path: root/src/main/java/org/apache/commons/math3/transform/FastFourierTransformer.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.math3.transform;

import org.apache.commons.math3.analysis.FunctionUtils;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.complex.Complex;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.MathIllegalStateException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.ArithmeticUtils;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;

import java.io.Serializable;
import java.lang.reflect.Array;

/**
 * Implements the Fast Fourier Transform for transformation of one-dimensional real or complex data
 * sets. For reference, see <em>Applied Numerical Linear Algebra</em>, ISBN 0898713897, chapter 6.
 *
 * <p>There are several variants of the discrete Fourier transform, with various normalization
 * conventions, which are specified by the parameter {@link DftNormalization}.
 *
 * <p>The current implementation of the discrete Fourier transform as a fast Fourier transform
 * requires the length of the data set to be a power of 2. This greatly simplifies and speeds up the
 * code. Users can pad the data with zeros to meet this requirement. There are other flavors of FFT,
 * for reference, see S. Winograd, <i>On computing the discrete Fourier transform</i>, Mathematics
 * of Computation, 32 (1978), 175 - 199.
 *
 * @see DftNormalization
 * @since 1.2
 */
public class FastFourierTransformer implements Serializable {

    /** Serializable version identifier. */
    static final long serialVersionUID = 20120210L;

    /**
     * {@code W_SUB_N_R[i]} is the real part of {@code exp(- 2 * i * pi / n)}: {@code W_SUB_N_R[i] =
     * cos(2 * pi/ n)}, where {@code n = 2^i}.
     */
    private static final double[] W_SUB_N_R = {
        0x1.0p0,
        -0x1.0p0,
        0x1.1a62633145c07p-54,
        0x1.6a09e667f3bcdp-1,
        0x1.d906bcf328d46p-1,
        0x1.f6297cff75cbp-1,
        0x1.fd88da3d12526p-1,
        0x1.ff621e3796d7ep-1,
        0x1.ffd886084cd0dp-1,
        0x1.fff62169b92dbp-1,
        0x1.fffd8858e8a92p-1,
        0x1.ffff621621d02p-1,
        0x1.ffffd88586ee6p-1,
        0x1.fffff62161a34p-1,
        0x1.fffffd8858675p-1,
        0x1.ffffff621619cp-1,
        0x1.ffffffd885867p-1,
        0x1.fffffff62161ap-1,
        0x1.fffffffd88586p-1,
        0x1.ffffffff62162p-1,
        0x1.ffffffffd8858p-1,
        0x1.fffffffff6216p-1,
        0x1.fffffffffd886p-1,
        0x1.ffffffffff621p-1,
        0x1.ffffffffffd88p-1,
        0x1.fffffffffff62p-1,
        0x1.fffffffffffd9p-1,
        0x1.ffffffffffff6p-1,
        0x1.ffffffffffffep-1,
        0x1.fffffffffffffp-1,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0,
        0x1.0p0
    };

