0 Introduction Let the matrix A C nn be given by all its n entries The problem is to compute the product Ab of a matrix A by input vector b C n Using

Transcription

1 Fast algorithms with preprocessing for matrix{vector multiplication problems IGohberg and VOlshevsky School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Ramat Aviv 998, Israel e{mail: Journal of Complexity, (99) Abstract In this paper the problem of complexity of multiplication of a matrix with a vector is studied for Toeplitz, Hankel, Vandermonde and Cauchy matrices and for matrices connected with them (ie for transpose, inverse and transpose to inverse matrices) The proposed algorithms have complexities at most O(n log n) ops and in a number of cases improve the known estimates In these algorithms, in a separate preprocessing phase, are singled out all the actions on the preparation of a given matrix, which aimed at the reduction of the complexity of the second stage of computations directly connected with the multiplication by an arbitrary vector Incidentally, the eective algorithms for computing the Vandermonde determinant and the determinant of a Cauchy matrix, are given

2 0 Introduction Let the matrix A C nn be given by all its n entries The problem is to compute the product Ab of a matrix A by input vector b C n Using the standard rule of a matrix times vector multiplication, the above problem can be solved in n ; n ops (ie oat point operations of addition, subtraction, multiplication and division) In the following three simple examples the special structure of a matrix enables faster computation of the product by avector Examples ) Band matrices Let A =(a ij ) i j=0 be a band matrix with the width of the band (ie, by denition, a ij =0 while j i ; j j> ; ) Obviously, in this case the computing the product Ab costs ( ; )n ops ) Small rank matrices Let matrix A C nn with rank R be given in the form of the outer sum of terms : A = X m= h m g T m (h m g m C n ): (0) The representation (0) shows that A can be multiplied by an arbitrary vector in (;)n; ops ) Semiseparable matrices Such a matrix A =(a ij ) i j=0 is dened by the equalities a ij = ( P m= f m i g m i i<j 0 i j where f m = (f m i ) g m = (g m i ) (m = ::: ) are given vectors from C n In this case X A = diag(f m ) m= diag(g m ) (0) 0 0 where by diag(f) is denoted the diagonal matrix, whose entries on the main diagonal equal the coordinates of the vector f C n The product of a semiseparable matrix by an arbitrary vector can be computed using (0) in (;)n; ops Obviously, the analogous estimate holds for the transpose to the matrices of the form (0) Let matrix A C nn be given The task is to compute the products Ab 0 Ab ::: of the matrix A by input vectors b 0 b ::: C n in the smallest possible time Such a situation arises naturally in a number of computational problems, for example, when we have to carry out the iterations with a given matrix The problem of computing the product of a matrix

3 by a matrix can also be solved in the above framework In the latter case the columns of a second matrix are interpreted as input vectors In accordance with the accepted scheme, we single out all the computations which are not dependent upon the input vectors b 0 b ::: Accordingly, the proposed algorithms will be divided into two separate phases : I Preprocessing for matrix A C nn, II Application to the vector, The rst phase also contains the preparation of the given matrix, which enables the second phase to be accomplished more eectively In the present paper we embody this scheme and propose a number of fast algorithms for matrices with a certain structure, namely for transposed Vandermonde matrix, for transpose to inverse of Vandermonde matrix, for Cauchy matrices and for matrices connected with them (ie for transpose, inverse and transpose to inverse matrices) Background Basic algorithms In this paper a limited number of well known algorithms is used intensively These algorithms are listed below and accompanied by the estimates of their complexities The particular implementations of these basic algorithms and complexity analysis for them can be found in various sources (see, for example, Aho, Hopcroft and Ullman (9)) (BA) Evaluation algorithm An algorithm of evaluation of a (n ; ) degree polynomial at n points The complexity of this algorithm will be denoted by "(n) It is well known that "(n) O(n log n): (BA) Interpolation algorithm An algorithm of interpolation of a (n ; ) degree polynomial from its values at n points The complexity of the interpolation algorithm is denoted by (n) and, as it is well known, (n) O(n log n): (BA) Fast Fourier Transform algorithm The discrete Fourier transform of the vector r =(r i ) C n is by denition the vector ( P k=0 r k! ik ) C n, where! is the primitive n-th root from unity The discrete Fourier transform can be computed by well known methods, collectively named Fast Fourier Transform (FFT) It is well known that for the complexity (n) of computing one FFT of order n, the following estimate holds (n) O(n log n):

