User:Thebigh/mysandbox: Difference between revisions
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this is my sandbox |
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The '''Minkowski question-mark function''' converts the continued fraction representation {{math|[a<sub>0</sub>; a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, ...]}} of a number into a binary decimal representation in which the integer part {{math|a<sub>0</sub>}} is unchanged and the {{math|a<sub>1</sub>, a<sub>2</sub>, ...}} become alternating runs of binary zeroes and ones of those lengths. The decimal point takes the place of the first zero. |
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Thus, {{math|?}}(31/7) = 71/16 because 31/17 has the continued fraction representation {{math|[4;2,3]}} giving the binary expansion {{math|4 + 0.0111<sub>2</sub>}}. |
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Among its interesting properties is that it maps roots of quadratic equations, which have repeating continued fractions, to rational numbers, which have repeating binary digits. |
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The question-mark function is continuous and monotonically increasing, so it has an inverse. |
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* Produce a function for {{math|?(x)}}. Be careful: rational numbers have two possible continued fraction representations- choose the one that will give a binary expansion ending with a 1. |
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* Produce the inverse function {{math|?<sup>-1</sup>(x)}} |
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* Verify that {{math|?(φ)}} = 5/3, where {{math|φ}} is the Greek golden ratio. |
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* Verify that {{math|?<sup>-1</sup>(-5/9)}} = (√13 - 7)/6 |
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* Verify that the two functions are inverses of each other by showing that {{math|?<sup>-1</sup>(?(x))}}={{math|x}} and {{math|?(?<sup>-1</sup>(y))}}={{math|y}} for {{math|x, y}} of your choice |
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Don't worry about precision error in the last few digits. |
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==header|FreeBASIC== |
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<lang freebasic>#define MAXITER 151 |
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#define MAXITER 151 |
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function minkowski( x as double ) as double |
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if x>1 or x<0 then return int(x)+minkowski(x-int(x)) |
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dim as ulongint p = int(x) |
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dim as ulongint q = 1, r = p + 1, s = 1, m, n |
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dim as double d = 1, y = p |
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while true |
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d = d / 2.0 |
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if y + d = y then exit while |
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m = p + r |
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if m < 0 or p < 0 then exit while |
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n = q + s |
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if n < 0 then exit while |
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if x < cast(double,m) / n then |
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r = m |
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s = n |
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else |
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y = y + d |
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p = m |
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q = n |
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end if |
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wend |
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return y + d |
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end function |
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function minkowski_inv( byval x as double ) as double |
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if x>1 or x<0 then return int(x)+minkowski_inv(x-int(x)) |
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if x=1 or x=0 then return x |
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redim as uinteger contfrac(0 to 0) |
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dim as uinteger curr=0, count=1, i = 0 |
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do |
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x *= 2 |
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if curr = 0 then |
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if x<1 then |
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count += 1 |
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else |
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i += 1 |
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redim preserve contfrac(0 to i) |
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contfrac(i-1)=count |
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count = 1 |
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curr = 1 |
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x=x-1 |
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endif |
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else |
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if x>1 then |
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count += 1 |
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x=x-1 |
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else |
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i += 1 |
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redim preserve contfrac(0 to i) |
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contfrac(i-1)=count |
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count = 1 |
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curr = 0 |
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endif |
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end if |
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if x = int(x) then |
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contfrac(i)=count |
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exit do |
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end if |
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loop until i = MAXITER |
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dim as double ret = 1.0/contfrac(i) |
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for j as integer = i-1 to 0 step -1 |
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ret = contfrac(j) + 1.0/ret |
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next j |
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return 1./ret |
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end function |
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print minkowski( 0.5*(1+sqr(5)) ), 5./3 |
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print minkowski_inv( -5./9 ), (sqr(13)-7)/6 |
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print minkowski(minkowski_inv(0.718281828)), minkowski_inv(minkowski(0.1213141516171819)) |
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</lang> |
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{{out}} |
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<pre> 1.666666666669698 1.666666666666667 |
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-0.5657414540893351 -0.5657414540893352 |
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0.7182818280000092 0.1213141516171819</pre> |
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Revision as of 14:10, 12 November 2020
this is my sandbox