    /**
     * {@code W_SUB_N_I[i]} is the imaginary part of {@code exp(- 2 * i * pi / n)}: {@code
     * W_SUB_N_I[i] = -sin(2 * pi/ n)}, where {@code n = 2^i}.
     */
    private static final double[] W_SUB_N_I = {
        0x1.1a62633145c07p-52,
        -0x1.1a62633145c07p-53,
        -0x1.0p0,
        -0x1.6a09e667f3bccp-1,
        -0x1.87de2a6aea963p-2,
        -0x1.8f8b83c69a60ap-3,
        -0x1.917a6bc29b42cp-4,
        -0x1.91f65f10dd814p-5,
        -0x1.92155f7a3667ep-6,
        -0x1.921d1fcdec784p-7,
        -0x1.921f0fe670071p-8,
        -0x1.921f8becca4bap-9,
        -0x1.921faaee6472dp-10,
        -0x1.921fb2aecb36p-11,
        -0x1.921fb49ee4ea6p-12,
        -0x1.921fb51aeb57bp-13,
        -0x1.921fb539ecf31p-14,
        -0x1.921fb541ad59ep-15,
        -0x1.921fb5439d73ap-16,
        -0x1.921fb544197ap-17,
        -0x1.921fb544387bap-18,
        -0x1.921fb544403c1p-19,
        -0x1.921fb544422c2p-20,
        -0x1.921fb54442a83p-21,
        -0x1.921fb54442c73p-22,
        -0x1.921fb54442cefp-23,
        -0x1.921fb54442d0ep-24,
        -0x1.921fb54442d15p-25,
        -0x1.921fb54442d17p-26,
        -0x1.921fb54442d18p-27,
        -0x1.921fb54442d18p-28,
        -0x1.921fb54442d18p-29,
        -0x1.921fb54442d18p-30,
        -0x1.921fb54442d18p-31,
        -0x1.921fb54442d18p-32,
        -0x1.921fb54442d18p-33,
        -0x1.921fb54442d18p-34,
        -0x1.921fb54442d18p-35,
        -0x1.921fb54442d18p-36,
        -0x1.921fb54442d18p-37,
        -0x1.921fb54442d18p-38,
        -0x1.921fb54442d18p-39,
        -0x1.921fb54442d18p-40,
        -0x1.921fb54442d18p-41,
        -0x1.921fb54442d18p-42,
        -0x1.921fb54442d18p-43,
        -0x1.921fb54442d18p-44,
        -0x1.921fb54442d18p-45,
        -0x1.921fb54442d18p-46,
        -0x1.921fb54442d18p-47,
        -0x1.921fb54442d18p-48,
        -0x1.921fb54442d18p-49,
        -0x1.921fb54442d18p-50,
        -0x1.921fb54442d18p-51,
        -0x1.921fb54442d18p-52,
        -0x1.921fb54442d18p-53,
        -0x1.921fb54442d18p-54,
        -0x1.921fb54442d18p-55,
        -0x1.921fb54442d18p-56,
        -0x1.921fb54442d18p-57,
        -0x1.921fb54442d18p-58,
        -0x1.921fb54442d18p-59,
        -0x1.921fb54442d18p-60
    };

    /** The type of DFT to be performed. */
    private final DftNormalization normalization;

    /**
     * Creates a new instance of this class, with various normalization conventions.
     *
     * @param normalization the type of normalization to be applied to the transformed data
     */
    public FastFourierTransformer(final DftNormalization normalization) {
        this.normalization = normalization;
    }

    /**
     * Performs identical index bit reversal shuffles on two arrays of identical size. Each element
     * in the array is swapped with another element based on the bit-reversal of the index. For
     * example, in an array with length 16, item at binary index 0011 (decimal 3) would be swapped
     * with the item at binary index 1100 (decimal 12).
     *
     * @param a the first array to be shuffled
     * @param b the second array to be shuffled
     */
    private static void bitReversalShuffle2(double[] a, double[] b) {
        final int n = a.length;
        assert b.length == n;
        final int halfOfN = n >> 1;

        int j = 0;
        for (int i = 0; i < n; i++) {
            if (i < j) {
                // swap indices i & j
                double temp = a[i];
                a[i] = a[j];
                a[j] = temp;

                temp = b[i];
                b[i] = b[j];
                b[j] = temp;
            }

            int k = halfOfN;
            while (k <= j && k > 0) {
                j -= k;
                k >>= 1;
            }
            j += k;
        }
    }

    /**
     * Applies the proper normalization to the specified transformed data.
     *
     * @param dataRI the unscaled transformed data
     * @param normalization the normalization to be applied
     * @param type the type of transform (forward, inverse) which resulted in the specified data
     */
    private static void normalizeTransformedData(
            final double[][] dataRI,
            final DftNormalization normalization,
            final TransformType type) {

        final double[] dataR = dataRI[0];
        final double[] dataI = dataRI[1];
        final int n = dataR.length;
        assert dataI.length == n;

        switch (normalization) {
            case STANDARD:
                if (type == TransformType.INVERSE) {
                    final double scaleFactor = 1.0 / ((double) n);
                    for (int i = 0; i < n; i++) {
                        dataR[i] *= scaleFactor;
                        dataI[i] *= scaleFactor;
                    }
                }
                break;
            case UNITARY:
                final double scaleFactor = 1.0 / FastMath.sqrt(n);
                for (int i = 0; i < n; i++) {
                    dataR[i] *= scaleFactor;
                    dataI[i] *= scaleFactor;
                }
                break;
            default:
                /*
                 * This should never occur in normal conditions. However this
                 * clause has been added as a safeguard if other types of
                 * normalizations are ever implemented, and the corresponding
                 * test is forgotten in the present switch.
                 */
                throw new MathIllegalStateException();
        }
    }

