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SingularityComputation
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  1. 8
      code/a.out
  2. 7
      code/a1.out
  3. BIN
      code/main
  4. 427
      code/main.cpp
  5. 56
      code/main2.cpp
  6. 93
      code/main3.cpp
  7. 37
      code/main4.cpp
  8. 7
      code/run.sh
  9. 130
      visualization/Coneplot.mw
  10. 160
      visualization/Cusp sphere plot.mw
  11. 238
      visualization/Enneper surface plot.mw
  12. 138
      visualization/Riemann surface plot.mw

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t*(-1+s^2)
2*s*t
t*(1+s^2)
1+s^2
2*t*(1+s^2)
...........
t
1+s^2

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#include <iostream>
#include <ginac/ginac.h>
#include <vector>
#include <cstdlib>
#include <ctime>
#include <chrono>
#include <cstdio>
using namespace std;
using namespace GiNaC;
const symbol & get_symbol(const string & s)
{
static map<string, symbol> directory;
map<string, symbol>::iterator i = directory.find(s);
if (i != directory.end())
return i->second;
else
return directory.insert(make_pair(s, symbol(s))).first->second;
}
// #define CREATE_VARIABLE_NAME(index) a##index
symbol x("x"), y("y"), z("z"),w("w");
symbol s("s"), t("t");
symbol a("a"), b("b"), c("c"), d("d");
static int deg;
vector<ex> f(4);
int get_idx(int i,int j){
return i*deg+j;
}
//组合数来生成计算minor
void generateCombination(vector<int>& candidates, int start, int p, vector<int>& combination, vector<vector<int>>& result) {
if (p == 0) {
result.push_back(combination);
return;
}
for (int i = start; i < candidates.size(); ++i) {
combination.push_back(candidates[i]);
generateCombination(candidates, i + 1, p - 1, combination, result);
combination.pop_back();
}
}
vector<vector<int> > getCombinations(int k, int p) {
vector<vector<int> > result;
vector<int> candidates;
for (int i = 0; i < k; ++i) {
candidates.push_back(i);
}
vector<int> combination;
generateCombination(candidates, 0, p - 1, combination, result);
return result;
}
matrix random_matrix(int n,int m,int k){
matrix cur(n,m);
int rem;//
if(k>10)
rem=2;
else
rem=10;
for(int i=0;i<n;i++)
for(int j=0;j<m;j++)
cur(i,j)=rand()%rem;
return cur;
}
ex KKS(matrix N,int p,int k)
{
srand(time(0));
matrix U1=random_matrix(p-1,p,k);
matrix U2=random_matrix(p-1,p,k);
matrix V1=random_matrix(k,p-1,k);
matrix V2=random_matrix(k,p-1,k);
matrix curU1=U1.mul(N);
matrix curU2=U2.mul(N);
matrix resU1=curU1.mul(V1);
matrix resU2=curU2.mul(V2);
// cout<<resU1<<endl;
// cout<<"....."<<endl;
// cout<<resU2<<endl;
// return 1;
ex g1=determinant(resU1);
// cout<<"123"<<endl;
ex g2=determinant(resU2);
return gcd(g1,g2);
}
vector<ex> factor_parser(ex f,parser &reader){
vector<ex> ans;
if(!is_a<mul>(f))
{
ans.push_back(f);
return ans;
}
int cnt=0;
string cur="";
int i=0;
stringstream ss;
ss<<f;
string curs;
getline(ss,curs);
while(i<curs.size()){// * ( )
if(curs[i]=='*'){
if(cnt==0)
{
ans.push_back(reader(cur));
cur="";
}
else
cur+=curs[i];
}
else if(curs[i]=='('){
cur+=curs[i];
cnt++;
}
else if(curs[i]==')'){
cur+=curs[i];
cnt--;
}
else
cur+=curs[i];
i++;
}
ans.push_back(reader(cur));
return ans;
}
void checktime(auto &st){
auto ed=std::chrono::high_resolution_clock::now();
auto duration = std::chrono::duration_cast<std::chrono::milliseconds>(ed-st);
st=ed;
std::cout << duration.count() << " ms" << std::endl;
}
int main()
{
freopen("a.out","w",stdout);
auto st = std::chrono::high_resolution_clock::now();
symtab table;
table["s"] = s;
table["t"] = t;
parser reader(table);
// random
//Ex 4.6
//Cone
f[0]=reader("-t*(-s^2 + 1)");
f[1]=reader("2*s*t");
f[2]=reader(" t*(s^2 + 1)");
f[3]=reader(" s^2 + 1");
//Ex 4.2
// f[0]=reader("(2*t^2 - 6*t + 4)*s^2 + 4*(t - 1)*s + 1");
// f[1]=reader(" -2*(t^2 - 3*t + 1)*s^2 + (4*t - 2)*s");
// f[2]=reader(" (t^2 - 4*t + 3)*s^2 - 2*(t - 1)^2 * s");
// f[3]=1;
// f[0]=(2*pow(t,2)-6*t+4)*pow(s,2)+4*(t-1)*s+1;
// f[1]=-2*(pow(t,2)-3*t+1)*pow(s,2)+(4*t-2)*s;
// f[2]=(pow(t,2)-4*t+3)*pow(s,2)-2*pow((t-1),2)*s;
// f[3]=1;
// Ex 6.1 Correct!
