diff --git a/contfrac.cc b/contfrac.cc
index 6c7eed1..565820f 100644
--- a/contfrac.cc
+++ b/contfrac.cc
@@ -20,6 +20,7 @@
 #include <stdio.h>
 #include <string.h>
 #include <math.h>
+#include <list>
 
 
 namespace LA {
@@ -132,6 +133,67 @@ for(int i=1; i<=n; ++i) //truncate if possible
 }
 
 
+template <typename T>
+ContFrac<T> Homographic<T>::value(const ContFrac<T>&x) const
+{
+Homographic<T> h(*this);
+
+std::list<T> l;
+for(int i=0; i<=x.length()+1; ++i) //scan all input
+	{
+	//digest next input term
+	Homographic<T> hnew;
+	if(i>x.length()||i!=0&&x[i]==0) //input is infinity
+		{
+		hnew.x[0][0]=hnew.x[0][1]=h.x[0][1];
+                hnew.x[1][0]=hnew.x[1][1]=h.x[1][1];
+		}
+	else
+		{
+		hnew.x[0][0]=h.x[0][1];
+		hnew.x[1][0]=h.x[1][1];
+		hnew.x[0][1]=h.x[0][0]+h.x[0][1]*x[i];
+        	hnew.x[1][1]=h.x[1][0]+h.x[1][1]*x[i];
+		}
+
+	//std::cout<<"hnew\n"<< hnew.x[0][0]<<" "<< hnew.x[0][1]<<"\n"<< hnew.x[1][0]<<" "<< hnew.x[1][1]<<"\n";
+
+	//check if we are ready to output
+	bool inf=0;	
+	T q0,q1;
+	if(hnew.x[1][0]==0) inf=1; else q0=hnew.x[0][0]/hnew.x[1][0];
+	if(hnew.x[1][1]==0) inf=1; else q1=hnew.x[0][1]/hnew.x[1][1];
+	while(!inf && q0==q1) //ready to output
+		{
+		l.push_back(q0);
+		//std::cout <<"out "<<q0<<std::endl;	
+		Homographic<T> hnew2;
+		hnew2.x[0][0]=hnew.x[1][0];	
+		hnew2.x[0][1]=hnew.x[1][1];	
+		hnew2.x[1][0]=hnew.x[0][0]-hnew.x[1][0]*q0;
+		hnew2.x[1][1]=hnew.x[0][1]-hnew.x[1][1]*q0;
+		//std::cout<<"hnew2\n"<< hnew2.x[0][0]<<" "<< hnew2.x[0][1]<<"\n"<< hnew2.x[1][0]<<" "<< hnew2.x[1][1]<<"\n";
+		hnew=hnew2;
+		inf=0;
+		if(hnew.x[1][0]==0) inf=1; else q0=hnew.x[0][0]/hnew.x[1][0];
+        	if(hnew.x[1][1]==0) inf=1; else q1=hnew.x[0][1]/hnew.x[1][1];
+		}
+	//termination
+	if(hnew.x[1][0]==0&&hnew.x[1][1]==0) //terminate
+		{
+		//std::cout<<"terminate at "<<i<<"\n";
+		if(i<=x.length()) laerror("unexpected termination in Homographic::value");
+		break;
+		}
+	h=hnew;
+	}
+//std::cout <<"size= "<<l.size()<<std::endl;
+return ContFrac<T>(l);
+}
+
+
+
+
 /***************************************************************************//**
  * forced instantization in the corresponding object file
  ******************************************************************************/
@@ -142,6 +204,22 @@ template class ContFrac<unsigned long>;
 template class ContFrac<long long>;
 template class ContFrac<unsigned long long>;
 
+template class Homographic<int>;
+template class Homographic<unsigned int>;
+template class Homographic<long>;
+template class Homographic<unsigned long>;
+template class Homographic<long long>;
+template class Homographic<unsigned long long>;
+
+template class BiHomographic<int>;
+template class BiHomographic<unsigned int>;
+template class BiHomographic<long>;
+template class BiHomographic<unsigned long>;
+template class BiHomographic<long long>;
+template class BiHomographic<unsigned long long>;
+
+
+
 
 #define INSTANTIZE(T) \
 
diff --git a/contfrac.h b/contfrac.h
index c870b5b..cfd038f 100644
--- a/contfrac.h
+++ b/contfrac.h
@@ -29,7 +29,12 @@ namespace LA {
 //NOTE: 0 on any position >0 means actually infinity; simplify()  shortens the vector
 //presently implements just conversion to/from rationals and floats
 //maybe implement arithmetic by Gosper's method cf. https://perl.plover.com/classes/cftalk/TALK
+//
 
+template <typename T>
+class ContFrac;
+
+//@@@basic rational arithmetics
 template <typename T>
 class Rational {
 public:
@@ -37,8 +42,26 @@ public:
 	T den;
 
