Bernoulli, Ramanujan, Toeplitz e le matrici triangolari Carmine Di - PDF document

Due Giorni di Algebra Lineare Numerica www.dima.unige.it/ ∼ dibenede/2gg/home.html Genova, 16–17 Febbraio 2012 Bernoulli, Ramanujan, Toeplitz e le matrici triangolari Carmine Di Fiore, Francesco Tudisco, Paolo Zellini Speaker: Carmine Di Fiore By using one of the definitions of the Bernoulli numbers, we observe that they solve particular odd and even lower triangular Toeplitz (l.t.T.) systems. In a paper Ramanujan writes down a sparse lower triangular system solved by Bernoulli numbers; we observe that such system is equivalent to a sparse l.t.T. system. The attempt to obtain the sparse l.t.T. Ramanujan system from the l.t.T. odd and even systems, leads us to study efficient methods for solving generic l.t.T. systems. 1

Bernoulli numbers are the rational numbers satisfying the following identity + ∞ + ∞ t B n (0) t n = − 1 B 2 k (0) � � (2 k )! t 2 k . e t − 1 = 2 t + n ! n =0 k =0 So, they satisfy the following linear equations � j [ j − 1 2 ] � − 1 � 2 j + B 2 k (0) = 0 , j = 2 , 3 , 4 , . . . , 2 k k =0 � 2  �  0   � 4 � 4  � �      B 0 (0) 1     0 2 B 2 (0) 2       � 6 � 6 � 6       � � � j even : B 4 (0) = 3 ,             0 2 4 B 6 (0) 4       � 8 � 8 � 8 � 8  � � � �  · ·     0 2 4 6   · · · · · � 1 �   0   � 3 � 3  � �      B 0 (0) 1     0 2 B 2 (0) 3 / 2       � 5 � 5 � 5  � � �      B 4 (0) 5 / 2 j odd : = .             0 2 4 B 6 (0) 7 / 2       � 7 � 7 � 7 � 7   � � � � · ·     0 2 4 6   · · · · · In other words, the Bernoulli numbers can be obtained by solving (by forward substitution) a lower triangular linear system (one of the above two). For example, by forward solving the first system, I have obtained the first Bernoulli numbers: B 0 (0) = 1 , B 2 (0) = 1 6 , B 4 (0) = − 1 30 , B 6 (0) = 1 42 , B 8 (0) = − 1 30 , B 10 (0) = 5 66 , B 12 (0) = − 691 2730 , B 14 (0) = 7 6 ≈ 1 . 16 , B 16 (0) = − 47021 6630 ≈ − 7 . 09 , . . . Bernoulli numbers appear in the Euler-Maclaurin summation formula, and, in particular, in the expression of the error of the trapezoidal quadrature rule as sum of even powers of the integration step h (the expression that justifies the efficiency of the Romberg-Trapezoidal quadrature method). Bernoulli numbers are also often involved when studying the Riemann-Zeta function. For example, well known is the following Euler formula: + ∞ ζ (2 n ) = 4 n | B 2 n (0) | π 2 n 1 � ζ ( s ) = k s , , n ∈ 1 , 2 , 3 , . . . 2(2 n )! k =1 (see also [Riemann’s Zeta Function, H. M. Edwards, 1974]). The Ramanujan’s paper we refer in the following is entitled “Some properties of Bernoulli’s numbers” (1911). 2

The coefficient matrices of the previous two lower triangular linear systems are submatrices of the matrix X displayed here below: � 0 �   0   � 1 � 1 � �       1 0   � 2 � 2 � 2  � � �    1     1 2 1 1 0     � 3 � 3 � 3 � 3  � � � �    1 2 1         X = 1 3 = 1 3 3 1 . 0 2     � 4 � 4 � 4 � 4 � 4     � � � � � 1 4 6 4 1         1 3 4 1 5 10 10 5 1  0 2    � 5 � 5 � 5 � 5 � 5 � 5   � � � � � � · · · · · · ·     1 3 5  0 2 4  � 6 � 6 � 6 � 6 � 6 � 6 � 6   � � � � � � �     1 3 5 6  0 2 4  · · · · · · · · One can easily observe that X can be rewritten as a power series:   0 1 0   + ∞   1 2 0 �   k ! Y k , X = Y =   3 0   k =0   4 0   · · � � 1 1 i − 1 ( i − j )![ Y i − j ] ij = Proof: [ X ] ij = ( i − j )! j · · · ( i − 2)( i − 1) = , 1 ≤ j ≤ i ≤ n. j − 1 This remark is the starting point in order to show that � 2 �   0 � 4 � 4   � � 2       12 0 2 + ∞   � 6 � 6 � 6 1   (2 k + 2)! φ k =   � � � � 30 ,         56 0 2 4   k =0   � 8 � 8 � 8 � 8   � � � �   ·   0 2 4 6   · · · · · � 1 �   0 � 3 � 3   � � 1       3 0 2 + ∞   � 5 � 5 � 5 1   (2 k + 1)! φ k =   � � � � 5 ,         7 0 2 4     k =0 � 7 � 7 � 7 � 7   � � � �   ·   0 2 4 6   · · · · ·   0 2 0     12 0   where φ = , 2 = 1 · 2, 12 = 3 · 4, 30 = 5 · 6, . . . .   30 0     56 0   · · 3

