldElement<T>> T[][] createBlocksLayout(final Field<T> field,
                                                                       final int rows, final int columns) {
        final int blockRows    = (rows    + BLOCK_SIZE - 1) / BLOCK_SIZE;
        final int blockColumns = (columns + BLOCK_SIZE - 1) / BLOCK_SIZE;

        final T[][] blocks = MathArrays.buildArray(field, blockRows * blockColumns, -1);
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart  = iBlock * BLOCK_SIZE;
            final int pEnd    = JdkMath.min(pStart + BLOCK_SIZE, rows);
            final int iHeight = pEnd - pStart;
            for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
                final int qStart = jBlock * BLOCK_SIZE;
                final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
                final int jWidth = qEnd - qStart;
                blocks[blockIndex] = MathArrays.buildArray(field, iHeight * jWidth);
                ++blockIndex;
            }
        }

        return blocks;
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> createMatrix(final int rowDimension,
                                       final int columnDimension)
        throws NotStrictlyPositiveException {
        return new BlockFieldMatrix<>(getField(), rowDimension,
                                       columnDimension);
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> copy() {

        // create an empty matrix
        BlockFieldMatrix<T> copied = new BlockFieldMatrix<>(getField(), rows, columns);

        // copy the blocks
        for (int i = 0; i < blocks.length; ++i) {
            System.arraycopy(blocks[i], 0, copied.blocks[i], 0, blocks[i].length);
        }

        return copied;
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> add(final FieldMatrix<T> m)
        throws MatrixDimensionMismatchException {
        if (m instanceof BlockFieldMatrix) {
            return add((BlockFieldMatrix<T>) m);
        }

        // safety check
        checkAdd(m);

        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, columns);

        // perform addition block-wise, to ensure good cache behavior
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < out.blockRows; ++iBlock) {
            for (int jBlock = 0; jBlock < out.blockColumns; ++jBlock) {

                // perform addition on the current block
                final T[] outBlock = out.blocks[blockIndex];
                final T[] tBlock   = blocks[blockIndex];
                final int      pStart   = iBlock * BLOCK_SIZE;
                final int      pEnd     = JdkMath.min(pStart + BLOCK_SIZE, rows);
                final int      qStart   = jBlock * BLOCK_SIZE;
                final int      qEnd     = JdkMath.min(qStart + BLOCK_SIZE, columns);
                int k = 0;
                for (int p = pStart; p < pEnd; ++p) {
                    for (int q = qStart; q < qEnd; ++q) {
                        outBlock[k] = tBlock[k].add(m.getEntry(p, q));
                        ++k;
                    }
                }

                // go to next block
                ++blockIndex;
            }
        }

        return out;
    }

    /**
     * Compute the sum of {@code this} and {@code m}.
     *
     * @param m matrix to be added
     * @return {@code this + m}
     * @throws MatrixDimensionMismatchException if {@code m} is not the same
     * size as {@code this}
     */
    public BlockFieldMatrix<T> add(final BlockFieldMatrix<T> m)
        throws MatrixDimensionMismatchException {

        // safety check
        checkAdd(m);

        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, columns);

        // perform addition block-wise, to ensure good cache behavior
        for (int blockIndex = 0; blockIndex < out.blocks.length; ++blockIndex) {
            final T[] outBlock = out.blocks[blockIndex];
            final T[] tBlock   = blocks[blockIndex];
            final T[] mBlock   = m.blocks[blockIndex];
            for (int k = 0; k < outBlock.length; ++k) {
                outBlock[k] = tBlock[k].add(mBlock[k]);
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> subtract(final FieldMatrix<T> m)
        throws MatrixDimensionMismatchException {
        if (m instanceof BlockFieldMatrix) {
            return subtract((BlockFieldMatrix<T>) m);
        }

        // safety check
        checkAdd(m);

        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, columns);

        // perform subtraction block-wise, to ensure good cache behavior
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < out.blockRows; ++iBlock) {
            for (int jBlock = 0; jBlock < out.blockColumns; ++jBlock) {

                // perform subtraction on the current block
                final T[] outBlock = out.blocks[blockIndex];
                final T[] tBlock   = blocks[blockIndex];
                final int      pStart   = iBlock * BLOCK_SIZE;
                final int      pEnd     = JdkMath.min(pStart + BLOCK_SIZE, rows);
                final int      qStart   = jBlock * BLOCK_SIZE;
                final int      qEnd     = JdkMath.min(qStart + BLOCK_SIZE, columns);
                int k = 0;
                for (int p = pStart; p < pEnd; ++p) {
                    for (int q = qStart; q < qEnd; ++q) {
                        outBlock[k] = tBlock[k].subtract(m.getEntry(p, q));
                        ++k;
                    }
                }

                // go to next block
                ++blockIndex;
            }
        }

        return out;
    }

    /**
     * Compute {@code this - m}.
     *
     * @param m matrix to be subtracted
     * @return {@code this - m}
     * @throws MatrixDimensionMismatchException if {@code m} is not the same
     * size as {@code this}
     */
    public BlockFieldMatrix<T> subtract(final BlockFieldMatrix<T> m) throws MatrixDimensionMismatchException {
        // safety check
        checkAdd(m);

        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, columns);

        // perform subtraction block-wise, to ensure good cache behavior
        for (int blockIndex = 0; blockIndex < out.blocks.length; ++blockIndex) {
            final T[] outBlock = out.blocks[blockIndex];
            final T[] tBlock   = blocks[blockIndex];
            final T[] mBlock   = m.blocks[blockIndex];
            for (int k = 0; k < outBlock.length; ++k) {
                outBlock[k] = tBlock[k].subtract(mBlock[k]);
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> scalarAdd(final T d) {
        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, columns);

