Continued fraction/Arithmetic/G(matrix NG, Contined Fraction N1, Contined Fraction N2)

From Rosetta Code
Continued fraction/Arithmetic/G(matrix NG, Contined Fraction N1, Contined Fraction N2) is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

This task performs the basic mathematical functions on 2 continued fractions. This requires the full version of matrix NG:

I may perform perform the following operations:

Input the next term of continued fraction N1
Input the next term of continued fraction N2
Output a term of the continued fraction resulting from the operation.

I output a term if the integer parts of and and and are equal. Otherwise I input a term from continued fraction N1 or continued fraction N2. If I need a term from N but N has no more terms I inject .

When I input a term t from continued fraction N1 I change my internal state:

is transposed thus

When I need a term from exhausted continued fraction N1 I change my internal state:

is transposed thus

When I input a term t from continued fraction N2 I change my internal state:

is transposed thus

When I need a term from exhausted continued fraction N2 I change my internal state:

is transposed thus

When I output a term t I change my internal state:

is transposed thus

When I need to choose to input from N1 or N2 I act:

if b and b2 are zero I choose N1
if b is zero I choose N2
if b2 is zero I choose N2
if abs( is greater than abs( I choose N1
otherwise I choose N2

When performing arithmetic operation on two potentially infinite continued fractions it is possible to generate a rational number. eg * should produce 2. This will require either that I determine that my internal state is approaching infinity, or limiting the number of terms I am willing to input without producing any output.

C++[edit]

/* Implement matrix NG
Nigel Galloway, February 12., 2013
*/

class NG_8 : public matrixNG {
private: int a12, a1, a2, a, b12, b1, b2, b, t;
double ab, a1b1, a2b2, a12b12;
const int chooseCFN(){return fabs(a1b1-ab) > fabs(a2b2-ab)? 0 : 1;}
const bool needTerm() {
if (b1==0 and b==0 and b2==0 and b12==0) return false;
if (b==0){cfn = b2==0? 0:1; return true;} else ab = ((double)a)/b;
if (b2==0){cfn = 1; return true;} else a2b2 = ((double)a2)/b2;
if (b1==0){cfn = 0; return true;} else a1b1 = ((double)a1)/b1;
if (b12==0){cfn = chooseCFN(); return true;} else a12b12 = ((double)a12)/b12;
thisTerm = (int)ab;
if (thisTerm==(int)a1b1 and thisTerm==(int)a2b2 and thisTerm==(int)a12b12){
t=a; a=b; b=t-b*thisTerm; t=a1; a1=b1; b1=t-b1*thisTerm; t=a2; a2=b2; b2=t-b2*thisTerm; t=a12; a12=b12; b12=t-b12*thisTerm;
haveTerm = true; return false;
}
cfn = chooseCFN();
return true;
}
void consumeTerm(){if(cfn==0){a=a1; a2=a12; b=b1; b2=b12;} else{a=a2; a1=a12; b=b2; b1=b12;}}
void consumeTerm(int n){
if(cfn==0){t=a; a=a1; a1=t+a1*n; t=a2; a2=a12; a12=t+a12*n; t=b; b=b1; b1=t+b1*n; t=b2; b2=b12; b12=t+b12*n;}
else{t=a; a=a2; a2=t+a2*n; t=a1; a1=a12; a12=t+a12*n; t=b; b=b2; b2=t+b2*n; t=b1; b1=b12; b12=t+b12*n;}
}
public:
NG_8(int a12, int a1, int a2, int a, int b12, int b1, int b2, int b): a12(a12), a1(a1), a2(a2), a(a), b12(b12), b1(b1), b2(b2), b(b){
}};

Testing[edit]

[3;7] + [0;2]

int main() {
NG_8 a(0,1,1,0,0,0,0,1);
r2cf n2(22,7);
r2cf n1(1,2);
for(NG n(&a, &n1, &n2); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
 
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;
}
Output:
3 1 1 1 4
3 1 1 1 4

[1:5,2] * [3;7]

int main() {
NG_8 b(1,0,0,0,0,0,0,1);
r2cf b1(13,11);
r2cf b2(22,7);
for(NG n(&b, &b1, &b2); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(NG n(&a, &b2, &b1); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf cf(286,77); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
std::cout << std::endl;
return 0;
}
Output:
3 1 2 2
3 1 2 2

