Continued fraction/Arithmetic/G(matrix ng, continued fraction n)

This task investigates mathmatical operations that can be performed on a single continued fraction. This requires only a baby version of NG:

{\begin{bmatrix}a_{1}&a\\b_{1}&b\end{bmatrix}}

I may perform perform the following operations:

Input the next term of N₁

Output a term of the continued fraction resulting from the operation.

I output a term if the integer parts of ${\frac {a}{b}}$ and ${\frac {a_{1}}{b_{1}}}$ are equal. Otherwise I input a term from N. If I need a term from N but N has no more terms I inject $\infty$ .

When I input a term t my internal state: ${\begin{bmatrix}a_{1}&a\\b_{1}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}a+a_{1}*t&a_{1}\\b+b_{1}*t&b_{1}\end{bmatrix}}$

When I output a term t my internal state: ${\begin{bmatrix}a_{1}&a\\b_{1}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}b_{1}&b\\a_{1}-b_{1}*t&a-b*t\end{bmatrix}}$

When I need a term t but there are no more my internal state: ${\begin{bmatrix}a_{1}&a\\b_{1}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}a_{1}&a_{1}\\b_{1}&b_{1}\end{bmatrix}}$

I am done when b₁ and b are zero.

Demonstrate your solution by calculating:

[1;5,2] + 1/2

[3;7] + 1/2

[3;7] divided by 4

Using a generator for ${\sqrt {2}}$ (e.g., from Continued fraction) calculate ${\frac {1}{\sqrt {2}}}$ . You are now at the starting line for using Continued Fractions to implement Arithmetic-geometric mean without ulps and epsilons.

The first step in implementing Arithmetic-geometric mean is to calculate ${\frac {1+{\frac {1}{\sqrt {2}}}}{2}}$ do this now to cross the starting line and begin the race.

C++

Uses ContinuedFraction and r2cf from Continued_fraction/Arithmetic/Construct_from_rational_number#C++ <lang cpp>/* Interface for all matrixNG classes

  Nigel Galloway, February 10th., 2013.

/

class matrixNG {

 private:
 virtual void consumeTerm(){}
 virtual void consumeTerm(int n){}
 virtual const bool needTerm(){}
 protected: int cfn = 0, thisTerm;
            bool haveTerm = false;
 friend class NG;

}; /* Implement the babyNG matrix

  Nigel Galloway, February 10th., 2013.

/

class NG_4 : public matrixNG {

 private: int a1, a, b1, b, t;
 const bool needTerm() {
   if (b1==0 and b==0) return false;
   if (b1==0 or b==0) return true; else thisTerm = a/b;
   if (thisTerm==(int)(a1/b1)){
     t=a; a=b; b=t-b*thisTerm; t=a1; a1=b1; b1=t-b1*thisTerm;
     haveTerm=true; return false;
   }
   return true;
 }
 void consumeTerm(){a=a1; b=b1;}
 void consumeTerm(int n){t=a; a=a1; a1=t+a1*n; t=b; b=b1; b1=t+b1*n;}
 public:
 NG_4(int a1, int a, int b1, int b): a1(a1), a(a), b1(b1), b(b){}

}; /* Implement a Continued Fraction which returns the result of an arithmetic operation on

  1 or more Continued Fractions (Currently 1 or 2).
  Nigel Galloway, February 10th., 2013.

/

class NG : public ContinuedFraction {

 private:
  matrixNG* ng;
  ContinuedFraction* n[2];
 public:
 NG(NG_4* ng, ContinuedFraction* n1): ng(ng){n[0] = n1;}
 NG(NG_8* ng, ContinuedFraction* n1, ContinuedFraction* n2): ng(ng){n[0] = n1; n[1] = n2;}
 const int nextTerm() {ng->haveTerm = false; return ng->thisTerm;}
 const bool moreTerms(){
   while(ng->needTerm()) if(n[ng->cfn]->moreTerms()) ng->consumeTerm(n[ng->cfn]->nextTerm()); else ng->consumeTerm();
   return ng->haveTerm;
 }

};</lang>

Testing

[1;5,2] + 1/2

<lang cpp>int main() {

 NG_4 a1(2,1,0,2);
 r2cf n1(13,11);
 for(NG n(&a1, &n1); n.moreTerms(); std::cout << n.nextTerm() << " ");
 std::cout << std::endl;
 return 0;

}</lang>

Output:

1 1 2 7

[3;7] * 7/22

<lang cpp>int main() {

 NG_4 a2(7,0,0,22);
 r2cf n2(22,7);
 for(NG n(&a2, &n2); n.moreTerms(); std::cout << n.nextTerm() << " ");
 std::cout << std::endl;
 return 0;

}</lang>

Output:

[3;7] + 1/22

<lang cpp>int main() {

 NG_4 a3(2,1,0,2);
 r2cf n3(22,7);
 for(NG n(&a3, &n3); n.moreTerms(); std::cout << n.nextTerm() << " ");
 std::cout << std::endl;
 return 0;

}</lang>

Output:

3 1 1 1 4

[3;7] divided by 4

<lang cpp>int main() {

 NG_4 a4(1,0,0,4);
 r2cf n4(22,7);
 for(NG n(&a4, &n4); n.moreTerms(); std::cout << n.nextTerm() << " ");
 std::cout << std::endl;
 return 0;

}</lang>

Output:

0 1 3 1 2

${\frac {1}{\sqrt {2}}}$

First I generate ${\frac {1}{\sqrt {2}}}$ as a continued fraction, then I obtain an approximate value using r2cf for comparison. <lang cpp>int main() {

 NG_4 a5(0,1,1,0);
 SQRT2 n5;
 int i = 0;
 for(NG n(&a5, &n5); n.moreTerms() and i++ < 20; std::cout << n.nextTerm() << " ");
 std::cout << "..." << std::endl;
 for(r2cf cf(10000000, 14142136); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
 std::cout << std::endl;
 return 0;

