《算法》BEYOND 部分程序 part 2

▶ 书中第六章部分程序，加上自己补充的代码，包括快速傅里叶变换，多项式表示

● 快速傅里叶变换，需要递归

 package package01;

 import edu.princeton.cs.algs4.StdOut;
 import edu.princeton.cs.algs4.StdRandom;
 import edu.princeton.cs.algs4.Complex;

 public class class01
 {
     private static final Complex ZERO = new Complex(0, 0);
     private static final double EPSILON = 1e-8;
     private class01() { }

     public static Complex[] fft(Complex[] x)// 正向傅里叶变换
     {
         int n = x.length;
         if (n == 1)
             return new Complex[] {x[0]};
         if (n % 2 != 0)
             throw new IllegalArgumentException("n != 2^k");

         Complex[] temp = new Complex[n/2], result = new Complex[n];
         for (int k = 0; k < n/2; k++)       // 偶数项和奇数分别递归
             temp[k] = x[2*k];
         Complex[] q = fft(temp);
         for (int k = 0; k < n/2; k++)
             temp[k] = x[2*k + 1];
         Complex[] r = fft(temp);
         for (int k = 0; k < n/2; k++)       // 合并
         {
             double kth = -2 * k * Math.PI / n;
             Complex wk = new Complex(Math.cos(kth), Math.sin(kth));
             result[k]       = q[k].plus(wk.times(r[k]));
             result[k + n/2] = q[k].minus(wk.times(r[k]));
         }
         return result;
     }

     public static Complex[] ifft(Complex[] x)// 傅里叶反变换，共轭、正变换、再共轭，最后除以项数
     {
         int n = x.length;
         Complex[] result = new Complex[n], temp;
         for (int i = 0; i < n; i++)
             result[i] = x[i].conjugate();
         result = fft(result);
         for (int i = 0; i < n; i++)
             result[i] = result[i].conjugate();
         for (int i = 0; i < n; i++)
             result[i] = result[i].scale(1.0 / n);
         return result;
     }

     public static Complex[] cconvolve(Complex[] x, Complex[] y)// 环形卷积
     {
         if (x.length != y.length)
             throw new IllegalArgumentException("x.length != y.length");
         int n = x.length;
         Complex[] a = fft(x), b = fft(y), c = new Complex[n];
         for (int i = 0; i < n; i++)
             c[i] = a[i].times(b[i]);
         return ifft(c);
     }

     public static Complex[] convolve(Complex[] x, Complex[] y)// 线性卷积，后一半垫 0
     {
         if (x.length != y.length)
             throw new IllegalArgumentException("x.length != y.length");
         Complex[] a = new Complex[2 * x.length], b = new Complex[2 * y.length];
         for (int i = 0; i < x.length; i++)
             a[i] = x[i];
         for (int i = x.length; i < 2*x.length; i++)
             a[i] = ZERO;
         for (int i = 0; i < y.length; i++)
             b[i] = y[i];
         for (int i = y.length; i < 2*y.length; i++)
             b[i] = ZERO;
         return cconvolve(a, b);
     }

     private static void show(Complex[] x, String title)
     {
         StdOut.printf("%s\n-------------------\n",title);
         for (int i = 0; i < x.length; i++)
             StdOut.println(x[i]);
         StdOut.println();
     }

     public static void main(String[] args)
     {
         int n = 16;
         Complex[] x = new Complex[n];
         for (int i = 0; i < n; i++)
             x[i] = new Complex(StdRandom.uniform(-1.0, 1.0), 0);

         show(x, "x");
         Complex[] y = fft(x);
         show(y, "y = fft(x)");
         Complex[] z = ifft(y);
         show(z, "z = ifft(y)");
         Complex[] c = cconvolve(x, x);
         show(c, "c = cconvolve(x, x)");
         Complex[] d = convolve(x, x);
         show(d, "d = convolve(x, x)");
     }
 }