[1:5,2] - [3;7]

int main() {
NG_8 c(0,1,-1,0,0,0,0,1);
r2cf c1(13,11);
r2cf c2(22,7);
for(NG n(&c, &c1, &c2); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf cf(-151,77); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
std::cout << std::endl;
return 0;
}
Output:
-1 -1 -24 -1 -2
-1 -1 -24 -1 -2

Divide [] by [3;7]

int main() {
NG_8 d(0,1,0,0,0,0,1,0);
r2cf d1(22*22,7*7);
r2cf d2(22,7);
for(NG n(&d, &d1, &d2); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
return 0;
}
Output:
3 7

([0;3,2] + [1;5,2]) * ([0;3,2] - [1;5,2])

int main() {
r2cf a1(2,7);
r2cf a2(13,11);
NG_8 na(0,1,1,0,0,0,0,1);
NG A(&na, &a1, &a2); //[0;3,2] + [1;5,2]
r2cf b1(2,7);
r2cf b2(13,11);
NG_8 nb(0,1,-1,0,0,0,0,1);
NG B(&nb, &b1, &b2); //[0;3,2] - [1;5,2]
NG_8 nc(1,0,0,0,0,0,0,1); //A*B
for(NG n(&nc, &A, &B); n.moreTerms(); std::cout << n.nextTerm() << " ");
std::cout << std::endl;
for(r2cf cf(2,7); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
std::cout << std::endl;
for(r2cf cf(13,11); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
std::cout << std::endl;
for(r2cf cf(-7797,5929); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
std::cout << std::endl;
return 0;
}

Go[edit]

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 ng8.go:

package cf
 
import "math"
 
// A 2×4 matix:
// [ a₁₂ a₁ a₂ a ]
// [ b₁₂ b₁ b₂ b ]
//
// which when "applied" to two continued fractions N1 and N2
// gives a new continued fraction z such that:
//
// a₁₂ * N1 * N2 + a₁ * N1 + a₂ * N2 + a
// z = -------------------------------------------
// b₁₂ * N1 * N2 + b₁ * N1 + b₂ * N2 + b
//
// Examples:
// NG8{0,1,1,0, 0,0,0,1} gives N1 + N2
// NG8{0,1,-1,0, 0,0,0,1} gives N1 - N2
// NG8{1,0,0,0, 0,0,0,1} gives N1 * N2
// NG8{0,1,0,0, 0,0,1,0} gives N1 / N2
// NG8{21,-15,28,-20, 0,0,0,1} gives 21*N1*N2 -15*N1 +28*N2 -20
// which is (3*N1 + 4) * (7*N2 - 5)
type NG8 struct {
A12, A1, A2, A int64
B12, B1, B2, B int64
}
 
// Basic identities as NG8 matrices.
var (
NG8Add = NG8{0, 1, 1, 0, 0, 0, 0, 1}
NG8Sub = NG8{0, 1, -1, 0, 0, 0, 0, 1}
NG8Mul = NG8{1, 0, 0, 0, 0, 0, 0, 1}
NG8Div = NG8{0, 1, 0, 0, 0, 0, 1, 0}
)
 
func (ng *NG8) needsIngest() bool {
if ng.B12 == 0 || ng.B1 == 0 || ng.B2 == 0 || ng.B == 0 {
return true
}
x := ng.A / ng.B
return ng.A1/ng.B1 != x || ng.A2/ng.B2 != x && ng.A12/ng.B12 != x
}
 
func (ng *NG8) isDone() bool {
return ng.B12 == 0 && ng.B1 == 0 && ng.B2 == 0 && ng.B == 0
}
 
func (ng *NG8) ingestWhich() bool { // true for N1, false for N2
if ng.B == 0 && ng.B2 == 0 {
return true
}
if ng.B == 0 || ng.B2 == 0 {
return false
}
x1 := float64(ng.A1) / float64(ng.B1)
x2 := float64(ng.A2) / float64(ng.B2)
x := float64(ng.A) / float64(ng.B)
return math.Abs(x1-x) > math.Abs(x2-x)
}
 
