RatioFromNumber

RatioFromNumber [/MERR=maxError /MITS=maxIterations /V[=v ]] num

The RatioFromNumber operation computes two integers whose ratio is equal to num +/- maxError. The ratio is returned in variables V_numerator and V_denominator.

Parameters

num is the number to approximate by V_numerator/V_denominator.

Flags

/MERR=maxError

Specifies the maximum tolerable error. The computed ratio differs from num by no more than maxError (default value is num*1e-6).

maxError must be a value between 0 and num. See Details about setting maxError to 0.

/MITS = maxIterations

Keeps returned values small by specifying a small number for maxIterations.

maxIterations must be a value between 1 and 32767 (default is 100).

/V[=v ]

Prints output variables to history.

v =1:	Prints variables (same as /V).
v =0:	Nothing printed (same as no /V).

Details

The ratio is computed by continued fraction expansion and recurrence relations for the convergents and checking num - (V_numerator/V_denominator) against maxError .

Setting maxError = 0 computes a maximally accurate ratio. The returned values can be surprisingly large:

RatioFromNumber/V/MERR=0 (1/1666)
  V_numerator= 4398046511104; V_denominator= 7.3271454874993e+15; 
  ratio= 0.00060024009603842; V_difference= 0;

Using the default /MERR returns the expected 1 and 1666. The difference is attributable to floating-point roundoff errors.

The ratio is computed by continued fraction expansion and recurrence relations for the convergents and checking num - (V_numerator/V_denominator) against /MERR.

Output Variables

RatioFromNumber sets the following output variables:

V_difference

V_numerator/V_denominator - num

(positive if the approximation is too big).

V_flag

0:	V_difference less than or equal to /MERR.
1:	V_difference greater than /MERR.

V_numerator, V_denominator

Values for the numerator and denominator. The ratio of V_numerator/V_denominator approximates num.

V_iterations

The number of iterations actually used.

RatioFromNumber prints the output variables if you specify /V or /V=1 but only when running in the main thread.

Examples

RatioFromNumber/V pi
  V_numerator= 355; V_denominator= 113; ratio= 3.141592920354; 
  V_difference= 2.6676418940497e-07; V_iterations= 3;

RatioFromNumber/V/MITS=2 pi
  V_numerator= 22; V_denominator= 7; ratio= 3.1428571428571; 
  V_difference= 0.0012644892673497; V_iterations= 1;