顺序表与链表实现多项式的加减乘与幂微分
作者：互联网
1.链表实现
分为三个文件进行编写,分别为自定义头文件,实现的cpp文件与main主程序文件.
整个过程中在实现乘的函数MutiplePoly和展示的Display函数上遇到较大困难.
1)MutiplePoly:由于一开始便默认数据是按低阶到高阶的,在设计乘算法时强迫自己尽量不使用双循环来计算.关键在于找到相乘后同阶的所有项并相加,,故将两个链表各循环一遍存入两个数组中方便访问.在将两个多项式各个系数相乘的结果规律排列成矩阵后,发现相乘后的阶数规律排列.
如两个数组大小分别为4,3.则下标分别为0,1,2,3与0,1,2,相乘后系数相加得到如下矩阵:
| 0 | 1 | 2 | 3 |
| 1 | 2 | 3 | 4 |
| 2 | 3 | 4 | 5 |
矩阵中相同数字代表的项阶数相同,则按斜线划分成6层,大循环从0层开始遍历到顶层.则计算可分为两部分,一部分层数0~2,a[i]+b[j],i+j=当前层数,当j<0时跳出小循环.另一部分3~5,当i>3时跳出小循环.通过小循环将相同阶数的系数全部相加.
2)Display:展示多项式的实现过程主要在于细节的繁多,如首项若为正项则不加符号,若为0则不显示,非首项若为0则不显示,为正项要添加正号等等.也因为这样最后的实现体跳转较多,代码显得繁琐.
顺序表实现虽然用的是vector,但基本也符合顺序表的定义,实现过程与链表相差无几.故不多赘述.
最后附上完整链表实现代码:
头文件:
#ifndef __POLY_H__
#define __POLY_H__
#include <iostream>
#include <vector>
using namespace std;
// 存储系数的结构体,采用双链表结构
typedef struct data
{
int coefficient;
struct data *next;
struct data *pre;
} Data;
typedef Data *PtrToNode;
// 存储链表首末位置的结构体
typedef struct upper
{
PtrToNode head;
PtrToNode end;
int items;
} list;
//声明链表
typedef list *List;
void InitList(List &); //初始化链表
void Add(List &); //添加数据
void AddNode(List &); //添加节点
void GetInfo(List &); //输入数据
void Calculate(List &); //运用霍纳法则进行计算
void AddPoly(List &, List &, List &); //将两个多项式相加
void SubtractPoly(List &, List &, List &); //将两个多项式相减
void MutiplePoly(List &, List &, List &); //将两个多项式相乘
// void DividePoly(List &, List &, List &);
void Differential(List &);
void Clear(List &); //清空列表
void Reset(List &, List &); //重新输入两个多项式
void menu1(); //展示菜单1
void menu2(); //展示菜单2
void Display(List &); //展示多项式
void ListToArray(List &, List &, vector<int> &, vector<int> &); //将链表数据转化为数组存储
void DeleteHead(List &); //删除首节点
#endif
实现文件:
#include "poly.h"
//给多项式添加信息
void Add(List &l)
{
int num;
cout << "Enter the number of terms:" << endl;
cin >> num;
cout << "Enter the coefficients in ascending power:" << endl; //提醒用户以幂次的升序输入系数
for (int i = 0; i < num; i++)
{
AddNode(l);
GetInfo(l);
}
}
// 初始化链表
void InitList(List &l)
{
l = (List) new list;
l->head = NULL;
l->end = NULL;
l->items = 0;
}
//多项式相加
void AddPoly(List &l1, List &l2, List &l3)
{
PtrToNode cur1 = l1->head;
PtrToNode cur2 = l2->head;
while (cur1 != NULL || cur2 != NULL)
{
AddNode(l3);
if (cur1 != NULL && cur2 != NULL)
{
l3->end->coefficient = cur1->coefficient + cur2->coefficient;
cur1 = cur1->next;
cur2 = cur2->next;
}
else if (cur1 == NULL) //处理多项式项数不同的情况
{
l3->end->coefficient = cur2->coefficient;
cur2 = cur2->next;
}
else
{
l3->end->coefficient = cur1->coefficient;
cur1 = cur1->next;
}
}
Display(l3);
}
// 多项式相减
void SubtractPoly(List &l1, List &l2, List &l3)
{
PtrToNode cur1 = l1->head;
PtrToNode cur2 = l2->head;
while (cur1 != NULL || cur2 != NULL)
{
AddNode(l3);
if (cur1 != NULL && cur2 != NULL)
{
l3->end->coefficient = cur1->coefficient - cur2->coefficient;
cur1 = cur1->next;
cur2 = cur2->next;
}
else if (cur1 == NULL)
{
l3->end->coefficient = -cur2->coefficient;
cur2 = cur2->next;
}
else
{
l3->end->coefficient = cur1->coefficient;
cur1 = cur1->next;
}
}
Display(l3);
}
void MutiplePoly(List &l1, List &l2, List &l3)
{
vector<int> a1;
vector<int> a2;
ListToArray(l1, l2, a1, a2);
//根据幂次相加后形成的规律矩阵查找每个幂次对应的项并相加,最后赋值
int top = (l1->items + l2->items - 2);
int floor = 0; //幂次为0开始
int border = a2.size() - 1;
int sum = 0; //存储每个幂次对应的系数
while (floor <= top)
{
AddNode(l3);
if (floor <= border)
for (int i = 0, j = floor - i; j >= 0; i++, j--)
{
sum += a1[i] * a2[j];
l3->end->coefficient = sum;
}
else
for (int j = border, i = floor - j; i < a1.size() && j >= 0; i++, j--)
{
sum += a1[i] * a2[j];
l3->end->coefficient = sum;
}
sum = 0;
floor++;
}
Display(l3);
}
// 求一阶导
void Differential(List &l)
{