// f[0]=s+3+t;
// f[1]=1+t*s*s+t;
// f[2]=s*s-3*s+2*s*t+1;
// f[3]=s+s*t+3*t;
//Ex 6.2 Correct!
// f[0]=s*s+t;
// f[1]=t*(s*s+1)+s*s+1;
// f[2]=-s*s*s+2*s*s*t+s*s+1;
// f[3]=t*(s*s*s+3)+1;
//Ex 6.3
// f[0]=-pow(s,3)*(t*t-1);
// f[1]=(s+2)*s*t;
// f[2]=-pow(t,3)*(pow(s,2)-4);
// f[3]=pow(s,3);
//Ex 6.4
// f[0]=reader("4*s^3 + s*t^2 + 4*s^2 - 12*s*t + t^2 + s + 1");
// f[1]=reader("4*s^4 + s^2*t^2 + s^2 + 6*t");
// f[2]=reader("6*t^2");
// f[3]=reader("4*s^2 + t^2 + 1");
//Ex 6.5
// f[0]=reader("4*s^3 + s*t^2 + 4*s^2 - 12*s*t + t^2 + s + 1");
// f[1]=reader("4*s^4 + s^2*t^2 + s^2 + 6*t");
// f[2]=reader("6*t^2");
// f[3]=reader("4*s^2 + t^2 + 1");
for(int i=0;i<4;i++)
cout<<f[i]<<endl;
cout<<".........."<<endl;
int d1=-10,d2=-10;
for(int i=0;i<f.size();i++){
d1=max(d1,f[i].degree(s));
d2=max(d2,f[i].degree(t));
}
// cout<<d1<<" "<<d2<<endl;
int v1,v2;
if(d1>=d2)//保证ex4.2是3，1，理论上应该是>=
v1=d1-1,v2=2*d2-1;//s v1,t v2
else
v1=2*d1-1,v2=d2-1;
// cout<<v1<<' '<<v2<<endl;
deg=2*d1*d2;
//basefunc of moving planes, also matrix
vector<ex> basefunc;
//O(2*d1*d2)
for(int i=0;i<=v2;i++)
{
for(int j=0;j<=v1;j++){
basefunc.push_back(pow(s,j)*pow(t,i));
}
}
// for(int i=0;i<deg;i++)
// cout<<basefunc[i]<<endl;
vector<ex> matbasefunc;//存储乘后的结果，用来解线性方程组
lst matbasefunc_lst;
//O((d2+v2)*(d2+v2)) 近似于 O(d1*d2)
for(int i=0;i<=(d2+v2);i++)
for(int j=0;j<=(d1+v1);j++)
{
matbasefunc.push_back(pow(s,j)*pow(t,i));
matbasefunc_lst.append(pow(s,j)*pow(t,i));
}
//生成用来求解的未知数变量
//O(d1*d2)
vector<ex> matvar;
for(int i=0;i<4;i++)
for(int j=0;j<2*d1*d2;j++){
string cur="a";
cur=cur+to_string(get_idx(i,j));
matvar.push_back(get_symbol(cur));
}
lst eqns={};
lst vars={};
//添加到vars当中
for(int i=0;i<matvar.size();i++)
vars.append(matvar[i]);
vector<ex> l;
for(int i=0;i<4;i++)
{
ex cur=0;
for(int j=0;j<deg;j++)
{
int idx=get_idx(i,j);
cur=cur+matvar[idx]*basefunc[j];
}
l.push_back(cur);
}
//多维的分离系数存在问题，不能直接使用coeff
ex mulres=0;
for(int i=0;i<4;i++)
mulres=mulres+l[i]*f[i];
mulres=expand(mulres);
// cout<<mulres<<endl;
mulres.collect(matbasefunc_lst,true);//collect也不好用，并不能按照规定合并，合并之后的结果包含a1*s*t+a2*s*t
//因此需要使用map累加所有的同类项
// cout<<mulres<<endl;
int cnt=0;
map<ex,ex,ex_is_less> mp;//ex不支持比较，声明map的时候必须像这样声明
for (const auto& term : mulres) {//遍历多项式中的每一项
ex cur;
//coeff无法求两个symbol对应的度数，因此需要用除法来求解
int curd1=term.degree(s),curd2=term.degree(t);
ex pl=pow(s,curd1)*pow(t,curd2);
cur=quo(term,pl,x);//计算商
if(mp.find(pl)!=mp.end())
mp[pl]=mp[pl]+cur;
else
mp[pl]=cur;
// cout<<cur<<"......"<<term<<endl;
// eqns.append(cur==0);
}
for(auto &i:mp){
eqns.append(i.second==0);
// cout<<i.second<<endl;
