链表的应用：单元多项式的加法、减法、乘法

使用链表来实现单元多项式的加法、减法、乘法。一个单元多项式的节点结构无非是这样的：系数域、指数域、链域。

如下图：

我们使用链表来模拟单元多项式的常见运算。其中，加法是其它运算的基础，减法：poly1-poly2=poly1+(-poly2)，乘法：poly1*poly2，可用poly1乘以poly2的每一项，相加其乘积结果。

单元多项式的节点结构类型是这样的：

typedef struct node
{
	float coef;   //系数 
	int expn;     //指数 
	struct node *next;
}PolyNode;      //多项式节点 polynomial node 

多项式的加法我们提供了两种：

1.Polynomial polyAdd(Polynomial poly1, Polynomial poly2),把poly1和poly2相加得到一个新的多项式，相加的过程中poly1和poly2保持不变，不会被破坏。

2.void add(Polynomial poly1, Polynomial poly2),把poly2加到poly1上，相加的过程中poly2的节点会被利用上。结束后，poly2不存在了。

提供第一种加法，是为了保持poly1和poly2不变，以便进行下一次的运算。提供第二种加法，是为了运算结束后，内存不会泄露。

其它具体细节得看代码了：

#include<stdio.h>
#include<stdlib.h>
typedef struct node
{
	float coef;   //系数 
	int expn;     //指数 
	struct node *next;
}PolyNode;      //多项式节点 polynomial node 
typedef PolyNode* Polynomial;
Polynomial createPolynomial()   //创建多项式 
{
	PolyNode *p, *q, *head = (PolyNode *)malloc(sizeof(PolyNode));   //头节点 
	head->next = NULL;
	float coef;
	int expn;
	printf("输入该多项式每一项的系数和指数，每项一行，输入0 0结束！\n");
	while (scanf("%f %d", &coef, &expn) && coef)    // 默认，按指数递减排列 
	{
		if (head->next)
		{
			p = head;
			while (p->next && expn < p->next->expn)
				p = p->next;
			if (p->next)
			{
				if (expn == p->next->expn)  //有相同指数的直接把系数加到原多项式 
				{
					p->next->coef += coef;
					//若是相加后系数为0，则舍弃该节点 
					if (p->next->coef > -0.000001 && p->next->coef < 0.000001)
					{
						q = p->next;
						p->next = q->next;
						free(q);
					}
				}
				else
				{
					q = (PolyNode*)malloc(sizeof(PolyNode));
					q->coef = coef;
					q->expn = expn;
					q->next = p->next;
					p->next = q;
				}
			}
			else
			{
				p->next = (PolyNode*)malloc(sizeof(PolyNode));
				p = p->next;
				p->coef = coef;
				p->expn = expn;
				p->next = NULL;
			}
		}
		else
		{
			head->next = (PolyNode*)malloc(sizeof(PolyNode));
			head->next->coef = coef;
			head->next->expn = expn;
			head->next->next = NULL;
		}
	}
	return head;
}
//多项式与指定单项式相乘，该单项式为 coefx^expn 
Polynomial multiply(Polynomial poly, float coef, int expn)
{
	PolyNode *p, *q, *Poly = (PolyNode*)malloc(sizeof(PolyNode));
	p = Poly;
	q = poly->next;
	while (q)
	{
		p->next = (PolyNode*)malloc(sizeof(PolyNode));
		p = p->next;
		p->coef = (q->coef*coef);
		p->expn = (q->expn + expn);
		q = q->next;
	}
	p->next = NULL;
	return Poly;
}
void add(Polynomial poly1, Polynomial poly2)   //把 poly2 加到 poly1 上
{
	PolyNode *p, *q, *r;
	r = poly1;
	p = poly1->next;  //指向第一个节点
	q = poly2->next;
	poly2->next = NULL;
	while (p && q)
	{
		if (p->expn > q->expn)
		{
			r->next = p;
			p = p->next;
			r = r->next;
		}
		else if (p->expn < q->expn)
		{
			r->next = q;
			q = q->next;
			r = r->next;
		}
		else
		{
			PolyNode *t;
			p->coef += q->coef;
			if (!(p->coef > -0.000001 && p->coef < 0.000001)) //系数不为0
			{
				r->next = p;
				r = r->next;
				p = p->next;
			}
			else
			{
				t = p;
				p = p->next;
				free(t);
			}
			t = q;
			q = q->next;
			free(t);
		}
	}
	if (p)
		r->next = p;
	if (q)
		r->next = q;
}
//多项式减法 poly1-poly2形成一个新的多项式
Polynomial polySubtract(Polynomial poly1, Polynomial poly2)