DeleteHead(l);
int time = 1;
PtrToNode cur = l->head;
while (cur != NULL)
{
cur->coefficient *= time;
cur = cur->next;
time++;
}
Display(l);
}
void Calculate(List &l)
{
//获得x值
int x;
cout << "Enter the value of x:" << endl;
cin >> x;
PtrToNode cur = l->end; //按照霍纳法则从后往前遍历
int poly = 0;
while (cur != NULL)
{
poly = x * poly + cur->coefficient; //将多项式进行拆分成相乘形式
cur = cur->pre;
}
cout << "The result is " << poly << endl;
}
//两个菜单
void menu1()
{
cout << "Choose one to execute:" << endl;
cout << "a)Add\nb)Subtract\nc)Mutiply" << endl;
}
void menu2()
{
cout << "Choose one to execute:" << endl;
cout << "a)Calculate\nb)Retype the two polynomials\nc)Differential\nq)quit" << endl;
}
//获得系数
void GetInfo(List &l)
{
cin >> l->end->coefficient;
}
void AddNode(List &l)
{
PtrToNode newNode = (PtrToNode) new Data;
newNode->next = NULL;
newNode->pre = l->end;
if (l->head == NULL)
{
l->head = newNode;
l->end = newNode;
}
else
{
l->end->next = newNode;
l->end = newNode;
}
l->items++;
}
//清除结果
void Clear(List &l)
{
PtrToNode cur = l->head;
PtrToNode temp;
while (cur != NULL)
{
temp = cur->next;
free(cur);
cur = temp;
}
l->head = NULL;
l->end = NULL;
l->items = 0;
}
//重设l1,l2
void Reset(List &l1, List &l2)
{
Clear(l1);
Clear(l2);
Add(l1);
Add(l2);
}
//将多项式以字符串形式展示
void Display(List &l)
{
string polynomial;
PtrToNode cur = l->head;
if (l->items == 0)
polynomial.append("0");
else //遍历以确定系数是否全为0
{
int cnt = 0;
while (cur != NULL)
{
if (cur->coefficient == 0)
cnt++;
cur = cur->next;
}
if (cnt == l->items)
{
polynomial.append("0");
cout << polynomial << endl;
return;
}
}
//通过不断将系数与符号append到string中从而展示
int time = 0; //记录幂的次数
cur = l->head;
while (cur != NULL)
{
if (cur == l->head)
{
if (cur->coefficient != 0)
polynomial.append(to_string(cur->coefficient));
cur = cur->next;
time++;
continue;
}
else
{
if (cur->coefficient > 0)
polynomial.append("+");
else if (cur->coefficient < 0)
polynomial.append("-");
else //系数为零则跳过不显示
{
cur = cur->next;
time++;
continue;
}
polynomial.append(to_string(abs(cur->coefficient)));
polynomial.append("x^");
polynomial.append(to_string(time));
time++;
cur = cur->next;
}
}
cout << polynomial << endl;
}
// 将链表存储到数组当中方便访问
void ListToArray(List &l1, List &l2, vector<int> &a1, vector<int> &a2)
{
PtrToNode p1 = l1->head;
PtrToNode p2 = l2->head;
//通过if,else语句让a1.a2分别固定存储长多项式于短多项式
if (l1->items > l2->items)
{
while (p1 != NULL)
{
a1.push_back(p1->coefficient);
p1 = p1->next;
}
while (p2 != NULL)
{
a2.push_back(p2->coefficient);
p2 = p2->next;
}
}
else
{
while (p1 != NULL)
{
a2.push_back(p1->coefficient);
p1 = p1->next;
}
while (p2 != NULL)
{
a1.push_back(p2->coefficient);
p2 = p2->next;
}
}
}
// 删除第一个节点
void DeleteHead(List &l)
{
PtrToNode temp = l->head;
if (l->items > 1) //项数大于1的情况
l->head->next->pre = NULL;
l->head = l->head->next;
delete temp;
l->items--;
}
主程序文件:
#include "poly.h"
int main(void)
{
List l1;//第一个多项式
List l2;//第二个多项式
List l3;//操作之后的多项式
char option;
//初始化三个链表
InitList(l1);
InitList(l2);
InitList(l3);
//输入多项式
Add(l1);
Add(l2);
do
{
menu1();
cin >> option;
switch (option)
{
case 'a':
AddPoly(l1, l2, l3);
break;
case 'b':
SubtractPoly(l1, l2, l3);
break;
case 'c':
MutiplePoly(l1, l2, l3);
break;
case 'd':
// DividePoly(l1, l2, l3);
break;
default:
cout << "Invalid input" << endl;
continue;
}
menu2();
cin >> option;
switch (option)
{
case 'a':
Calculate(l3);
break;
case 'b':
Reset(l1, l2);
break;
case 'c':
Differential(l3);
break;
case 'q':
break;
default:
cout << "Invalid input" << endl;
break;
}
Clear(l3);
} while (option != 'q');
}
标签：coefficient,cur,多项式,void,List,链表,表与,l3,NULL 来源： https://www.cnblogs.com/roc-t/p/13795207.html