        // perform subtraction block-wise, to ensure good cache behavior
        for (int blockIndex = 0; blockIndex < out.blocks.length; ++blockIndex) {
            final T[] outBlock = out.blocks[blockIndex];
            final T[] tBlock   = blocks[blockIndex];
            for (int k = 0; k < outBlock.length; ++k) {
                outBlock[k] = tBlock[k].add(d);
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> scalarMultiply(final T d) {

        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, columns);

        // perform subtraction block-wise, to ensure good cache behavior
        for (int blockIndex = 0; blockIndex < out.blocks.length; ++blockIndex) {
            final T[] outBlock = out.blocks[blockIndex];
            final T[] tBlock   = blocks[blockIndex];
            for (int k = 0; k < outBlock.length; ++k) {
                outBlock[k] = tBlock[k].multiply(d);
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> multiply(final FieldMatrix<T> m)
        throws DimensionMismatchException {
        if (m instanceof BlockFieldMatrix) {
            return multiply((BlockFieldMatrix<T>) m);
        }

        // safety check
        checkMultiply(m);

        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, m.getColumnDimension());
        final T zero = getField().getZero();

        // perform multiplication block-wise, to ensure good cache behavior
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < out.blockRows; ++iBlock) {

            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);

            for (int jBlock = 0; jBlock < out.blockColumns; ++jBlock) {

                final int qStart = jBlock * BLOCK_SIZE;
                final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, m.getColumnDimension());

                // select current block
                final T[] outBlock = out.blocks[blockIndex];

                // perform multiplication on current block
                for (int kBlock = 0; kBlock < blockColumns; ++kBlock) {
                    final int kWidth      = blockWidth(kBlock);
                    final T[] tBlock = blocks[iBlock * blockColumns + kBlock];
                    final int rStart      = kBlock * BLOCK_SIZE;
                    int k = 0;
                    for (int p = pStart; p < pEnd; ++p) {
                        final int lStart = (p - pStart) * kWidth;
                        final int lEnd   = lStart + kWidth;
                        for (int q = qStart; q < qEnd; ++q) {
                            T sum = zero;
                            int r = rStart;
                            for (int l = lStart; l < lEnd; ++l) {
                                sum = sum.add(tBlock[l].multiply(m.getEntry(r, q)));
                                ++r;
                            }
                            outBlock[k] = outBlock[k].add(sum);
                            ++k;
                        }
                    }
                }

                // go to next block
                ++blockIndex;
            }
        }

        return out;
    }

    /**
     * Returns the result of postmultiplying {@code this} by {@code m}.
     *
     * @param m matrix to postmultiply by
     * @return {@code this * m}
     * @throws DimensionMismatchException if the matrices are not compatible.
     */
    public BlockFieldMatrix<T> multiply(BlockFieldMatrix<T> m)
        throws DimensionMismatchException {

        // safety check
        checkMultiply(m);

        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, m.columns);
        final T zero = getField().getZero();

        // perform multiplication block-wise, to ensure good cache behavior
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < out.blockRows; ++iBlock) {

            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);

            for (int jBlock = 0; jBlock < out.blockColumns; ++jBlock) {
                final int jWidth = out.blockWidth(jBlock);
                final int jWidth2 = jWidth  + jWidth;
                final int jWidth3 = jWidth2 + jWidth;
                final int jWidth4 = jWidth3 + jWidth;

                // select current block
                final T[] outBlock = out.blocks[blockIndex];

                // perform multiplication on current block
                for (int kBlock = 0; kBlock < blockColumns; ++kBlock) {
                    final int kWidth = blockWidth(kBlock);
                    final T[] tBlock = blocks[iBlock * blockColumns + kBlock];
                    final T[] mBlock = m.blocks[kBlock * m.blockColumns + jBlock];
                    int k = 0;
                    for (int p = pStart; p < pEnd; ++p) {
                        final int lStart = (p - pStart) * kWidth;
                        final int lEnd   = lStart + kWidth;
                        for (int nStart = 0; nStart < jWidth; ++nStart) {
                            T sum = zero;
                            int l = lStart;
                            int n = nStart;
                            while (l < lEnd - 3) {
                                sum = sum.
                                      add(tBlock[l].multiply(mBlock[n])).
                                      add(tBlock[l + 1].multiply(mBlock[n + jWidth])).
                                      add(tBlock[l + 2].multiply(mBlock[n + jWidth2])).
                                      add(tBlock[l + 3].multiply(mBlock[n + jWidth3]));
                                l += 4;
                                n += jWidth4;
                            }
                            while (l < lEnd) {
                                sum = sum.add(tBlock[l++].multiply(mBlock[n]));
                                n += jWidth;
                            }
                            outBlock[k] = outBlock[k].add(sum);
                            ++k;
                        }
                    }
                }

                // go to next block
                ++blockIndex;
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public T[][] getData() {

        final T[][] data = MathArrays.buildArray(getField(), getRowDimension(), getColumnDimension());
        final int lastColumns = columns - (blockColumns - 1) * BLOCK_SIZE;

        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);
            int regularPos   = 0;
            int lastPos      = 0;
            for (int p = pStart; p < pEnd; ++p) {
                final T[] dataP = data[p];
                int blockIndex = iBlock * blockColumns;
                int dataPos    = 0;
                for (int jBlock = 0; jBlock < blockColumns - 1; ++jBlock) {
                    System.arraycopy(blocks[blockIndex++], regularPos, dataP, dataPos, BLOCK_SIZE);
                    dataPos += BLOCK_SIZE;
                }
                System.arraycopy(blocks[blockIndex], lastPos, dataP, dataPos, lastColumns);
                regularPos += BLOCK_SIZE;
                lastPos    += lastColumns;
            }
        }

        return data;
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> getSubMatrix(final int startRow, final int endRow,
                                       final int startColumn,
                                       final int endColumn)
        throws OutOfRangeException, NumberIsTooSmallException {
        // safety checks
        checkSubMatrixIndex(startRow, endRow, startColumn, endColumn);

