Continued fraction/Arithmetic/G(matrix ng, continued fraction n)
You are encouraged to solve this task according to the task description, using any language you may know.
This task investigates mathmatical operations that can be performed on a single continued fraction. This requires only a baby version of NG:
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 and are equal. Otherwise I input a term from N. If I need a term from N but N has no more terms I inject .
When I input a term t my internal state: is transposed thus
When I output a term t my internal state: is transposed thus
When I need a term t but there are no more my internal state: is transposed thus
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 (e.g., from Continued fraction) calculate . 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 do this now to cross the starting line and begin the race.
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:
1
[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
First I generate 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
First I generate 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
Perl 6
All the important stuff takes place in the NG object. Everything else is helper subs for testing and display. <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)" ]
); </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
The cf for (1+√2)/2 in the testing routine is an approximation. 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. Here are the first 100 terms from the infinite cf (1+√2)/2 and its Rational representation. <lang perl6>my @continued-fraction = NG.new( 1,1,0,2 ).apply( ( 1, 2 xx * ), limit => 100 ); say @continued-fraction.&ppcf.comb(/ . ** 1..80/).join("\n"); say @continued-fraction.&cf2r.&pprat;</lang>
[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
Python
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
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 ...
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.
<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