Vectorize This Loop

x xxxxxxxxxx xxxxx xxx xxxx xxxxxxx xxxx x xxxxx-xxxxxxxxxxx xxxxxxxx. xx xxxxxxxx, xxx xxxx xxxxxxxxx xxxxx xxxxxx, xxx xxxxx xxxxxxx xxxxxxxx xx x xxxx xxxxxx, xxx xx xx xxx xxxxxxx xxxx xx xxx xxxxx:

xxx xxxxx_xxxxx(xxxxx, xxxxxx):
    xxx = []
    xxx x xx xxxxxx:
        xxxxxxxx = x.x
        xxx x xx xxxxx:
            xxxxxxxx = xxxxxxxx * x + x
        xxx.xxxxxx(xxxxxxxx)
    xxxxxx xxx

xxxx xxxxxxxxxx x xxxx xxxx xxx xxxxxxx xxxxx-xxxx xxxxxxxxx, xxxx-xxxxx xxxxxxxxx xxxxxxxx, xxx xxxxxx xxxx-xxxx xxxx: ( x_x = x_{x-x} xxxx x + x_x ), xxxx ( x_{-x} = x ), xxx xxxx xxx xxxxx ( x_{x-x} ) xx xxxx xxx xxxxx.

xxxx xxxx: xxxxxxxxx xxxxx_xxxxx(xxxxx, xxxxxx) xxxx xxxxxxxxx xxxxxxx (xxxx xxxxx xxxxxxxx xxxxxx (xx^{-x}); xxx xxxxx xxxxx xxxxxxxxxx, xxxxx xxxxx xxxxx xxxxxxxx) xxxx xxxxxx xxx xxx xxxxxxxxxxx xxxxx. xxxxx xx xxxxxxxxx, xxx xxx xxxxxxxx xxxxxxxxx xxxx xxx xxxxx xxxxxxxxxx xxxx xxxxx xxxxxxxxxx; x xxxx-xxxxxxxxx xxxx xxx xxxx xxxx xx xxx xxxx xxx xxx-xxxx xxxx xxxxxxx. xxxxxx x xxxx xx xxxxx xxxxxx xxxxxx, xxx xxx xxxxx, xx xxxxx xxxxx (xxxxx xxxx xxx xx xxxxxx).

xxxxxxxx

xxx(xxxxx_xxxxx([x.x, x.x, x.x], [x.x])[x] - x.xx) < xx-x
# xxxx

xxx(xxx(x - x) < xx-x xxx x, x xx xxx(xxxxx_xxxxx([xx.x, -x.x, x.x, x.x], [x.x, x.x]), [x.x, xx.x]))
# xxxx
# x = x xxxxx xxxx xxx xxxx xxxx; x = x xxxxx xxxxxxx xxx xxxx xxxxxxxxxx.

xxxxxxxxxxx

• x <= xxx(xxxxx) <= x * xx^x, xx xx xx xxxxxx, xxxx x <= x <= x.
• xxx xxxxxx xxxxxxxxxxx xxxx xx xxx,xxx xxxxx xxxxx xx xxxxxx: (xx^x) xxxxxx-xxxxx xxxx xxxxx xxx xxxxx xx xxx xxxx xx xxx xxxxx'x xxxxxx, xx xxxxxxxxxxx xxx xxxxxxxxxx xxxxxxx xx xxxxxxx xxx xxxx xxxx.
• xxxxx xxx xxxxxxxx xx xxx xxxxxx xxx xxxxxxxxxx xxxx xxxxxx xx x: (x^{xxxxxx}) xxxxxxxxxx xx xxxx, xxxxx xx xxxxxxx xxx xx xx xxxx (xxxxx xxxxx xxx xxxxx (xx^{-xxx})), xxx xxxxxxxx xx xxxx xxxxxx xxxxxxx xx xxxxxxxxxxx xx xxx.