}</lang>

Output:

0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ...
0 1 2 2 2 2 2 2 2 2 2 6 1 2 4 1 1 2

${\frac {1+{\sqrt {2}}}{2}}$

First I generate ${\frac {1+{\sqrt {2}}}{2}}$ as a continued fraction, then I obtain an approximate value using r2cf for comparison. <lang cpp>int main() {

 int i = 0;
 NG_4 a6(1,1,0,2);
 SQRT2 n6;
 for(NG n(&a6, &n6); n.moreTerms() and i++ < 20; std::cout << n.nextTerm() << " ");
 std::cout << "..." << std::endl;
 for(r2cf cf(24142136, 20000000); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
 std::cout << std::endl;
 return 0;

}</lang>

Output:

1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 ...
1 4 1 4 1 4 1 4 1 4 3 2 1 9 5

Go

Adding to the existing package from the Continued_fraction/Arithmetic/Construct_from_rational_number#Go task, re-uses cf.go and rat.go as given in that task.

File ng4.go: <lang Go>package cf

// A 2×2 matix: // [ a₁ a ] // [ b₁ b ] // // which when "applied" to a continued fraction representing x // gives a new continued fraction z such that: // // a₁ * x + a // z = ---------- // b₁ * x + b // // Examples: // NG4{0, 1, 0, 4}.ApplyTo(x) gives 0*x + 1/4 -> 1/4 = [0; 4] // NG4{0, 1, 1, 0}.ApplyTo(x) gives 1/x // NG4{1, 1, 0, 2}.ApplyTo(x) gives (x+1)/2 // // Note that several operations (e.g. addition and division) // can be efficiently done with a single matrix application. // However, each matrix application may require // several calculations for each outputed term. type NG4 struct { A1, A int64 B1, B int64 }

func (ng NG4) needsIngest() bool { if ng.isDone() { panic("b₁==b==0") } return ng.B1 == 0 || ng.B == 0 || ng.A1/ng.B1 != ng.A/ng.B }

func (ng NG4) isDone() bool { return ng.B1 == 0 && ng.B == 0 }

func (ng *NG4) ingest(t int64) { // [ a₁ a ] becomes [ a + a₁×t a₁ ] // [ b₁ b ] [ b + b₁×t b₁ ] ng.A1, ng.A, ng.B1, ng.B = ng.A+ng.A1*t, ng.A1, ng.B+ng.B1*t, ng.B1 }

func (ng *NG4) ingestInfinite() { // [ a₁ a ] becomes [ a₁ a₁ ] // [ b₁ b ] [ b₁ b₁ ] ng.A, ng.B = ng.A1, ng.B1 }

func (ng *NG4) egest(t int64) { // [ a₁ a ] becomes [ b₁ b ] // [ b₁ b ] [ a₁ - b₁×t a - b×t ] ng.A1, ng.A, ng.B1, ng.B = ng.B1, ng.B, ng.A1-ng.B1*t, ng.A-ng.B*t }

// ApplyTo "applies" the matrix `ng` to the continued fraction `cf`, // returning the resulting continued fraction. func (ng NG4) ApplyTo(cf ContinuedFraction) ContinuedFraction { return func() NextFn { next := cf() done := false return func() (int64, bool) { if done { return 0, false } for ng.needsIngest() { if t, ok := next(); ok { ng.ingest(t) } else { ng.ingestInfinite() } } t := ng.A1 / ng.B1 ng.egest(t) done = ng.isDone() return t, true } } }</lang> File ng4_test.go: <lang Go>package cf

import ( "fmt" "reflect" "testing" )

func Example_NG4() { cases := [...]struct { r Rat ng NG4 }{ {Rat{13, 11}, NG4{2, 1, 0, 2}}, {Rat{22, 7}, NG4{2, 1, 0, 2}}, {Rat{22, 7}, NG4{1, 0, 0, 4}}, } for _, tc := range cases { cf := tc.r.AsContinuedFraction() fmt.Printf("%5s = %-9s with %v gives %v\n", tc.r, cf.String(), tc.ng, tc.ng.ApplyTo(cf), ) }

invSqrt2 := NG4{0, 1, 1, 0}.ApplyTo(Sqrt2) fmt.Println(" 1/√2 =", invSqrt2) result := NG4{1, 1, 0, 2}.ApplyTo(Sqrt2) fmt.Println("(1+√2)/2 =", result)

// Output: // 13/11 = [1; 5, 2] with {2 1 0 2} gives [1; 1, 2, 7] // 22/7 = [3; 7] with {2 1 0 2} gives [3; 1, 1, 1, 4] // 22/7 = [3; 7] with {1 0 0 4} gives [0; 1, 3, 1, 2] // 1/√2 = [0; 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...] // (1+√2)/2 = [1; 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, ...] }</lang>

Output:

13/11 = [1; 5, 2] with {2 1 0 2} gives [1; 1, 2, 7]
 22/7 = [3; 7]    with {2 1 0 2} gives [3; 1, 1, 1, 4]
 22/7 = [3; 7]    with {1 0 0 4} gives [0; 1, 3, 1, 2]
    1/√2 = [0; 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]
(1+√2)/2 = [1; 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, ...]

J

Note that the continued fraction representation used here differs from those implemented in the Continued_fraction fraction task. In that task, we alternated a and b values. Here, we only work with a values -- b is implicitly always 1.

