A Tiny (WP34S) program does a lot

[Dieter]

(11-29-2014 12:34 AM)BarryMead Wrote:  What method/software did you use to obtain that result.

Wolfram Alpha. Simply enter "Minimum of Gamma(x+1) near 1.5". ;-)

(11-29-2014 12:34 AM)BarryMead Wrote:  I did notice that the value you quoted for the factorial was the same as my findings. Perhaps the result was in error because it could not be discriminated from the correct value in the SLV function which probably stops searching when it sees no change in the value returned from the derivative function.

It will stop as soon as the solved function becomes zero. Depending on the second derivative, there may a more or less wide range of 16-digit values that return the same function result near or equal to zero.

However, this is not quite the case here – the second derivative is roughly 1 at this point, so changing x by 1 ULP = 10^–16 will change the first derivative by a similar amount. Actually the exact first derivative (i.e. the digamma function) of your original result is not quite zero:
digamma(1.4616321449682268) = –1.31 · 10^–13

The sign actually changes between
digamma(1.4616321449683623) = –3,99 · 10^–17
and
digamma(1.4616321449683624) = +5,68 · 10^–17

The difference probably is due to the fact that the 34s f'(x) function provides just an approximation of the first derivative.

Since you posted a more precise result: Wolfram Alpha returns
x = 0.46163 21449 68362 34126 26595 42325 72132 84682
y = 0.88560 31944 10888 70027 88159 00582 58873 32080

Dieter