    /**
     * Computes the standard transform of the specified complex data. The computation is done in
     * place. The input data is laid out as follows
     *
     * <ul>
     *   <li>{@code dataRI[0][i]} is the real part of the {@code i}-th data point,
     *   <li>{@code dataRI[1][i]} is the imaginary part of the {@code i}-th data point.
     * </ul>
     *
     * @param dataRI the two dimensional array of real and imaginary parts of the data
     * @param normalization the normalization to be applied to the transformed data
     * @param type the type of transform (forward, inverse) to be performed
     * @throws DimensionMismatchException if the number of rows of the specified array is not two,
     *     or the array is not rectangular
     * @throws MathIllegalArgumentException if the number of data points is not a power of two
     */
    public static void transformInPlace(
            final double[][] dataRI,
            final DftNormalization normalization,
            final TransformType type) {

        if (dataRI.length != 2) {
            throw new DimensionMismatchException(dataRI.length, 2);
        }
        final double[] dataR = dataRI[0];
        final double[] dataI = dataRI[1];
        if (dataR.length != dataI.length) {
            throw new DimensionMismatchException(dataI.length, dataR.length);
        }

        final int n = dataR.length;
        if (!ArithmeticUtils.isPowerOfTwo(n)) {
            throw new MathIllegalArgumentException(
                    LocalizedFormats.NOT_POWER_OF_TWO_CONSIDER_PADDING, Integer.valueOf(n));
        }

        if (n == 1) {
            return;
        } else if (n == 2) {
            final double srcR0 = dataR[0];
            final double srcI0 = dataI[0];
            final double srcR1 = dataR[1];
            final double srcI1 = dataI[1];

            // X_0 = x_0 + x_1
            dataR[0] = srcR0 + srcR1;
            dataI[0] = srcI0 + srcI1;
            // X_1 = x_0 - x_1
            dataR[1] = srcR0 - srcR1;
            dataI[1] = srcI0 - srcI1;

            normalizeTransformedData(dataRI, normalization, type);
            return;
        }

        bitReversalShuffle2(dataR, dataI);

        // Do 4-term DFT.
        if (type == TransformType.INVERSE) {
            for (int i0 = 0; i0 < n; i0 += 4) {
                final int i1 = i0 + 1;
                final int i2 = i0 + 2;
                final int i3 = i0 + 3;

                final double srcR0 = dataR[i0];
                final double srcI0 = dataI[i0];
                final double srcR1 = dataR[i2];
                final double srcI1 = dataI[i2];
                final double srcR2 = dataR[i1];
                final double srcI2 = dataI[i1];
                final double srcR3 = dataR[i3];
                final double srcI3 = dataI[i3];

                // 4-term DFT
                // X_0 = x_0 + x_1 + x_2 + x_3
                dataR[i0] = srcR0 + srcR1 + srcR2 + srcR3;
                dataI[i0] = srcI0 + srcI1 + srcI2 + srcI3;
                // X_1 = x_0 - x_2 + j * (x_3 - x_1)
                dataR[i1] = srcR0 - srcR2 + (srcI3 - srcI1);
                dataI[i1] = srcI0 - srcI2 + (srcR1 - srcR3);
                // X_2 = x_0 - x_1 + x_2 - x_3
                dataR[i2] = srcR0 - srcR1 + srcR2 - srcR3;
                dataI[i2] = srcI0 - srcI1 + srcI2 - srcI3;
                // X_3 = x_0 - x_2 + j * (x_1 - x_3)
                dataR[i3] = srcR0 - srcR2 + (srcI1 - srcI3);
                dataI[i3] = srcI0 - srcI2 + (srcR3 - srcR1);
            }
        } else {
            for (int i0 = 0; i0 < n; i0 += 4) {
                final int i1 = i0 + 1;
                final int i2 = i0 + 2;
                final int i3 = i0 + 3;

                final double srcR0 = dataR[i0];
                final double srcI0 = dataI[i0];
                final double srcR1 = dataR[i2];
                final double srcI1 = dataI[i2];
                final double srcR2 = dataR[i1];
                final double srcI2 = dataI[i1];
                final double srcR3 = dataR[i3];
                final double srcI3 = dataI[i3];

