[原创]【首帖】求函数微分的程序+函数化简
话说从来没在计算机智能板块发过贴。。

这几天没事干敲程序
把很久以前的某个愿望实现了——求一个函数的导数即微分

从前我用的算法(猥琐版):
①输入字符串..
②直接对字符串处理,用求导的各种法则
③输出

结果处理到一半发现函数的乘积和商的导数上面弄不出来了。。
放弃

两年之后的今天。。
在学习了OOP和STL之后。。

新算法:
①输入字符串(必须的)
②构建类的数据结构
类:

class expression{
            vector<pair<expression,double> > seg;
            int status;
            ...
}

其中status存放这一部分表达式的种类
seg存放各个子表达式及其系数或者次数(请看下文)
status=-1:表达式是一个常数 比如说是5.1
status=0:表达式是一个变量 比如说是y
status=1:表达式是由和和差组成 比如说 sin(x) + x^2 + 5
此时seg中存放的就是各个子表达式(由加减号分割)以及其系数sin(x) x^2 5
status=2: 表达式是由乘积和商组成 比如说??tan(x)*x*2
此时seg中存放的就是各个子表达式(由乘除号分割)以及其次数tan(x) x 2
status=3; 表达式是一个幂 比如说 sin(x)^x
此时seg中存放的就是底数和指数sin(x)、x
status=4:表达式是一个函数(复合)比如说 ln(e^a)
此时seg中存放的就是函数里面的那一部分(e^a)

由这种结构递归定义形成一个表达式对象来储存这个表达式(函数)

③接下来求导的任务就简单多了(不解释)
④对表达式进行化简
求导出来的表达式看起来是很恶心的。。(由一大堆东西加减乘除在一起,其中还有很多是0和1的)
所以需要化简、合并同类项
关于辨别哪个是同类项的问题:
我是通过求值来实现的
就是说取随机数,值相等的话两个表达式就相等(辨别多次)

⑤输出结果


这样输出的导函数就比较像样了[s:252]    ??

下面附上巨大的程序(600+line)
附件在下面
math.rar4.00k1次

1、math.cpp?? (程序,很短小的)

#include"function.h"
int main() {

    val['@'] = 3.14159265358979323846;//"@"refers to pi,will be replaced at the predeal()
    val['e'] = 2.71828182845904523536;
    expr exprrr;
    string s1;
    while (1) {
        cin >> s1;
        exprrr = convert(s1);
        output(exprrr);
        cout << endl;
        simplify(exprrr);
        output(exprrr);
        cout << endl;

        exprrr = derivative(exprrr, 'x');
        output(exprrr);
        cout << endl;
        simplify(exprrr);
        output(exprrr);
        cout << endl;
        cout << value(exprrr) << endl;
    }
}



2、function.h (头文件,恐怖。。)


#include<iostream>
#include<cmath>
#include<vector>
#include<algorithm>
#include<map>
#include<string>
#include<cctype>
#include<sstream>
#include<iomanip>
#include<ctime>
using namespace std;

map<char, double> val;

class expression;
typedef expression expr;
expression strtoexp(string);
expression convert(string&);
double value(expr);
class expression {
public:
    vector<pair<expr, double> > segment;
    int status;//-1 number 0 variable 1 items 2 factors 3 exponents(2) 4 single function f(x)
    string funcname;
    double value;
    char variable;
    expression();
    expression(string);
};
expression::expression() {
    status = -1;
    funcname = "";
}
expression::expression(string s) {
    (*this) = convert(s);
}
double proportion(expr thsi, expr that) {
  char i;
    srand(time(0));
    map<char, double> temp = val;
    for (i = 65; i <= 122; i++)
        val = rand() / (double) RAND_MAX;
    double k = value(thsi) / value(that);
    for (i = 65; i <= 122; i++)
        val = rand() / (double) RAND_MAX;
    if (fabs(value(thsi) / value(that) - k) > 1e-5) {
        val = temp;
        return -1e11;
    }
    val = temp;
    return k;
}
bool iszero(expr a) {
    srand(time(0));
    map<char, double> temp = val;
    for (char i = 65; i <= 122; i++)
        val = rand() / (double) RAND_MAX;
    if (fabs(value(a)) < 1e-5) {
        val = temp;
        return true;
    } else {
        val = temp;
        return false;
    }
}
bool lessthan(pair<expr, double> a, pair<expr, double> b) {
    return a.first.status > b.first.status;
}
template<class T>
int round(T x) {
    return (((ceil(double(x)) - x) > (x - floor(double(x)))) ? (floor(double(x)))
            : (ceil(double(x))));
}

