Rational approximations

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Rational approximations*From*: cfw2 at po.cwru.edu (Charles F. Wells)*Date*: Wed, 6 Jan 93 17:29:56 -0500

The following code defines a command Approximate[r,k] that calculates the rational number that best approximates the real number r, under the constraint that the numerator and denominator of the rational number in lowest terms each has no more than k digits (base 10). In[1] := RelativelyPrime[{a_,b_}] := GCD[a,b] == 1; In[2] := RelativelyPrime[a_,b_] := RelativelyPrime[{a,b}] In[3] := Approximate[r_, digits_Integer?Positive] := Module[{pairlist, lgth, bound = 10^digits-1, start, a, b, error}, start = {1,1,1-r // N // Abs}; pairlist = Select[ Flatten[Table[{a,b},{a,1,bound},{b,1,bound}],1], RelativelyPrime]; lgth = Length[pairlist]; Do[(a = pairlist[[k]][[1]]; b = pairlist[[k]][[2]]; error = r- (a/b) // N // Abs; If[ error < start[[3]], start = {a,b,error} ] ), {k,1,lgth}]; {start[[1]]/start[[2]], start[[3]]} ] In[4] := Approximate[Pi,2] 22 Out[4] = {--, 0.00126449} 7 In[5] := Approximate[E,2] 87 Out[5] = {--, 0.000468172} 32 In[6] := Approximate[Sqrt[2],2] // Timing 99 Out[6] = {39.6 Second, {--, 0.0000721519}} 70 (This was on a 486-33 Gateway machine). I have two questions: (1) Is there a more functional way to define Approximate? (2) I wrote this message using EditPro, a Windows editor I just bought so that I could cut and paste between Mathematica and text files. The cutting and pasting worked very well, except that I was unable to copy the "In[n]" and "Out[n]" part of the notebook to put them into this message, so I had to type them by hand. Is there a way to get that part of the notebook to copy too? Charles Wells Department of Mathematics Case Western Reserve University 10900 Euclid Avenue Cleveland, OH 44106-7058 216-368-2893