{
	//把poly2的系数取相反数，形成一个新的多项式
	Polynomial poly = (PolyNode*)malloc(sizeof(PolyNode)); //构造头节点
	PolyNode *p, *q;
	p = poly;
	q = poly2->next;
	while (q)
	{
		p->next = (PolyNode*)malloc(sizeof(PolyNode));
		p = p->next;
		p->coef = -(q->coef);  //系数取反
		p->expn = q->expn;
		q = q->next;
	}
	p->next = NULL;
	add(poly, poly1);  //利用加法
	return poly;
}
//多项式相加 poly1+poly2形成一个新的多项式 
Polynomial polyAdd(Polynomial poly1, Polynomial poly2)
{
	Polynomial poly = (PolyNode*)malloc(sizeof(PolyNode));  //和多项式的头节点 
	poly->next = NULL;
	PolyNode *p, *q, *r;
	r = poly;
	p = poly1->next;
	q = poly2->next;
	while (p&&q)
	{
		if (p->expn > q->expn)
		{
			r->next = (PolyNode*)malloc(sizeof(PolyNode));
			r = r->next;
			r->coef = p->coef;
			r->expn = p->expn;
			p = p->next;
		}
		else if (p->expn < q->expn)
		{
			r->next = (PolyNode*)malloc(sizeof(PolyNode));
			r = r->next;
			r->coef = q->coef;
			r->expn = q->expn;
			q = q->next;
		}
		else
		{
			float m = p->coef + q->coef;
			if (!(m > -0.000001 && m < 0.000001))
			{
				r->next = (PolyNode*)malloc(sizeof(PolyNode));
				r = r->next;
				r->coef = m;
				r->expn = p->expn;
			}
			q = q->next;
			p = p->next;
		}
	}
	while (p)
	{
		r->next = (PolyNode*)malloc(sizeof(PolyNode));
		r = r->next;
		r->coef = p->coef;
		r->expn = p->expn;
		p = p->next;
	}
	while (q)
	{
		r->next = (PolyNode*)malloc(sizeof(PolyNode));
		r = r->next;
		r->coef = q->coef;
		r->expn = q->expn;
		q = q->next;
	}
	r->next = NULL;
	return poly;
}
Polynomial polyMultiply(Polynomial poly1, Polynomial poly2)   //多项式相乘  
{
	Polynomial poly = (PolyNode*)malloc(sizeof(PolyNode));  //创建多项式和的头节点 
	poly->next = NULL;
	PolyNode *p;
	p = poly2->next;
	while (p)
	{
		add(poly, multiply(poly1, p->coef, p->expn));
		p = p->next;
	}
	return poly;
}
void printPoly(Polynomial poly)    //打印多项式 
{
	if (poly && poly->next)
	{
		PolyNode *p = poly->next;  //p指向第一个节点
		while (p->next)
		{
			printf("%gx^%d", p->coef, p->expn);
			p = p->next;
			if (p && (p->coef > 0))
				printf("+");
		}
		if (p->expn == 0)
			printf("%g", p->coef);   //打印常数项 
		else
			printf("%gx^%d", p->coef, p->expn);
		printf("\n");
	}
}
void clear(Polynomial poly)   //释放内存
{
	if (poly && poly->next)
	{
		PolyNode *p, *q;
		p = poly;
		while (p)
		{
			q = p->next;
			free(p);
			p = q;
		}
	}
	poly = NULL;
}

调用方法：

int main()
{
	printf("用链表实现多项式的加减法\n");
	Polynomial poly1, poly2, poly;
	printf("创建多项式一\n");
	poly1 = createPolynomial();
	printf("多项式一：\n");
	printPoly(poly1);
	printf("创建多项式二\n");
	poly2 = createPolynomial();
	printf("多项式二：\n");
	printPoly(poly2);
	printf("两多项式相加，和为：\n");
	poly = polyAdd(poly1, poly2);
	printPoly(poly);
	clear(poly);
	printf("两个多项式相乘，积为：\n");
	poly = polyMultiply(poly1, poly2);
	printPoly(poly);
	clear(poly);
	printf("两多项式相减，差为：\n");
	poly = polySubtract(poly1, poly2);
	printPoly(poly);
	clear(poly1);
	clear(poly2);
	clear(poly);
	system("pause");
	return 0;
}

调用中，调用次序是加法、乘法、减法，减法放最后。这是因为减法的过程中poly2会被破坏掉。仔细看看add()方法就可明白。

运行：

代码比较长，逻辑有些复杂，得反复地看。

完整代码下载：一元多项式的加法、减法、乘法

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