template<class T>
bool isint(T x, double prs = 1e-10) {
    return (fabs(double(x - round(x))) < prs);
}
string::iterator find_bracket(const string& s, string::iterator b) {
    int i = 1;
    b++;
    while (b != s.end() && i) {
        if (*b == '(')
            i++;
        else if (*b == ')')
            i--;
        b++;
    }
    return b;
}
string::iterator rfind_bracket(const string & s, string::iterator b) {
    int i = -1;
    b--;
    while (b != s.rend().base() && i) {
        if (*b == '(')
            i++;
        else if (*b == ')')
            i--;
        b--;
    }
    return b;
}

template<class T>
T atox(string s) {
    stringstream ss;
    T ans;
    ss << s;
    ss >> ans;
    return ans;
}
template<class T>
string xtoa(T x) {
    stringstream ss;
    ss << x;
    return ss.str();
}

expression convert(string& s) {
    expression ans;
    int i;
    while ((i = s.find("pi")) != -1)
        s.replace(i, 2, "@");
    ans = strtoexp(s);
    return ans;
}

expression strtoexp(string s) {
    expression ans, temp;
    string::iterator it1, it2;
    string t, s1;
    int sign = 1;

    ans.status = 0;
    //wipe the outer bracket
    while (*s.begin() == '(' && *s.rbegin() == ')' && s.rbegin().base()
            == find_bracket(s, s.begin())) {
        s.erase(0, 1);
        s.erase(s.length() - 1, 1);
    }
    s1 = s;
    //deal with the items separated by '+' || '-'
    if (s[0] == '-')
        sign = -1, ans.status = 1;
    else if (s[0] == '+')
        
    else
        s = '+' + s;
    s += '+';

    it1 = s.begin() + 1;
    it2 = s.begin();
    while (it1 != s.end()) {
        if (*it1 == '+' || *it1 == '-') {
            if (it1 != s.end() - 1)
                ans.status = 1;
            if (!ans.status)
                break;
            t.assign(it2 + 1, it1);
            temp = strtoexp(t);
            ans.segment.push_back(pair<expr, double> (temp, sign));
            sign = (*it1 == '+' ? 1 : -1);
            it2 = it1;
        }
        if (*it1 == '(')
            it1 = find_bracket(s, it1);
        else
            it1++;
    }
    if (ans.status)
        return ans;

    //deal with the factors separated by '*' || '/'
    s = s1;
    sign = 1;
    s = '*' + s;
    s += '*';
    it1 = s.begin() + 1;
    it2 = s.begin();
    while (it1 != s.end()) {
        if (*it1 == '*' || *it1 == '/') {
            if (it1 != s.end() - 1)
                ans.status = 2;
            if (!ans.status)
                break;
            t.assign(it2 + 1, it1);
            temp = strtoexp(t);
            ans.segment.push_back(pair<expr, double> (temp, sign));
            sign = (*it1 == '*' ? 1 : -1);
            it2 = it1;
        }
        if (*it1 == '(')
            it1 = find_bracket(s, it1);
        else
            it1++;
    }
    if (ans.status)
        return ans;
    //deal with base & exponent
    s = s1;
    it1 = s.rbegin().base();
    while (it1 != s.rend().base() && *it1 != '^') {
        if (*it1 == ')')
            it1 = rfind_bracket(s, it1);
        else
            it1--;
    }
    if (it1 != s.rend().base()) {
        ans.status = 3;
        t.assign(s.begin(), it1);
        temp = strtoexp(t);
        ans.segment.push_back(pair<expr, double> (temp, 0));
        t.assign(it1 + 1, s.end());
        temp = strtoexp(t);
        ans.segment.push_back(pair<expr, double> (temp, 0));
        return ans;
    }