func (ng *NG8) ingest(isN1 bool, t int64) {
if isN1 {
// [ a₁₂ a₁ a₂ a ] becomes [ a₂+a₁₂*t a+a₁*t a₁₂ a₁]
// [ b₁₂ b₁ b₂ b ] [ b₂+b₁₂*t b+b₁*t b₁₂ b₁]
ng.A12, ng.A1, ng.A2, ng.A,
ng.B12, ng.B1, ng.B2, ng.B =
ng.A2+ng.A12*t, ng.A+ng.A1*t, ng.A12, ng.A1,
ng.B2+ng.B12*t, ng.B+ng.B1*t, ng.B12, ng.B1
} else {
// [ a₁₂ a₁ a₂ a ] becomes [ a₁+a₁₂*t a₁₂ a+a₂*t a₂]
// [ b₁₂ b₁ b₂ b ] [ b₁+b₁₂*t b₁₂ b+b₂*t b₂]
ng.A12, ng.A1, ng.A2, ng.A,
ng.B12, ng.B1, ng.B2, ng.B =
ng.A1+ng.A12*t, ng.A12, ng.A+ng.A2*t, ng.A2,
ng.B1+ng.B12*t, ng.B12, ng.B+ng.B2*t, ng.B2
}
}
 
func (ng *NG8) ingestInfinite(isN1 bool) {
if isN1 {
// [ a₁₂ a₁ a₂ a ] becomes [ a₁₂ a₁ a₁₂ a₁ ]
// [ b₁₂ b₁ b₂ b ] [ b₁₂ b₁ b₁₂ b₁ ]
ng.A2, ng.A, ng.B2, ng.B =
ng.A12, ng.A1,
ng.B12, ng.B1
} else {
// [ a₁₂ a₁ a₂ a ] becomes [ a₁₂ a₁₂ a₂ a₂ ]
// [ b₁₂ b₁ b₂ b ] [ b₁₂ b₁₂ b₂ b₂ ]
ng.A1, ng.A, ng.B1, ng.B =
ng.A12, ng.A2,
ng.B12, ng.B2
}
}
 
func (ng *NG8) egest(t int64) {
// [ a₁₂ a₁ a₂ a ] becomes [ b₁₂ b₁ b₂ b ]
// [ b₁₂ b₁ b₂ b ] [ a₁₂-b₁₂*t a₁-b₁*t a₂-b₂*t a-b*t ]
ng.A12, ng.A1, ng.A2, ng.A,
ng.B12, ng.B1, ng.B2, ng.B =
ng.B12, ng.B1, ng.B2, ng.B,
ng.A12-ng.B12*t, ng.A1-ng.B1*t, ng.A2-ng.B2*t, ng.A-ng.B*t
}
 
// ApplyTo "applies" the matrix `ng` to the continued fractions
// `N1` and `N2`, returning the resulting continued fraction.
// After ingesting `limit` terms without any output terms the resulting
// continued fraction is terminated.
func (ng NG8) ApplyTo(N1, N2 ContinuedFraction, limit int) ContinuedFraction {
return func() NextFn {
next1, next2 := N1(), N2()
done := false
sinceEgest := 0
return func() (int64, bool) {
if done {
return 0, false
}
for ng.needsIngest() {
sinceEgest++
if sinceEgest > limit {
done = true
return 0, false
}
isN1 := ng.ingestWhich()
next := next2
if isN1 {
next = next1
}
if t, ok := next(); ok {
ng.ingest(isN1, t)
} else {
ng.ingestInfinite(isN1)
}
}
sinceEgest = 0
t := ng.A / ng.B
ng.egest(t)
done = ng.isDone()
return t, true
}
}
}