// cnt++;
}
// cout<<cnt<<endl;
// checktime(st);
ex ans=lsolve(eqns,vars);//存储答案
// checktime(st);
// cout<<ans<<endl;
//判断自由变量
vector<ex> freevar;
for (int i=0;i<matvar.size();i++)
{
if(lhs(ans[i])==rhs(ans[i]))
freevar.push_back(lhs(ans[i]));
}
//计算这一向量空间的基，也就是计算无穷组解组成的向量空间的基
//最多有 自由变量+1个基（1是自由变量全0的时候，不过有可能自由变量全0的时候解为0向量，需要特判）
lst cureqns=eqns;
for(int j=0;j<freevar.size();j++)
{
cureqns.append(freevar[j]==0);
}
ex curans=lsolve(cureqns,vars);
int flag=0;
for(int i=0;i<matvar.size();i++){
if(rhs(curans[i])!=0)
flag=1;
}
// checktime(st);
int p=2*d1*d2,k;//p*k维矩阵
//特判k的大小
if(flag==1)
{
k=freevar.size()+1;
}
else
k=freevar.size();
matrix M(p,k);//全0,从0开始计算idx
matrix N(p,k);//直接求，不做替换
// cout<<M<<endl;
//如果为1，则要加入答案当中
vector<ex> ff;
ff.push_back(x);
ff.push_back(y);
ff.push_back(z);
ff.push_back(w);
if(flag==1)
{
for(int i=0;i<4;i++)
{
for(int j=0;j<2*d1*d2;j++)
{
ex varans=rhs(curans[get_idx(i,j)]);
if(varans!=0)//取出答案
{
M(j,p-1)=M(j,p-1)+varans*ff[i];
N(j,p-1)=N(j,p-1)+varans*f[i];
}
}
}
}
// cout<<p<<" "<<k<<endl;
for(int i=0;i<freevar.size();i++){
lst cureqns=eqns;
for(int j=0;j<freevar.size();j++)
{
if(j!=i)
cureqns.append(freevar[j]==0);
}
cureqns.append(freevar[i]==1);
ex curans=lsolve(cureqns,vars);
for(int ii=0;ii<4;ii++)
{
for(int j=0;j<2*d1*d2;j++)
{
ex varans=rhs(curans[get_idx(ii,j)]);
if(varans!=0)//取出答案
{
N(j,i)=N(j,i)+varans*f[ii];
M(j,i)=M(j,i)+varans*ff[ii];
}
}
}
}
// checktime(st);
int testflag;
testflag=0;
if(testflag)
{ ex H=0;
vector<ex> ni;//用于存储所有可能的minor
//计算H，相当于计算矩阵所有可能的p-1 *p-1维度的minor，然后求解GCD
//先递归找到所有的列组合
vector<vector<int> > combinations = getCombinations(k, p);
// matrix calmat=ex_to<matrix>(sub_matrix(N,0,p-2,0,p-2));
// cout<<determinant(calmat)<<endl;
for(int i=0;i<p;i++)
{
for(auto &nums:combinations)
{
//由于不支持选取不连续的几行几列，并且reduce去除矩阵的某行某列时必须同时提供行列
//先构建一个p*p的矩阵，p行是为了构建完矩阵后删除某一行，因此需要对应的多加一列
//因为最后一列是全0列，直接删就行
matrix curmat(p,p);
for(int j=0;j<nums.size();j++)
{
int col=nums[j];//选取出某一列
for(int k=0;k<p;k++){
curmat(k,j)=N(k,col);
}
}
matrix calmat=ex_to<matrix>(reduced_matrix(curmat,i,p-1));
ni.push_back(determinant(calmat));
}
cout<<i<<endl;
}
//求解GCD,即H
for(int i=0;i<ni.size();i++)
{
if(i==0)
H=ni[0];
else
H=gcd(H,ni[i]);
}
H=expand(H);
//对H因式分解
cout<<H<<endl;
cout<<"..........."<<endl;
ex curh=factor(H);
cout<<curh<<endl;
cout<<"-------------"<<endl;
}
//尝试先不带入xyzw的表达式，来加速,但是发现带入后计算的更快？？可能存在消项
//因为在计算GCD的时候先带入表达式再计算GCD和先计算GCD再带入表达式求出结果完全一样
ex quickH=KKS(N,p,k);
quickH=expand(quickH);
quickH=factor(quickH);
// lst factors=ex_to<lst>(factor(quickH));
cout<<quickH<<endl;
cout<<"..........."<<endl;
// checktime(st);