    //deal with function
    s = s1;
    it1 = s.begin();
    while (it1 != s.end() && *it1 != '(' && (isalnum(*it1) || *it1 == '@'))
        it1++;
    if (it1 != s.end() && *it1 == '(') {
        ans.status = 4;
        ans.funcname.assign(s.begin(), it1);
        t.assign(it1 + 1, s.end() - 1);
        temp = strtoexp(t);
        ans.segment.push_back(pair<expr, double> (temp, 0));
        return ans;
    }
    //deal with single number
    s = s1;
    if (isdigit(s[0]) || s[0] == '+' || s[0] == '-') {
        ans.status = -1;
        ans.value = atox<double> (s);
    } else {
        ans.status = 0;
        ans.variable = s[0];
    }
    return ans;
}

void output(expression exp) {
    vector<pair<expr, double> >::iterator i;
    switch (exp.status) {
    case -1:
        cout << setprecision(5) << exp.value;
        break;
    case 0:
        cout << exp.variable;
        break;
    case 1:
        cout << '(';
        for (i = exp.segment.begin(); i != exp.segment.end(); i++) {
            if (i->second > 0)
                cout << '+';
            else
                cout << '-';
            if (fabs(i->second) != 1)
                cout << fabs(i->second);
            output(i->first);
        }
        cout << ')';
        break;
    case 2:
        cout << '(';
        for (i = exp.segment.begin(); i != exp.segment.end(); i++) {
            if (i->second > 0)
                cout << '*';
            else
                cout << '/';
            output(i->first);
            if (fabs(i->second) != 1)
                cout << '^' << fabs(i->second);
        }
        cout << ')';
        break;
    case 3:
        cout << '(';
        output(exp.segment.front().first);
        cout << '^';
        output(exp.segment.back().first);
        cout << ')';
        break;
    case 4:
        cout << exp.funcname << '(';
        output(exp.segment[0].first);
        cout << ')';
    }
}

double value(expression exp) {
    double ans = 0, k;
    vector<pair<expr, double> >::iterator i;
    map<char, double>::iterator t = val.find(exp.variable);
    switch (exp.status) {
    case -1:
        return exp.value;
    case 0:
        if (t == val.end()) {
            cout << "Input " << exp.variable << ":\n";
            cin >> k;
            val[exp.variable] = k;
        }
        return val[exp.variable];
    case 1:
        for (i = exp.segment.begin(); i != exp.segment.end(); i++)
            ans += value(i->first) * i->second;
        return ans;
    case 2:
        ans = 1;
        for (i = exp.segment.begin(); i != exp.segment.end(); i++) {
            ans *= pow(value(i->first), i->second);
        }
        return ans;
    case 3:
        k = value(exp.segment[0].first);
        ans = value(exp.segment[1].first);
        if (k > 0 || k == 0 && ans != 0 || k < 0 && (isint(ans) || isint(1
                / ans)))
            return pow(k, ans);
    case 4:
        k = value(exp.segment[0].first);
        if (exp.funcname == "sin")
            return sin(k);
        else if (exp.funcname == "cos")
            return cos(k);
        else if (exp.funcname == "tan") {
            if (fabs(cos(k)) < 1e-10)
                
            return tan(k);
        } else if (exp.funcname == "cot") {
            if (fabs(sin(k)) < 1e-10)
                
            return cos(k) / sin(k);
        } else if (exp.funcname == "lg") {
            if (k <= 0)
                
            return log10(k);
        } else if (exp.funcname == "ln") {
            if (k <= 0)
                
            return log(k);
        } else if (exp.funcname.substr(0, 3) == "log") {
            ans = atox<double> (exp.funcname.substr(3, exp.funcname.length()
                    - 3));
            if (ans <= 0 || k <= 0)
                
            return log(k) / log(ans);
        } else if (exp.funcname == "arcsin") {
            if (fabs(k) > 1)
                
            return asin(k);
        } else if (exp.funcname == "arccos") {
            if (fabs(k) > 1)
                
            return acos(k);
        } else if (exp.funcname == "arctan")
            return atan(k);
    }
}