File ng8_test.go:

package cf
 
import "fmt"
 
func ExampleNG8() {
cases := [...]struct {
op string
r1, r2 Rat
ng NG8
}{
{"+", Rat{22, 7}, Rat{1, 2}, NG8Add},
{"*", Rat{13, 11}, Rat{22, 7}, NG8Mul},
{"-", Rat{13, 11}, Rat{22, 7}, NG8Sub},
{"/", Rat{22 * 22, 7 * 7}, Rat{22, 7}, NG8Div},
}
for _, tc := range cases {
n1 := tc.r1.AsContinuedFraction()
n2 := tc.r2.AsContinuedFraction()
z := tc.ng.ApplyTo(n1, n2, 1000)
fmt.Printf("%v %s %v is %v %v %v gives %v\n",
tc.r1, tc.op, tc.r2,
tc.ng, n1, n2, z,
)
}
 
z := NG8Mul.ApplyTo(Sqrt2, Sqrt2, 1000)
fmt.Println("√2 * √2 =", z)
 
// Output:
// 22/7 + 1/2 is {0 1 1 0 0 0 0 1} [3; 7] [0; 2] gives [3; 1, 1, 1, 4]
// 13/11 * 22/7 is {1 0 0 0 0 0 0 1} [1; 5, 2] [3; 7] gives [3; 1, 2, 2]
// 13/11 - 22/7 is {0 1 -1 0 0 0 0 1} [1; 5, 2] [3; 7] gives [-1; -1, -24, -1, -2]
// 484/49 / 22/7 is {0 1 0 0 0 0 1 0} [9; 1, 7, 6] [3; 7] gives [3; 7]
// √2 * √2 = [1; 0, 1]
}
Output:

(Note, [1; 0, 1] = 1 + 1 / (0 + 1/1 ) = 2, so the answer is correct, it should however be normalised to the more reasonable form of [2].)

22/7 + 1/2 is {0 1 1 0 0 0 0 1} [3; 7] [0; 2] gives [3; 1, 1, 1, 4]
13/11 * 22/7 is {1 0 0 0 0 0 0 1} [1; 5, 2] [3; 7] gives [3; 1, 2, 2]
13/11 - 22/7 is {0 1 -1 0 0 0 0 1} [1; 5, 2] [3; 7] gives [-1; -1, -24, -1, -2]
484/49 / 22/7 is {0 1 0 0 0 0 1 0} [9; 1, 7, 6] [3; 7] gives [3; 7]
√2 * √2 = [1; 0, 1]

Kotlin[edit]

Translation of: C++

The C++ entry uses a number of classes which have been coded in other "Continued Fraction" tasks. I've pulled all these into my Kotlin translation and unified the tests so that the whole thing can, hopefully, be understood and run as a single program.

// version 1.2.10
 
import kotlin.math.abs
 
abstract class MatrixNG {
var cfn = 0
var thisTerm = 0
var haveTerm = false
 
abstract fun consumeTerm()
abstract fun consumeTerm(n: Int)
abstract fun needTerm(): Boolean
}
 
class NG4(
var a1: Int, var a: Int, var b1: Int, var b: Int
) : MatrixNG() {
 
private var t = 0
 
override fun needTerm(): Boolean {
if (b1 == 0 && b == 0) return false
if (b1 == 0 || b == 0) return true
thisTerm = a / b
if (thisTerm == a1 / b1) {
t = a; a = b; b = t - b * thisTerm
t = a1; a1 = b1; b1 = t - b1 * thisTerm
haveTerm = true
return false
}
return true
}
 
override fun consumeTerm() {
a = a1
b = b1
}
 
override fun consumeTerm(n: Int) {
t = a; a = a1; a1 = t + a1 * n
t = b; b = b1; b1 = t + b1 * n
}
}
 
class NG8(
var a12: Int, var a1: Int, var a2: Int, var a: Int,
var b12: Int, var b1: Int, var b2: Int, var b: Int
) : MatrixNG() {
 
private var t = 0
private var ab = 0.0
private var a1b1 = 0.0
private var a2b2 = 0.0
private var a12b12 = 0.0
 
fun chooseCFN() = if (abs(a1b1 - ab) > abs(a2b2-ab)) 0 else 1
 
override fun needTerm(): Boolean {
if (b1 == 0 && b == 0 && b2 == 0 && b12 == 0) return false
if (b == 0) {
cfn = if (b2 == 0) 0 else 1
return true
}
else ab = a.toDouble() / b
 