● 多项式表示

 package package01;

 import edu.princeton.cs.algs4.StdOut;

 public class class01
 {
     private int[] coef;                 // 系数列表
     private int degree;                 // 次数

     public class01(int a, int b)        // 建立单项式 a*x^b
     {
         coef = new int[b + 1];
         coef[b] = a;
         reduce();
     }

     private void reduce()               // 记录多项式的次数
     {
         degree = -1;
         for (int i = coef.length - 1; i >= 0; i--)
         {
             if (coef[i] != 0)
             {
                 degree = i;
                 return;
             }
         }
     }

     public int degree()
     {
         return degree;
     }

     public class01 plus(class01 that)   // p(x) + q(x)，把系数从低次项向高次项各加一遍
     {
         class01 poly = new class01(0, Math.max(degree, that.degree));
         for (int i = 0; i <= degree; i++)
             poly.coef[i] = coef[i];
         for (int i = 0; i <= that.degree; i++)
             poly.coef[i] += that.coef[i];
         poly.reduce();
         return poly;
     }

     public class01 minus(class01 that)
     {
         class01 poly = new class01(0, Math.max(degree, that.degree));
         for (int i = 0; i <= degree; i++)
             poly.coef[i] = coef[i];
         for (int i = 0; i <= that.degree; i++)
             poly.coef[i] -= that.coef[i];
         poly.reduce();
         return poly;
     }

     public class01 times(class01 that)
     {
         class01 poly = new class01(0, degree + that.degree);
         for (int i = 0; i <= degree; i++)
         {
             for (int j = 0; j <= that.degree; j++)
                 poly.coef[i + j] += (coef[i] * that.coef[j]);
         }
         poly.reduce();
         return poly;
     }

     public class01 compose(class01 that)// p(q(x))，用了秦九韶
     {
         class01 poly = new class01(0, 0);
         for (int i = degree; i >= 0; i--)
         {
             class01 term = new class01(coef[i], 0);
             poly = term.plus(that.times(poly));
         }
         return poly;
     }

     public class01 differentiate()      // p'(x)
     {
         if (degree == 0)
             return new class01(0, 0);
         class01 poly = new class01(0, degree - 1);
         poly.degree = degree - 1;
         for (int i = 0; i < degree; i++)
             poly.coef[i] = (i + 1) * coef[i + 1];
         return poly;
     }

     public int evaluate(int x)          // p(x) /. {x->a}
     {
         int p = 0;
         for (int i = degree; i >= 0; i--)
             p = coef[i] + (x * p);
         return p;
     }

     public int compareTo(class01 that)
     {
         if (degree < that.degree)
             return -1;
         if (degree > that.degree)
             return +1;
         for (int i = degree; i >= 0; i--)
         {
             if (coef[i] < that.coef[i])
                 return -1;
             if (coef[i] > that.coef[i])
                 return +1;
         }
         return 0;
     }

     @Override
         public String toString()
     {
         if (degree == -1)
             return "0";
         if (degree == 0)
             return "" + coef[0];
         if (degree == 1)
             return coef[1] + "x + " + coef[0];
         String s = coef[degree] + "x^" + degree;
         for (int i = degree - 1; i >= 0; i--)
         {
             if (coef[i] == 0)
                 continue;
             if (coef[i]  > 0)
                 s += " + " + (coef[i]);
             if (coef[i]  < 0)
                 s += " - " + (-coef[i]);
             s += (i == 1) ? "x" : ("x^" + i);
         }
         return s;
     }

     public static void main(String[] args)
     {
         class01 zero = new class01(0, 0);
         class01 p1 = new class01(4, 3);
         class01 p2 = new class01(3, 2);
         class01 p3 = new class01(1, 0);
         class01 p4 = new class01(2, 1);
         class01 p = p1.plus(p2).plus(p3).plus(p4);   // 4x^3 + 3x^2 + 1

         class01 q1 = new class01(3, 2);
         class01 q2 = new class01(5, 0);
         class01 q = q1.plus(q2);                     // 3x^2 + 5

         StdOut.println("zero(x)     = " + zero);
         StdOut.println("p(x)        = " + p);
         StdOut.println("q(x)        = " + q);
         StdOut.println("p(x) + q(x) = " + p.plus(q));
         StdOut.println("p(x) * q(x) = " + p.times(q));
         StdOut.println("p(q(x))     = " + p.compose(q));
         StdOut.println("p(x) - p(x) = " + p.minus(p));
         StdOut.println("0 - p(x)    = " + zero.minus(p));
         StdOut.println("p(3)        = " + p.evaluate(3));
         StdOut.println("p'(x)       = " + p.differentiate());
         StdOut.println("p''(x)      = " + p.differentiate().differentiate());
     }
 }