用Java划分多項式？

Question

我試圖通過創建多項式的除法類來找到多項式的GCD。 我了解如何相加，相減，相乘和創建多項式，但是我不知道如何將兩者相除（使用代碼）。 有人可以幫我嗎？

到目前為止，這是我的代碼：

public class Polynomial2 {
private int[] coef;  // coefficients
private int deg;     // degree of polynomial (0 for the zero polynomial)

// a * x^b
public Polynomial2(int a, int b) {
    coef = new int[b+1];
    coef[b] = a;
    deg = degree();
}

// return the degree of this polynomial (0 for the zero polynomial)
public int degree() {
    int d = 0;
    for (int i = 0; i < coef.length; i++)
        if (coef[i] != 0) d = i;
    return d;
}

// return c = a + b
public Polynomial2 plus(Polynomial2 b) {
    Polynomial2 a = this;
    Polynomial2 c = new Polynomial2(0, Math.max(a.deg, b.deg));
    for (int i = 0; i <= a.deg; i++) c.coef[i] += a.coef[i];
    for (int i = 0; i <= b.deg; i++) c.coef[i] += b.coef[i];
    c.deg = c.degree();
    return c;
}

// return (a - b)
public Polynomial2 minus(Polynomial2 b) {
    Polynomial2 a = this;
    Polynomial2 c = new Polynomial2(0, Math.max(a.deg, b.deg));
    for (int i = 0; i <= a.deg; i++) c.coef[i] += a.coef[i];
    for (int i = 0; i <= b.deg; i++) c.coef[i] -= b.coef[i];
    c.deg = c.degree();
    return c;
}

// return (a * b)
public Polynomial2 times(Polynomial2 b) {
    Polynomial2 a = this;
    Polynomial2 c = new Polynomial2(0, a.deg + b.deg);
    for (int i = 0; i <= a.deg; i++)
        for (int j = 0; j <= b.deg; j++)
            c.coef[i+j] += (a.coef[i] * b.coef[j]);
    c.deg = c.degree();
    return c;
}

// return a(b(x))  - compute using Horner's method
public Polynomial2 compose(Polynomial2 b) {
    Polynomial2 a = this;
    Polynomial2 c = new Polynomial2(0, 0);
    for (int i = a.deg; i >= 0; i--) {
        Polynomial2 term = new Polynomial2(a.coef[i], 0);
        c = term.plus(b.times(c));
    }
    return c;
}


// do a and b represent the same polynomial?
public boolean eq(Polynomial2 b) {
    Polynomial2 a = this;
    if (a.deg != b.deg) return false;
    for (int i = a.deg; i >= 0; i--)
        if (a.coef[i] != b.coef[i]) return false;
    return true;
}


// use Horner's method to compute and return the polynomial evaluated at x
public int evaluate(int x) {
    int p = 0;
    for (int i = deg; i >= 0; i--)
        p = coef[i] + (x * p);
    return p;
}

// differentiate this polynomial and return it
public Polynomial2 differentiate() {
    if (deg == 0) return new Polynomial2(0, 0);
    Polynomial2 deriv = new Polynomial2(0, deg - 1);
    deriv.deg = deg - 1;
    for (int i = 0; i < deg; i++)
        deriv.coef[i] = (i + 1) * coef[i + 1];
    return deriv;
}

// convert to string representation
public String toString() {
    if (deg ==  0) return "" + coef[0];
    if (deg ==  1) return coef[1] + "x + " + coef[0];
    String s = coef[deg] + "x^" + deg;
    for (int i = deg-1; i >= 0; i--) {
        if      (coef[i] == 0) continue;
        else if (coef[i]  > 0) s = s + " + " + ( coef[i]);
        else if (coef[i]  < 0) s = s + " - " + (-coef[i]);
        if      (i == 1) s = s + "x";
        else if (i >  1) s = s + "x^" + i;
    }
    return s;
}

