So sánh và kiểm định độ phân kỳ của các dãy và tập điểm tựa ngẫu nhiên một chiều

Discrepencies of quasi-random sequences depend on bases. The choice of a suitable base has more experimental significance than theoretical. The main object of this paper is onedimensional sequences of two simple forms: Van der Corput with base p and Richtmyer {(p)}, where pis a prime number. | ’ Tap ch´ Tin hoc v` Diˆu khiˆn hoc, , (2007), 251–259 ı e e . . a ` . ’ ´ ` ˆ ˆ ˆ ` ’ ´ ˜ SO SANH VA KIEM DINH DO PHAN KY CUA CAC DAY . . . ’ ˜ ` ˆ ˆ ˆ ˆ ˆ ` ˆ VA TAP DIEM TU A-NGAU-NHIEN MOT CHIEU . . . . . ˜ ˆ ˆ ˜ ` ˆ VU HOAI CHU O NG, NGUYEN CONG DI` U E Viˆn Cˆng nghˆ thˆng tin, Viˆn Khoa hoc v` Cˆng nghˆ Viˆt Nam e o e o e e e . . . . a o . . Abstract. Discrepencies of quasi-random sequences depend on bases. The choice of a suitable base has more experimental significance than theoretical. The main object of this paper is onedimensional sequences of two simple forms: van der Corput with base p and Richtmyer {(p)}, √ where p is a prime number. Besides the case α(p) = p is well-known, we tested following cases α(p) = ln(p), α(p) = sin(p), α(p) = cos(p), α(p) = arctan(p) and α are irrational mathematical constants. We also introduced equidistant (k/N , midpoint, V ) point-sets and two pseudo-random sequences R for comparison. Criteria for ranking is a mean discrepancy computed from 21 different values N . Computational results show that modified forms generate numerical sequences with discrepancy in the extreme: very small or very big depending on a choice of concrete base. The ranking table lets us define priority using this sequences in certain quasi-Monte Carlo schemes. ´ ˜ ´ ´ T´m t˘t. Dˆ phˆn k` cua c´c d˜y phu thuˆc v`o co. sˆ d`ng dˆn. Viˆc chon o a a e o a o o a y ’ a a . e e . . . u . . . . sˆ th´ l` vˆn dˆ c´ ´ ngh˜ nghiˆm l` l´ thuyˆt. Dˆi khao s´t ch´ ´ ´ e ´ ´ ’ a co o ıch . a a ` o y e a y e ınh ıa o . . . ` ´ ’ trong b`i n`y l` c´c d˜y mˆt chiˆu thuˆc hai dang gian: van der Corput co. sˆ p v` Richtmyer a a a a a o e o o a . . . √ . th`.a nhˆn, ch´ng tˆi thu. kiˆm ’ ´ ´ ’ e u a u o {(p)}, trong d´ p l` sˆ nguyˆn tˆ. Bˆn canh α(p) = p d˜ du . o a o e o e . a . ’ dinh c´c dang cai biˆn α(p) = ln(p), α(p) = sin(p), α(p) = cos(p), α(p) = arctan(p) v` α l` a e a a . . ’ ’ ` ng sˆ to´n .