expr derivative(expression exp, char variable) {
    expr dc, prd, itm, temp;
    vector<pair<expr, double> >::iterator it1, it2;
    if (exp.status == -1)
        return expr("0");
    else if (exp.status == 0) {
        if (exp.variable == variable)
            return expr("1");
        else
            return expr("0");
    } else if (exp.status == 1) {
        dc.status = 1;
        for (it1 = exp.segment.begin(); it1 != exp.segment.end(); it1++)
            dc.segment.push_back(pair<expr, double> (derivative(it1->first,
                    variable), it1->second));
    } else if (exp.status == 2) {
        dc.status = 1;
        for (it1 = exp.segment.begin(); it1 != exp.segment.end(); it1++) {
            prd.status = 2;
            prd.segment.clear();
            for (it2 = exp.segment.begin(); it2 != it1; it2++)
                prd.segment.push_back(*it2);
            if (it1->second == 1) {
                prd.segment.push_back(pair<expr, double> (derivative(
                        it1->first, variable), it1->second));
            } else {
                prd.segment.push_back(pair<expr, double> (it1->first, -1));
                prd.segment.push_back(pair<expr, double> (it1->first, -1));
                prd.segment.push_back(pair<expr, double> (derivative(
                        it1->first, variable), it1->second));
            }
            for (it2 = it2 + 1; it2 != exp.segment.end(); it2++)
                prd.segment.push_back(*it2);
            dc.segment.push_back(pair<expr, double> (prd, it1->second));
        }
    } else if (exp.status == 3) {
        if (exp.segment[1].first.status == -1 && exp.segment[0].first.status
                == -1)
            return expr("0");
        else if (exp.segment[0].first.status == -1) {
            dc.status = 2;
            dc.segment.push_back(pair<expr, double> (expr("ln(" + xtoa(
                    exp.segment[0].first.value) + ")"), 1));
            dc.segment.push_back(pair<expr, double> (exp, 1));
            dc.segment.push_back(pair<expr, double> (derivative(
                    exp.segment[1].first, variable), 1));
        } else if (exp.segment[1].first.status == -1) {
            dc.status = 2;
            dc.segment.push_back(pair<expr, double> (exp.segment[1].first, 1));
            prd.status = 3;
            prd = expr(exp);
            prd.segment[1].first.value--;
            dc.segment.push_back(pair<expr, double> (prd, 1));
            dc.segment.push_back(pair<expr, double> (derivative(
                    exp.segment[0].first, variable), 1));
        } else {
            dc.status = 2;//product

            //No.1 f^g The same as the initial state

            dc.segment.push_back(pair<expr, double> (exp, 1));

            //No.2 g'lnf+gf'/f

            itm.status = 1;//sum


            //1. g'*lnf

            prd.status = 2;//product
            prd.segment.push_back(pair<expr, double> (derivative(
                    exp.segment[1].first, variable), 1));

            temp.status = 4;//function
            temp.funcname = "ln";
            temp.segment.push_back(exp.segment[0]);
            prd.segment.push_back(pair<expr, double> (temp, 1));

            itm.segment.push_back(pair<expr, double> (prd, 1));

            //2.gf'/f
            prd.status = 2;//product;
            prd.segment.clear();

            prd.segment.push_back(pair<expr, double> (exp.segment[1].first, 1));
            prd.segment.push_back(pair<expr, double> (derivative(
                    exp.segment[0].first, variable), 1));
            prd.segment.push_back(pair<expr, double> (exp.segment[0].first, -1));

            itm.segment.push_back(pair<expr, double> (prd, 1));