 	Rational(const T p, const T q) : num(p),den(q) {};
+	explicit Rational(const T (&a)[2]) :num(a[0]), den(a[1]) {};
+	Rational(const ContFrac<T> &cf) {cf.convergent(&num,&den);};
 };
 
+template <typename T>
+std::ostream & operator<<(std::ostream &s, const Rational<T> &x) 
+{
+s<<x.num<<"/"<<x.den;
+return s;
+}
+
+template <typename T>
+class Homographic;
+
+template <typename T>
+class BiHomographic;
+
+
+//@@@implement iterator and rewrite  Homographic<T>::value
+
 template <typename T>
 class ContFrac : public NRVec<T> {
 private:
@@ -50,7 +73,7 @@ public:
 	ContFrac(const int n) : NRVec<T>(n+1) {};
 	ContFrac(double x, const int n, const T thres=0); //might yield a non-canonical form
 	ContFrac(const T p, const T q); //should yield a canonical form
-	ContFrac(const Rational<T> r) : ContFrac(r.num,r.den) {};
+	ContFrac(const Rational<T> &r) : ContFrac(r.num,r.den) {};
 
 	void canonicalize();
 	void convergent(T *p, T*q, const int trunc= -1) const;
@@ -64,9 +87,39 @@ public:
 		NRVec<T>::resize(n+1,preserve);
 		if(preserve) for(int i=nold+1; i<=n;++i) (*this)[i]=0;
 		} 
+
+	ContFrac operator+(const Rational<T> &rhs) const {Homographic<T> h({{rhs.num,rhs.den},{rhs.den,0}}); return h.value(*this);};
+	ContFrac operator-(const Rational<T> &rhs) const {Homographic<T> h({{-rhs.num,rhs.den},{rhs.den,0}}); return h.value(*this);};
+	ContFrac operator*(const Rational<T> &rhs) const {Homographic<T> h({{0,rhs.num},{rhs.den,0}}); return h.value(*this);};
+	ContFrac operator/(const Rational<T> &rhs) const {Homographic<T> h({{0,rhs.den},{rhs.num,0}}); return h.value(*this);};
+
 	
 };
 
+//for Gosper's arithmetic
+
+template <typename T>
+class Homographic {
+public:
+T x[2][2]; //{{a,b},{c,d}} for (a+b.z)/(c+d.z)
+
+	Homographic(){};
+	explicit Homographic(const T (&a)[2][2]) {memcpy(x,a,2*2*sizeof(T));};
+	ContFrac<T> value(const ContFrac<T>&x) const;
+	
+};
+
+
+template <typename T>
+class BiHomographic {
+public:
+T x[2][2][2];
+	
+	BiHomographic(){};
+	explicit BiHomographic(const T (&a)[2][2][2]) {memcpy(x,a,2*2*2*sizeof(T));};
+	ContFrac<T> value(const ContFrac<T>&x, const ContFrac<T>&y) const;
+};
+
 
 
 }//namespace
diff --git a/permutation.cc b/permutation.cc
index 4b0c4a7..b83b455 100644
--- a/permutation.cc
+++ b/permutation.cc
@@ -19,6 +19,7 @@
 #include "permutation.h"
 #include <stdio.h>
 #include <string.h>
+#include <list>
 
 
 namespace LA {
diff --git a/t.cc b/t.cc
index fc48b46..d9d61b0 100644
--- a/t.cc
+++ b/t.cc
@@ -2431,7 +2431,7 @@ NRMat<double> mm=m.permuted_rows(p);
 cout <<mm;
 }
 
-if(1)
+if(0)
 {
 double x;
 cin >>x;
@@ -2463,5 +2463,15 @@ double zzz=zz.value();
 cout <<z<<" "<<zzz<<endl;
 }
 
+if(1)
+{
+Rational<int> r({11,101});
+ContFrac<int> x(r);
+ContFrac<int> y= x+Rational<int>({2,3});
+cout<<Rational<int>(y)<<endl;
+ContFrac<int> z= x*Rational<int>({2,3});
+cout<<Rational<int>(z)<<endl;
+}
+
 
 }
diff --git a/vec.cc b/vec.cc
index 9a38566..1e46a9a 100644
--- a/vec.cc
+++ b/vec.cc
@@ -899,9 +899,8 @@ return -1;
 
 
 template<typename T>
-NRVec<T>::NRVec(const std::list<T> l)
+NRVec<T>::NRVec(const std::list<T> l) : NRVec<T>(l.size())
 {
-resize(l.size());
 int ii=0;
 for(typename std::list<T>::const_iterator i=l.begin(); i!=l.end(); ++i) (*this)[ii++] = *i;
 }