Implementation:

<lang J>ng4cf=: 4 : 0

 cf=. 1000{.!._ y
 ng=. x
 r=.i. ndx=.0
 while. +./0~:{:ng do.
   if.=/<.%/ng do.
     r=.r, t=.{.<.%/ng
     ng=. t (|.@] - ]*0,[) ng 
   else.
     if. _=t=.ndx{cf do.
       ng=. ng+/ .*2 2$1 1 0 0
     else.
       ng=. ng+/ .*2 2$t,1 1 0
     end.
     if. (#cf)=ndx=. ndx+1 do. r return. end.
   end.
 end.
 r

)</lang>

Notes:

we arbitrarily stop processing continued fractions after 1000 elements. That's more than enough precision for most purposes.

we can convert a continued fraction to a rational number using (+%)/ though if we want the full represented precision we should instead use (+%)/@x: (which is slower).

we can convert a rational number to a continued fraction using 1 1 {."1@}. ({: , (0 , {:) #: {.)^:(*@{:)^:a: but also this expects a numerator,denominator pair so if you have only a single number use ,&1 to give it a denominator. This works equally well with floating point and arbitrary precision numbers.

Some arbitrary continued fractions and their floating point representations

<lang J> arbs=:(,1);(,3);?~&.>3+i.10

  ":@>arbs

1 3 1 2 0 0 2 3 1 1 0 3 2 4 0 2 3 5 1 4 2 5 0 1 6 3 4 7 5 6 3 0 4 1 2 7 0 1 2 6 3 8 4 5 8 0 5 6 3 7 4 9 1 2 0 9 8 1 3 10 2 5 6 7 4 1 7 3 4 5 8 9 10 6 11 0 2

  (+%)/@>arbs

1 3 1 0.444444 4.44444 0.431925 2.16238 7.19368 8.46335 13.1583 0.109719 1.13682</lang>

Some NG based cf functions, verifying their behavior against our test set:

<lang J> plus1r2=: (2 1,:0 2)&ng4cf

  (plus1r2 each  -&((+%)/@>) ]) arbs

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5</lang>

For every one of our arbitrary continued fractions, the 2 1,:0 2 NG matrix gives us a new continued fraction whose rational value is the original rational value + 1r2.

<lang J> times7r22=: (7 0,:0 22)&ng4cf

  (times7r22 each %&((+%)/@>) ]) arbs

0.318182 0.318182 0.318182 0.318182 0.318182 0.318182 0.318182 0.318182 0.318182 0.318182 0.318182 0.318182

  (times7r22 each %&((+%)/@x:@>) ]) arbs

7r22 7r22 7r22 7r22 7r22 7r22 7r22 7r22 7r22 7r22 7r22 7r22</lang>

For every one of our arbitrary continued fractions, the 7 0,:0 22 NG matrix gives us a new continued fraction whose rational value is 7r22 times the original rational value.

<lang J> times1r4=:(1 0,:0 4)&ng4cf

  (times1r4 each %&((+%)/@>) ]) arbs

0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25</lang>

It seems like a diagonal matrix has the effect of multiplying the numerator by the upper left element and the denominator by the lower right element. And our first experiment suggests that an upper right element in NG with a 0 for the bottom left will add the top right divided by bottom right to our continued fraction.

<lang J> reciprocal=:(0 1,:1 0)&ng4cf

  (reciprocal each *&((+%)/@>) ]) arbs

1 1 1 1 1 1 1 1 1 1 1 1</lang>

Looks like we can also divide by our continued fraction...

<lang J> plus1r2times1r2=: (1 1,:0 2)&ng4cf

  (plus1r2times1r2 each (= 0.5+0.5*])&((+%)/@>) ]) arbs

1 1 1 1 1 1 1 1 1 1 1 1</lang>

We can add and multiply using a single "ng4" operation.

Task examples:

1r2 + 13r11 <lang J> (+%)/1 5 2 1.18182

  plus1r2 1 5 2

1 1 2 7

  (+%)/plus1r2 1 5 2

1.68182</lang>

7r22 * 22r7 <lang J> (+%)/3 7x 22r7

  times7r22 3 7x

1</lang>

1r2 + 22r7 <lang J> plus1r2 3 7x 3 1 1 1 4

  (+%)/plus1r2 3 7x

3.64286

  (+%)/x:plus1r2 3 7x

51r14</lang>

1r4 * 22r7 <lang J> times1r4 3 7x 0 1 3 1 2

  (+%)/x:times1r4 3 7x

11r14</lang>

${\frac {1}{\sqrt {2}}}$ <lang J> reciprocal 1,999$2 0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ...

  (+%)/1,999$2

1.41421

  (+%)/reciprocal 1,999$2

0.707107</lang>

${\frac {1+{\sqrt {2}}}{2}}$ <lang J> plus1r2times1r2 1,999$2 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 ...

  (+%)/plus1r2times1r2 1,999$2

1.20711</lang>

${\frac {1+{\frac {1}{\sqrt {2}}}}{2}}$ <lang J> plus1r2times1r2 0 1,999$2 0 1 5 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 ...

  (+%)/plus1r2times1r2 0 1,999$2

0.853553</lang>

Julia

Translation of: Ruby

<lang julia>function r2cf(n1::Integer, n2::Integer)

   ret = Int[]
   while n2 != 0
       n1, (t1, n2) = n2, divrem(n1, n2)
       push!(ret, t1)
   end
   ret

end r2cf(r::Rational) = r2cf(numerator(r), denominator(r))

function r2cf(n1, n2, maxiter=20)

   ret = Int[]
   while n2 != 0 && maxiter > 0
       n1, (t1, n2) = n2, divrem(n1, n2)
       push!(ret, t1)
       maxiter -= 1
   end
   ret

end

mutable struct NG

   a1::Int
   a::Int
   b1::Int
   b::Int

end

function ingress(ng, n)

   ng.a, ng.a1= ng.a1, ng.a + ng.a1 * n
   ng.b, ng.b1 = ng.b1, ng.b + ng.b1 * n

end

needterm(ng) = ng.b == 0 || ng.b1 == 0 || !(ng.a // ng.b == ng.a1 // ng.b1)

function egress(ng)

   n = ng.a // ng.b
   ng.a, ng.b = ng.b, ng.a - ng.b * n
   ng.a1, ng.b1 = ng.b1, ng.a1 - ng.b1 * n
   r2cf(n)

end

egress_done(ng) = (if needterm(ng) ng.a, ng.b = ng.a1, ng.b1 end; egress(ng))

done(ng) = ng.b == 0 && ng.b1 == 0

function testng()

   data = [["[1;5,2] + 1/2",      [2,1,0,2], [13,11]],
       ["[3;7] + 1/2",        [2,1,0,2], [22, 7]],
       ["[3;7] divided by 4", [1,0,0,4], [22, 7]],
       ["[1;1] divided by sqrt(2)", [0,1,1,0], [1,sqrt(2)]]]

   for d in data
       str, ng, r = d[1], NG(d[2]...), d[3]
       print(rpad(str, 25), "->")
       for n in r2cf(r...)
           if !needterm(ng)
               print(" $(egress(ng))")
           end
           ingress(ng, n)
       end
       while true
           print(" $(egress_done(ng))")
           if done(ng)
               println()
               break
           end
       end
   end

end

testng()