4 Basic matrices In this subsection some matrices, which are known to have alower than O(n ) ops complexity ofmultiplication with a vector, are considered Note that the methods for fast computing of the products of these basic matrices by vectors are based on the algorithms (BA) - (BA) (BM) Vandermonde matrix By denition, the Vandermonde matrix V (t) is the matrix of the form V (t) = t 0 t 0 t t with t =(t i) C n : t t Obviously, the problem of computing the product V (t)b of the matrix V (t)byavector b =(b i ) is equivalent to the problem of evaluation of p() = P b i i at n points t 0 t ::: t Thus, the product V (t)b can be computed using the algorithm (BA) in "(n) ops (BM) Inverse of Vandermonde matrix Consider the problem of application to a vector of the matrix A C nn, which is dened as the inverse of the given Vandermonde matrix V (t) (t C n ) It is easy to see that the product Ab can be computed using the algorithm (BA) Thus, for the inverse of Vandermonde matrix, the application to a vector costs (n) ops (BM) Fourier matrix and its inverse Let! = e i n be primitive n-th root from the unity Consider the Fourier matrix F = p n V ( ) with =(! i ) and its inverse F ; = F, where superscript means conjugate transpose The products Fb and F b can be computed using the algorithm (BA) in (n) ops (BM) Factor circulant By Circ ' (r) will be denoted '-circulant with the rst column r =(r i ), ie matrix of the form r 0 'r 'r Circ ' (r) = r r 0 'r r 'r r r r 0 The matrix Circ (r) is referred to as a circulant It is known (see Cline, Plemmons and Worm (9)) that the matrix Circ ' (r) admits the following decomposition : Circ ' (r) =D ; ' F ' (r) FD ' () where F is the Fourier matrix, ' (r) = diag(fd ' r) and D ' = diag(( i ) ) with C satisfying the condition n = ' From () it follows that if the coordinates of :

5 the vector =( i ) C n are given, then the product of the matrix Circ ' (r) by an arbitrary vector can be computed in (n)+o(n) ops In this case the preprocessing phase consists of computation of the central factor ' (r) intheright-hand side of () and costs ops (n)+o(n) (BM) Toeplitz matrices The product of Toeplitz matrix A = (a i;j ) i j=0 by a vector can be computed by embedding of the matrix A in the circulant ofdoublesize Correspondingly, the amount of operations is (n)+o(n) =(n)+o(n) ops after preprocessing (n)+o(n) =(n)+o(n) (BM) Inverses of Toeplitz matrices In computations with the inverses of Toeplitz matrices, the Gohberg-Semencul formula (see Gohberg and Semencul (9) or Gohberg and Feldman (9) ) is useful This formula represents the inverse of a Toeplitz matrix in the form of the sum of products of triangular Toeplitz matrices Namely, if for the Toeplitz matrix T the equations T x = e 0 T y = e () h i h i T T with e 0 = 0 0, e = 0 0 have solutions x = (xi ), y =(y i ) and x 0 =0,then T is invertible and x y y n; y 0 T ; = x x 0 ( 0 y x 0 x y n; ; 0 ; x x x y 0 0 y 0 y n; y 0 0 y n; 0 0 y 0 x x 0 0 x x 0 0 ): ()

6 On the basis of formula () the number of fast algorithms for the inversion of Toeplitz matrices was elaborated (see Brent, Gustavson and Yun (980), de Hoog (98), Chun and Kailath (99), where the methods for solving the equations of the form () are suggested) Denote by (n) the complexity of solving one equation of the form () According to Brent, Gustavson and Yun (980), de Hoog (98), Chun and Kailath (99) (n) O(n log n): Thus, if matrix A C nn is the inverse of the given Toeplitz matrix, then from the above arguments follows that after (n) + 8(n) ops preprocessing the product of A by an arbitrary vector can be computed by () in (n) + O(n) ops Below we show how this complexity can be reduced If for Toeplitz matrix T C nn the equations () have solutions x = (x i ), y = (y i ) and x 0 =0,then T ; = x 0 (' ; ) (Circ (x) Circ ' (Z ' y) ; Circ (Z y) Circ ' (x)) () where numbers ' and (= ') are arbitrary and Z ' = 0 0 ' (' = 0) is the '-cyclic lower shift matrix Formula () was obtained in Ammar and Gader (990) for positive denite Toeplitz matrices and ' = = ;, and was extended to the general case in Gohberg and Olshevsky (99) Note, in the latter paper one can also nd other factor circulant decompositions for the inverses of Toeplitz matrices, which are useful in the fast computations with Toeplitz matrices Furthermore, representing each factor circulant in the left-hand side of () in the form of (), we nally get T ; = x 0 (' ; ) D; F ( (x) F D D ; ' F ' (Z ' y); ; (Z y) FD D ; ' F ' (x)) FD ' () where F is a Fourier matrix and the diagonal matrices D ' and ' (r) (r C n ) are dened as in (BM) From () it follows that after (n)+(n)+o(n) () ops preprocessing, the product of the inverse of the Toeplitz matrix by an arbitrary vector can be computed in (n)+o(n) ()

