【大数四则运算】——线性表 [习题描述] 设计—个实现任意长的整数进行四则运算和幂次运算的演示程序。 [基本要求] 利用双向循环链表实现大数的存储，每个结点含一个整型变量。 [实现提示] 实现原理：任何一个数都可以表示成指数形式，如下所示： N=nEe (0=<n 的绝对值<=1,e 为 10 的指数幂) 例如 100 可以表示成 1E2,1001 可以表示成 1.01E3

这是一道关于大数四则运算的题目，需要使用双向循环链表来存储大数，每个节点包含一个整型变量，实现指数形式的表示法。

具体实现思路如下：

1. 定义一个结构体表示链表的节点，包含一个整型变量和两个指针分别指向前驱节点和后继节点。

2. 定义一个函数来创建链表，该函数使用一个字符串表示大数，将其转化为链表的形式存储。

3. 定义四个函数分别实现大数的加、减、乘、除四种运算，这四个函数的实现都需要对链表进行操作。

4. 定义一个函数实现大数的幂次运算，该函数使用快速幂算法实现，也需要对链表进行操作。

5. 定义一个函数实现链表的输出，将链表转换为指数形式输出。

具体实现过程中需要注意链表的初始化、插入、删除等操作，以及处理进位和借位的情况。同时，为了方便计算，可以将大数转化为以10的指数幂为底的指数形式，这样可以方便地进行数位的对齐和运算。

以下是一个可能的C语言实现示例：

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define BASE 10000
#define MAX_DIGITS 100

typedef struct Node {
    int val;
    struct Node *prev, *next;
} Node;

typedef struct {
    Node *head, *tail;
    int len;
} BigInt;

Node *new_node(int val) {
    Node *node = (Node *)malloc(sizeof(Node));
    node->val = val;
    node->prev = node->next = NULL;
    return node;
}

BigInt *new_bigint() {
    BigInt *bigint = (BigInt *)malloc(sizeof(BigInt));
    bigint->head = bigint->tail = NULL;
    bigint->len = 0;
    return bigint;
}

void insert_front(BigInt *bigint, int val) {
    Node *node = new_node(val);
    if (bigint->head == NULL) {
        bigint->head = bigint->tail = node;
    } else {
        node->next = bigint->head;
        bigint->head->prev = node;
        bigint->head = node;
    }
    bigint->len++;
}

void insert_back(BigInt *bigint, int val) {
    Node *node = new_node(val);
    if (bigint->tail == NULL) {
        bigint->head = bigint->tail = node;
    } else {
        node->prev = bigint->tail;
        bigint->tail->next = node;
        bigint->tail = node;
    }
    bigint->len++;
}

void remove_front(BigInt *bigint) {
    if (bigint->head == NULL) return;
    Node *node = bigint->head;
    bigint->head = node->next;
    if (bigint->head != NULL) bigint->head->prev = NULL;
    else bigint->tail = NULL;
    bigint->len--;
    free(node);
}

void remove_back(BigInt *bigint) {
    if (bigint->tail == NULL) return;
    Node *node = bigint->tail;
    bigint->tail = node->prev;
    if (bigint->tail != NULL) bigint->tail->next = NULL;
    else bigint->head = NULL;
    bigint->len--;
    free(node);
}

void clear(BigInt *bigint) {
    while (bigint->len > 0) remove_front(bigint);
}

BigInt *from_str(char *str) {
    BigInt *bigint = new_bigint();
    int len = strlen(str);
    int i, j, n;
    for (i = len - 1; i >= 0; i -= 4) {
        int val = 0;
        for (j = 0, n = 1; j < 4 && i - j >= 0; j++, n *= 10) {
            val += (str[i - j] - '0') * n;
        }
        insert_front(bigint, val);
    }
    while (bigint->len > 1 && bigint->head->val == 0) remove_front(bigint);
    return bigint;
}

char *to_str(BigInt *bigint) {
    char *str = (char *)malloc(sizeof(char) * (bigint->len * 4 + 1));
    int i = 0;
    Node *node = bigint->head;
    while (node != NULL) {
        if (node != bigint->head) {
            sprintf(str + i, "%04d", node->val);
            i += 4;
        } else {
            sprintf(str + i, "%d", node->val);
            i += (node->val == 0 ? 1 : (int)log10(node->val) + 1);
        }
        node = node->next;
    }
    str[i] = '\0';
    return str;
}

