Due Giori di Algebra Lieare Numerica www.dima.uige.it/ dibeede/gg/home.html Geova, 6 7 Febbraio Beroulli, Ramauja, Toeplitz e le matrici triagolari Carmie Di Fiore, Fracesco Tudisco, Paolo Zellii Speaker: Carmie Di Fiore By usig oe of the defiitios of the Beroulli umbers, we observe that they solve particular odd ad eve lower triagular Toeplitz (l.t.t.) systems. I a paper Ramauja writes dow a sparse lower triagular system solved by Beroulli umbers; we observe that such system is equivalet to a sparse l.t.t. system. The attempt to obtai the sparse l.t.t. Ramauja system from the l.t.t. odd ad eve systems, leads us to study efficiet methods for solvig geeric l.t.t. systems.
Beroulli umbers are the ratioal umbers satisfyig the followig idetity t + e t So, they satisfy the followig liear equatios j ] [ j + B () t +! t + B k () (k)! tk. k ( ) j B k k (), j, 3, 4,..., k ( ) j eve : ( ) ( ) 4 4 ( ) ( ) ( ) 6 6 6 ( ) ( ) ( 4 ) ( ) 8 8 8 8 4 6 ( ) j odd : ( ) ( ) 3 3 ( ) ( ) ( ) 5 5 5 ( ) ( ) ( 4 ) ( ) 7 7 7 7 4 6 B () B () B 4() B 6() B () B () B 4() B 6() I other words, the Beroulli umbers ca be obtaied by solvig (by forward substitutio) a lower triagular liear system (oe of the above two). For example, by forward solvig the first system, I have obtaied the first Beroulli umbers: 3 4 3/ 5/ 7/ B (), B () 6, B 4() 3, B 6() 4, B 8() 3, B () 5 66, B () 69 73, B 4() 7 6.6, B 6() 47 663 7.9,... Beroulli umbers appear i the Euler-Maclauri summatio formula, ad, i particular, i the expressio of the error of the trapezoidal quadrature rule as sum of eve powers of the itegratio step h (the expressio that justifies the efficiecy of the Romberg-Trapezoidal quadrature method). Beroulli umbers are also ofte ivolved whe studyig the Riema-Zeta fuctio. For example, well kow is the followig Euler formula:,. ζ(s) + k k s, ζ() 4 B () π,,, 3,... ()! (see also [Riema s Zeta Fuctio, H. M. Edwards, 974]). The Ramauja s paper we refer i the followig is etitled Some properties of Beroulli s umbers (9).
The coefficiet matrices of the previous two lower triagular liear systems are submatrices of the matrix X displayed here below: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 X ( ) ( ) ( ) ( 3 ) ( ) 3 3. 4 4 4 4 4 4 6 4 3 4 ( ) ( ) ( ) ( ) ( ) ( ) 5 5 5 5 5 5 5 5 3 4 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 6 6 6 6 6 6 3 4 5 6 Oe ca easily observe that X ca be rewritte as a power series: Proof: [X] ij X + k (i j)! [Y i j ] ij 3 56 k! Y k, Y 3 4 j (i )(i ) (i j)! This remark is the startig poit i order to show that ( ) where φ 3 56 3 5 7 ( ) i, j i. j ( ) ( ) 4 4 + ( ) ( ) ( ) (k + )! φk 6 6 6, k 4 ( ) ( ) ( ) ( ) 8 8 8 8 4 6 ( ) ( ) ( ) 3 3 + ( ) ( ) ( ) (k + )! φk 5 5 5, k 4 ( ) ( ) ( ) ( ) 7 7 7 7 4 6,, 3 4, 3 5 6,.... 3
