How to resolve the algorithm Continued fraction/Arithmetic/G(matrix ng, continued fraction n) step by step in the J programming language
How to resolve the algorithm Continued fraction/Arithmetic/G(matrix ng, continued fraction n) step by step in the J programming language
Table of Contents
Problem Statement
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: I output a term if the integer parts of
a b
{\displaystyle {\frac {a}{b}}}
and
a
1
b
1
{\displaystyle {\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
∞
{\displaystyle \infty }
. When I input a term t my internal state:
[
a
1
a
b
1
b
]
{\displaystyle {\begin{bmatrix}a_{1}&a\b_{1}&b\end{bmatrix}}}
is transposed thus
[
a +
a
1
∗ t
a
1
b +
b
1
∗ t
b
1
]
{\displaystyle {\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:
[
a
1
a
b
1
b
]
{\displaystyle {\begin{bmatrix}a_{1}&a\b_{1}&b\end{bmatrix}}}
is transposed thus
[
b
1
b
a
1
−
b
1
∗ t
a − b ∗ t
]
{\displaystyle {\begin{bmatrix}b_{1}&b\a_{1}-b_{1}t&a-bt\end{bmatrix}}}
When I need a term t but there are no more my internal state:
[
a
1
a
b
1
b
]
{\displaystyle {\begin{bmatrix}a_{1}&a\b_{1}&b\end{bmatrix}}}
is transposed thus
[
a
1
a
1
b
1
b
1
]
{\displaystyle {\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: Using a generator for
2
{\displaystyle {\sqrt {2}}}
(e.g., from Continued fraction) calculate
1
2
{\displaystyle {\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
1 +
1
2
2
{\displaystyle {\frac {1+{\frac {1}{\sqrt {2}}}}{2}}}
do this now to cross the starting line and begin the race.
Let's start with the solution:
Step by Step solution about How to resolve the algorithm Continued fraction/Arithmetic/G(matrix ng, continued fraction n) step by step in the J programming language
Source code in the j programming language
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
)
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
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
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
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
reciprocal=:(0 1,:1 0)&ng4cf
(reciprocal each *&((+%)/@>) ]) arbs
1 1 1 1 1 1 1 1 1 1 1 1
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
(+%)/1 5 2
1.18182
plus1r2 1 5 2
1 1 2 7
(+%)/plus1r2 1 5 2
1.68182
(+%)/3 7x
22r7
times7r22 3 7x
1
plus1r2 3 7x
3 1 1 1 4
(+%)/plus1r2 3 7x
3.64286
(+%)/x:plus1r2 3 7x
51r14
times1r4 3 7x
0 1 3 1 2
(+%)/x:times1r4 3 7x
11r14
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
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
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
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