if (b2 == 0) {
cfn = 1
return true
}
else a2b2 = a2.toDouble() / b2
 
if (b1 == 0) {
cfn = 0
return true
}
else a1b1 = a1.toDouble() / b1
 
if (b12 == 0) {
cfn = chooseCFN()
return true
}
else a12b12 = a12.toDouble() / b12
 
thisTerm = ab.toInt()
if (thisTerm == a1b1.toInt() && thisTerm == a2b2.toInt() &&
thisTerm == a12b12.toInt()) {
t = a; a = b; b = t - b * thisTerm
t = a1; a1 = b1; b1 = t - b1 * thisTerm
t = a2; a2 = b2; b2 = t - b2 * thisTerm
t = a12; a12 = b12; b12 = t - b12 * thisTerm
haveTerm = true
return false
}
cfn = chooseCFN()
return true
}
 
override fun consumeTerm() {
if (cfn == 0) {
a = a1; a2 = a12
b = b1; b2 = b12
}
else {
a = a2; a1 = a12
b = b2; b1 = b12
}
}
 
override fun consumeTerm(n: Int) {
if (cfn == 0) {
t = a; a = a1; a1 = t + a1 * n
t = a2; a2 = a12; a12 = t + a12 * n
t = b; b = b1; b1 = t + b1 * n
t = b2; b2 = b12; b12 = t + b12 * n
}
else {
t = a; a = a2; a2 = t + a2 * n
t = a1; a1 = a12; a12 = t + a12 * n
t = b; b = b2; b2 = t + b2 * n
t = b1; b1 = b12; b12 = t + b12 * n
}
}
}
 
interface ContinuedFraction {
fun nextTerm(): Int
fun moreTerms(): Boolean
}
 
class R2cf(var n1: Int, var n2: Int) : ContinuedFraction {
 
override fun nextTerm(): Int {
val thisTerm = n1 /n2
val t2 = n2
n2 = n1 - thisTerm * n2
n1 = t2
return thisTerm
}
 
override fun moreTerms() = abs(n2) > 0
}
 
class NG : ContinuedFraction {
val ng: MatrixNG
val n: List<ContinuedFraction>
 
constructor(ng: NG4, n1: ContinuedFraction) {
this.ng = ng
n = listOf(n1)
}
 
constructor(ng: NG8, n1: ContinuedFraction, n2: ContinuedFraction) {
this.ng = ng
n = listOf(n1, n2)
}
 
override fun nextTerm(): Int {
ng.haveTerm = false
return ng.thisTerm
}
 
override fun moreTerms(): Boolean {
while (ng.needTerm()) {
if (n[ng.cfn].moreTerms())
ng.consumeTerm(n[ng.cfn].nextTerm())
else
ng.consumeTerm()
}
return ng.haveTerm
}
}
 
fun test(desc: String, vararg cfs: ContinuedFraction) {
println("TESTING -> $desc")
for (cf in cfs) {
while (cf.moreTerms()) print ("${cf.nextTerm()} ")
println()
}
println()
}
 
fun main(args: Array<String>) {
val a = NG8(0, 1, 1, 0, 0, 0, 0, 1)
val n2 = R2cf(22, 7)
val n1 = R2cf(1, 2)
val a3 = NG4(2, 1, 0, 2)
val n3 = R2cf(22, 7)
test("[3;7] + [0;2]", NG(a, n1, n2), NG(a3, n3))
 
val b = NG8(1, 0, 0, 0, 0, 0, 0, 1)
val b1 = R2cf(13, 11)
val b2 = R2cf(22, 7)
test("[1;5,2] * [3;7]", NG(b, b1, b2), R2cf(286, 77))
 
val c = NG8(0, 1, -1, 0, 0, 0, 0, 1)
val c1 = R2cf(13, 11)
val c2 = R2cf(22, 7)
test("[1;5,2] - [3;7]", NG(c, c1, c2), R2cf(-151, 77))
 
val d = NG8(0, 1, 0, 0, 0, 0, 1, 0)
val d1 = R2cf(22 * 22, 7 * 7)
val d2 = R2cf(22,7)
test("Divide [] by [3;7]", NG(d, d1, d2))
 