It follows that the liear systems solved by the Beroulli umbers, ca be rewritte as follows, i terms of the matrix φ: / / B () + B () / /6 (k + )! φk B 4 () k B 6 () 3/3 4/56 / /4 : q e, (almosteve) 5/9 /8 + k (k + )! φk B () B () B 4 () B 6 () / (3/)/3 (5/)/5 (7/)/7 (9/)/9 / / / / Now we trasform φ ito a Toeplitz matrix. We have that d d DφD d 3 d 4 3 56 d d d3 d xz, Z 3 d4 d 3 56 d5 d 4 : q o. (almostodd) d d d 3 d 4, iff d k xk d, k,, 3,..., iff (k )! D d D x, D x x! x 4! x ( )!. We are ready to itroduce the two eve ad odd lower triagular Toeplitz (l.t.t.) systems solved by the Beroulli umbers. Set B () b B () B 4 () where the B i (), i,,,..., are the Beroulli umbers. 4
The the (almosteve) system + k (k+)! φk b q e is equivalet to the system + k (k+)! (D xφd x ) k (D x b) D x q e, i.e. to the followig l.t.t. eve system: + k Idem, the (almostodd) system + k D x q o, i.e. to the followig l.t.t. odd system: x k (k + )! Zk (D x b) D x q e. (eve) (k+)! φk b q o is equivalet to the system + k + k (k+)! (D xφd x ) k (D x b) x k (k + )! Zk (D x b) D x q o. (odd) So, Beroulli umbers ca be computed by usig a l.t.t. liear system solver. Such solver yields the followig vector z: B () x z D x b! B () x 4! B 4() x s (s)! B s() x ( )! B () from which oe obtais the vector of the first Beroulli umbers: {b} {D x z}. Why x positive differet from may be useful A suitable choice of x ca make possible ad more stable the computatio via a l.t.t. solver of the etries z i of z for very large i. I fact, sice i i, x i (i)! B i() ( ) i+ p i, p i xi (i)! 4 πi (πe) i, p i+ x p i 4π, we have that xi (i)! B i() (+ ) if x < 4π (x > 4π ), both bad situatios. Istead, for x 4π 39.47.. the sequece xi (i)! B i(), i,,,..., should be lower ad upper bouded.... x (4)! B 4() iff x 6.84 x4 (8)! B 8() iff x 33. x8 (6)! B 6() iff x 36. x6 (3)! B 3() about iff x 6 (8.54) 3 93 47.9 iff x 37.8 xs (s)! B s() about iff xs (s)! 4 πs ss (πe) iff x s (s)! (πe) s s s s 4... πs More geerally, the parameter x should be used to make more stable the l.t.t. solver. 5
Ramauja i a paper states that the Beroulli umbers B (), B 4 (), B 6 (),..., satisfy the followig lower triagular sparse system of liear equatios: 3 5 43 5 4 86 4 3 4 5 938 33 7 7 7 76 3553 5 4 B () B 4 () B 6 () B 8 () B () B () B 4 () B 6 () B 8 () B () B () /6 /3 /4 /45 /3 4/455 / /36 3/665 /3 /55 (actually, sice Ramauja-Beroulli umbers are the moduli of ours, his equatios, obtaied by a aalytical proof, are a bit differet). So, for example, from the above equatios I have easily computed the Beroulli umbers B 8 (), B (), B () (from the oes already computed): B 8 () 43867 798 54.97, B () 746 59., B () 85453 69.. 33 38 Problem. Is it possible to obtai such Ramauja lower triagular sparse system of equatios from our odd ad eve l.t.t. liear systems Is it possible to obtai other sparse equatios (hopefully more sparse tha Ramauja oes) defiig the Beroulli umbers Note that the Ramauja matrix, say R, has ozero etries exactly i the places where the followig Toeplitz matrix + γ k (Z 3 ) k, Z k has ozero etries. But R it is ot a Toeplitz matrix!... break for some moths... Note that the vector of the idetermiates i the Ramauja system is Z T times our vector b: B () B () B 4 () B 6 () B () B 4 () ZT b. So, the Ramauja system ca be rewritte as follows R(Z T b) f, f [ f f f 3 ] T. 6