</lang>

Output:

[1;5,2] + 1/2            -> [1, 1, 2, 7]
[3;7] + 1/2              -> [3, 1, 1, 1, 4]
[3;7] divided by 4       -> [0, 1, 3, 1, 2]
[1;1] divided by sqrt(2) -> [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

Kotlin

This is based on the Python entry but has been expanded to deal with the '√2' calculations: <lang scala>// version 1.1.3 // compile with -Xcoroutines=enable flag from command line

import kotlin.coroutines.experimental.*

typealias CFGenerator = (Pair<Int, Int>) -> Sequence<Int>

data class CFData(

   val str: String, 
   val ng: IntArray,
   val r: Pair<Int,Int>,
   val gen: CFGenerator

)

fun r2cf(frac: Pair<Int, Int>) =

   buildSequence {
       var num = frac.first
       var den = frac.second
       while (Math.abs(den) != 0) {
           val div = num / den
           val rem = num % den
           num = den
           den = rem
           yield(div)
       }
   }

fun d2cf(d: Double) =

   buildSequence {
       var dd  = d
       while (true) {
           val div = Math.floor(dd)
           val rem = dd - div
           yield(div.toInt())
           if (rem == 0.0) break
           dd = 1.0 / rem
       }
   }

@Suppress("UNUSED_PARAMETER") fun root2(dummy: Pair<Int, Int>) =

   buildSequence {
       yield(1)
       while (true) yield(2)
   }

@Suppress("UNUSED_PARAMETER") fun recipRoot2(dummy: Pair<Int, Int>) =

   buildSequence {
      yield(0)
      yield(1)
      while (true) yield(2)
   }

class NG(var a1: Int, var a: Int, var b1: Int, var b: Int) {

   fun ingress(n: Int) {
       var t = a
       a = a1
       a1 = t + a1 * n
       t = b
       b = b1
       b1 = t + b1 * n
   }

   fun egress(): Int {
       val n = a / b
       var t = a
       a = b
       b = t - b * n
       t = a1
       a1 = b1
       b1 = t - b1 * n
       return n
   }

   val needTerm get() = (b == 0 || b1 == 0) || ((a / b) != (a1 / b1))
   
   val egressDone: Int
       get() {
           if (needTerm) {
               a = a1
               b = b1
           }
           return egress()
       }
       
   val done get() = b == 0 &&  b1 == 0

}

fun main(args: Array<String>) {

   val data = listOf(
       CFData("[1;5,2] + 1/2        ", intArrayOf(2, 1, 0, 2), 13 to 11, ::r2cf),
       CFData("[3;7] + 1/2          ", intArrayOf(2, 1, 0, 2), 22 to 7,  ::r2cf),
       CFData("[3;7] divided by 4   ", intArrayOf(1, 0, 0, 4), 22 to 7,  ::r2cf),
       CFData("sqrt(2)              ", intArrayOf(0, 1, 1, 0),  0 to 0,  ::recipRoot2),
       CFData("1 / sqrt(2)          ", intArrayOf(0, 1, 1, 0),  0 to 0,  ::root2),
       CFData("(1 + sqrt(2)) / 2    ", intArrayOf(1, 1, 0, 2),  0 to 0,  ::root2),
       CFData("(1 + 1 / sqrt(2)) / 2", intArrayOf(1, 1, 0, 2),  0 to 0,  ::recipRoot2)       
   )
   println("Produced by NG class:")
   for ((str, ng, r, gen) in data) {
       print("$str -> ")
       val (a1, a, b1, b) = ng
       val op = NG(a1, a, b1, b)        
       for (n in gen(r).take(20)) {
           if (!op.needTerm) print(" ${op.egress()} ") 
           op.ingress(n) 
       } 
       while (true) {
           print(" ${op.egressDone} ")
           if (op.done) break
       }
       println()
   }
   println("\nProduced by direct calculation:")
   val data2 = listOf(
       Pair("(1 + sqrt(2)) / 2    ", (1 + Math.sqrt(2.0)) / 2), 
       Pair("(1 + 1 / sqrt(2)) / 2", (1 + 1 / Math.sqrt(2.0)) / 2)
   )
   for ((str, d) in data2) {
       println("$str ->  ${d2cf(d).take(20).joinToString("  ")}")
   }

}</lang>

Output:

Produced by NG class:
[1;5,2] + 1/2         ->  1  1  2  7 
[3;7] + 1/2           ->  3  1  1  1  4 
[3;7] divided by 4    ->  0  1  3  1  2 
sqrt(2)               ->  1  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2 
1 / sqrt(2)           ->  0  1  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2 
(1 + sqrt(2)) / 2     ->  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4 
(1 + 1 / sqrt(2)) / 2 ->  0  1  5  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  5 

Produced by direct calculation:
(1 + sqrt(2)) / 2     ->  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4
(1 + 1 / sqrt(2)) / 2 ->  0  1  5  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1  4  1