val na = NG8(0, 1, 1, 0, 0, 0, 0, 1)
val a1 = R2cf(2, 7)
val a2 = R2cf(13, 11)
val aa = NG(na, a1, a2)
val nb = NG8(0, 1, -1, 0, 0, 0, 0, 1)
val b3 = R2cf(2, 7)
val b4 = R2cf(13, 11)
val bb = NG(nb, b3, b4)
val nc = NG8(1, 0, 0, 0, 0, 0, 0, 1)
val desc = "([0;3,2] + [1;5,2]) * ([0;3,2] - [1;5,2])"
test(desc, NG(nc, aa, bb), R2cf(-7797, 5929))
}
Output:
TESTING -> [3;7] + [0;2]
3 1 1 1 4 
3 1 1 1 4 

TESTING -> [1;5,2] * [3;7]
3 1 2 2 
3 1 2 2 

TESTING -> [1;5,2] - [3;7]
-1 -1 -24 -1 -2 
-1 -1 -24 -1 -2 

TESTING -> Divide [] by [3;7]
3 7 

TESTING -> ([0;3,2] + [1;5,2]) * ([0;3,2] - [1;5,2])
-1 -3 -5 -1 -2 -1 -26 -3 
-1 -3 -5 -1 -2 -1 -26 -3 

Perl 6[edit]

Works with: Rakudo version 2016.01

The NG2 object can work with infinitely long continued fractions, it does lazy evaluation. By default, it is limited to returning the first 30 terms. Pass in a limit value if you want something other than default.

class NG2 {
has ( $!a12, $!a1, $!a2, $!a, $!b12, $!b1, $!b2, $!b );
 
# Public methods
method operator($!a12, $!a1, $!a2, $!a, $!b12, $!b1, $!b2, $!b ) { self }
 
method apply(@cf1, @cf2, :$limit = 30) {
my @cfs = [@cf1], [@cf2];
gather {
while @cfs[0] or @cfs[1] {
my $term;
(take $term if $term = self!extract) unless self!needterm;
my $from = self!from;
$from = @cfs[$from] ?? $from !! $from +^ 1;
self!inject($from, @cfs[$from].shift);
}
take self!drain while $!b;
}[ ^$limit ].grep: *.defined;
}
 
# Private methods
method !inject ($n, $t) {
multi sub xform(0, $t, $x12, $x1, $x2, $x) { $x2 + $x12 * $t, $x + $x1 * $t, $x12, $x1 }
multi sub xform(1, $t, $x12, $x1, $x2, $x) { $x1 + $x12 * $t, $x12, $x + $x2 * $t, $x2 }
( $!a12, $!a1, $!a2, $!a ) = xform($n, $t, $!a12, $!a1, $!a2, $!a );
( $!b12, $!b1, $!b2, $!b ) = xform($n, $t, $!b12, $!b1, $!b2, $!b );
}
method !extract {
my $t = $!a div $!b;
( $!a12, $!a1, $!a2, $!a, $!b12, $!b1, $!b2, $!b ) =
$!b12, $!b1, $!b2, $!b,
$!a12 - $!b12 * $t,
$!a1 - $!b1 * $t,
$!a2 - $!b2 * $t,
$!a - $!b * $t;
$t;
}
method !from {
return $!b == $!b2 == 0 ?? 0 !!
$!b == 0 || $!b2 == 0 ?? 1 !!
abs($!a1*$!b*$!b2 - $!a*$!b1*$!b2) > abs($!a2*$!b*$!b1 - $!a*$!b1*$!b2) ?? 0 !! 1;
}
method !needterm {
so !([&&] $!b12, $!b1, $!b2, $!b) or $!a/$!b != $!a1/$!b1 != $!a2/$!b2 != $!a12/$!b1;
}
method !noterms($which) {
$which ?? (($!a1, $!a, $!b1, $!b ) = $!a12, $!a2, $!b12, $!b2)
!! (($!a2, $!a, $!b2, $!b ) = $!a12, $!a1, $!b12, $!b1);
}
method !drain {
self!noterms(self!from) if self!needterm;
self!extract;
}
}
 
sub r2cf(Rat $x is copy) { # Rational to continued fraction
gather loop {
$x -= take $x.floor;
last unless $x;
$x = 1 / $x;
}
}
 
sub cf2r(@a) { # continued fraction to Rational
my $x = @a[* - 1].FatRat; # Use FatRats for arbitrary precision
$x = @a[$_- 1] + 1 / $x for reverse 1 ..^ @a;
$x
}
 