Remark The Ramauja matrix R satisfies the followig idetity ivolvig a sparse lower triagular Toeplitz matrix R: R! x 4! x 6! x 3! x 4! x 6! x 3 R, R + s x 3s (6s + )!(s + ) Z3s. RZ T b f, f [f f f 3 ] T iff R! x R! x x! 4! x 6! x 3 4! x 6! x 3 x 4! x 3 6! R x! x! x 4! x 4! B () B 4 () B 6 () x 3 6! x 3 6! x! ZT b f ZT b f So, the Ramauja system is is equivalet to the followig sparse l.t.t. system: R L(a R ) R (Z T D x b) x! x 4! x 3 6! f f f 3, x 4! where x 3 6! iff iff 8!3 x3 8!3 x3 8!3 x3 4!5 x6 8!3 x3 4!5 x6 8!3 x3 4!5 x6 8!3 x3!7 x9 4!5 x6 8!3 x3!7 x9 4!5 x6 8!3 x3 Note the ew otatio : a a a a L(a) f f f 3. a a a a a a, a R. 8!3 x3 4!5 x6!7 x9. 7
THEOREM Notatios: Z is the lower shift matrix Z L(a) is the lower triagular Toeplitz matrix with first colum a, i.e. a a a a + a, L(a) a a a i Z i a a a i a 3 a a a d(z) is the diagoal matrix with z i as diagoal etries. Set b B () B () B 4 (), where B i (), i,,,..., are the Beroulli umbers.,, D x diag ( xi, i,,,...), x R, (i)! The the vectors D x b ad Z T D x b solve the followig l.t.t. liear systems L(a) (D x b) D x q, L(a) (Z T D x b) d(z)z T D x q, where the vectors a (a i ) + i, q (q i) + i, ad z (z i) + i, ca assume respectively the values: a R x i i δ i mod 3 (i + )!( 3 i + ), qr i (i + )(i + ) ( δ 3 i mod ), i,,, 3,... 3 zi R δ i mod 3 i,, 3,..., 3i +, a o i a e i xi (i + )!, qe i z e i, i,,, 3,... i + i, i,, 3,..., i + x i (i + )!, i,,, 3,..., qo, qi o, i,, 3,... zi o i, i,, 3,.... i + 8
Problem (regardig the computatio of the Beroulli umbers). Ca the Ramauja l.t.t. sparse system L(a R )D x b D x q R, be obtaied as a cosequece of the eve ad odd l.t.t. system L(a e )D x b D x q e, L(a o )D x b D x q o fid â e, â o such that L(â e )L(a e ) L(a () ) L(â o )L(a o ), i.e. such that L(a e )â e a () L(a o )â o, with a () a R or a() more sparse tha a R. Importat: the computatio of such vectors â e, â o ad a () should be cheaper tha solvig the origial eve ad odd (dese) systems. A more geeral problem: is it possible to trasform efficietly a geeric l.t.t. matrix ito a more sparse l.t.t. matrix Questio: give a i, i,, 3,..., is it possible to obtai cheaply â i ad a () i a a a a 3 a a a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a â â â 3 â 4 â 5 â â â 3 â â â 3 â 4 â 5 a () a () a () ; a () ; γ, R b such that If the aswer is yes, the a dese l.t.t system ca be trasformed efficietly ito a sparse l.t.t. system: L(a)z c iff L(â)L(a)z L(â)c iff L(γ)z L(â)c. 9
DEFINITIONS: u E u u LEMMA: If u a a a, Eu u u PROBLEM: Give a, v u u C+, L(a), b, E s, b, L(Eu) v v, the a a a,, bs, I u T u I I u T u T L(Eu)Ev EL(u)L(v), L(E s u)e s v E s L(u)v, s N. a a, fid â L(a)â Ea () â â a () ad a() a () a () such that, b, L(a)L(â) L(Ea () )., b. Questios: Is it possible to obtai cheaply â i ad a () i There exist explicit formulas for the â i ad a () i At the momet, let us see i detail, with two examples, how the solutios of the above Problem ca lead to efficiet methods for solvig geeric l.t.t. liear systems.