Phix

Library: Phix/Class

Requires 0.8.2+. Self-contained. The supporting cast of r2cf(), cf2s(), cf2r() and d2cf() ended up as more code than the task itself. <lang Phix>class baby_matrix

 integer a1, a, b1, b

 --
 -- used by apply_baby_matrix to yield (a1*cf+a)/(b1*cf+b)
 --
 -- examples: (a1 a  b1 b) => above, simplified:
 -- ========   =  =  =  =
 --           {2, 1, 0, 2} => (2*cf+1)/2, aka cf+1/2
 --           {1, 0, 0, 4} => cf/4
 --           {1, 0, 0, 1} => cf/1, aka cf
 --           {0, 1, 1, 0} => 1/cf
 --           {1, 1, 0, 2} => (cf+1)/2
 --

 function need_term()
   return b==0 or b1==0 or ((a/b)!=(a1/b1))
 end function

 function next_term()
   integer n = floor(a/b)
   {a1,a,b1,b} = {b1,b,a1-b1*n,a-b*n}
   return n
 end function

 procedure in_term(object n={})
   if integer(n) then
       {a1,a,b,b1} = {a+a1*n,a1,b1,b+b1*n}
   else
       {a,b} = {a1,b1}
   end if
 end procedure

 function done()
   return b=0 and b1=0
 end function

end class

function apply_baby_matrix(sequence m, cf) -- -- for m of integer {a1,a,b1,b}, return (a1*cf+a)/(b1*cf+b): --

   baby_matrix bm = new(m)
   sequence res = {}
   for i=1 to length(cf) do
       if not bm.need_term() then
           res &= bm.next_term()
       end if  
       bm.in_term(cf[i])
   end for
   while true do   
       if bm.need_term() then
           bm.in_term()
       end if
       res &= bm.next_term()
       if bm.done() then exit end if
   end while
   return res

end function

function r2cf(sequence rat, integer count=20)

   sequence s = {}
   atom {num,den} = rat
   while den!=0 and length(s)<count do
       s &= trunc(num/den)
       {num,den} = {den,num-s[$]*den}
   end while
   return s

end function

function root2(integer count=20)

   return {1}&repeat(2,count-1)

end function

function recip_root2(integer count=20)

   return {0,1}&repeat(2,count-2)

end function

function cf2s(sequence cf)

   sequence s = join(apply(cf,sprint),",") -- eg "1,5,2"
   return "["&substitute(s,",",";",1)&"]"  -- => "[1;5,2]"

end function

include mpfr.e

function cf2r(sequence cf)

   mpq res = mpq_init(), -- 0/1
       cfn = mpq_init()
   for n=length(cf) to 1 by -1 do
       mpq_set_si(cfn,cf[n])
       mpq_add(res,res,cfn)
       if n=1 then exit end if
       mpq_inv(res,res)
   end for
   mpz num = mpz_init(),
       den = mpz_init()
   mpq_get_num(num,res)
   mpq_get_den(den,res)
   mpfr x = mpfr_init()
   mpfr_set_q(x,res)
   string xs = mpfr_sprintf("%.15Rf",x),
          ns = mpz_get_str(num),
          ds = mpz_get_str(den),
           s = sprintf("%s (%s/%s)",{xs,ns,ds})
   return s

end function

function d2cf(atom d, integer count=20)

   string res = "["
   integer sep = ';'
   while count do
       atom div = floor(d),
            rem = d - div
       res &= sprintf("%d%c",{div,sep})
       if rem==0 then exit end if
       d = 1/rem
       count -= 1
       sep = ','
   end while
   res[$] = ']'
   return res

end function

constant tests = {

   {"[1;5,2] + 1/2  ", {2, 1, 0, 2}, r2cf({13,11}), 37/22},
   {"[3;7] + 1/2    ", {2, 1, 0, 2}, r2cf({22, 7}), 3+1/7+1/2},
   {"[3;7] / 4      ", {1, 0, 0, 4}, r2cf({22, 7}), (3+1/7)/4},
   {"sqrt(2)        ", {1, 0, 0, 1}, root2(),       sqrt(2)},
   {"sqrt(2) (inv)  ", {0, 1, 1, 0}, recip_root2(), 1/(1/sqrt(2))},
   {"1/sqrt(2)      ", {1, 0, 0, 1}, recip_root2(), 1/sqrt(2)},
   {"1/sqrt(2) (inv)", {0, 1, 1, 0}, root2(),       1/sqrt(2)},
   {"(1+sqrt(2))/2  ", {1, 1, 0, 2}, root2(),       (1+sqrt(2))/2},
   {"(1+1/sqrt(2))/2", {1, 1, 0, 2}, recip_root2(), (1+1/sqrt(2))/2}}

for i=1 to length(tests) do

   {string str, sequence bm, sequence cf, atom eres} = tests[i]
   sequence res = apply_baby_matrix(bm, cf)
   printf(1,"%s ->  %s --> %s\n",{str,cf2s(res),cf2r(res)})
   printf(1,"           direct:  %s ==> %.15f\n",{d2cf(eres,length(res)),eres})

end for</lang>

Output:

[1;5,2] + 1/2   ->  [1;1,2,7] --> 1.681818181818182 (37/22)
           direct:  [1;1,2,6] ==> 1.681818181818182
[3;7] + 1/2     ->  [3;1,1,1,4] --> 3.642857142857143 (51/14)
           direct:  [3;1,1,1,3] ==> 3.642857142857143
[3;7] / 4       ->  [0;1,3,1,2] --> 0.785714285714286 (11/14)
           direct:  [0;1,3,1,2] ==> 0.785714285714286
sqrt(2)         ->  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] --> 1.414213562373096 (22619537/15994428)
           direct:  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] ==> 1.414213562373095
sqrt(2) (inv)   ->  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] --> 1.414213562373087 (9369319/6625109)
           direct:  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] ==> 1.414213562373095
1/sqrt(2)       ->  [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] --> 0.707106781186552 (6625109/9369319)
           direct:  [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] ==> 0.707106781186547
1/sqrt(2) (inv) ->  [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] --> 0.707106781186547 (15994428/22619537)
           direct:  [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] ==> 0.707106781186547
(1+sqrt(2))/2   ->  [1;4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4] --> 1.207106781186548 (38613965/31988856)
           direct:  [1;4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4] ==> 1.207106781186547
(1+1/sqrt(2))/2 ->  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,5] --> 0.853553390593276 (7997214/9369319)
           direct:  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4] ==> 0.853553390593274

