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会别破坏掉。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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专栏完整目录：数据结构与算法目录
链表的应用：单元多项式的加法、减法、乘法,布布扣,bubuko.com
时间： 2024-12-16 03:01:40