//method for division
public Polynomial2 divides(Polynomial2 b) {

}

// test client
public static void main(String[] args) { 

    /*first polynomial*/
    Polynomial2 p1   = new Polynomial2(1, 13);
    Polynomial2 p2   = new Polynomial2(-12, 12);
    Polynomial2 p3   = new Polynomial2(55, 11);
    Polynomial2 p4   = new Polynomial2(-148, 10);
    Polynomial2 p5   = new Polynomial2(337, 9);
    Polynomial2 p6   = new Polynomial2(-460, 8);
    Polynomial2 p7   = new Polynomial2(55, 7);
    Polynomial2 p8   = new Polynomial2(-148, 6);
    Polynomial2 p9   = new Polynomial2(336, 5);
    Polynomial2 p10   = new Polynomial2(-448, 4);

    Polynomial2 p    = p1.plus(p2).plus(p3).plus(p4).plus(p5).plus(p6).plus(p7).
            plus(p8).plus(p9).plus(p10);   
    //x^13 - 12x^12 - 55x^11 - 148x^10 + 337x^9 + 460x^8 + 55x^7 - 148x^5 + 336x^5 - 448x^4

    /*second polynomial*/
    Polynomial2 q1   = new Polynomial2(1, 20);
    Polynomial2 q2   = new Polynomial2(-12, 19);
    Polynomial2 q3   = new Polynomial2(55, 18);
    Polynomial2 q4   = new Polynomial2(-148, 17);
    Polynomial2 q5   = new Polynomial2(335, 16);
    Polynomial2 q6   = new Polynomial2(-436, 15);
    Polynomial2 q7   = new Polynomial2(-55, 14);
    Polynomial2 q8   = new Polynomial2(148, 13);
    Polynomial2 q9   = new Polynomial2(-336, 12);
    Polynomial2 q10   = new Polynomial2(448, 11);
    Polynomial2 q11  = new Polynomial2(1, 7);
    Polynomial2 q12   = new Polynomial2(-12, 6);
    Polynomial2 q13   = new Polynomial2(55, 5);
    Polynomial2 q14   = new Polynomial2(-148, 4);
    Polynomial2 q15   = new Polynomial2(336, 3);
    Polynomial2 q16   = new Polynomial2(-448, 2);
    Polynomial2 q    = q1.plus(q2).plus(q3).plus(q4).plus(q5).plus(q6).plus(q7).
            plus(q8).plus(q9).plus(q10).plus(q11).plus(q12).plus(q13).
            plus(q14).plus(q15).plus(q16);  
    //x^20 - 12x^19 + 55x^18 - 148x^17 + 335x^16 - 436x^15 - 55x^14 + 148x^13 - 336x^12
    //448x^11 + x^7 - 12x^6 + 55x^5 - 148x^4 + 336x^3 - 448x^2;

    System.out.println("p(x) =        " + p);
    System.out.println("q(x) =        " + q);
   }

 }

Answer 1

您需要在兩個多項式中除以匹配度的系數，例如：

public Polynomial2 divides(Polynomial2 b) {
    Polynomial2 a = this;
    if ((b.deg == 0) && (b.coef[0] == 0))
        throw new RuntimeException("Divide by zero polynomial"); //Zero polynomial is the one having coeff and degree both zero.

    if (a.deg < b.deg) return new Polynomial2(0,0);

    int coefficient = a.coef[a.deg]/(b.coef[b.deg]);
    int exponent = a.deg - b.deg;
    Polynomial2 c = new Polynomial2(coefficient, exponent);
    return c.plus( (a.minus(b.times(c)).divides(b)) );
}

此代碼應執行您想要的操作。 您可能需要根據需要進行一些小的改進。 上面的代碼被slightely改性的版本此 。

Answer 2

這是普林斯頓檔案中的一些偽代碼

    // return (a / b)
    public Polynomial2 divides(Polynomial2 b) {
        Polynomial2 a = this;
        if ((b.deg == 0) && (b.coef[0] == 0))
            throw new RuntimeException("Divide by zero polynomial");

        if (a.deg < b.deg) return ZERO;

        int coefficient = a.coef[a.deg].divides(b.coef[b.deg]);
        int exponent = a.deg - b.deg;
        Polynomial2 c = new Polynomial2(coefficient, exponent);
        return c.plus( (a.minus(b.times(c)).divides(b)) );
    }