It follows that the linear systems solved by the Bernoulli numbers, can be rewritten as follows, in terms of the matrix φ :     1 / 2 1 / 2   B 0 (0) 2 / 12 1 / 6     B 2 (0) + ∞       1 3 / 30 1 / 10 �       (2 k + 2)! φ k =: q e , 2 B 4 (0) = 2 = 2 (almosteven)       4 / 56 1 / 14       B 6 (0) k =0       5 / 90 1 / 18     · · ·     1 / 1 1   B 0 (0) (3 / 2) / 3 1 / 2     B 2 (0) + ∞       1 � (5 / 2) / 5 1 / 2       (2 k + 1)! φ k =: q o . B 4 (0) = = (almostodd)       (7 / 2) / 7 1 / 2       B 6 (0)     k =0   (9 / 2) / 9 1 / 2     · · · Now we transform φ into a Toeplitz matrix. We have that   0     d − 1 d 1 1 2 0   d − 1 d 2       2 12 0 DφD − 1 =       d − 1 d 3       3 30 0       d − 1 d 4       4 56 0   · · · ·     0 0 2 d 2 0   1 0   d 1     12 d 3  0  1 0    d 2  = = xZ, Z = ,    30 d 4  0 1 0     d 3     56 d 5 1 0 0     d 4 · · · · iff d k = x k − 1 d 1 (2 k − 2)!, k = 1 , 2 , 3 , . . . , iff   1 x   2!   x 2    4!  D = d 1 D x , D x = .  ·    x n − 1     (2 n − 2)! · We are ready to introduce the two even and odd lower triangular Toeplitz (l.t.T.) systems solved by the Bernoulli numbers. Set   B 0 (0) B 2 (0)   b =   B 4 (0)   · where the B 2 i (0), i = 0 , 1 , 2 , . . . , are the Bernoulli numbers. 4

Then the (almosteven) system � + ∞ (2 k +2)! φ k b = q e is equivalent to the system � + ∞ 1 (2 k +2)! ( D x φD − 1 1 x ) k ( D x b ) = k =0 2 k =0 2 D x q e , i.e. to the following l.t.T. even system: + ∞ x k � (2 k + 2)! Z k ( D x b ) = D x q e . 2 (even) k =0 Idem, the (almostodd) system � + ∞ (2 k +1)! φ k b = q o is equivalent to the system � + ∞ 1 (2 k +1)! ( D x φD − 1 1 x ) k ( D x b ) = k =0 k =0 D x q o , i.e. to the following l.t.T. odd system: + ∞ x k � (2 k + 1)! Z k ( D x b ) = D x q o . (odd) k =0 So, Bernoulli numbers can be computed by using a l.t.T. linear system solver . Such solver yields the following vector z :   1 · B 0 (0) x 2! B 2 (0)     x 2 4! B 4 (0)     ·     z = D x b = , x s  (2 s )! B 2 s (0)      ·    x n − 1  (2 n − 2)! B 2 n − 2 (0)   · from which one obtains the vector of the first n Bernoulli numbers: { b } n = { D − 1 x z } n . Why x positive different from 1 may be useful? A suitable choice of x can make possible and more stable the computation via a l.t.T. solver of the entries z i of z for very large i . In fact, since √ i 2 i x i x i p i +1 x (2 i )! B 2 i (0) ≈ ( − 1) i +1 p i , p i = (2 i )!4 πi ( πe ) 2 i , → 4 π 2 , p i (2 i )! B 2 i (0) | → 0 (+ ∞ ) if x < 4 π 2 ( x > 4 π 2 ), both bad situations. Instead, for x ≈ 4 π 2 = we have that | x i 39 . 47 .. the sequence | x i (2 i )! B 2 i (0) | , i = 0 , 1 , 2 , . . . , should be lower and upper bounded. . . . | x 2 (4)! B 4 (0) | ≤ 1 iff | x | ≤ 26 . 84 | x 4 (8)! B 8 (0) | ≤ 1 iff | x | ≤ 33 . 2 | x 8 (16)! B 16 (0) | ≤ 1 iff | x | ≤ 36 . 2 (32)! B 32 (0) | ≤ 1 about iff | x | 16 ≤ (8 . 54) 32 | x 16 1 iff | x | ≤ 37 . 82 1293 4 · 7 . 09 (2 s )! 4 √ πs ( πe ) 2 s | ≤ 1 iff | x | s ≤ (2 s )! ( πe ) 2 s | x s (2 s )! B 2 s (0) | ≤ 1 about iff | x s s 2 s 4 √ πs . . . s 2 s More generally, the parameter x should be used to make more stable the l.t.T. solver. 5