EXAMPLE: 8 ( b k, b, k 3) a a () a a a 3 a 4 a 5 a 6 a 7, L(a) a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a, E. step : From a a () fid â â () such that L(a)â a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a â â â 3 â 4 â 5 â 6 â 7 : Ea () a () a () a () 3, L(a)L(â) L(Ea () ); step : From a () fid â () such that L(Ea () )Eâ () a () a () a () a () a () a () a () 3 a () a () a () 3 a () a () â () â () â () 3 : E a () a (), L(Ea () )L(Eâ () ) L(E a () ); step 3 log 8: From a () fid â () such that L(E a () )E â () a () a () a () a () â () : E 3 a (3), L(E a () )L(E â () ) L(E 3 a (3) ). L(E â () )L(Eâ () )L(â) [ L(a) ] L(E 3 a (3) ) [ I8 O ], so, oe realizes that we have performed a kid of Gaussia elimiatio.
How may arithmetic operatios (a.o.) Actually, give a i a () i, i,..., 7, we have to compute â i â () i, a () i â () i, a () a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a â () a () a () a () a () 3 a () a () a () â () â () â () â () 3 â â â 3 â 4 â 5 â 6 â 7 a (), ϕ a.o.. a () a () a () 3, ϕ 4 a.o.,, ϕ 8 a.o., The geeral case: k Give a i a () i, i,..., k, we have to compute â (j) i, a (j+) i a (j) a (j) a (j) a (j) j a (j) a (j) }{{} j j j,,..., k, k (j k : oly â (j) i ) â (j) â (j) â (j) j a (j+) a (j+) j+, ϕ j a.o., Total cost: k j ϕ j Remark. Note that, at step j, the a (j+) i are the j+ ozero etries of a matrix j j ( k j k j ) l.t.t. by vector product. So, if we assume such matrix by vector product computable i at most c k j (k j) a.o., for some costat c, the k ϕ j k c j j k k j (k j) + j j CostCompOf ( â (j) i ) k ( O( k k) + CostCompOf â (j) i, i,..., ) j
Computig the first colum of L(a) ( 3 8) Let v [v v v ] T be ay vector. From the idetity L(E â () )L(Eâ () )L(â) [ L(a) ] L(E 3 a (3) ) [ I8 O ] ad from the Lemma, it follows that L(a)z E v iff L(E 3 a (3) )z L(â)L(Eâ () )L(E â () )E v L(â)L(Eâ () )E L(â () )v L(â)EL(â () )EL(â () )v. So, the system L(a)z E v is equivalet to the system [ ] [ ] I8 O {z}8 L(E 3 a (3) )z L(â)EL(â () )EL(â () )v {z} 8 {L(â)} 8 {E} 8 {L(â () )} 8 {E} 8 {L(â () )} 8 {v} 8 {L(â)} 8 {E} 8,4 {L(â () )} 4 {E} 4, {L(â () )} {v}. Thus the vector z z â z â â z 3 z 4 â 3 â â â 4 â 3 â â z 5 â 5 â 4 â 3 â â z 6 â 6 â 5 â 4 â 3 â â z 7 â 7 â 6 â 5 â 4 â 3 â â â () â () â () â () 3 â () â () [ â () ] [ v v ] is such that How may arithmetic operatios (a.o.) Case k a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a z z z z 3 z 4 z 5 z 6 z 7 It is clear that the above procedure requires the computatio of matrix j j l.t.t. by vector products, with j,..., k (the vectors are sparse for j,..., k). So, if we assume such matrix by vector product computable i at most c j j a.o., for some costat c, the the above procedure requires at most arithmetic operatios. c k j j O( k k) j v v. 3
EXAMPLE: 9 ( b k, b 3, k ) a a () a a a 3 a 4 a 5 a 6 a 7 a 8, L(a) a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a, E. step : From a a () fid â â () such that L(a)â a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a â â â 3 â 4 â 5 â 6 â 7 â 8 : Ea () a () a (), L(a)L(â) L(Ea () ); step log 3 9: From a () fid â () such that L(Ea () )Eâ () a () a () a () a () a () a () a () a () a () â () â () : E a (), L(Ea () )L(Eâ () ) L(E a () ); L(Eâ () )L(â) [ L(a) ] L(E a () ) [ I9 O ], so, oe realizes that we have performed a kid of Gaussia elimiatio. 4