# format continued fraction for pretty printing
sub ppcf(@cf) { "[{ @cf.join(',').subst(',',';') }]" }
 
# format Rational for pretty printing. Use FatRats for arbitrary precision
sub pprat($a) { $a.FatRat.denominator == 1 ?? $a !! $a.FatRat.nude.join('/') }
 
my %ops = ( # convenience hash of NG matrix operators
'+' => (0,1,1,0,0,0,0,1),
'-' => (0,1,-1,0,0,0,0,1),
'*' => (1,0,0,0,0,0,0,1),
'/' => (0,1,0,0,0,0,1,0)
);
 
sub test_NG2 ($rat1, $op, $rat2) {
my @cf1 = $rat1.&r2cf;
my @cf2 = $rat2.&r2cf;
my @result = NG2.new.operator(|%ops{$op}).apply( @cf1, @cf2 );
say "{$rat1.&pprat} $op {$rat2.&pprat} => {@cf1.&ppcf} $op ",
"{@cf2.&ppcf} = {@result.&ppcf} => {@result.&cf2r.&pprat}\n";
}
 
# Testing
test_NG2(|$_) for
[ 22/7, '+', 1/2 ],
[ 23/11, '*', 22/7 ],
[ 13/11, '-', 22/7 ],
[ 484/49, '/', 22/7 ];
 
 
# Sometimes you may want to limit the terms in the continued fraction to something other than default.
# Here a lazy infinite continued fraction for √2, then multiply it by itself. We'll limit the result
# to 6 terms for brevity’s' sake. We'll then convert that continued fraction back to an arbitrary precision
# FatRat Rational number. (Perl 6 stores FatRats internally as a ratio of two arbitrarily long integers.
# We need to exercise a little caution because they can eat up all of your memory if allowed to grow unchecked,
# hence the limit of 6 terms in continued fraction.) We'll then convert that number to a normal precision
# Rat, which is accurate to the nearest 1 / 2^64,
 
say "√2 expressed as a continued fraction, then squared: ";
my @root2 = lazy flat 1, 2 xx *;
my @result = NG2.new.operator(|%ops{'*'}).apply( @root2, @root2, limit => 6 );
say @root2.&ppcf, "² = \n";
say @result.&ppcf;
say "\nConverted back to an arbitrary (ludicrous) precision Rational: ";
say @result.&cf2r.nude.join(" /\n");
say "\nCoerced to a standard precision Rational: ", @result.&cf2r.Num.Rat;
Output:
22/7 + 1/2 => [3;7] + [0;2] = [3;1,1,1,4] => 51/14

23/11 * 22/7 => [2;11] * [3;7] = [6;1,1,3] => 46/7

13/11 - 22/7 => [1;5,2] - [3;7] = [-2;25,1,2] => -151/77

484/49 / 22/7 => [9;1,7,6] / [3;7] = [3;7] => 22/7

√2 expressed as a continued fraction, then squared: 
[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 = 

[1;1,-58451683124983302025,-1927184886226364356176,-65467555105469489418600,-2223969688699736275876224]

Converted back to an arbitrary (ludicrous) precision Rational: 
32802382178012409621354320392819425499699206367450594986122623570838188983519955166754002 /
16401191089006204810536863200564985394427741343927508600629139291039556821665755787817601

Coerced to a standard precision Rational: 2

Tcl[edit]