7 ops The last estimate is obtained here without the additional requirement for matrix A to be Hermitian In the Hermitian case this result is obtained in the earlier paper Ammar and Gader (990) Note, () and () imply that the product of the inverse of Toeplitz matrix by an arbitrary matrix can be computed in (n)n + O(n ) ops (BM) Hankel matrices Let us recall that an arbitrary Hankel matrix H = (a i+j ) i j=0 can be transformed in the Toeplitz matrix by means of multiplication from the right by the reverse identity matrix J = : Therefore the complexity of the multiplication with a vector for the class of Hankel matrices coincides with the same complexity for the class of Toeplitz matrices Transpose to Vandermonde and to inverse of Vandermonde matrices Relations between Vandermonde matrices and Bezoutians Let n R be the maximal degree of two polynomials f() andg(), then the bilinear form B f g ( ) = f() g() ; f() g() ; = X i j=0 b ij i j is called the Bezoutian of f() and g() The matrix B f g =(b ij ) i j=0 whose entries are determined by the coecients of the bilinear form B f g ( ), will be referred to as a Bezout matrix which corresponds to the polynomials f() andg() Obviously, h i Bf g = B f g ( ): () The equality () yields the following useful property of the Bezout matrix : V (s) B f g V (t) T =(B f g (s i t j )) i j=0 ()

8 where s =(s i ), t =(t i ) C n From () and obvious equality B f g ( ) =f 0 () g() ; f() g 0 () it follows that if t 0 t ::: t are n simple roots of the polynomial f(), then V (t) B f g V (t) T = diag((f 0 (t i ) g(t i )) ): () The latter formula appeared in Lander (9) and can also be found in Heinig and Rost (98) On the basis of () in the next subsection the fast algorithm for multiplication of transpose Vandermonde matrix by a vector is presented Transpose to Vandermonde matrix and its inverse For vector t =(t i ) (t i = t j while i = j) set f() = Y ( ; t i )= n + X r i i and g() =' ; n () where ' is such that ' = t n i (i = 0 ::: n; ) Under these conditions the matrix in the right-hand side of () is invertible Hence, all the matrices in the left-hand side of () are also invertible, and moreover V (t) T = B ; f g V (t) ; diag((f 0 (t i ) g(t i )) ): () Furthermore, matrix B f g is the Bezout matrix of two polynomials, one of them having the special form g() = ' ; n This fact allows to receive for B f g and for its inverse a representation involving a factor circulant To prove this, let us compute the Bezoutian B f g ( ) of the polynomials f() and g(), dened by () : = ' nx i= f() ; f() B f g ( ) =' ; f() n ; n f() = ; ; r i ( i; + i; + ::: + i; )+ X r i ( i + i+ n; + ::: + i ): with r n = From the last identity it can be easily seen that the corresponding Bezout matrix B f g has the special form B f g = 'r 'r 'r r 0 + ' 'r r 'r r n; r 0 + ' r r n; r =Circ ' (r + 'e 0 ) J () where J is, as above, the reverse identity matrix Substituting () in (), we have V (t) T = J Circ ' (r + 'e 0 ) ; V (t) ; diag((f 0 (t i ) g(t i )) ): () 8

