continued fraction
Mathworld





a

0


+









b

1












a

1


+









b

2












a

2


+









b

3












a

3


+
⋱

















{\displaystyle a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+\ddots }}}}}}}


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, using the following coefficients:

For the square root of 2, use 




a

0


=
1


{\displaystyle a_{0}=1}

 then 




a

N


=
2


{\displaystyle a_{N}=2}

. 




b

N




{\displaystyle b_{N}}

 is always 



1


{\displaystyle 1}

.







2


=
1
+








1









2
+








1









2
+








1









2
+
⋱

















{\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}}


For Napier's Constant, use 




a

0


=
2


{\displaystyle a_{0}=2}

, then 




a

N


=
N


{\displaystyle a_{N}=N}

. 




b

1


=
1


{\displaystyle b_{1}=1}

 then 




b

N


=
N
−
1


{\displaystyle b_{N}=N-1}

.





e
=
2
+








1









1
+








1









2
+








2









3
+








3









4
+
⋱






















{\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+\ddots }}}}}}}}}


For Pi, use 




a

0


=
3


{\displaystyle a_{0}=3}

 then 




a

N


=
6


{\displaystyle a_{N}=6}

. 




b

N


=
(
2
N
−
1

)

2




{\displaystyle b_{N}=(2N-1)^{2}}

.





π
=
3
+








1









6
+








9









6
+








25









6
+
⋱

















{\displaystyle \pi =3+{\cfrac {1}{6+{\cfrac {9}{6+{\cfrac {25}{6+\ddots }}}}}}}




See also

   Continued fraction/Arithmetic for tasks that do arithmetic over continued fractions.


package main
 
import "fmt"
 
type cfTerm struct {
    a, b int
}
 
// follows subscript convention of mathworld and WP where there is no b(0).
// cf[0].b is unused in this representation.
type cf []cfTerm
 
func cfSqrt2(nTerms int) cf {
    f := make(cf, nTerms)
    for n := range f {
        f[n] = cfTerm{2, 1}
    }
    f[0].a = 1
    return f
}
 
func cfNap(nTerms int) cf {
    f := make(cf, nTerms)
    for n := range f {
        f[n] = cfTerm{n, n - 1}
    }
    f[0].a = 2
    f[1].b = 1
    return f
}
 
func cfPi(nTerms int) cf {
    f := make(cf, nTerms)
    for n := range f {
        g := 2*n - 1
        f[n] = cfTerm{6, g * g}
    }
    f[0].a = 3
    return f
}
 
func (f cf) real() (r float64) {
    for n := len(f) - 1; n > 0; n-- {
        r = float64(f[n].b) / (float64(f[n].a) + r)
    }
    return r + float64(f[0].a)
}
 
func main() {
    fmt.Println("sqrt2:", cfSqrt2(20).real())
    fmt.Println("nap:  ", cfNap(20).real())
    fmt.Println("pi:   ", cfPi(20).real())
}