        // create the output matrix
        final BlockFieldMatrix<T> out =
            new BlockFieldMatrix<>(getField(), endRow - startRow + 1, endColumn - startColumn + 1);

        // compute blocks shifts
        final int blockStartRow    = startRow    / BLOCK_SIZE;
        final int rowsShift        = startRow    % BLOCK_SIZE;
        final int blockStartColumn = startColumn / BLOCK_SIZE;
        final int columnsShift     = startColumn % BLOCK_SIZE;

        // perform extraction block-wise, to ensure good cache behavior
        int pBlock = blockStartRow;
        for (int iBlock = 0; iBlock < out.blockRows; ++iBlock) {
            final int iHeight = out.blockHeight(iBlock);
            int qBlock = blockStartColumn;
            for (int jBlock = 0; jBlock < out.blockColumns; ++jBlock) {
                final int jWidth = out.blockWidth(jBlock);

                // handle one block of the output matrix
                final int      outIndex = iBlock * out.blockColumns + jBlock;
                final T[] outBlock = out.blocks[outIndex];
                final int      index    = pBlock * blockColumns + qBlock;
                final int      width    = blockWidth(qBlock);

                final int heightExcess = iHeight + rowsShift - BLOCK_SIZE;
                final int widthExcess  = jWidth + columnsShift - BLOCK_SIZE;
                if (heightExcess > 0) {
                    // the submatrix block spans on two blocks rows from the original matrix
                    if (widthExcess > 0) {
                        // the submatrix block spans on two blocks columns from the original matrix
                        final int width2 = blockWidth(qBlock + 1);
                        copyBlockPart(blocks[index], width,
                                      rowsShift, BLOCK_SIZE,
                                      columnsShift, BLOCK_SIZE,
                                      outBlock, jWidth, 0, 0);
                        copyBlockPart(blocks[index + 1], width2,
                                      rowsShift, BLOCK_SIZE,
                                      0, widthExcess,
                                      outBlock, jWidth, 0, jWidth - widthExcess);
                        copyBlockPart(blocks[index + blockColumns], width,
                                      0, heightExcess,
                                      columnsShift, BLOCK_SIZE,
                                      outBlock, jWidth, iHeight - heightExcess, 0);
                        copyBlockPart(blocks[index + blockColumns + 1], width2,
                                      0, heightExcess,
                                      0, widthExcess,
                                      outBlock, jWidth, iHeight - heightExcess, jWidth - widthExcess);
                    } else {
                        // the submatrix block spans on one block column from the original matrix
                        copyBlockPart(blocks[index], width,
                                      rowsShift, BLOCK_SIZE,
                                      columnsShift, jWidth + columnsShift,
                                      outBlock, jWidth, 0, 0);
                        copyBlockPart(blocks[index + blockColumns], width,
                                      0, heightExcess,
                                      columnsShift, jWidth + columnsShift,
                                      outBlock, jWidth, iHeight - heightExcess, 0);
                    }
                } else {
                    // the submatrix block spans on one block row from the original matrix
                    if (widthExcess > 0) {
                        // the submatrix block spans on two blocks columns from the original matrix
                        final int width2 = blockWidth(qBlock + 1);
                        copyBlockPart(blocks[index], width,
                                      rowsShift, iHeight + rowsShift,
                                      columnsShift, BLOCK_SIZE,
                                      outBlock, jWidth, 0, 0);
                        copyBlockPart(blocks[index + 1], width2,
                                      rowsShift, iHeight + rowsShift,
                                      0, widthExcess,
                                      outBlock, jWidth, 0, jWidth - widthExcess);
                    } else {
                        // the submatrix block spans on one block column from the original matrix
                        copyBlockPart(blocks[index], width,
                                      rowsShift, iHeight + rowsShift,
                                      columnsShift, jWidth + columnsShift,
                                      outBlock, jWidth, 0, 0);
                    }
               }
                ++qBlock;
            }
            ++pBlock;
        }

        return out;
    }

    /**
     * Copy a part of a block into another one
     * <p>This method can be called only when the specified part fits in both
     * blocks, no verification is done here.</p>
     * @param srcBlock source block
     * @param srcWidth source block width ({@link #BLOCK_SIZE} or smaller)
     * @param srcStartRow start row in the source block
     * @param srcEndRow end row (exclusive) in the source block
     * @param srcStartColumn start column in the source block
     * @param srcEndColumn end column (exclusive) in the source block
     * @param dstBlock destination block
     * @param dstWidth destination block width ({@link #BLOCK_SIZE} or smaller)
     * @param dstStartRow start row in the destination block
     * @param dstStartColumn start column in the destination block
     */
    private void copyBlockPart(final T[] srcBlock, final int srcWidth,
                               final int srcStartRow, final int srcEndRow,
                               final int srcStartColumn, final int srcEndColumn,
                               final T[] dstBlock, final int dstWidth,
                               final int dstStartRow, final int dstStartColumn) {
        final int length = srcEndColumn - srcStartColumn;
        int srcPos = srcStartRow * srcWidth + srcStartColumn;
        int dstPos = dstStartRow * dstWidth + dstStartColumn;
        for (int srcRow = srcStartRow; srcRow < srcEndRow; ++srcRow) {
            System.arraycopy(srcBlock, srcPos, dstBlock, dstPos, length);
            srcPos += srcWidth;
            dstPos += dstWidth;
        }
    }

