On the properties and the construction of finite-row ( t , s - PowerPoint PPT Presentation

On the properties and the construction of finite-row ( t , s )-sequences 1 Roswitha Hofer 2 Institute of Financial Mathematics, University of Linz, Austria 13.02.12, MCQMC12, Sydney, Australia 1 Partially joint work with Pirsic and with Larcher 2 supported by the Austrian Science Fund (FWF), Project P21943. Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 1 / 19

Overview of my talk Definition of finite-row ( t , s )-sequences Existence and examples of such sequences finite-row ( t , s )-sequences and Niederreiter-Halton sequences ... Experiments Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 2 / 19

Definition (digital sequences in base q by Niederreiter 1987) s ≥ 1 , q ∈ P . Let C 1 , . . . , C s be N × N -matrices over the finite field Z q . � � x (1) n , . . . , x ( s ) ( x n ) n ≥ 0 , x n = n x ( i ) is generated as follows: n = n 0 + n 1 q + n 2 q 2 + · · · n � � ⊤ C i · ( n 0 , n 1 , . . . ) ⊤ =: y ( i ) 0 , y ( i ) ∈ Z N 1 , . . . q and := y ( i ) + y ( i ) q 2 + y ( i ) x ( i ) 0 1 2 q 3 + · · · ∈ [0 , 1) . n q If the generator matrices satisfy that each row contains just finitely many nonzero entries ... “finite-row (digital) sequence”. Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 3 / 19

digital ( t , s )-sequences – condition on the rank structure! Here the generator matrices fulfill for all m ∈ N and all d 1 + . . . + d s = m − t , ( d i ≥ 0) that     m m � �� � � �� � gggggg } d 1 gggggg } d s     C 1 =  , . . . , C s =        m � �� � } gggggggggg d 1 the matrix has rank m − t . . . } d s Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 4 / 19

Example (van der Corput sequence = finite-row (0 , 1)-sequence) The van der Corput sequence in base q is a finite-row (digital) (0 , 1)-sequence, since the generator matrix,   1 0 0 0 . . . 0 1 0 0   . . .   0 0 1 0 ∈ Z N × N   . . . ,   q 0 0 0 1   . . .   . . . . ... . . . . . . . . satisfies the condition on the rank structure and is a finite-row matrix. Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 5 / 19

Example (digital (0 , s )-sequences by Faure 1982) For prime base q , the Pascal matrices P ( i ) defined by � 1 � � 2 � � 3 �   i 1 i 2 i 3 1 . . . 0 0 0 � 2 � � 3 � i 1 i 2 0 1   . . . P ( i ) := 1 1  ∈ Z N × N   � 3 � , i 1  0 0 1  q . . . 2  . . . . ... . . . . . . . . i ∈ { 0 , 1 . . . , q − 1 } generate a digital (0 , q )-sequence in base q . For q = 2 (Sobol 1967) the matrices are sketched: 1 10 20 32 1 10 20 32 1 1 1 1 10 10 10 10 20 20 20 20 32 32 32 32 1 10 20 32 1 10 20 32 Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 6 / 19

Research Question Let s > 1 . Can finite rows satisfy for all m ∈ N and d 1 , . . . , d s ≥ 0 with d 1 + . . . + d s = m m � �� � · · · · · · · · · · · · } d 1 . has rank m ? . . · · · · · · · · · · · · } d s Do there exist multi-dimensional finite-row (0 , s ) -sequences? Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 7 / 19

Lower bounds on the lengths: For s = 2 it is not so hard to check that     x 0 0 0 0 0 0 x 0 0 0 0 0 x . . . . . . x x x 0 0 0 0 x x x x 0 0 0     . . . . . .     x 0 0  and x 0 x x x x x x x x x     . . . . . .    . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . where the ‘ x ’ entries are nonzero have “lowest possible row lengths”. Theorem (Faure & Tezuka 2000) If - C 1 , . . . , C s ∈ Z N × N generate a digital (0 , s ) -sequence in prime base q q ≥ s and - M is a NUT matrix in Z q . ( “Scrambling Matrix” ) Then the matrices C 1 M , . . . , C s M generate a digital (0 , s ) -sequence. Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 8 / 19