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

Works with: Tcl version 8.6
oo::class create NG2 {
variable a b a1 b1 a2 b2 a12 b12 cf1 cf2
superclass Generator
constructor {args} {
lassign $args a12 a1 a2 a b12 b1 b2 b
next
}
method operands {N1 N2} {
set cf1 $N1
set cf2 $N2
return [self]
}
 
method Ingress1 t {
lassign [list [expr {$a2+$a12*$t}] [expr {$a+$a1*$t}] $a12 $a1 \
[expr {$b2+$b12*$t}] [expr {$b+$b1*$t}] $b12 $b1] \
a12 a1 a2 a b12 b1 b2 b
}
method Exhaust1 {} {
lassign [list $a12 $a1 $a12 $a1 $b12 $b1 $b12 $b1] \
a12 a1 a2 a b12 b1 b2 b
}
method Ingress2 t {
lassign [list [expr {$a1+$a12*$t}] $a12 [expr {$a+$a2*$t}] $a2 \
[expr {$b1+$b12*$t}] $b12 [expr {$b+$b2*$t}] $b2] \
a12 a1 a2 a b12 b1 b2 b
}
method Exhaust2 {} {
lassign [list $a12 $a12 $a2 $a2 $b12 $b12 $b2 $b2] \
a12 a1 a2 a b12 b1 b2 b
}
method Egress {} {
set t [expr {$a/$b}]
lassign [list $b12 $b1 $b2 $b \
[expr {$a12 - $b12*$t}] [expr {$a1 - $b1*$t}] \
[expr {$a2 - $b2*$t}] [expr {$a - $b*$t}]] \
a12 a1 a2 a b12 b1 b2 b
return $t
}
 
method DoIngress1 {} {
try {tailcall my Ingress1 [$cf1]} on break {} {}
oo::objdefine [self] forward DoIngress1 my Exhaust1
set cf1 ""
tailcall my Exhaust1
}
method DoIngress2 {} {
try {tailcall my Ingress2 [$cf2]} on break {} {}
oo::objdefine [self] forward DoIngress2 my Exhaust2
set cf2 ""
tailcall my Exhaust2
}
method Ingress {} {
if {$b==0} {
if {$b2 == 0} {
tailcall my DoIngress1
} else {
tailcall my DoIngress2
}
}
if {!$b2} {
tailcall my DoIngress2
}
if {!$b1} {
tailcall my DoIngress1
}
if {[my FirstSource?]} {
tailcall my DoIngress1
} else {
tailcall my DoIngress2
}
}
 
method FirstSource? {} {
expr {abs($a1*$b*$b2 - $a*$b1*$b2) > abs($a2*$b*$b1 - $a*$b1*$b2)}
}
method NeedTerm? {} {
expr {
($b*$b1*$b2*$b12==0) ||
!($a/$b == $a1/$b1 && $a/$b == $a2/$b2 && $a/$b == $a12/$b12)
}
}
method Done? {} {
expr {$b==0 && $b1==0 && $b2==0 && $b12==0}
}
 
method Produce {} {
# Until we've drained both continued fractions...
while {$cf1 ne "" || $cf2 ne ""} {
if {[my NeedTerm?]} {
my Ingress
} else {
yield [my Egress]
}
}
# Drain our internal state
while {![my Done?]} {
yield [my Egress]
}
}
}

Demonstrating:

set op [[NG2 new 0 1 1 0 0 0 0 1] operands [R2CF new 1/2] [R2CF new 22/7]]
printcf "\[3;7\] + \[0;2\]" $op
 
set op [[NG2 new 1 0 0 0 0 0 0 1] operands [R2CF new 13/11] [R2CF new 22/7]]
printcf "\[1:5,2\] * \[3;7\]" $op
 
set op [[NG2 new 0 1 -1 0 0 0 0 1] operands [R2CF new 13/11] [R2CF new 22/7]]
printcf "\[1:5,2\] - \[3;7\]" $op
 
set op [[NG2 new 0 1 0 0 0 0 1 0] operands [R2CF new 484/49] [R2CF new 22/7]]
printcf "div test" $op
 
set op1 [[NG2 new 0 1 1 0 0 0 0 1] operands [R2CF new 2/7] [R2CF new 13/11]]
set op2 [[NG2 new 0 1 -1 0 0 0 0 1] operands [R2CF new 2/7] [R2CF new 13/11]]
set op3 [[NG2 new 1 0 0 0 0 0 0 1] operands $op1 $op2]
printcf "layered test" $op3
Output:
[3;7] + [0;2]  -> 3,1,1,1,4
[1:5,2] * [3;7]-> 3,1,2,2
[1:5,2] - [3;7]-> -2,25,1,2
div test       -> 3,7
layered test   -> -2,1,2,5,1,2,1,26,3