java多项式相乘fft

FFT（快速傅里叶变换）是一种高效的算法，可用于多项式乘法。以下是Java中实现多项式相乘的FFT算法的代码示例：

import java.util.Arrays;

public class FFT {

    private Complex[] omega;

    public Complex[] fft(Complex[] A) {
        int n = A.length;

        if (n == 1)
            return new Complex[] { A[0] };

        Complex[] A0 = new Complex[n / 2];
        Complex[] A1 = new Complex[n / 2];
        for (int i = 0; i < n / 2; i++) {
            A0[i] = A[2 * i];
            A1[i] = A[2 * i + 1];
        }

        Complex[] y0 = fft(A0);
        Complex[] y1 = fft(A1);
        Complex[] y = new Complex[n];

        for (int k = 0; k < n / 2; k++) {
            Complex w = omega[n][k];
            y[k] = y0[k].plus(w.times(y1[k]));
            y[k + n / 2] = y0[k].minus(w.times(y1[k]));
        }

        return y;
    }

    public Complex[] inverseFFT(Complex[] y) {
        int n = y.length;

        Complex[] yInv = new Complex[n];
        for (int i = 0; i < n; i++) {
            yInv[i] = y[i].conjugate();
        }

        Complex[] aInv = fft(yInv);
        for (int i = 0; i < n; i++) {
            aInv[i] = aInv[i].conjugate().times(1.0 / n);
        }

        return aInv;
    }

    public void initOmega(int n) {
        omega = new Complex[n + 1];
        for (int i = 0; i <= n; i++) {
            double angle = 2 * Math.PI * i / n;
            omega[i] = new Complex(Math.cos(angle), Math.sin(angle));
        }
    }

    public Complex[] multiply(Complex[] a, Complex[] b) {
        int n = Integer.highestOneBit(Math.max(a.length, b.length) - 1) << 1;
        initOmega(n);

        Complex[] A = Arrays.copyOf(a, n);
        Complex[] B = Arrays.copyOf(b, n);

        Complex[] y1 = fft(A);
        Complex[] y2 = fft(B);
        Complex[] y = new Complex[n];

        for (int k = 0; k < n; k++) {
            y[k] = y1[k].times(y2[k]);
        }

        Complex[] c = inverseFFT(y);
        return c;
    }

    public static void main(String[] args) {
        FFT fft = new FFT();
        Complex[] a = new Complex[] { new Complex(1), new Complex(2), new Complex(3) };
        Complex[] b = new Complex[] { new Complex(4), new Complex(5), new Complex(6) };
        Complex[] c = fft.multiply(a, b);

        for (int i = 0; i < c.length; i++) {
            System.out.print(c[i] + " ");
        }
    }
}

class Complex {
    public double re;
    public double im;

    public Complex(double re, double im) {
        this.re = re;
        this.im = im;
    }

    public Complex plus(Complex other) {
        return new Complex(re + other.re, im + other.im);
    }

    public Complex minus(Complex other) {
        return new Complex(re - other.re, im - other.im);
    }

    public Complex times(Complex other) {
        return new Complex(re * other.re - im * other.im, re * other.im + im * other.re);
    }

    public Complex conjugate() {
        return new Complex(re, -im);
    }

    public String toString() {
        return String.format("(%f,%f)", re, im);
    }
}

该代码使用Complex类来表示复数，实现了FFT算法和多项式相乘的操作。在主方法中，可以看到如何使用该类来计算多项式a和b的乘积。