    /** {@inheritDoc} */
    @Override
    public void setSubMatrix(final T[][] subMatrix, final int row,
                             final int column)
        throws DimensionMismatchException, OutOfRangeException,
        NoDataException, NullArgumentException {
        // safety checks
        NullArgumentException.check(subMatrix);
        final int refLength = subMatrix[0].length;
        if (refLength == 0) {
            throw new NoDataException(LocalizedFormats.AT_LEAST_ONE_COLUMN);
        }
        final int endRow    = row + subMatrix.length - 1;
        final int endColumn = column + refLength - 1;
        checkSubMatrixIndex(row, endRow, column, endColumn);
        for (final T[] subRow : subMatrix) {
            if (subRow.length != refLength) {
                throw new DimensionMismatchException(refLength, subRow.length);
            }
        }

        // compute blocks bounds
        final int blockStartRow    = row / BLOCK_SIZE;
        final int blockEndRow      = (endRow + BLOCK_SIZE) / BLOCK_SIZE;
        final int blockStartColumn = column / BLOCK_SIZE;
        final int blockEndColumn   = (endColumn + BLOCK_SIZE) / BLOCK_SIZE;

        // perform copy block-wise, to ensure good cache behavior
        for (int iBlock = blockStartRow; iBlock < blockEndRow; ++iBlock) {
            final int iHeight  = blockHeight(iBlock);
            final int firstRow = iBlock * BLOCK_SIZE;
            final int iStart   = JdkMath.max(row,    firstRow);
            final int iEnd     = JdkMath.min(endRow + 1, firstRow + iHeight);

            for (int jBlock = blockStartColumn; jBlock < blockEndColumn; ++jBlock) {
                final int jWidth      = blockWidth(jBlock);
                final int firstColumn = jBlock * BLOCK_SIZE;
                final int jStart      = JdkMath.max(column,    firstColumn);
                final int jEnd        = JdkMath.min(endColumn + 1, firstColumn + jWidth);
                final int jLength     = jEnd - jStart;

                // handle one block, row by row
                final T[] block = blocks[iBlock * blockColumns + jBlock];
                for (int i = iStart; i < iEnd; ++i) {
                    System.arraycopy(subMatrix[i - row], jStart - column,
                                     block, (i - firstRow) * jWidth + (jStart - firstColumn),
                                     jLength);
                }
            }
        }
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> getRowMatrix(final int row)
        throws OutOfRangeException {
        checkRowIndex(row);
        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), 1, columns);

        // perform copy block-wise, to ensure good cache behavior
        final int iBlock  = row / BLOCK_SIZE;
        final int iRow    = row - iBlock * BLOCK_SIZE;
        int outBlockIndex = 0;
        int outIndex      = 0;
        T[] outBlock = out.blocks[outBlockIndex];
        for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
            final int jWidth     = blockWidth(jBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            final int available  = outBlock.length - outIndex;
            if (jWidth > available) {
                System.arraycopy(block, iRow * jWidth, outBlock, outIndex, available);
                outBlock = out.blocks[++outBlockIndex];
                System.arraycopy(block, iRow * jWidth, outBlock, 0, jWidth - available);
                outIndex = jWidth - available;
            } else {
                System.arraycopy(block, iRow * jWidth, outBlock, outIndex, jWidth);
                outIndex += jWidth;
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public void setRowMatrix(final int row, final FieldMatrix<T> matrix)
        throws MatrixDimensionMismatchException, OutOfRangeException {
        if (matrix instanceof BlockFieldMatrix) {
            setRowMatrix(row, (BlockFieldMatrix<T>) matrix);
        } else {
            super.setRowMatrix(row, matrix);
        }
    }

    /**
     * Sets the entries in row number <code>row</code>
     * as a row matrix.  Row indices start at 0.
     *
     * @param row the row to be set
     * @param matrix row matrix (must have one row and the same number of columns
     * as the instance)
     * @throws MatrixDimensionMismatchException if the matrix dimensions do
     * not match one instance row.
     * @throws OutOfRangeException if the specified row index is invalid.
     */
    public void setRowMatrix(final int row, final BlockFieldMatrix<T> matrix)
        throws MatrixDimensionMismatchException, OutOfRangeException {
        checkRowIndex(row);
        final int nCols = getColumnDimension();
        if (matrix.getRowDimension() != 1 ||
            matrix.getColumnDimension() != nCols) {
            throw new MatrixDimensionMismatchException(matrix.getRowDimension(),
                                                       matrix.getColumnDimension(),
                                                       1, nCols);
        }

        // perform copy block-wise, to ensure good cache behavior
        final int iBlock = row / BLOCK_SIZE;
        final int iRow   = row - iBlock * BLOCK_SIZE;
        int mBlockIndex  = 0;
        int mIndex       = 0;
        T[] mBlock  = matrix.blocks[mBlockIndex];
        for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
            final int jWidth     = blockWidth(jBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            final int available  = mBlock.length - mIndex;
            if (jWidth > available) {
                System.arraycopy(mBlock, mIndex, block, iRow * jWidth, available);
                mBlock = matrix.blocks[++mBlockIndex];
                System.arraycopy(mBlock, 0, block, iRow * jWidth, jWidth - available);
                mIndex = jWidth - available;
            } else {
                System.arraycopy(mBlock, mIndex, block, iRow * jWidth, jWidth);
                mIndex += jWidth;
           }
        }
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> getColumnMatrix(final int column)
        throws OutOfRangeException {
        checkColumnIndex(column);
        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), rows, 1);

        // perform copy block-wise, to ensure good cache behavior
        final int jBlock  = column / BLOCK_SIZE;
        final int jColumn = column - jBlock * BLOCK_SIZE;
        final int jWidth  = blockWidth(jBlock);
        int outBlockIndex = 0;
        int outIndex      = 0;
        T[] outBlock = out.blocks[outBlockIndex];
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int iHeight = blockHeight(iBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            for (int i = 0; i < iHeight; ++i) {
                if (outIndex >= outBlock.length) {
                    outBlock = out.blocks[++outBlockIndex];
                    outIndex = 0;
                }
                outBlock[outIndex++] = block[i * jWidth + jColumn];
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public void setColumnMatrix(final int column, final FieldMatrix<T> matrix)
        throws MatrixDimensionMismatchException, OutOfRangeException {
        if (matrix instanceof BlockFieldMatrix) {
            setColumnMatrix(column, (BlockFieldMatrix<T>) matrix);
        } else {
            super.setColumnMatrix(column, matrix);
        }
    }