vector<ex> h=factor_parser(quickH,reader);
for(auto &i:h)
if(!is_a<numeric>(i))//skip numeric
cout<<i<<endl;
//ALG2
vector<int> order;
for(auto &i:h){//处理每一个h
int s_deg=i.degree(s);
int t_deg=i.degree(t);
}
return 0;
}

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#include <iostream>
#include <ginac/ginac.h>
#include <chrono>
using namespace std;
using namespace GiNaC;
vector<ex> factor_parser(ex f){
vector<ex> ans;
int cnt=0;
string cur="";
return ans;
}
int main() {
symbol s("s");
symbol t("t"); // Declare a GiNaC symbol 'x'
symtab table;
table["s"] = s;
table["t"] = t;
parser reader(table);
ex e=reader("4*s^4 + s^2*t^2 + s^2 + 6*t");
matrix A(17,17);
for(int i=0;i<17;i++)
for(int j=0;j<17;j++)
A(i,j)=rand()%5-2;
// cout<<determinant(A)<<endl;
lst t1={s==0.1,t==0.1};
double sss=ex_to<numeric>(subs(e,t1)).to_double();
cout<<sss<<endl;
// cout<<ex_to<double>(subs(e,t1))<<endl;
// 获取当前时间点，作为程序开始运行的时间
// auto start_time = std::chrono::high_resolution_clock::now();
// // 模拟程序运行，这里用一个简单的循环做示例
// for (int i = 0; i < 10000000; ++i) {
// // do something
// }
// // 获取当前时间点，作为程序结束运行的时间
// auto end_time = std::chrono::high_resolution_clock::now();
// // 计算程序运行时间
// auto duration = std::chrono::duration_cast<std::chrono::milliseconds>(end_time - start_time);
// // 输出程序运行时间（以秒为单位）
// std::cout << "程序运行时间：" << duration.count() << " 秒" << std::endl;
// return 0;
// cout<<e<<endl;
// ex curh=quo(e,s*s+4*s-4,s);
// ex r=rem(e,s*s+4*s-4,s);
// cout<<e<<endl;
// cout<<r<<endl;
// curh=expand(curh);
// cout<<curh<<endl;
}

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code/main3.cpp
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#include <iostream>
#include <vector>
#include <ginac/ginac.h>
#include<string>
#include<sstream>
#include<cstdio>
using namespace std;
using namespace GiNaC;
symbol s("s");
symbol t("t");
vector<ex> factor_parser(ex f,parser &reader){
vector<ex> ans;
int cnt=0;
string cur="";
int i=0;
stringstream ss;
ss<<f;
string curs;
getline(ss,curs);
while(i<curs.size()){// * ( )
if(curs[i]=='*'){
if(cnt==0)
{
ans.push_back(reader(cur));
cur="";
}
else
cur+=curs[i];
}
else if(curs[i]=='('){
cur+=curs[i];
cnt++;
}
else if(curs[i]==')'){
cur+=curs[i];
cnt--;
}
else
cur+=curs[i];
i++;
}
ans.push_back(reader(cur));
return ans;
}
int main() {
// symbol s("s");
// symbol t("t");
symtab table;
table["s"] = s;
table["t"] = t;
parser reader(table);
ex expression = reader("1/4*(33+t^4*(1+26*s^3+18*s^2+5*s^6+s^4-2*s^5)+14*s^3-98*s^2+2*(-7+4*s^3+8*s^2-17*s^6+25*s^4-6*s^5)*t^2-8*(-2+8*s^3+8*s^2+s^6)*t^3-51*s^6+8*t*(-6+18*s^2+11*s^6-14*s^4+6*s^5)+89*s^4-50*s^5)*s^2");