                // 4-term DFT
                // X_0 = x_0 + x_1 + x_2 + x_3
                dataR[i0] = srcR0 + srcR1 + srcR2 + srcR3;
                dataI[i0] = srcI0 + srcI1 + srcI2 + srcI3;
                // X_1 = x_0 - x_2 + j * (x_3 - x_1)
                dataR[i1] = srcR0 - srcR2 + (srcI1 - srcI3);
                dataI[i1] = srcI0 - srcI2 + (srcR3 - srcR1);
                // X_2 = x_0 - x_1 + x_2 - x_3
                dataR[i2] = srcR0 - srcR1 + srcR2 - srcR3;
                dataI[i2] = srcI0 - srcI1 + srcI2 - srcI3;
                // X_3 = x_0 - x_2 + j * (x_1 - x_3)
                dataR[i3] = srcR0 - srcR2 + (srcI3 - srcI1);
                dataI[i3] = srcI0 - srcI2 + (srcR1 - srcR3);
            }
        }

        int lastN0 = 4;
        int lastLogN0 = 2;
        while (lastN0 < n) {
            int n0 = lastN0 << 1;
            int logN0 = lastLogN0 + 1;
            double wSubN0R = W_SUB_N_R[logN0];
            double wSubN0I = W_SUB_N_I[logN0];
            if (type == TransformType.INVERSE) {
                wSubN0I = -wSubN0I;
            }

            // Combine even/odd transforms of size lastN0 into a transform of size N0 (lastN0 * 2).
            for (int destEvenStartIndex = 0; destEvenStartIndex < n; destEvenStartIndex += n0) {
                int destOddStartIndex = destEvenStartIndex + lastN0;

                double wSubN0ToRR = 1;
                double wSubN0ToRI = 0;

                for (int r = 0; r < lastN0; r++) {
                    double grR = dataR[destEvenStartIndex + r];
                    double grI = dataI[destEvenStartIndex + r];
                    double hrR = dataR[destOddStartIndex + r];
                    double hrI = dataI[destOddStartIndex + r];

                    // dest[destEvenStartIndex + r] = Gr + WsubN0ToR * Hr
                    dataR[destEvenStartIndex + r] = grR + wSubN0ToRR * hrR - wSubN0ToRI * hrI;
                    dataI[destEvenStartIndex + r] = grI + wSubN0ToRR * hrI + wSubN0ToRI * hrR;
                    // dest[destOddStartIndex + r] = Gr - WsubN0ToR * Hr
                    dataR[destOddStartIndex + r] = grR - (wSubN0ToRR * hrR - wSubN0ToRI * hrI);
                    dataI[destOddStartIndex + r] = grI - (wSubN0ToRR * hrI + wSubN0ToRI * hrR);

                    // WsubN0ToR *= WsubN0R
                    double nextWsubN0ToRR = wSubN0ToRR * wSubN0R - wSubN0ToRI * wSubN0I;
                    double nextWsubN0ToRI = wSubN0ToRR * wSubN0I + wSubN0ToRI * wSubN0R;
                    wSubN0ToRR = nextWsubN0ToRR;
                    wSubN0ToRI = nextWsubN0ToRI;
                }
            }

            lastN0 = n0;
            lastLogN0 = logN0;
        }

        normalizeTransformedData(dataRI, normalization, type);
    }

    /**
     * Returns the (forward, inverse) transform of the specified real data set.
     *
     * @param f the real data array to be transformed
     * @param type the type of transform (forward, inverse) to be performed
     * @return the complex transformed array
     * @throws MathIllegalArgumentException if the length of the data array is not a power of two
     */
    public Complex[] transform(final double[] f, final TransformType type) {
        final double[][] dataRI =
                new double[][] {MathArrays.copyOf(f, f.length), new double[f.length]};

        transformInPlace(dataRI, normalization, type);

        return TransformUtils.createComplexArray(dataRI);
    }

    /**
     * Returns the (forward, inverse) transform of the specified real function, sampled on the
     * specified interval.
     *
     * @param f the function to be sampled and transformed
     * @param min the (inclusive) lower bound for the interval
     * @param max the (exclusive) upper bound for the interval
     * @param n the number of sample points
     * @param type the type of transform (forward, inverse) to be performed
     * @return the complex transformed array
     * @throws org.apache.commons.math3.exception.NumberIsTooLargeException if the lower bound is
     *     greater than, or equal to the upper bound
     * @throws org.apache.commons.math3.exception.NotStrictlyPositiveException if the number of
     *     sample points {@code n} is negative
     * @throws MathIllegalArgumentException if the number of sample points {@code n} is not a power
     *     of two
     */
    public Complex[] transform(
            final UnivariateFunction f,
            final double min,
            final double max,
            final int n,
            final TransformType type) {

        final double[] data = FunctionUtils.sample(f, min, max, n);
        return transform(data, type);
    }

