If you have an approximation x=a/b,then the error is less than abs(x- a%b)%((b^2)SQR(5)). Another more exact estimate gives for two sequential ratios c/d and f/g for c/d an error estimate 1%(d*g). Leo Vohandu
> Members of the Forum - > > If I'm approximating, e.g. the square root of 3, with a matrix method > which returns an extended precision numerator and denominator, when I > work out the decimal equivalent of this, at what point do I run out of > significant digits? > > For example, > > (1 3,:1 1)&(+/ .*)^:(5+i.5)],.1 0 NB. Successive approximations of > %:3 > 76 > 44 > > 208 > 120 > > 568 > 328 > > 1552 > 896 > > 4240 > 2448 > (%:3)-%/(1 3,:1 1)&(+/ .*)^:10],.x: 1 0 NB. How far off are > successive > approximations? > _6.6086991e_6 > (%:3)-%/(1 3,:1 1)&(+/ .*)^:20],.x: 1 0 > _1.2607915e_11 > (%:3)-%/(1 3,:1 1)&(+/ .*)^:30],.x: 1 0 > _2.220446e_16 > (%:3)-%/(1 3,:1 1)&(+/ .*)^:40],.x: 1 0 NB. More than 16 digits > 0 > (1 3,:1 1)&(+/ .*)^:40],.x: 1 0 NB. How precise is this in > decimal? > 144052522725670912 > 83168762773110784 > 2^.(1 3,:1 1)&(+/ .*)^:40],.x: 1 0 NB. Bits/number - relevant? > 56.999373 > 56.206891 > > From the progression of exponents in the cases for 10, 20, and 30 > iterations, I'm guessing the answer should be about 21 significant > digits but is there a way to better quantify this? > > Thanks, > > Devon > -- > Devon McCormick, CFA > ^me^ at acm. > org is my > preferred e-mail > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