    /**
     * Sets the entries in column number {@code column}
     * as a column matrix.  Column indices start at 0.
     *
     * @param column Column to be set.
     * @param matrix Column matrix (must have one column and the same number of rows
     * as the instance).
     * @throws MatrixDimensionMismatchException if the matrix dimensions do
     * not match one instance column.
     * @throws OutOfRangeException if the specified column index is invalid.
     */
    void setColumnMatrix(final int column, final BlockFieldMatrix<T> matrix)
        throws MatrixDimensionMismatchException, OutOfRangeException {
        checkColumnIndex(column);
        final int nRows = getRowDimension();
        if (matrix.getRowDimension() != nRows ||
            matrix.getColumnDimension() != 1) {
            throw new MatrixDimensionMismatchException(matrix.getRowDimension(),
                                                       matrix.getColumnDimension(),
                                                       nRows, 1);
        }

        // perform copy block-wise, to ensure good cache behavior
        final int jBlock  = column / BLOCK_SIZE;
        final int jColumn = column - jBlock * BLOCK_SIZE;
        final int jWidth  = blockWidth(jBlock);
        int mBlockIndex = 0;
        int mIndex      = 0;
        T[] mBlock = matrix.blocks[mBlockIndex];
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int iHeight = blockHeight(iBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            for (int i = 0; i < iHeight; ++i) {
                if (mIndex >= mBlock.length) {
                    mBlock = matrix.blocks[++mBlockIndex];
                    mIndex = 0;
                }
                block[i * jWidth + jColumn] = mBlock[mIndex++];
            }
        }
    }

    /** {@inheritDoc} */
    @Override
    public FieldVector<T> getRowVector(final int row)
        throws OutOfRangeException {
        checkRowIndex(row);
        final T[] outData = MathArrays.buildArray(getField(), columns);

        // perform copy block-wise, to ensure good cache behavior
        final int iBlock  = row / BLOCK_SIZE;
        final int iRow    = row - iBlock * BLOCK_SIZE;
        int outIndex      = 0;
        for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
            final int jWidth     = blockWidth(jBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            System.arraycopy(block, iRow * jWidth, outData, outIndex, jWidth);
            outIndex += jWidth;
        }

        return new ArrayFieldVector<>(getField(), outData, false);
    }

    /** {@inheritDoc} */
    @Override
    public void setRowVector(final int row, final FieldVector<T> vector)
        throws MatrixDimensionMismatchException, OutOfRangeException {
        if (vector instanceof ArrayFieldVector) {
            setRow(row, ((ArrayFieldVector<T>) vector).getDataRef());
        } else {
            super.setRowVector(row, vector);
        }
    }

    /** {@inheritDoc} */
    @Override
    public FieldVector<T> getColumnVector(final int column)
        throws OutOfRangeException {
        checkColumnIndex(column);
        final T[] outData = MathArrays.buildArray(getField(), rows);

        // perform copy block-wise, to ensure good cache behavior
        final int jBlock  = column / BLOCK_SIZE;
        final int jColumn = column - jBlock * BLOCK_SIZE;
        final int jWidth  = blockWidth(jBlock);
        int outIndex      = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int iHeight = blockHeight(iBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            for (int i = 0; i < iHeight; ++i) {
                outData[outIndex++] = block[i * jWidth + jColumn];
            }
        }

        return new ArrayFieldVector<>(getField(), outData, false);
    }

    /** {@inheritDoc} */
    @Override
    public void setColumnVector(final int column, final FieldVector<T> vector)
        throws OutOfRangeException, MatrixDimensionMismatchException {
        if (vector instanceof ArrayFieldVector) {
            setColumn(column, ((ArrayFieldVector<T>) vector).getDataRef());
        } else {
            super.setColumnVector(column, vector);
        }
    }

    /** {@inheritDoc} */
    @Override
    public T[] getRow(final int row) throws OutOfRangeException {
        checkRowIndex(row);
        final T[] out = MathArrays.buildArray(getField(), columns);

        // perform copy block-wise, to ensure good cache behavior
        final int iBlock  = row / BLOCK_SIZE;
        final int iRow    = row - iBlock * BLOCK_SIZE;
        int outIndex      = 0;
        for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
            final int jWidth     = blockWidth(jBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            System.arraycopy(block, iRow * jWidth, out, outIndex, jWidth);
            outIndex += jWidth;
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public void setRow(final int row, final T[] array)
        throws OutOfRangeException, MatrixDimensionMismatchException {
        checkRowIndex(row);
        final int nCols = getColumnDimension();
        if (array.length != nCols) {
            throw new MatrixDimensionMismatchException(1, array.length, 1, nCols);
        }

        // perform copy block-wise, to ensure good cache behavior
        final int iBlock  = row / BLOCK_SIZE;
        final int iRow    = row - iBlock * BLOCK_SIZE;
        int outIndex      = 0;
        for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
            final int jWidth     = blockWidth(jBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            System.arraycopy(array, outIndex, block, iRow * jWidth, jWidth);
            outIndex += jWidth;
        }
    }

    /** {@inheritDoc} */
    @Override
    public T[] getColumn(final int column) throws OutOfRangeException {
        checkColumnIndex(column);
        final T[] out = MathArrays.buildArray(getField(), rows);

