Content: complete the addition operation of two polynomials. It is known that there are two polynomials PM (x) and QM (x). Design an algorithm to realize the operation of Pm(x)+Qm(x) and Pm(x)-Qm(x). Moreover, the wig operation does not reopen the storage space, and it is required to be realized by chain storage structure.

Steps:

1. algorithm analysis

The implementation of the addition algorithm of two polynomials is to store the two polynomials in a linked list. Set two pointers LAI and LBI to move the first node of equations Pm(x) and Qm(x) respectively, and compare the exponential direction of the node referred to by LAI and LBI, which can be divided into the following three cases:

1. If LAI - > exp < LBI - > exp, the node referred to by LAI is one of the polynomials, and the LAI pointer moves back one position on the original basis.
2. If LAI - > exp = LBI - > exp, add the coefficients of the corresponding terms, and then deal with them in two cases: if the sum of the coefficient terms is zero, release the node pointed to by LAI and LBI; if the sum of the coefficient terms is not zero, modify the coefficient field of the node pointed to by LAI and release the LBI node.
3. If Lai - > exp > LBI - > exp, the node referred to by LBI is one of the polynomials, and the LBI pointer moves back one position on the original basis.

Two polynomial subtraction algorithm, the first is to store the two polynomials in a linked list. Set two pointers LAI and LBI to move the first node of equations Pm(x) and Qm(x) respectively, and compare the exponential direction of the node referred to by LAI and LBI, which can be divided into the following three cases:

1. If LAI - > exp < LBI - > exp, the node referred to by LAI is one of the polynomials, and the LAI pointer moves back one position on the original basis.
2. If LAI - > exp = LBI - > exp, subtract the coefficients of the corresponding term, and then deal with it in two cases: if the sum of the coefficient terms is zero, release the node pointed to by LAI and LBI; if the sum of the coefficient terms is not zero, modify the coefficient field of the node pointed to by LAI and release the LBI node.
3. If Lai - > exp > LBI - > exp, the node referred to by LBI is one of the polynomials, and the LBI pointer moves back one position on the original basis.

1. Algorithm design

Five functions are designed in the program:

1. Init() is used to initialize the linked list;
2. CreatFromTail() creates a linked list by tail interpolation;
3. Polyadd() is used to realize the addition algorithm of two polynomials;
4. Polysub() is used to realize the subtraction of two polynomials;
5. Print() is used to output polynomials.

   #include<stdio.h>
   #include<stdlib.h>
   #include<malloc.h>
   typedef struct poly
   {
   	int exp; 	//index 
   	int coef;	//coefficient 
   	struct poly *next;
   	
   }PNode,*PLinklist;
   /*Linked list initialization*/
   int Init(PLinklist *head)
   {
   	*head=(PLinklist)malloc(sizeof(PNode));
   	if(*head)
   	{
   		(*head)->next=NULL;
   		return 1;
   	}
   	else
   		return 0;		
   }
   /*Creating linked list by tail interpolation*/ 
   int CreateFromTail(PLinklist*head)
   {
   	PNode *pTemp,*pHead;
   	int c;		//Storage factor 
   	int exp;	//Storage index 
   	int i=1	;	//Counter 
   	pHead=*head;
   	scanf("%d,%d",&c,&exp);
   	while(c!=0)//When the coefficient is zero, end the input
   	{
   		pTemp=(PLinklist)malloc(sizeof(PNode));
   		if(pTemp)
   		{
   			pTemp->exp=exp;		//Acceptance index 
   			pTemp->coef=c;		//Acceptance coefficient 
   			pTemp->next=NULL;
   			pHead->next=pTemp;
   			pHead=pTemp;
   			scanf("%d,%d",&c,&exp);
   		 } 
   		 else
   		 return 0;
   	 } 
   	 return 1;
   }
   /*Add two polynomials*/ 
   void Polyadd(PLinklist LA,PLinklist LB)
   {
   	PNode*LAI=LA->next;		//Pointer LAI moves in polynomial A 
   	PNode*LBI=LB->next;		 
   	PNode*temp;				//Pointer temp saves the node to be deleted 
   	int sum=0;				//Sum of preservation factors 
   	/*Compare the exponential terms of the nodes referred to by LAI and LBI*/ 
   	while(LAI&&LBI)
   	{
   		if(LAI->exp<LBI->exp){		  
   			LA->next=LAI;
   			LA=LA->next;
   			LAI=LAI->next;
   		}
   		else if(LAI->exp==LBI->exp)
   		{
   			sum=LAI->coef+LBI->coef;
   			if(sum)
   			{
   				LAI->coef=sum;
   				LA->next=LAI;
   				LA=LA->next;
   				LAI=LAI->next;
   				temp=LBI;
   				LBI=LBI->next;
   				free(temp);
   				