            dc.segment.push_back(pair<expr, double> (itm, 1));
        }
    } else if (exp.status == 4) {
        dc.status = 2;
        prd.status = 4;
        prd.segment.push_back(exp.segment[0]);
        if (exp.funcname == "sin") {
            prd.status = 4;
            prd.segment.push_back(exp.segment[0]);
            prd.funcname = "cos";
            dc.segment.push_back(pair<expr, double> (prd, 1));
        } else if (exp.funcname == "cos") {
            itm.status = 1;
            prd.status = 4;
            prd.segment.push_back(exp.segment[0]);
            prd.funcname = "sin";
            itm.segment.push_back(pair<expr, double> (prd, -1));
            dc.segment.push_back(pair<expr, double> (itm, 1));
        } else if (exp.funcname == "tan") {
            itm.status = 3;
            prd.status = 4;
            prd.segment.push_back(exp.segment[0]);
            prd.funcname = "cos";
            itm.segment.push_back(pair<expr, double> (prd, 0));
            itm.segment.push_back(pair<expr, double> (expr("-2"), 0));
            dc.segment.push_back(pair<expr, double> (itm, 1));
        }
        dc.segment.push_back(pair<expr, double> (derivative(
                exp.segment[0].first, variable), 1));
    }
    return dc;
}
void simplify(expr& exp) {//鍖栫畝
    int i, j;
    double k;
    expr t;
    switch (exp.status) {
    case 1://瀵逛簬鍔犳硶
        for (i = 0; i < exp.segment.size(); i++) {
            simplify(exp.segment.first);//閫掑綊鍖栫畝
        }
        for (i = 0; i < exp.segment.size(); i++)//鎶婂瓙琛ㄨ揪寮忎腑鐨勫姞褰掑苟杩涙潵
            if (exp.segment.first.status == 1) {
                for (j = 0; j < exp.segment.first.segment.size(); j++)
                    exp.segment.push_back(exp.segment.first.segment[j]);
                exp.segment.erase(exp.segment.begin() + i);
                i--;
            }
        sort(exp.segment.begin(), exp.segment.end(), lessthan);//鎺掑簭(鏁板瓧鍦ㄦ渶鍚?
        for (i = 0; i < exp.segment.size(); i++)//鎶婃瘡椤圭殑绯绘暟鎻愬彇鍑烘潵
            if (exp.segment.first.status == 2
                    && exp.segment.first.segment.back().first.status == -1) {
                exp.segment.second
                        *= exp.segment.first.segment.back().first.value;
                exp.segment.first.segment.pop_back();
            }
        //Merging
        for (i = 0; i < exp.segment.size(); i++) {
            if (iszero(exp.segment.first)) {
                exp.segment.erase(exp.segment.begin() + i);
                i--;
                continue;
            }
            for (j = i + 1; j < exp.segment.size(); j++) {
                if (iszero(exp.segment[j].first)) {
                    exp.segment.erase(exp.segment.begin() + j);
                    j--;
                    continue;
                }
                k = proportion(exp.segment[j].first, exp.segment.first);
                cout<<"k:"<<k<<endl;
                if (k < -1e10)
                    continue;
                exp.segment.second += k * exp.segment[j].second;
                exp.segment.erase(exp.segment.begin() + j);
                j--;
            }
            if(exp.segment.first.status==-1)
                exp.segment.first.value*=fabs(exp.segment.second),
                exp.segment.second/=fabs(exp.segment.second);
        }
        //鑻ュ彧鏈変竴椤?涓旀槸鏁板瓧鍒欏彧闃惰瘽瑙?
        if (exp.segment.size() == 1 && exp.segment[0].first.status == -1) {
            exp.status = -1;
            exp.value = exp.segment[0].first.value * exp.segment[0].second;
            return;
        }
        //鑻ュ彧鏈変竴椤逛笖绯绘暟涓?鍒欑洿鎺ュ寲鍑?
        if (exp.segment.size() == 1 && fabs(exp.segment[0].second - 1) < 1e-10) {
            t = exp.segment[0].first;
            exp = t;
            return;
        }
        if (exp.segment.size() == 0) {
            exp = expr("0");
            return;
        }

        break;

    case 2:
        for (i = 0; i < exp.segment.size(); i++)
            simplify(exp.segment.first);//閫掑綊鍖栫畝
        output(exp);
        cout << endl;
        for (i = 0; i < exp.segment.size(); i++)
            if (exp.segment.first.status == 2) {//鍚堝苟
                for (j = 0; j < exp.segment.first.segment.size(); j++)
                    exp.segment.push_back(exp.segment.first.segment[j]);
                exp.segment.erase(exp.segment.begin() + i);
                i--;
            }

