Continued fraction: Difference between revisions

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A number may be represented as a continued fraction (see http://mathworld.wolfram.com/ContinuedFraction.html) as follows:

cf=a_{1}+{\cfrac {b_{1}}{a_{2}+{\cfrac {b_{2}}{a_{3}+{\cfrac {b_{3}}{\mathrm {etc\ldots } }}}}}}

The task is to write a program which generates such a number and prints a real representation of it. The code should be tested by calculating and printing the square root of 2, Napier's Constant, and Pi.

For the square root of 2, $b_{N}$ is always $1$ . $a_{1}$ is $1$ then $a_{N}$ is $2$ .

For Napier's Constant, $a_{1}$ is $2$ , then $a_{N}$ is $N-1$ . $b_{1}$ is $1$ then $b_{N}$ is $N-1$ .

For Pi, $a_{1}$ is 3 then $a_{N}$ is always $6$ . $b_{N}$ is $(2N-1)(2N-1)$ .

Python

<lang python># The continued Fraction def CF(a, b, t):

 if 0 < t:
   a1 = next(a)
   b1 = next(b)
   z  = CF(a, b, t-1)
   return (a1*z[0] + b1*z[1], z[0])
 return (1,1)

Convert the continued fraction to a string

def pRes(cf, d):

 res = str(cf[0] // cf[1])
 res += "."
 x = cf[0] % cf[1]
 while 0 < d:
   x *= 10
   res += str(x // cf[1])
   x = x % cf[1]
   d -= 1
 return res

Test the Continued Fraction for sqrt2

def sqrt2_a():

 yield 1
 while (True):
   yield 2

def sqrt2_b():

 while (True):
   yield 1

cf = CF(sqrt2_a(), sqrt2_b(), 950) print(pRes(cf, 200))

1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147

Test the Continued Fraction for Napier's Constant

def Napier_a():

 yield 2
 n = 1
 while (True):
   yield n
   n += 1

def Napier_b():

 n=1
 yield n
 while (True):
   yield n
   n += 1

cf = CF(Napier_a(), Napier_b(), 950) print(pRes(cf, 200))

2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746639193200305992181741359662904357290033429526059563073813232862794349076323382988075319525101901

Test the Continued Fraction for Pi

def Pi_a():

 yield 3
 while (True):
   yield 6

def Pi_b():

 x = 1
 while (True):
   yield x*x
   x += 2

cf = CF(Pi_a(), Pi_b(), 950) print(pRes(cf, 10))

3.1415926532</lang>

@@ Line 1: / Line 1: @@
 {{draft task}}A number may be represented as a continued fraction (see http://mathworld.wolfram.com/ContinuedFraction.html) as follows:
+:<math>cf = a_1 + \cfrac{b_1}{a_2 + \cfrac{b_2}{a_3 + \cfrac{b_3}{\mathrm{etc\ldots}}}}</math>
-[[Image:Continued_Fraction.PNG|center|Continued Fraction|200px]]
 The task is to write a program which generates such a number and prints a real representation of it.
 The code should be tested by calculating and printing the square root of 2, Napier's Constant, and Pi.
-For the square root of 2 bN is always 1. a1 is 1 then aN is 2.
+For the square root of 2, <math>b_N</math> is always <math>1</math>. <math>a_1</math> is <math>1</math> then <math>a_N</math> is <math>2</math>.
-For Napier's Constant a1 is 2, then aN is N-1. b1 is 1 then bN is N-1.
+For Napier's Constant, <math>a_1</math> is <math>2</math>, then <math>a_N</math> is <math>N-1</math>. <math>b_1</math> is <math>1</math> then <math>b_N</math> is <math>N-1</math>.
-For Pi a1 is 3 then aN is always 6. bN is (2N-1)(2N-1).
+For Pi, <math>a_1</math> is 3 then <math>a_N</math> is always <math>6</math>. <math>b_N</math> is <math>(2N-1)(2N-1)</math>.
 =={{header|Python}}==
+<lang python># The continued Fraction
-<lang python>
-# The continued Fraction
 def CF(a, b, t):
   if 0 < t:
@@ Line 86: / Line 82: @@
 cf = CF(Pi_a(), Pi_b(), 950)
 print(pRes(cf, 10))
-#3.1415926532
+#3.1415926532</lang>
-</lang>