   			}
   			else
   			{
   				temp=LAI;
   				LAI=LAI->next;
   				free(temp);
   				temp=LBI;
   				LBI=LBI->next;
   				free(temp);
   				
   			}
   		}
   		else
   		{
   			LA->next=LBI;
   			LA=LA->next;
   			LBI=LBI->next;
   		}
   		
   	}
   	if(LAI)
   		LA->next=LAI;
   	else
   		LA->next=LBI; 
   }
   /*Subtraction of two polynomials*/ 
   void Polysub(PLinklist LA,PLinklist LB)
   {
   	PNode*LAI=LA->next;		//Pointer LAI moves in polynomial A 
   	PNode*LBI=LB->next;		 
   	PNode*temp;				//Pointer temp saves the node to be deleted 
   	int difference=0;				//Preservation factor
   	/*Compare the exponential terms of the nodes referred to by LAI and LBI*/ 
   	while(LAI&&LBI)
   	{
   		if(LAI->exp<LBI->exp){		  
   			LA->next=LAI;
   			LA=LA->next;
   			LAI=LAI->next;
   		}
   		else if(LAI->exp==LBI->exp)
   		{
   			difference=LAI->coef-LBI->coef;
   			if(difference)
   			{
   				LAI->coef=difference;
   				LA->next=LAI;
   				LA=LA->next;
   				LAI=LAI->next;
   				temp=LBI;
   				LBI=LBI->next;
   				free(temp);
   				
   			}
   			else
   			{
   				temp=LAI;
   				LAI=LAI->next;
   				free(temp);
   				temp=LBI;
   				LBI=LBI->next;
   				free(temp);
   				
   			}
   		}
   		else
   		{
   			LA->next=LBI;
   			LA=LA->next;
   			LBI=LBI->next;
   		}
   		
   	}
   	if(LAI)
   		LA->next=LAI;
   	else
   		LA->next=LBI; 
   }
   void Print(PLinklist head)
   {
   	head=head->next;
   	while(head)
   	{
   		if(head->exp)
   			printf("(%dx^%d)",head->coef,head->exp);
   		else
   			printf("%d",head->coef);
   		if(head->next)
   			printf("+");
   		else
   			break;
   		head=head->next;
   	}
   }
   
   int main(void)
   {
   	PLinklist LA;
   	PLinklist LB;
   	Init(&LA);
   	Init(&LB);
   	printf("Please enter the coefficient of the first polynomial,index,Enter 0,0 End input\n");
   	CreateFromTail(&LA);
   	printf("Please enter the coefficient of the second polynomial,index,Enter 0,0 End input\n");
   	CreateFromTail(&LB);
   	Print(LA);
   	printf("\n");
   	Print(LB);
   	printf("\n");
   	
   	int i; 
   	
   	printf("(Please enter 1 for addition polynomial and 0 for subtraction polynomial)\n"); 
   	scanf("%d",&i);
   	
   	if(1==i){
   		Polyadd(LA,LB);
   		printf("The result of adding two polynomials:\n");
   		Print(LA);
   		printf("\n");
   	}
   	else if(0==i){
   		Polysub(LA,LB);
   		printf("The result of subtracting two polynomials:\n");
   		Print(LA);
   		printf("\n");
   	}
   	else 
   		printf("Please enter the correct characters");
   	return 0;
   	
   }