int compare(BigInt *a, BigInt *b) {
    if (a->len != b->len) return (a->len > b->len ? 1 : -1);
    Node *a_node = a->head, *b_node = b->head;
    while (a_node != NULL && b_node != NULL) {
        if (a_node->val != b_node->val) return (a_node->val > b_node->val ? 1 : -1);
        a_node = a_node->next;
        b_node = b_node->next;
    }
    return 0;
}

void add(BigInt *a, BigInt *b, BigInt *result) {
    int carry = 0;
    Node *a_node = a->tail, *b_node = b->tail;
    while (a_node != NULL || b_node != NULL || carry != 0) {
        int sum = carry;
        if (a_node != NULL) sum += a_node->val;
        if (b_node != NULL) sum += b_node->val;
        insert_front(result, sum % BASE);
        carry = sum / BASE;
        if (a_node != NULL) a_node = a_node->prev;
        if (b_node != NULL) b_node = b_node->prev;
    }
}

void subtract(BigInt *a, BigInt *b, BigInt *result) {
    int borrow = 0;
    Node *a_node = a->tail, *b_node = b->tail;
    while (a_node != NULL || b_node != NULL || borrow != 0) {
        int diff = borrow;
        if (a_node != NULL) diff += a_node->val;
        if (b_node != NULL) diff -= b_node->val;
        if (diff < 0) {
            diff += BASE;
            borrow = -1;
        } else {
            borrow = 0;
        }
        insert_front(result, diff);
        if (a_node != NULL) a_node = a_node->prev;
        if (b_node != NULL) b_node = b_node->prev;
    }
    while (result->len > 1 && result->head->val == 0) remove_front(result);
}

void multiply(BigInt *a, BigInt *b, BigInt *result) {
    clear(result);
    int i, j;
    for (i = 0; i < a->len; i++) {
        int carry = 0;
        BigInt *temp = new_bigint();
        for (j = 0; j < i; j++) insert_back(temp, 0);
        Node *b_node = b->tail;
        while (b_node != NULL || carry != 0) {
            int prod = carry;
            if (b_node != NULL) prod += b_node->val * a->head->val;
            insert_front(temp, prod % BASE);
            carry = prod / BASE;
            if (b_node != NULL) b_node = b_node->prev;
        }
        add(result, temp, result);
        clear(temp);
    }
}

void divide(BigInt *a, BigInt *b, BigInt *quo, BigInt *rem) {
    clear(quo);
    clear(rem);
    BigInt *dividend = new_bigint();
    Node *node = a->head;
    while (node != NULL) {
        insert_back(dividend, node->val);
        node = node->next;
        int i;
        for (i = 0; i < BASE; i++) {
            multiply(b, rem, rem);
            add(rem, new_bigint(), rem);
            add(rem, new_node(i), rem);
            if (compare(rem, b) >= 0) break;
        }
        insert_front(quo, i);
        subtract(rem, b, rem);
    }
    while (quo->len > 1 && quo->head->val == 0) remove_front(quo);
    while (rem->len > 1 && rem->head->val == 0) remove_front(rem);
    free(dividend);
}

void pow_helper(BigInt *a, int n, BigInt *result) {
    if (n == 0) {
        insert_front(result, 1);
        return;
    }
    pow_helper(a, n / 2, result);
    multiply(result, result, result);
    if (n % 2 == 1) {
        BigInt *temp = new_bigint();
        copy(a, temp);
        multiply(result, temp, result);
        clear(temp);
    }
}

void pow(BigInt *a, int n, BigInt *result) {
    clear(result);
    pow_helper(a, n, result);
}

int main() {
    BigInt *a = from_str("123456789012345678901234567890");
    BigInt *b = from_str("987654321098765432109876543210");
    BigInt *c = new_bigint();
    BigInt *d = new_bigint();
    BigInt *e = new_bigint();
    add(a, b, c);
    subtract(a, b, d);
    multiply(a, b, e);
    char *a_str = to_str(a);
    char *b_str = to_str(b);
    char *c_str = to_str(c);
    char *d_str = to_str(d);
    char *e_str = to_str(e);
    printf("%s + %s = %s\n", a_str, b_str, c_str);
    printf("%s - %s = %s\n", a_str, b_str, d_str);
    printf("%s * %s = %s\n", a_str, b_str, e_str);
    pow(a, 3, c);
    char *c_pow_str = to_str(c);
    printf("%s ^ 3 = %s\n", a_str, c_pow_str);
    free(a_str);
    free(b_str);
    free(c_str);
    free(d_str);
    free(e_str);
    free(c_pow_str);
    clear(a);
    clear(b);
    clear(c);
    clear(d);
    clear(e);
    free(a);
    free(b);
    free(c);
    free(d);
    free(e);
    return 0;
}