9 On the basis of the last formula the following algorithm is proposed Algorithm Computing the product of transpose Vandermonde matrix with an arbitrary vector I Preprocessing Complexity : (n)+"(n)+(n)+n log n + O(n) ops Input : Vector t =(t i ) C n (t i = t j while i = j): Output : Parameters of the representation V (t) T in the form V (t) T = J D ' F ' (r + 'e 0 ) ; F D ; ' V (t) ; D f g : (8) Compute the coecients of the polynomial f() = Q (;t i)= n + P r i i in (n) ops Compute in O(n) ops the coecients of the polynomial f 0 () Evaluate in "(n) ops the values f 0 (t i ) (i =0 ::: n; ) Evaluate in at most n log n+n ops the values g(t i )=';t n i (i =0 ::: ) using, for example, the algorithms from x in Knuth (99) Compute in n ops the entries of the matrix D f g =diag((f 0 (t i ) g(t i )) ) Set = + max 0i t i Compute in O(n) ops the numbers i (i =0 ::: n) Set ' = n and D ' = diag(( i ) ) Compute in O(n) ops the entries of the matrix D ; ' 8 Compute the coordinates of the vector FD ' y with y = r + 'e 0 in (n)+o(n) ops using (BM), where r =(r i ) 9 Compute in O(n) ops the inverse of the diagonal matrix ' (r + 'e 0 ) = diag(fd ' y) II Application to the vector Complexity : (n) +(n) +O(n) ops Input : Vector b C n Output : Vector V (t) T b C n Compute vector V (t) T b by (8) using (BM) and (BM) The product of transposed Vandermonde matrix with an arbitrary matrix can be computed using algorithm in (n)n +(n)n + O(n ) ops 9

10 Note that representations of the matrix V (t) T in the form of the product of matrix V (t) ; by some matrices with lower complexity of multiplication with vectors, could be found in Heinig and Rost (98) and Canny, Kaltofen and Yagati (989) In Canny, Kaltofen and Yagati (989), for example, the matrix V (t) T is represented in the form V (t) T = H V (t) ; (9) where matrix H = V (t) T V (t) turned out to be a Hankel matrix For computing the entries of H it is proposed in Canny, Kaltofen and Yagati (989) to interpolate the n-th degree polynomial from its zeros at n points, and then to solve an equation of the type T x = b with Toeplitz matrix T from C nn Let (n) beasabove the complexity ofsolving one Toeplitz equation of the form () Thus, computing the entries of the matrix H in (9) by the scheme in Canny, Kaltofen and Yagati (989) and afterwards the preparation of the Hankel matrix H according to (BM) costs (n) + (n) + (n) + O(n) ops This complexity exceeds the complexity of preprocessing in the algorithm Furthermore, the appearance of the factor circulant in() instead of the Hankel matrix in (9) is reected in the additional economy in (n) ops at the stage of application to the vector Formula () also enables the proposition of fast algorithm for computing the product of matrix V (t) ;T byavector Moreover, the preprocessing phase consists of computing the parameters of the representation of V (t) ;T in the form analogous to the representation (8) of V (t) T and costs (n)+"(n)+(n)+nlog n + O(n) ops Afterwards, the application to the vector costs "(n)+(n)+o(n) ops The product of the matrix V (t) ;T by an arbitrary matrix can be computed in "(n)n +(n)n + O(n ) ops Formulas () and () yield the following representation for the inverse to a Vandermonde matrix : V (t) ; =Circ ' (r + 'e 0 ) J V (t) T diag(( f 0 (t i ) (' ; t n i ) ) ): Here, as above, J stands for the reverse identity matrix, f() = Q (;t i )= n + P r i i and r = (r i ) The last formula can be used for computing the entries of the inverse of Vandermonde matrix in O(n ) ops Cauchy matrices and matrices connected with them By denition, the Cauchy matrix C(s t) has the form C(s t) =( ) s i ; t j 0 i j=0

11 where s =(s i ) and t =(t i ) are given vectors from C n such thatt i s i (i =0 ::: ) are n dierent complex numbers As in the previous section set f() = Y ( ; t i )= n + X r i i g() =' ; n where ' is such that ' = t n i (i =0 ::: n; ) Formula () yields V (s) B f g V (t) T = ( f(s i) g(t j ) ) s i ; t j From here follows i j=0 =diag((f(s i )) ) C(s t) diag((g(t i )) ): C(s t) = diag(( f(s i ) ) ) V (s) B f g V (t) T diag(( g(t i ) ) ): This equality and () yields the following representation of the Cauchy matrix : C(s t) = diag(( f(s i ) ) ) V (s) V (t) ; diag((f 0 (t i )) ): () Formula () enables us to propose of the following algorithm Algorithm Computing the product of the Cauchy matrix by a vector I Preprocessing Complexity : (n) +"(n) +O(n) ops Input : Vectors s =(s i ) and t =(t i ) (t i = t j while i = j): Output : Parameters of the representation C(s t) intheform () Compute the coecients of the polynomial f() = Q ( ; t i )in(n) ops Evaluate in "(n)+ O(n) ops the values f (s i ) (i =0 ::: n; ) Compute in O(n) ops the coecients of the polynomial f 0 () Evaluate in "(n) ops the values f 0 (t i ) (i =0 ::: n; ) II Application to the vector Complexity : (n)+"(n)+o(n) ops Input : Vector b C n Output : Vector C(s t)b C n Compute C(s t)b by () using (BM) and (BM)