    /**
     * Returns the (forward, inverse) transform of the specified complex data set.
     *
     * @param f the complex data array to be transformed
     * @param type the type of transform (forward, inverse) to be performed
     * @return the complex transformed array
     * @throws MathIllegalArgumentException if the length of the data array is not a power of two
     */
    public Complex[] transform(final Complex[] f, final TransformType type) {
        final double[][] dataRI = TransformUtils.createRealImaginaryArray(f);

        transformInPlace(dataRI, normalization, type);

        return TransformUtils.createComplexArray(dataRI);
    }

    /**
     * Performs a multi-dimensional Fourier transform on a given array. Use {@link
     * #transform(Complex[], TransformType)} in a row-column implementation in any number of
     * dimensions with O(N&times;log(N)) complexity with N = n<sub>1</sub> &times; n<sub>2</sub>
     * &times;n<sub>3</sub> &times; ... &times; n<sub>d</sub>, where n<sub>k</sub> is the number of
     * elements in dimension k, and d is the total number of dimensions.
     *
     * @param mdca Multi-Dimensional Complex Array, i.e. {@code Complex[][][][]}
     * @param type the type of transform (forward, inverse) to be performed
     * @return transform of {@code mdca} as a Multi-Dimensional Complex Array, i.e. {@code
     *     Complex[][][][]}
     * @throws IllegalArgumentException if any dimension is not a power of two
     * @deprecated see MATH-736
     */
    @Deprecated
    public Object mdfft(Object mdca, TransformType type) {
        MultiDimensionalComplexMatrix mdcm =
                (MultiDimensionalComplexMatrix) new MultiDimensionalComplexMatrix(mdca).clone();
        int[] dimensionSize = mdcm.getDimensionSizes();
        // cycle through each dimension
        for (int i = 0; i < dimensionSize.length; i++) {
            mdfft(mdcm, type, i, new int[0]);
        }
        return mdcm.getArray();
    }

    /**
     * Performs one dimension of a multi-dimensional Fourier transform.
     *
     * @param mdcm input matrix
     * @param type the type of transform (forward, inverse) to be performed
     * @param d index of the dimension to process
     * @param subVector recursion subvector
     * @throws IllegalArgumentException if any dimension is not a power of two
     * @deprecated see MATH-736
     */
    @Deprecated
    private void mdfft(
            MultiDimensionalComplexMatrix mdcm, TransformType type, int d, int[] subVector) {

        int[] dimensionSize = mdcm.getDimensionSizes();
        // if done
        if (subVector.length == dimensionSize.length) {
            Complex[] temp = new Complex[dimensionSize[d]];
            for (int i = 0; i < dimensionSize[d]; i++) {
                // fft along dimension d
                subVector[d] = i;
                temp[i] = mdcm.get(subVector);
            }

            temp = transform(temp, type);

            for (int i = 0; i < dimensionSize[d]; i++) {
                subVector[d] = i;
                mdcm.set(temp[i], subVector);
            }
        } else {
            int[] vector = new int[subVector.length + 1];
            System.arraycopy(subVector, 0, vector, 0, subVector.length);
            if (subVector.length == d) {
                // value is not important once the recursion is done.
                // then an fft will be applied along the dimension d.
                vector[d] = 0;
                mdfft(mdcm, type, d, vector);
            } else {
                for (int i = 0; i < dimensionSize[subVector.length]; i++) {
                    vector[subVector.length] = i;
                    // further split along the next dimension
                    mdfft(mdcm, type, d, vector);
                }
            }
        }
    }

    /**
     * Complex matrix implementation. Not designed for synchronized access may eventually be
     * replaced by jsr-83 of the java community process http://jcp.org/en/jsr/detail?id=83 may
     * require additional exception throws for other basic requirements.
     *
     * @deprecated see MATH-736
     */
    @Deprecated
    private static class MultiDimensionalComplexMatrix implements Cloneable {