Idea (Construct a proper NUT scrambling matrix)   1 0 0 0     . . . 1 0 0 0 x x 1 0 0 0 0 0 0 1 0 0 . . . . . .   . . . 0 1 1 0 x x 0 1 1 0 0 0       0 0 1 0 . . . . . .       ·  = . . . 0 0 1 1 x x 0 0 1 1 x 0       0 0 0 1 . . . . . .      . . . . . . . . . . . . . . . ... ...   . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . .   1 1 1 1     1 0 0 0 1 1 0 0 0 0 . . . x x 0 1 0 1 . . . . . .   . . . 0 1 1 0 x x 0 1 1 1 0 0       0 0 1 1 . . . . . .       · 0 0 1 1  = 0 0 1 0 . . . x x x x       0 0 0 1 . . . . . .      . . . . . . . . . . . . . . . ... ...   . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 9 / 19

Theorem (H.& Larcher 2009) Let s ∈ N and q ∈ P . For all generator matrices C 1 , . . . , C s ∈ Z N × N of a q digital (0 , s ) -sequence in base q there exists a NUT matrix M ∈ Z N × N q such that C 1 M , . . . , C s M ∈ Z N × N are generator matrices of a finite-row q (0 , s ) -sequence in base q and they have even lowest possible row lengths . Figure: The Pascal matrices in base 2 and the modified matrices with lowest possible row lengths. 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 32 32 32 32 32 32 32 32 1 10 20 32 1 10 20 32 1 10 20 32 1 10 20 32 Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 10 / 19

Figure: The Pascal matrices in base 5: 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 40 40 50 50 50 50 50 50 50 50 50 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 Figure: The modified matrices in base 5: 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 40 40 50 50 50 50 50 50 50 50 50 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 1 10 20 30 40 50 Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 11 / 19

A formula for the scrambling matrix? Theorem (H. & Pirsic 2011) Let q ∈ P . Then the following matrix is a suitable scrambling matrix for the Pascal matrices in base q. �� �� r S = , j − 1 j ≥ 1 , r ≥ 0 � n � where is the Karamata notation for the unsigned Stirling numbers of m the first kind. Furthermore the new generator matrices P (0) S , P (1) S , . . . , P ( q − 1) S , satisfy   1 − 1 0 0 0 · · · 0 1 − 2 0 0 · · ·   P ( i ) S = SQ i   , with Q =  . 0 0 1 − 3 0 · · ·    . ... ... ... . . · · · Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 12 / 19

Further results on finite-row ( t , s )-sequences (H.& Pirsic, unp.) A formula for the scrambling matrix which goes along with the generator matrices of classical Niederreiter sequences. (H., 2012) Explicit construction of finite-row (0 , s )-sequences over finite fields F q . (H., unp.) Explicit construction of finite-row ( t , s )-sequences over finite fields F q . Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 13 / 19

Motivation of finite-row sequences ... Koksma-Hlawka Inequality . � � N − 1 � � � [0 , 1] s f ( x ) d x − 1 � � ≤ V ( f ) D ∗ � � f ( x n ) � � N N � n =0 Need “ low-discrepancy sequence ”. Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 14 / 19

Examples of low-discrepancy sequences digital ( t , s )-sequences (Sobol sequences, Faure sequences, Niederreiter sequences, ...) Halton sequences: ... Definition (Halton-sequence, Halton 1960) Take s different primes q 1 , . . . , q s and juxtapose the van der Corput sequences (a digital (0 , 1) -sequence) in the different bases q 1 , . . . , q s . Observation We take s one-dimensional low-discrepancy sequences and get an s-dimensional low-discrepancy sequence! Roswitha Hofer (Linz) finite-row ( t , s )-sequences MCQMC12 15 / 19