        // perform copy block-wise, to ensure good cache behavior
        final int jBlock  = column / BLOCK_SIZE;
        final int jColumn = column - jBlock * BLOCK_SIZE;
        final int jWidth  = blockWidth(jBlock);
        int outIndex      = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int iHeight = blockHeight(iBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            for (int i = 0; i < iHeight; ++i) {
                out[outIndex++] = block[i * jWidth + jColumn];
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public void setColumn(final int column, final T[] array)
        throws MatrixDimensionMismatchException, OutOfRangeException {
        checkColumnIndex(column);
        final int nRows = getRowDimension();
        if (array.length != nRows) {
            throw new MatrixDimensionMismatchException(array.length, 1, nRows, 1);
        }

        // perform copy block-wise, to ensure good cache behavior
        final int jBlock  = column / BLOCK_SIZE;
        final int jColumn = column - jBlock * BLOCK_SIZE;
        final int jWidth  = blockWidth(jBlock);
        int outIndex      = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int iHeight = blockHeight(iBlock);
            final T[] block = blocks[iBlock * blockColumns + jBlock];
            for (int i = 0; i < iHeight; ++i) {
                block[i * jWidth + jColumn] = array[outIndex++];
            }
        }
    }

    /** {@inheritDoc} */
    @Override
    public T getEntry(final int row, final int column)
        throws OutOfRangeException {
        checkRowIndex(row);
        checkColumnIndex(column);

        final int iBlock = row    / BLOCK_SIZE;
        final int jBlock = column / BLOCK_SIZE;
        final int k      = (row    - iBlock * BLOCK_SIZE) * blockWidth(jBlock) +
            (column - jBlock * BLOCK_SIZE);

        return blocks[iBlock * blockColumns + jBlock][k];
    }

    /** {@inheritDoc} */
    @Override
    public void setEntry(final int row, final int column, final T value)
        throws OutOfRangeException {
        checkRowIndex(row);
        checkColumnIndex(column);

        final int iBlock = row    / BLOCK_SIZE;
        final int jBlock = column / BLOCK_SIZE;
        final int k      = (row    - iBlock * BLOCK_SIZE) * blockWidth(jBlock) +
            (column - jBlock * BLOCK_SIZE);

        blocks[iBlock * blockColumns + jBlock][k] = value;
    }

    /** {@inheritDoc} */
    @Override
    public void addToEntry(final int row, final int column, final T increment)
        throws OutOfRangeException {
        checkRowIndex(row);
        checkColumnIndex(column);

        final int iBlock = row    / BLOCK_SIZE;
        final int jBlock = column / BLOCK_SIZE;
        final int k      = (row    - iBlock * BLOCK_SIZE) * blockWidth(jBlock) +
            (column - jBlock * BLOCK_SIZE);
        final T[] blockIJ = blocks[iBlock * blockColumns + jBlock];

        blockIJ[k] = blockIJ[k].add(increment);
    }

    /** {@inheritDoc} */
    @Override
    public void multiplyEntry(final int row, final int column, final T factor)
        throws OutOfRangeException {
        checkRowIndex(row);
        checkColumnIndex(column);

        final int iBlock = row    / BLOCK_SIZE;
        final int jBlock = column / BLOCK_SIZE;
        final int k      = (row    - iBlock * BLOCK_SIZE) * blockWidth(jBlock) +
            (column - jBlock * BLOCK_SIZE);
        final T[] blockIJ = blocks[iBlock * blockColumns + jBlock];

        blockIJ[k] = blockIJ[k].multiply(factor);
    }

    /** {@inheritDoc} */
    @Override
    public FieldMatrix<T> transpose() {
        final int nRows = getRowDimension();
        final int nCols = getColumnDimension();
        final BlockFieldMatrix<T> out = new BlockFieldMatrix<>(getField(), nCols, nRows);

        // perform transpose block-wise, to ensure good cache behavior
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < blockColumns; ++iBlock) {
            for (int jBlock = 0; jBlock < blockRows; ++jBlock) {

                // transpose current block
                final T[] outBlock = out.blocks[blockIndex];
                final T[] tBlock   = blocks[jBlock * blockColumns + iBlock];
                final int      pStart   = iBlock * BLOCK_SIZE;
                final int      pEnd     = JdkMath.min(pStart + BLOCK_SIZE, columns);
                final int      qStart   = jBlock * BLOCK_SIZE;
                final int      qEnd     = JdkMath.min(qStart + BLOCK_SIZE, rows);
                int k = 0;
                for (int p = pStart; p < pEnd; ++p) {
                    final int lInc = pEnd - pStart;
                    int l = p - pStart;
                    for (int q = qStart; q < qEnd; ++q) {
                        outBlock[k] = tBlock[l];
                        ++k;
                        l+= lInc;
                    }
                }

                // go to next block
                ++blockIndex;
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public int getRowDimension() {
        return rows;
    }

    /** {@inheritDoc} */
    @Override
    public int getColumnDimension() {
        return columns;
    }

    /** {@inheritDoc} */
    @Override
    public T[] operate(final T[] v) throws DimensionMismatchException {
        if (v.length != columns) {
            throw new DimensionMismatchException(v.length, columns);
        }
        final T[] out = MathArrays.buildArray(getField(), rows);
        final T zero = getField().getZero();

        // perform multiplication block-wise, to ensure good cache behavior
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);
            for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
                final T[] block  = blocks[iBlock * blockColumns + jBlock];
                final int      qStart = jBlock * BLOCK_SIZE;
                final int      qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
                int k = 0;
                for (int p = pStart; p < pEnd; ++p) {
                    T sum = zero;
                    int q = qStart;
                    while (q < qEnd - 3) {
                        sum = sum.
                              add(block[k].multiply(v[q])).
                              add(block[k + 1].multiply(v[q + 1])).
                              add(block[k + 2].multiply(v[q + 2])).
                              add(block[k + 3].multiply(v[q + 3]));
                        k += 4;
                        q += 4;
                    }
                    while (q < qEnd) {
                        sum = sum.add(block[k++].multiply(v[q++]));
                    }
                    out[p] = out[p].add(sum);
                }
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public T[] preMultiply(final T[] v) throws DimensionMismatchException {

        if (v.length != rows) {
            throw new DimensionMismatchException(v.length, rows);
        }
        final T[] out = MathArrays.buildArray(getField(), columns);
        final T zero = getField().getZero();