How may arithmetic operatios (a.o.) Actually, give a i a () i, i,..., 8, we have to compute â i â () i, a () i a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a â () i a () a () a () â () â (), â â â 3 â 4 â 5 â 6 â 7 â 7 â 8 ϕ 3 a.o.. a () a (), ϕ 9 a.o., The geeral case: 3 k Give a i a () i, i,..., 3 k, we have to compute â (j) i, a (j+) i a (j) a (j) a (j) a (j) 3 j a (j) a (j) }{{} 3 j 3 j j,,..., k, k (j k : oly â (j) i ) â (j) â (j) â (j) 3 j a (j+) a (j+) 3j+, ϕ 3 j a.o., Total cost: k j ϕ 3 j Remark. Note that, at step j, the a (j+) i are the 3 j+ ozero etries of a matrix 3 j 3 j (3 k j 3 k j ) l.t.t. by vector product. So, if we assume such matrix by vector product computable i at most c 3 k j (k j) a.o., for some costat c, the k ϕ j k c 3 j j k 3 k j (k j) + j j CostCompOf ( â (j) i ) k ( O(3 k k) + CostCompOf â (j) i, i,..., ) 3 j 5
Computig the first colum of L(a) (case 3 9): Let v [v v v ] T be ay vector. From the idetity L(Eâ () )L(â) [ L(a) ] L(E a () ) [ I9 O ] ad from the Lemma, it follows that L(a)z Ev iff L(E a () )z L(â)L(Eâ () )Ev L(â)EL(â () )v. So, the system L(a)z Ev is equivalet to the system [ ] [ ] I9 O {z}9 L(E a () )z L(â)EL(â () )v Thus the vector z z z z 3 z 4 z 5 z 6 z 7 z 8 {z} 9 {L(â)} 9 {E} 9 {L(â () )} 9 {v} 9 {L(â)} 9 {E} 9,3 {L(â () )} 3 {v} 3. â â â â 3 â â â 4 â 3 â â â 5 â 4 â 3 â â â 6 â 5 â 4 â 3 â â â 7 â 6 â 5 â 4 â 3 â â â 8 â 7 â 6 â 5 â 4 â 3 â â â () â () â () v v v is such that How may arithmetic operatios (a.o.) Case 3 k a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a z z z z 3 z 4 z 5 z 6 z 7 z 8 It is clear that the above procedure requires the computatio of matrix 3 j 3 j l.t.t. by vector products, with j,..., k (the vectors are sparse for j,..., k). So, if we assume such matrix by vector product computable i at most c 3 j j a.o., for some costat c, the the above procedure requires at most arithmetic operatios. For the geeral case b k see the Appedix. c k 3 j j O(3 k k) j v v v. 6
PROBLEM Aswer to the quotatio marks i the followig equality: L(a)â a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a Ea (), E. - There is ot a uique aswer to the. - There exists a aswer that allows to obtai from the eve ad odd systems, a system solved by Beroulli umbers where i the coefficiet matrix ull diagoals alterate with the o ull oes. Fid it... - If there exists a aswer such that the first j etries of â ca be computed i at most O( j j) arithmetic operatios, for all j k, the we have a algorithm of complexity O( k k) for solvig geeric k k lower triagular Toeplitz systems. Here below is a aswer such that the first j etries of â ca be computed with zero arithmetic operatios: L(a)â a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a a a 3 a 4 a 5 a a a 4 a a 3 + a a 6 a a 5 + a a 4 a 3 a 8 a a 7 + a a 6 a 3 a 5 + a 4 Ea (), L(a) ( e + + i ( ) i a i e i+ ) e + + i δ i mod ( ai + i j ( ) j a j a i j ) ei+. 7