12 In an earlier paper of Gerasoulis (98) another algorithm for computing the product of Cauchy matrix with a vector was proposed Moreover, this algorithm does not restrict itself to Cauchy matrices and is valid for more general classes of matrices Gerasoulis algorithm applied for computing the product C(s t)b requires (n) +"(n) +O(n) ops of which (n) + "(n) + O(n) can be incorporated into a preprocessing stage Note, the product of C(s t) by an arbitrary matrix can be computed using algorithm in (n)n + "(n)n + O(n ) ops Formula () also enables the fast algorithms of multiplication with vectors for matrices C(s t) T C(s t) ; and C(s t) ;T to be elaborated For example, formula () implies the following representation for the inverse of Cauchy matrix : C(s t) ; =diag(( f 0 (t i ) ) ) V (t) V (s) ; diag((f(s i )) ): This formula yields that product of the inverse of Cauchy matrix by a vector can be computed in (n)+"(n)+o(n) ops after the same preprocessing as in algorithm In fact the latter proposition can be formulated as follows The system of linear equations with invertible Cauchy matrix can be solved in O(n log n) ops Determinants of Vandermonde and Cauchy matrices In the recent paper Pan (990) an outline of some algorithms for computing up to a sign the determinants of Vandermonde and Cauchy matrices with real entries is suggested For computing the Vandermonde determinant it is proposed therein rst to compute using the algorithm from Canny, Kaltofen and Yagati (989) the entries of the Hankel matrix H in (9) This step costs (n) +(n) +O(n) ops Then compute det H = (det V (t)) This seems to be quite expensive For the determinant of a Cauchy matrix in Pan (990) a complicated procedure of reducing this problem to the analogous problem for some closeto-toeplitz matrix is proposed It turned out that well-known formulas lead to simple and signicantly more eective algorithms for computing the determinants of Vandermonde and Cauchy matrices with complex entries Indeed, as is well known, det V (t) = Y 0i<j (t j ; t i ): () The square of the expression in the right-hand side of () is, by denition, the discriminant D(f) of the polynomial f() = Q ( ; t i ) It is well known (see, for example, van der Q Warden (9)) and can easily be seen, that D(f) =(;) n() f 0 (t i ) Thus, ( det V (t)) =(;) n() Y f 0 (t i ):

13 This formula enables the following algorithm to be proposed Algorithm Computing the Vandermonde determinant Complexity: (n) +"(n) + O(n) ops plus one extraction of a square root Input : Vector t =(t i ) C n (t i = t j while i = j): Output : Value det V (t) uptoasign Compute in (n) ops the coecients of the polynomial f() = Q ( ; t i ) Compute in O(n) ops the coecients of the polynomial f 0 () Evaluate the values f 0 (t i )(i =0 ::: n; ) in "(n) ops Q Compute det (V (t)) =(;) n() f 0 (t i )ino(n) ops Recover det V (t) up to a sign in one operation of the extraction of a square root Furthermore, if all the coordinates of the vector t are real numbers (this situation was considered in Pan (990) ), then the explicit expression in the right-hand side of () allows in O(n log n) time to come to a conclusion about the sign of V (t) Indeed, in this case the sign of V (t) is determined by the quantity of the pairs (t i t j ) satisfying the condition t i > t j while i < j Thus, the sign of det V (t) is determined by the number of inversions in the permutation k =(k 0 k ::: k ), where k i is the position of the element t i in the sequence, which is derived from ft i g by the rearrangement of all the elements in increasing order To compute this number of inversions, we have to rearrange the two-component sequence f(t i i)g in the increasing order of the rst components and then extract the second components into the separate sequence m =(m 0 m ::: m ) It is easy to see that m is the inverse of the permutation k and consequently has the same number of inversions Therefore, to determine the sign of det V (t) it remains to compute the number In(m) of inversions in the permutation m The algorithm based on these arguments is proposed below Note that here we deviate from the rule accepted in the present paper, and estimate the complexity in terms time, and not in ops The reason for this is that we use here the algorithms of sorting and of computing the number of inversions in a permutation These two algorithms do not involve oatpoint operations and involve cheaper operations of comparison and exchange Algorithm Determination of the sign of Vandermonde determinant Complexity : O(n log n) time Input : Vector t =(t i ) R n (t i = t j while i = j): Output : Sign of det V (t) Sort in O(n log n) time the sequence of the pairs = f(t i i)g in the increasing order of the rst components, using, for example, an algorithm from x in Knuth (9)