        // perform multiplication block-wise, to ensure good cache behavior
        for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
            final int jWidth  = blockWidth(jBlock);
            final int jWidth2 = jWidth  + jWidth;
            final int jWidth3 = jWidth2 + jWidth;
            final int jWidth4 = jWidth3 + jWidth;
            final int qStart = jBlock * BLOCK_SIZE;
            final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
            for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
                final T[] block  = blocks[iBlock * blockColumns + jBlock];
                final int      pStart = iBlock * BLOCK_SIZE;
                final int      pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);
                for (int q = qStart; q < qEnd; ++q) {
                    int k = q - qStart;
                    T sum = zero;
                    int p = pStart;
                    while (p < pEnd - 3) {
                        sum = sum.
                              add(block[k].multiply(v[p])).
                              add(block[k + jWidth].multiply(v[p + 1])).
                              add(block[k + jWidth2].multiply(v[p + 2])).
                              add(block[k + jWidth3].multiply(v[p + 3]));
                        k += jWidth4;
                        p += 4;
                    }
                    while (p < pEnd) {
                        sum = sum.add(block[k].multiply(v[p++]));
                        k += jWidth;
                    }
                    out[q] = out[q].add(sum);
                }
            }
        }

        return out;
    }

    /** {@inheritDoc} */
    @Override
    public T walkInRowOrder(final FieldMatrixChangingVisitor<T> visitor) {
        visitor.start(rows, columns, 0, rows - 1, 0, columns - 1);
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);
            for (int p = pStart; p < pEnd; ++p) {
                for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
                    final int jWidth = blockWidth(jBlock);
                    final int qStart = jBlock * BLOCK_SIZE;
                    final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
                    final T[] block = blocks[iBlock * blockColumns + jBlock];
                    int k = (p - pStart) * jWidth;
                    for (int q = qStart; q < qEnd; ++q) {
                        block[k] = visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
             }
        }
        return visitor.end();
    }

    /** {@inheritDoc} */
    @Override
    public T walkInRowOrder(final FieldMatrixPreservingVisitor<T> visitor) {
        visitor.start(rows, columns, 0, rows - 1, 0, columns - 1);
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);
            for (int p = pStart; p < pEnd; ++p) {
                for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
                    final int jWidth = blockWidth(jBlock);
                    final int qStart = jBlock * BLOCK_SIZE;
                    final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
                    final T[] block = blocks[iBlock * blockColumns + jBlock];
                    int k = (p - pStart) * jWidth;
                    for (int q = qStart; q < qEnd; ++q) {
                        visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
             }
        }
        return visitor.end();
    }

    /** {@inheritDoc} */
    @Override
    public T walkInRowOrder(final FieldMatrixChangingVisitor<T> visitor,
                            final int startRow, final int endRow,
                            final int startColumn, final int endColumn)
        throws OutOfRangeException, NumberIsTooSmallException {
        checkSubMatrixIndex(startRow, endRow, startColumn, endColumn);
        visitor.start(rows, columns, startRow, endRow, startColumn, endColumn);
        for (int iBlock = startRow / BLOCK_SIZE; iBlock < 1 + endRow / BLOCK_SIZE; ++iBlock) {
            final int p0     = iBlock * BLOCK_SIZE;
            final int pStart = JdkMath.max(startRow, p0);
            final int pEnd   = JdkMath.min((iBlock + 1) * BLOCK_SIZE, 1 + endRow);
            for (int p = pStart; p < pEnd; ++p) {
                for (int jBlock = startColumn / BLOCK_SIZE; jBlock < 1 + endColumn / BLOCK_SIZE; ++jBlock) {
                    final int jWidth = blockWidth(jBlock);
                    final int q0     = jBlock * BLOCK_SIZE;
                    final int qStart = JdkMath.max(startColumn, q0);
                    final int qEnd   = JdkMath.min((jBlock + 1) * BLOCK_SIZE, 1 + endColumn);
                    final T[] block = blocks[iBlock * blockColumns + jBlock];
                    int k = (p - p0) * jWidth + qStart - q0;
                    for (int q = qStart; q < qEnd; ++q) {
                        block[k] = visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
             }
        }
        return visitor.end();
    }

    /** {@inheritDoc} */
    @Override
    public T walkInRowOrder(final FieldMatrixPreservingVisitor<T> visitor,
                            final int startRow, final int endRow,
                            final int startColumn, final int endColumn)
        throws OutOfRangeException, NumberIsTooSmallException {
        checkSubMatrixIndex(startRow, endRow, startColumn, endColumn);
        visitor.start(rows, columns, startRow, endRow, startColumn, endColumn);
        for (int iBlock = startRow / BLOCK_SIZE; iBlock < 1 + endRow / BLOCK_SIZE; ++iBlock) {
            final int p0     = iBlock * BLOCK_SIZE;
            final int pStart = JdkMath.max(startRow, p0);
            final int pEnd   = JdkMath.min((iBlock + 1) * BLOCK_SIZE, 1 + endRow);
            for (int p = pStart; p < pEnd; ++p) {
                for (int jBlock = startColumn / BLOCK_SIZE; jBlock < 1 + endColumn / BLOCK_SIZE; ++jBlock) {
                    final int jWidth = blockWidth(jBlock);
                    final int q0     = jBlock * BLOCK_SIZE;
                    final int qStart = JdkMath.max(startColumn, q0);
                    final int qEnd   = JdkMath.min((jBlock + 1) * BLOCK_SIZE, 1 + endColumn);
                    final T[] block = blocks[iBlock * blockColumns + jBlock];
                    int k = (p - p0) * jWidth + qStart - q0;
                    for (int q = qStart; q < qEnd; ++q) {
                        visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
             }
        }
        return visitor.end();
    }