PROBLEM Aswer to the quotatio marks i the followig equality: a a a L(a)â a 3 a a Ea (), E a 4 a 3 a a a 5 a 4 a 3 a a. - There is ot a uique aswer to the. - There exists a aswer that allows to obtai the Ramauja system solved by Beroulli umbers as a cosequece of the odd (eve) system. Fid it... - If there exists a aswer such that the first 3 j etries of â ca be computed i at most O(3 j j) arithmetic operatios, for all j k, the we have a algorithm of complexity O(3 k k) for solvig geeric 3 k 3 k lower triagular Toeplitz systems. Here below is a aswer: L(a)â a a a a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a + a a 3 a a a 4 a a 3 + a a 5 + a a 4 a a 3 a 6 a a 5 a a 4 + a 3 a 7 a a 6 + a a 5 a 3 a 4 a 8 + a a 7 a a 6 a 3 a 5 + a 4 a 9 a a 8 a a 7 + a 3 a 6 a 4 a 5 a a a 9 + a a 8 a 3 a 7 a 4 a 6 + a 5 a + a a a a 9 a 3 a 8 + a 4 a 7 a 5 a 6 a () a () a () 3 Ea (), a () 3a 3 3a a + a 3, a () 3a 6 3a a 5 3a a 4 + 3a 3 3a a a 3 + 3a a 4 + a 3, a () 3 3a 9 3a a 8 3a a 7 + 6a 3a 6 3a a a 6 3a a 3a 5 3a a 3a 4 + 3a a 7 + 3a a 4 3a 4a 5 + 3a 5a + a 3 3,... â i i r a ra i r + δ i mod a i + 3 { s 3 i 6 s 6 i 6 a i 3 i+3 +3sa 3s i odd a i 6 i+6 +3sa 3s i eve Ca such â i, i,,..., 3 j, be computed i at most O(3 j j) arithmetic operatios, i,,, 3, 4, 5,.... 8
( a 3 a 3a a 3 3 3a a 4 3a 3a 3 3 a 5 3a 3a 4 3a a 6 3a 3 3 3a 5 3a a 7 3a 4 3a 3a 6 3a 3 3 a 8 3a 5 3a 3a 7 3a 4 3a a 9 3a 6 3a 3 3 3a 8 3a 5 3a a 3a 7 3a 4 3a 3a 9 3a 6 3a 3 3 a 3a 8 3a 5 3a 3a 3a 7 3a 4 3a a 3a 9 3a 6 3a 3 3 3a 3a 8 3a 5 3a a 3 3a 3a 7 3a 4 3a a a a a a a a 3 a a a 3 a a a 4 a 3 a a a 4 a 3 a a a 5 a 4 a 3 a a a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a a 8 a 7 a 6 a 5 a 4 a 3 a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a a a a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a ) a a a 3 a 4 a 5 a 6 a 7 a 8 a 9 a a a a 3 a 4 a 5 a 6 a 7 a 8 a 9 a a a a 3 a 4 a 5 a 6 Ca the above 3 j 3 j (7 7) matrix by vector product be computed i at most O(3 j j) arithmetic operatios If yes, the we would have a method which solves 3 k 3 k lower triagular Toeplitz liear systems i at most O(3 k k) arithmetic operatios. If o, the look for aother solutio â of the system L(a)â [ ] T such that { â } is computable 3 j from { a } i at most O(3 j j) arithmetic operatios... 3 j THE END 9
APPENDIX Itroduce low complexity l.t.t. liear system solvers: L(a) a a a, a() : a Fid â (), a () such that L(a () )â () Ea (), b, so that Fid â (), a () such that L(a () )L(â () ) L(Ea () ). L(a () )â () Ea (), b, so that L(Ea () )Eâ () E a (), b, L(Ea () )L(Eâ () ) L(E a () ). (use Lemma). Fid â (), a (3) such that... Fid â (k ), a (k ) such that L(a () )â () Ea (3) L(E a () )E â () E 3 a (3), b, L(E a () )L(E â () ) L(E 3 a (3) ). L(a (k ) )â (k ) Ea (k ) so that, b3,, b, L(E k a (k ) )E k â (k ) E k a (k ) L(E k a (k ) )L(E k â (k ) ) L(E k a (k ) ). so that, b k, Fid â (k ), a (k) such that L(a (k ) )â (k ) Ea (k), b, so that L(E k a (k ) )E k â (k ) E k a (k), bk, L(E k a (k ) )L(E k â (k ) ) L(E k a (k) ).