Python

Translation of: Ruby

Python: NG

<lang python>class NG:

 def __init__(self, a1, a, b1, b):
   self.a1, self.a, self.b1, self.b = a1, a, b1, b

 def ingress(self, n):
   self.a, self.a1 = self.a1, self.a + self.a1 * n
   self.b, self.b1 = self.b1, self.b + self.b1 * n

 @property
 def needterm(self):
   return (self.b == 0 or self.b1 == 0) or not self.a//self.b == self.a1//self.b1

 @property
 def egress(self):
   n = self.a // self.b
   self.a,  self.b  = self.b,  self.a  - self.b  * n
   self.a1, self.b1 = self.b1, self.a1 - self.b1 * n
   return n

 @property
 def egress_done(self):
   if self.needterm: self.a, self.b = self.a1, self.b1
   return self.egress

 @property
 def done(self):
   return self.b == 0 and self.b1 == 0</lang>

Python: Testing

Uses r2cf method from here. <lang python>data = [["[1;5,2] + 1/2", [2,1,0,2], [13,11]],

       ["[3;7] + 1/2",        [2,1,0,2], [22, 7]],
       ["[3;7] divided by 4", [1,0,0,4], [22, 7]]]

for string, ng, r in data:

 print( "%-20s->" % string, end= )
 op = NG(*ng)
 for n in r2cf(*r):
   if not op.needterm: print( " %r" % op.egress, end= )
   op.ingress(n)
 while True:
   print( " %r" % op.egress_done, end= )
   if op.done: break
 print()</lang>

Output:

[1;5,2] + 1/2       -> 1 1 2 7
[3;7] + 1/2         -> 3 1 1 1 4
[3;7] divided by 4  -> 0 1 3 1 2

Racket

Translation of: Python

Translation of: C++

Main part of the NG-baby matrices. They are implemented as mutable structs. <lang Racket>#lang racket/base

(struct ng (a1 a b1 b) #:transparent #:mutable)

(define (ng-ingress! v t)

 (define a (ng-a v))
 (define a1 (ng-a1 v))
 (define b (ng-b v))
 (define b1 (ng-b1 v))
 (set-ng-a! v a1)
 (set-ng-a1! v (+ a (* a1 t)))
 (set-ng-b! v b1)
 (set-ng-b1! v (+ b (* b1 t))))

(define (ng-needterm? v)

 (or (zero? (ng-b v)) 
     (zero? (ng-b1 v)) 
     (not (= (quotient (ng-a v) (ng-b v)) (quotient (ng-a1 v) (ng-b1 v))))))

(define (ng-egress! v)

 (define t (quotient (ng-a v) (ng-b v)))
 (define a (ng-a v))
 (define a1 (ng-a1 v))
 (define b (ng-b v))
 (define b1 (ng-b1 v))
 (set-ng-a! v b)
 (set-ng-a1! v b1)
 (set-ng-b! v (- a (* b t)))
 (set-ng-b1! v (- a1 (* b1 t)))
 t)

(define (ng-infty! v)

 (when (ng-needterm? v)
   (set-ng-a! v (ng-a1 v))
   (set-ng-b! v (ng-b1 v))))

(define (ng-done? v)

 (and (zero? (ng-b v)) (zero? (ng-b1 v))))</lang>

Auxiliary functions to create producers of well known continued fractions. The function rational->cf is copied from r2cf task. <lang Racket>(define ((rational->cf n d))

 (and (not (zero? d))
      (let-values ([(q r) (quotient/remainder n d)])
        (set! n d)
        (set! d r)
        q)))

(define (sqrt2->cf)

 (define first? #t)
 (lambda ()
   (if first?
       (begin 
         (set! first? #f)
         1)
       2)))</lang>

The function combine-ng-cf->cf combines a ng-matrix and a cf- producer and creates a cf-producer. The cf-producers can represent infinitely long continued fractions. The function cf-showln shows the first coefficients of a continued fraction represented in a cf-producer. <lang Racket>(define (combine-ng-cf->cf ng cf)

 (define empty-producer? #f)
 (lambda ()
   (let loop ()
     (cond 
       [(not empty-producer?) (define t (cf))
                              (cond 
                                  [t (ng-ingress! ng t)
                                     (if (ng-needterm? ng)
                                         (loop)
                                         (ng-egress! ng))]
                                  [else (set! empty-producer? #t)
                                        (loop)])]
       [(ng-done? ng) #f]
       [(ng-needterm? ng) (ng-infty! ng) 
                          (loop)]
       [else (ng-egress! ng)]))))

(define (cf-showln cf n)

 (for ([i (in-range n)])
   (define val (cf))
   (when val
     (printf " ~a" val)))
 (when (cf)
   (printf " ..."))
 (printf "~n"))</lang>

Some test <lang Racket>(display "[1;5,2] + 1/2 ->") (cf-showln (combine-ng-cf->cf (ng 2 1 0 2) (rational->cf 13 11)) 20)

(display "[3;7] + 1/2 ->") (cf-showln (combine-ng-cf->cf (ng 2 1 0 2) (rational->cf 22 7)) 20)

(display "[3;7] / 4 ->") (cf-showln (combine-ng-cf->cf (ng 1 0 0 4) (rational->cf 22 7)) 20)

(display "sqrt(2)/2 ->") (cf-showln (combine-ng-cf->cf (ng 1 0 0 2) (sqrt2->cf)) 20)

(display "1/sqrt(2) ->") (cf-showln (combine-ng-cf->cf (ng 0 1 1 0) (sqrt2->cf)) 20)

(display "(1+sqrt(2))/2 ->") (cf-showln (combine-ng-cf->cf (ng 1 1 0 2) (sqrt2->cf)) 20)</lang>

Sample output:

[1;5,2] + 1/2 -> 1 1 2 7
[3;7] + 1/2 -> 3 1 1 1 4
[3;7] / 4 -> 0 1 3 1 2
sqrt(2)/2 -> 0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ...
1/sqrt(2) -> 0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ...
(1+sqrt(2))/2 -> 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 ...