        /** Size in all dimensions. */
        protected int[] dimensionSize;

        /** Storage array. */
        protected Object multiDimensionalComplexArray;

        /**
         * Simple constructor.
         *
         * @param multiDimensionalComplexArray array containing the matrix elements
         */
        MultiDimensionalComplexMatrix(Object multiDimensionalComplexArray) {

            this.multiDimensionalComplexArray = multiDimensionalComplexArray;

            // count dimensions
            int numOfDimensions = 0;
            for (Object lastDimension = multiDimensionalComplexArray;
                    lastDimension instanceof Object[]; ) {
                final Object[] array = (Object[]) lastDimension;
                numOfDimensions++;
                lastDimension = array[0];
            }

            // allocate array with exact count
            dimensionSize = new int[numOfDimensions];

            // fill array
            numOfDimensions = 0;
            for (Object lastDimension = multiDimensionalComplexArray;
                    lastDimension instanceof Object[]; ) {
                final Object[] array = (Object[]) lastDimension;
                dimensionSize[numOfDimensions++] = array.length;
                lastDimension = array[0];
            }
        }

        /**
         * Get a matrix element.
         *
         * @param vector indices of the element
         * @return matrix element
         * @exception DimensionMismatchException if dimensions do not match
         */
        public Complex get(int... vector) throws DimensionMismatchException {

            if (vector == null) {
                if (dimensionSize.length > 0) {
                    throw new DimensionMismatchException(0, dimensionSize.length);
                }
                return null;
            }
            if (vector.length != dimensionSize.length) {
                throw new DimensionMismatchException(vector.length, dimensionSize.length);
            }

            Object lastDimension = multiDimensionalComplexArray;

            for (int i = 0; i < dimensionSize.length; i++) {
                lastDimension = ((Object[]) lastDimension)[vector[i]];
            }
            return (Complex) lastDimension;
        }

        /**
         * Set a matrix element.
         *
         * @param magnitude magnitude of the element
         * @param vector indices of the element
         * @return the previous value
         * @exception DimensionMismatchException if dimensions do not match
         */
        public Complex set(Complex magnitude, int... vector) throws DimensionMismatchException {

            if (vector == null) {
                if (dimensionSize.length > 0) {
                    throw new DimensionMismatchException(0, dimensionSize.length);
                }
                return null;
            }
            if (vector.length != dimensionSize.length) {
                throw new DimensionMismatchException(vector.length, dimensionSize.length);
            }

            Object[] lastDimension = (Object[]) multiDimensionalComplexArray;
            for (int i = 0; i < dimensionSize.length - 1; i++) {
                lastDimension = (Object[]) lastDimension[vector[i]];
            }

            Complex lastValue = (Complex) lastDimension[vector[dimensionSize.length - 1]];
            lastDimension[vector[dimensionSize.length - 1]] = magnitude;

            return lastValue;
        }

        /**
         * Get the size in all dimensions.
         *
         * @return size in all dimensions
         */
        public int[] getDimensionSizes() {
            return dimensionSize.clone();
        }

        /**
         * Get the underlying storage array.
         *
         * @return underlying storage array
         */
        public Object getArray() {
            return multiDimensionalComplexArray;
        }

        /** {@inheritDoc} */
        @Override
        public Object clone() {
            MultiDimensionalComplexMatrix mdcm =
                    new MultiDimensionalComplexMatrix(
                            Array.newInstance(Complex.class, dimensionSize));
            clone(mdcm);
            return mdcm;
        }

        /**
         * Copy contents of current array into mdcm.
         *
         * @param mdcm array where to copy data
         */
        private void clone(MultiDimensionalComplexMatrix mdcm) {

            int[] vector = new int[dimensionSize.length];
            int size = 1;
            for (int i = 0; i < dimensionSize.length; i++) {
                size *= dimensionSize[i];
            }
            int[][] vectorList = new int[size][dimensionSize.length];
            for (int[] nextVector : vectorList) {
                System.arraycopy(vector, 0, nextVector, 0, dimensionSize.length);
                for (int i = 0; i < dimensionSize.length; i++) {
                    vector[i]++;
                    if (vector[i] < dimensionSize[i]) {
                        break;
                    } else {
                        vector[i] = 0;
                    }
                }
            }

            for (int[] nextVector : vectorList) {
                mdcm.set(get(nextVector), nextVector);
            }
        }
    }
}