        sort(exp.segment.begin(), exp.segment.end(), lessthan);//鎺掔华
        output(exp);
        cout << endl;
        for (i = 0; i < exp.segment.size(); i++)//鎶婂悇涓?箓鐨勬?鏁版彁鍙栧嚭鏉?
            if (exp.segment.first.status == 3
                    && exp.segment.first.segment.back().first.status == -1) {
                exp.segment.second
                        *= exp.segment.first.segment.back().first.value;
                exp.segment.first.segment.pop_back();
                exp.segment.first.status = 2;
                exp.segment.first.segment[0].second = 1;
                simplify(exp.segment.first);
            }

        output(exp);
        cout << 'r' << endl;
        k = 1;//鍚堝苟涓轰笉浣嗘兂涔﹀瓙
        while (exp.segment.size() > 0 && exp.segment.back().first.status == -1) {
            k *= pow(exp.segment.back().first.value, exp.segment.back().second);
            exp.segment.pop_back();
        }
        if (fabs(k) < 1e-10) {
            exp = expr("0");
            return;
        } else if (fabs(k - 1) > 1e-10) {
            t.status = -1;
            t.value = k;
            exp.segment.push_back(pair<expr, double> (t, 1));
        }
        if (exp.segment.size() == 1 && fabs(exp.segment[0].second - 1) < 1e-10) {
            exp.status--;
            simplify(exp);
            return;
        }
        output(exp);
        cout << "mul" << endl;
        cout << exp.segment.size() << endl;
        if (exp.segment.size() == 0) {
            exp = expr("1");
            return;
        }
        break;
    case 3:
        simplify(exp.segment[0].first);
        simplify(exp.segment[1].first);
    case 4:
        simplify(exp.segment[0].first);
        if (exp.funcname == "ln") {
            if (exp.segment[0].first.status == 0
                    && exp.segment[0].first.variable == 'e') {
                exp = expr("1");
                return;
            }
            if (exp.segment[0].first.status == 3
                    && exp.segment[0].first.segment[0].first.status == 0
                    && exp.segment[0].first.segment[0].first.variable == 'e') {
                exp = exp.segment[0].first.segment[1].first;
                return;
            }
        }
    }
}



THE END
哦还有我是在Ubuntu里用GCC编译的
用其他编译器的朋友可能遇到编译问题
+1  学术分    novakon   2010-09-26   自主学习
+500  科创币    joyeep   2010-09-28   希望重构,把解析功能进行封装,参考Vistor模式。算法还能继续精简,容器的大量应用,降低了代码优雅。
来自 软件综合
 
2010-9-24 14:22:52
caoyuan9642(作者)
1楼
第一行是函数,最后一行是导数
效果图
55625

55626

55627
55628

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2楼
似乎有些代码被解析成WindCode而变形了,想用坛子里的code标签修饰一下,但是失败了

于是。。。麻烦LZ打包成附件上传吧
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caoyuan9642(作者)
3楼
注释全部杯具掉了。。。
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4楼
diff(expression_string) 搞定
matlab爽~
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2010-9-26 11:43:44
2010-9-26 11:43:44
caoyuan9642(作者)
5楼
话说我又不是不知道有软件可以用。。用mathematica的飘过。。
我只是想自己实现实现底层代码罢了
练习编程技术~~
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2010-9-28 10:42:31
2010-9-28 10:42:31
6楼
在算法使用STL 并不是个好方法,容器会管理内存,需要消耗一定的资源,对于纯算法而言,最好使用数组。

LZ 没有大量运用STL 的算法,却使用了大量的容器。在后期的学习中要加强对STL 的研究。

(小提示: 函数对象,仿函数等,对编写算法非常有利)

另外,这个程序里面OPP思想也没有体现出来,单纯的面对过程的方式。

如果,这个程序仔细琢磨,重构,封装,是非常容易编写成为一个通过的算法库的。

另外,建议LZ买本数值计算的书研究下。
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2010-9-29 15:42:41
2010-9-29 15:42:41
caoyuan9642(作者)
7楼
谢谢LS提出改进意见!
我会慢慢改进的
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