// vector<ex> ans=factor_parser(expression,reader);
// cout<<ans.size()<<endl;
// for(auto i:ans)
// cout<<i<<endl;
cout<<ex_to<lst>(expression)[1];
// expression=factor(expression);
// stringstream ss;
// ss<<expression;
// string curs;
// getline(ss,curs);
// cout<<curs<<endl;
// for(int i=0;i<curs.size();i++)
// {
// }
// lst factors=ex_to<lst>(factor(expression));
// cout<<ex_to<lst>(factor(expression)).nops()<<endl;
// for(auto &i:ex_to<lst>(factor(expression)))
// cout<<i<<endl;
// const lst &factors = ex_to<lst>(expression);
// if (is_a<mul>(expression)) {
// const lst &factors = ex_to<lst>(expression);
// vector<ex> factors_vector;
// for (const auto &factor : factors) {
// factors_vector.push_back(factor);
// }
// cout << "Factors:" << endl;
// for (const auto &factor : factors_vector) {
// cout << factor << endl;
// }
// } else {
// // The expression is not a product, so it's already factored or a single term.
// cout << "The expression is not factored or is a single term." << endl;
// }
return 0;
}

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code/main4.cpp
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#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
// 递归函数，将表达式中的所有子项存储到列表中
void collect_sub_expressions(const ex& expr, lst& sub_expressions) {
if (is_a<add>(expr) || is_a<mul>(expr)) {
for (const auto& op : expr) {
collect_sub_expressions(op, sub_expressions);
}
} else {
sub_expressions.append(expr);
}
}
int main() {
int cnt=0;
for(double i=-10;i<=10;i+=0.1)
for(double j=-10;j<=10;j=j+0.1)
{
// lst t1={s==i,t==j};
// double curw=ex_to<numeric>(subs(f[3],t1)).to_double();
// if(curw>=-0.01&&curw<=0.01)
// continue;
// double curx=ex_to<numeric>(subs(f[0],t1)).to_double();
// double cury=ex_to<numeric>(subs(f[1],t1)).to_double();
// double curz=ex_to<numeric>(subs(f[2],t1)).to_double();
// points(cnt,0)=curx/curw;
// points(cnt,1)=cury/curw;
// points(cnt,2)=curz/curw;
cnt++;
}
cout<<cnt<<endl;
return 0;
}

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g++ main.cpp -o main -lginac -lcln
# if [ $? -eq 0 ]; then
# # echo "编译成功，运行程序..."
# ./main
# else
# echo "Compliation fail"
# fi

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visualization/Coneplot.mw
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visualization/Cusp sphere plot.mw
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visualization/Enneper surface plot.mw
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visualization/Riemann surface plot.mw
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