Raku

(formerly Perl 6)

Works with: Rakudo version 2018.03

All the important stuff takes place in the NG object. Everything else is helper subs for testing and display. The NG object is capable of working with infinitely long continued fractions, but displaying them can be problematic. You can pass in a limit to the apply method to get a fixed maximum number of terms though. See the last example: 100 terms from the infinite cf (1+√2)/2 and its Rational representation. <lang perl6>class NG {

   has ( $!a1, $!a, $!b1, $!b );
   submethod BUILD ( :$!a1, :$!a, :$!b1, :$!b ) { }

   # Public methods
   method new( $a1, $a, $b1, $b ) { self.bless( :$a1, :$a, :$b1, :$b ) }
   method apply(@cf, :$limit = Inf) {
       (gather {
           map { take self!extract unless self!needterm; self!inject($_) }, @cf;
           take self!drain until self!done;
       })[ ^ $limit ]
   }

   # Private methods
   method !inject ($n) {
       sub xform($n, $x, $y) { $x, $n * $x + $y }
       ( $!a, $!a1 ) = xform( $n, $!a1, $!a );
       ( $!b, $!b1 ) = xform( $n, $!b1, $!b );
   }
   method !extract {
       sub xform($n, $x, $y) { $y, $x - $y * $n }
       my $n = $!a div $!b;
       ($!a,  $!b ) = xform( $n, $!a,  $!b  );
       ($!a1, $!b1) = xform( $n, $!a1, $!b1 );
       $n
   }
   method !drain { $!a = $!a1, $!b = $!b1 if self!needterm; self!extract }
   method !needterm { so [||] !$!b, !$!b1, $!a/$!b != $!a1/$!b1 }
   method !done { not [||] $!b, $!b1 }

}

sub r2cf(Rat $x is copy) { # Rational to continued fraction

   gather loop {

$x -= take $x.floor; last if !$x; $x = 1 / $x;

}

sub cf2r(@a) { # continued fraction to Rational

   my $x = @a[* - 1]; # Use FatRats for arbitrary precision
   $x = ( @a[$_- 1] + 1 / $x ).FatRat for reverse 1 ..^ @a;
   $x

}

sub ppcf(@cf) { # format continued fraction for pretty printing

   "[{ @cf.join(',').subst(',',';') }]"

}

sub pprat($a) { # format Rational for pretty printing

  # Use FatRats for arbitrary precision
  $a.FatRat.denominator == 1 ?? $a !! $a.FatRat.nude.join('/')

}

sub test_NG ($rat, @ng, $op) {

   my @cf = $rat.Rat(1e-18).&r2cf;
   my @op = NG.new( |@ng ).apply( @cf );
   say $rat.perl, ' as a cf: ', @cf.&ppcf, " $op = ",
       @op.&ppcf, "\tor ", @op.&cf2r.&pprat, "\n";

}

Testing

test_NG(|$_) for (

   [ 13/11, [<2 1 0 2>], '+ 1/2 '    ],
   [ 22/7,  [<2 1 0 2>], '+ 1/2    ' ],
   [ 22/7,  [<1 0 0 4>], '/ 4      ' ],
   [ 22/7,  [<7 0 0 22>], '* 7/22   ' ],
   [ 2**.5, [<1 1 0 2>], "\n(1+√2)/2 (approximately)" ]

);

say '100 terms of (1+√2)/2 as a continued fraction and as a rational value:';

my @continued-fraction = NG.new( 1,1,0,2 ).apply( (lazy flat 1, 2 xx * ), limit => 100 ); say @continued-fraction.&ppcf.comb(/ . ** 1..80/).join("\n"); say @continued-fraction.&cf2r.&pprat; </lang>

Output:

<13/11> as a cf: [1;5,2] + 1/2  = [1;1,2,7]	or 37/22

<22/7> as a cf: [3;7] + 1/2     = [3;1,1,1,4]	or 51/14

<22/7> as a cf: [3;7] / 4       = [0;1,3,1,2]	or 11/14

<22/7> as a cf: [3;7] * 7/22    = [1]	or 1

1.4142135623731e0 as a cf: [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] 
(1+√2)/2 (approximately) = [1;4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4]	or 225058681/186444716

100 terms of (1+√2)/2 and its rational value
[1;4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4
,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4
,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4]
161733217200188571081311986634082331709/133984184101103275326877813426364627544

Ruby

NG

<lang ruby># I define a class to implement baby NG class NG

 def initialize(a1, a, b1, b)
   @a1, @a, @b1, @b = a1, a, b1, b
 end
 def ingress(n)
   @a, @a1 = @a1, @a + @a1 * n
   @b, @b1 = @b1, @b + @b1 * n
 end
 def needterm?
   return true if @b == 0 or @b1 == 0
   return true unless @a/@b == @a1/@b1
   false
 end
 def egress
   n = @a / @b
   @a,  @b  = @b,  @a  - @b  * n
   @a1, @b1 = @b1, @a1 - @b1 * n
   n
 end
 def egress_done
   @a, @b = @a1, @b1 if needterm?
   egress
 end
 def done?
   @b == 0 and @b1 == 0
 end

end</lang>

Testing

Uses r2cf method from here. <lang ruby>data = [["[1;5,2] + 1/2", [2,1,0,2], [13,11]],

       ["[3;7] + 1/2",        [2,1,0,2], [22, 7]],
       ["[3;7] divided by 4", [1,0,0,4], [22, 7]]]

data.each do |str, ng, r|

 printf "%-20s->", str
 op = NG.new(*ng)
 r2cf(*r) do |n|
   print " #{op.egress}" unless op.needterm?
   op.ingress(n)
 end
 print " #{op.egress_done}" until op.done?
 puts

end</lang>

Output:

[1;5,2] + 1/2       -> 1 1 2 7
[3;7] + 1/2         -> 3 1 1 1 4
[3;7] divided by 4  -> 0 1 3 1 2

Tcl

This uses the Generator class, R2CF class and printcf procedure from the r2cf task.