    /** {@inheritDoc} */
    @Override
    public T walkInOptimizedOrder(final FieldMatrixChangingVisitor<T> visitor) {
        visitor.start(rows, columns, 0, rows - 1, 0, columns - 1);
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);
            for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
                final int qStart = jBlock * BLOCK_SIZE;
                final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
                final T[] block = blocks[blockIndex];
                int k = 0;
                for (int p = pStart; p < pEnd; ++p) {
                    for (int q = qStart; q < qEnd; ++q) {
                        block[k] = visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
                ++blockIndex;
            }
        }
        return visitor.end();
    }

    /** {@inheritDoc} */
    @Override
    public T walkInOptimizedOrder(final FieldMatrixPreservingVisitor<T> visitor) {
        visitor.start(rows, columns, 0, rows - 1, 0, columns - 1);
        int blockIndex = 0;
        for (int iBlock = 0; iBlock < blockRows; ++iBlock) {
            final int pStart = iBlock * BLOCK_SIZE;
            final int pEnd   = JdkMath.min(pStart + BLOCK_SIZE, rows);
            for (int jBlock = 0; jBlock < blockColumns; ++jBlock) {
                final int qStart = jBlock * BLOCK_SIZE;
                final int qEnd   = JdkMath.min(qStart + BLOCK_SIZE, columns);
                final T[] block = blocks[blockIndex];
                int k = 0;
                for (int p = pStart; p < pEnd; ++p) {
                    for (int q = qStart; q < qEnd; ++q) {
                        visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
                ++blockIndex;
            }
        }
        return visitor.end();
    }

    /** {@inheritDoc} */
    @Override
    public T walkInOptimizedOrder(final FieldMatrixChangingVisitor<T> visitor,
                                  final int startRow, final int endRow,
                                  final int startColumn, final int endColumn)
        throws OutOfRangeException, NumberIsTooSmallException {
        checkSubMatrixIndex(startRow, endRow, startColumn, endColumn);
        visitor.start(rows, columns, startRow, endRow, startColumn, endColumn);
        for (int iBlock = startRow / BLOCK_SIZE; iBlock < 1 + endRow / BLOCK_SIZE; ++iBlock) {
            final int p0     = iBlock * BLOCK_SIZE;
            final int pStart = JdkMath.max(startRow, p0);
            final int pEnd   = JdkMath.min((iBlock + 1) * BLOCK_SIZE, 1 + endRow);
            for (int jBlock = startColumn / BLOCK_SIZE; jBlock < 1 + endColumn / BLOCK_SIZE; ++jBlock) {
                final int jWidth = blockWidth(jBlock);
                final int q0     = jBlock * BLOCK_SIZE;
                final int qStart = JdkMath.max(startColumn, q0);
                final int qEnd   = JdkMath.min((jBlock + 1) * BLOCK_SIZE, 1 + endColumn);
                final T[] block = blocks[iBlock * blockColumns + jBlock];
                for (int p = pStart; p < pEnd; ++p) {
                    int k = (p - p0) * jWidth + qStart - q0;
                    for (int q = qStart; q < qEnd; ++q) {
                        block[k] = visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
            }
        }
        return visitor.end();
    }

    /** {@inheritDoc} */
    @Override
    public T walkInOptimizedOrder(final FieldMatrixPreservingVisitor<T> visitor,
                                  final int startRow, final int endRow,
                                  final int startColumn, final int endColumn)
        throws OutOfRangeException, NumberIsTooSmallException {
        checkSubMatrixIndex(startRow, endRow, startColumn, endColumn);
        visitor.start(rows, columns, startRow, endRow, startColumn, endColumn);
        for (int iBlock = startRow / BLOCK_SIZE; iBlock < 1 + endRow / BLOCK_SIZE; ++iBlock) {
            final int p0     = iBlock * BLOCK_SIZE;
            final int pStart = JdkMath.max(startRow, p0);
            final int pEnd   = JdkMath.min((iBlock + 1) * BLOCK_SIZE, 1 + endRow);
            for (int jBlock = startColumn / BLOCK_SIZE; jBlock < 1 + endColumn / BLOCK_SIZE; ++jBlock) {
                final int jWidth = blockWidth(jBlock);
                final int q0     = jBlock * BLOCK_SIZE;
                final int qStart = JdkMath.max(startColumn, q0);
                final int qEnd   = JdkMath.min((jBlock + 1) * BLOCK_SIZE, 1 + endColumn);
                final T[] block = blocks[iBlock * blockColumns + jBlock];
                for (int p = pStart; p < pEnd; ++p) {
                    int k = (p - p0) * jWidth + qStart - q0;
                    for (int q = qStart; q < qEnd; ++q) {
                        visitor.visit(p, q, block[k]);
                        ++k;
                    }
                }
            }
        }
        return visitor.end();
    }

    /**
     * Get the height of a block.
     * @param blockRow row index (in block sense) of the block
     * @return height (number of rows) of the block
     */
    private int blockHeight(final int blockRow) {
        return (blockRow == blockRows - 1) ? rows - blockRow * BLOCK_SIZE : BLOCK_SIZE;
    }

    /**
     * Get the width of a block.
     * @param blockColumn column index (in block sense) of the block
     * @return width (number of columns) of the block
     */
    private int blockWidth(final int blockColumn) {
        return (blockColumn == blockColumns - 1) ? columns - blockColumn * BLOCK_SIZE : BLOCK_SIZE;
    }
}