The b k {}}{ I O [ a (k) ] L(Ek a (k) ) L(E k â (k ) )L(E k â (k ) ) L(Eâ () )L(â () )L(a () ). This implies that Moreover, if L(a () )z c iff L(E k a (k) )z L(â () )L(Eâ () ) L(E k â (k ) )L(E k â (k ) )c. c E k v v v v, b, k where v (v i ) + i is ay vector (for example v e ), the by usig the Lemma, we obtai the followig result: L(a () )z c iff I b k O [ a (k) ] z L(Ek a (k) )z L(â () )EL(â () )E EL(â (k ) )EL(â (k ) )v. I other words, the vector { z }, bk, such that { } { } L(a) z (for example { L(a) } { } e { } { } { } z L(â () ) { } L(â () ) a a b k a { } z v v v b, b k, v, v i i ), ca be represeted as follows { } { } { } { } { } E L(â () ) E L(â (k ) ) E L(â (k ) ) {v} { } { } { } { } { } E L(â () ) E L(â (k ) ) E b b, b, b b k b k, b k { L(â (k ) ) } {v} b b k FIRST: Compute the first etries of â () ad the first b etries of a() (cost ϕ b k); compute the first b etries of â () ad the first b etries of a () (cost ϕ b k );... compute the first etries of â (k ) ad the b k first etries of a (k ) (cost ϕ b k b ); compute the first etries of â (k ) (cost ϕ b k b ). Total cost of this FIRST operatio: k j ϕ. b j SECOND: To such cost add k j cost( (b j b j l.t.t ) (b j vector) ) (the vector is sparse if j,..., k; the cost for j is zero if v e ). See also the ext page.
Amout of operatios. I the followig b k ad b : FIRST: For j,..., k compute, by performig ϕ b j I a(j+), i.e. scalars â (j) b j+ i ad a (j+) i such that a (j) (ote that there is o a (k) i a (j) a (j) a (j) b j a (j) a (j) }{{} b j b j Case b. I this case, sice â (j) i to be computed (the a (j+) i are the b j+ â (j) â (j) â (j) b j arithmetic operatios, the vectors I b j â(j) ad a (j+) a (j+) bj+ SECOND: Compute the b b l.t.t. by vector product { L(â (k ) ) } b k vector products of type COMMENTS, j,..., k to be computed). ( ) i a (j) i, oly b j b l.t.t. by vector products, j,..., k, eed j ozero etries of the resultig vectors). v v b, ad b j b j l.t.t. by { L(â (j) ) } b }{{ j } b j b j, j k,...,,. So, i case b, we have to perform j j l.t.t. by vector products, for j,..., k, twice. If we assume the cost of a j j l.t.t. by vector product bouded by c j j (c costat), the the total cost of the above operatios is smaller tha O( k k) O( log ). As a cosequece we have obtaied, i particular, a l.t.t. liear system solver of complexity O( log ) Aalogously, for b 3, if we assume both ϕ 3 j ad the cost of a 3 j 3 j l.t.t. by vector product bouded by c3 j j, the the total cost of the above operatios is smaller tha O(3 k k) O( log 3 ).... But is ϕ 3 j bouded by c3 j j... For me: http://www.imsc.res.i/ rao/ramauja/collectedidex.html http://mathworld.wolfram.com/beroullinumber.html http://umbers.computatio.free.fr/costats/miscellaeous/beroulli.html