Works with: Tcl version 8.6

Translation of: Ruby

<lang tcl># The single-operand version of the NG operator, using our little generator framework oo::class create NG1 {

   superclass Generator

   variable a1 a b1 b cf
   constructor args {

next lassign $args a1 a b1 b

   }
   method Ingress n {

lassign [list [expr {$a + $a1*$n}] $a1 [expr {$b + $b1*$n}] $b1] \ a1 a b1 b

   }
   method NeedTerm? {} {

expr {$b1 == 0 || $b == 0 || $a/$b != $a1/$b1}

   }
   method Egress {} {

set n [expr {$a/$b}] lassign [list $b1 $b [expr {$a1 - $b1*$n}] [expr {$a - $b*$n}]] \ a1 a b1 b return $n

   }
   method EgressDone {} {

if {[my NeedTerm?]} { set a $a1 set b $b1 } tailcall my Egress

   }
   method Done? {} {

expr {$b1 == 0 && $b == 0}

   method operand {N} {

set cf $N return [self]

   }
   method Produce {} {

while 1 { set n [$cf] if {![my NeedTerm?]} { yield [my Egress] } my Ingress $n } while {![my Done?]} { yield [my EgressDone] }

}</lang> Demonstrating: <lang tcl># The square root of 2 as a continued fraction in the framework oo::class create Root2 {

   superclass Generator
   method apply {} {

yield 1 while {[self] ne ""} { yield 2 }

}

set op [[NG1 new 2 1 0 2] operand [R2CF new 13/11]] printcf "\[1;5,2\] + 1/2" $op

set op [[NG1 new 7 0 0 22] operand [R2CF new 22/7]] printcf "\[3;7\] * 7/22" $op

set op [[NG1 new 2 1 0 2] operand [R2CF new 22/7]] printcf "\[3;7\] + 1/2" $op

set op [[NG1 new 1 0 0 4] operand [R2CF new 22/7]] printcf "\[3;7\] / 4" $op

set op [[NG1 new 0 1 1 0] operand [Root2 new]] printcf "1/\u221a2" $op 20

set op [[NG1 new 1 1 0 2] operand [Root2 new]] printcf "(1+\u221a2)/2" $op 20 printcf "approx val" [R2CF new 24142136 20000000]</lang>

Output:

[1;5,2] + 1/2  -> 1,1,2,7
[3;7] * 7/22   -> 1
[3;7] + 1/2    -> 3,1,1,1,4
[3;7] / 4      -> 0,1,3,1,2
1/√2           -> 0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,…
(1+√2)/2       -> 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,…
approx val     -> 1,4,1,4,1,4,1,4,1,4,3,2,1,9,5

zkl

Translation of: Python

<lang zkl>class NG{

  fcn init(_a1,_a, _b1,_b){ var a1=_a1,a=_a, b1=_b1,b=_b; }
  var [proxy] done    =fcn{ b==0 and b1==0 };
  var [proxy] needterm=fcn{ (b==0 or b1==0) or (a/b!=a1/b1) };
  fcn ingress(n){
     t:=self.copy(True);  // tmp copy of vars for eager vs late evaluation 
     a,a1=t.a1, t.a + n*t.a1;
     b,b1=t.b1, t.b + n*t.b1;
  }
  fcn egress{
     n,t:=a/b,self.copy(True);
     a,b  =t.b, t.a  - n*t.b;
     a1,b1=t.b1,t.a1 - n*t.b1;
     n
  }
  fcn egress_done{
     if(needterm) a,b=a1,b1;
     egress()
  }

}</lang> <lang zkl> // from task: Continued fraction/Arithmetic/Construct from rational number fcn r2cf(nom,dnom){ // -->Walker (iterator)

  Walker.tweak(fcn(_,state){
     nom,dnom:=state;
     if(dnom==0) return(Void.Stop);
     n,d:=nom.divr(dnom);
     state.clear(dnom,d);
     n
  }.fp1(List(nom,dnom)))

}</lang> <lang zkl>data:=T(T("[1;5,2] + 1/2", T(2,1,0,2), T(13,11)),

       T("[3;7] + 1/2",        T(2,1,0,2), T(22, 7)),
       T("[3;7] divided by 4", T(1,0,0,4), T(22, 7)));

foreach string,ng,r in (data){

  print("%-20s-->".fmt(string));
  op:=NG(ng.xplode());
  foreach n in (r2cf(r.xplode())){
     if(not op.needterm) print(" %s".fmt(op.egress()));
     op.ingress(n);
  }
  do{ print(" ",op.egress_done()) }while(not op.done);
  println();

}</lang>

Output:

[1;5,2] + 1/2       --> 1 1 2 7
[3;7] + 1/2         --> 3 1 1 1 4
[3;7] divided by 4  --> 0 1 3 1 2

Testing

[1;5,2] + 1/2

[3;7] * 7/22

[3;7] + 1/22

[3;7] divided by 4

1 2 {\displaystyle {\frac {1}{\sqrt {2}}}}

1 + 2 2 {\displaystyle {\frac {1+{\sqrt {2}}}{2}}}

Python: NG

Python: Testing

NG

Testing

${\frac {1}{\sqrt {2}}}$

${\frac {1+{\sqrt {2}}}{2}}$