14 For the permutation m, formed from the second components of the sorted sequence, compute in O(n log n) time the number In(m) of its inversions, using, for example, the algorithm from Knuth (9), problem Set sign V (t) =ifthenumber In(m) iseven and sign V (t) =; in the opposite case Furthermore, for the determinant of Cauchy matrix the following expression is wellknown : Q 0i<j(s j ; s i ) Q 0i<j(t i ; t j ) det C(s t) = Q 0i j(s i ; t j ) (see, for example Polya and Szego (9), where it is attributed to Cauchy (89)) Using (), the latter formula can be rewritten in the form : det C(s t) =(;) n() det V (s) det V (t) Q f(s () i) where as above f() = Q ( ; t i ) Using this formula and the algorithm, one can compute the value det C(s t) upto a sign in (n)+"(n)+o(n) ops plus one operation of the extraction of a square root Furthermore, in the case when s i and t i are real, the formula () and the algorithm allow to determine the sign of the value det C(s t) ino(n log n) time References Aho,AV, Hopcroft,JE and Ullman,JD (9), \The design and analysis of computer algorithms", Addison-Wesley, Reading, Mass Ammar,G and Gader,P (990), New decompositions of the inverse of a Toeplitz matrix, in \Signal processing, Scattering and Operator Theory, and Numerical Methods, Proc Int Symp MTNS-89, vol III", (MAKaashoek, JH van Schruppen and ACMRan Eds), Birkhauser, Boston, pp { 8 Brent,R, Gustavson,F and Yun,D (980), Fast solutions of Toeplitz systems of equations and computation of Pade approximants, Journal of Algorithms,, 9 { 9 Canny,JF, Kaltofen,E and Yagati,L (989), Solving systems of non-linear equations faster, in \Proc ACM-SIGSAM 989 Internat Symp Symbolic Algebraic Comput", ACM, New York, pp { Chun,J and Kailath,T (99), Divide-and-conquer solutions of least-squares problems for matrices with displacement structure, SIAM Journal of Matrix Analysis Appl, (No ), 8 {

15 Cline,RE, Plemmons,RJ and Worm,G (9), Generalized inverses of certain Toeplitz matrices, Linear Algebra Appl, 8, { Gerasoulis,A (98), A fast algorithm for the multiplication of generalized Hilbert matrices with vectors, Math of Computation, 0 (No 8), 9 { 88 8 Gohberg,I and Feldman,I (9), \Convolution equations and projection methods for their solutions", Translations of Mathematical Monographs,, Amer Math Soc 9 Gohberg,I and Olshevsky, V (99), Circulants, displacements and decompositions of matrices, Integral Equations and Operator Theory, (No ), 0 { 0 Gohberg,I and Semencul,A (9), On the inversion of nite Toeplitz matrices and their continuous analogs (in Russian), Matem Issled, Kishinev, (No ), 0 { Heinig,G, Rost, K (98), \Algebraic methods for Toeplitz-like matrices and operators", Operator Theory, vol, Birkauser Verlag, Basel de Hoog,F (98), A new algorithm for solving Toeplitz system of equations, Linear Algebra Appl, 88/89, { 8 Knuth,D (99), "The art of computer programming", vol, \Seminumerical algorithms", Addison-Wesley, Reading, Mass Knuth,D (9), "The art of computer programming", vol, "Sorting and searching", Addison-Wesley, Reading, Mass Lander,FI (9), The Bezoutian and the inversion of Hankel and Toeplitz matrices (in Russian), Matem Issled, Kishinev, 9 (No ), 9{ 8 Pan,V (990), On computations with dense structured matrices, Math of Computation, (No 9), 9 { 90 Polya,G and Szego, G (9), \Problems and theorems in analysis", Springer Verlag, Berlin, New York 8 Van der Warden,BL (9), \Algebra I", Springer Verlag, Berlin, New York