2.8 Vandermonde Matrices and Toeplitz Matrices 93 a recursive procedure that solves the M-dimensional Toeplitz problem M R-= (i=1,,M0 (2.8.10) j=1 intum for M1,2..until M-N,the desired result,is finally reached.The vector) is the result at the Mth stage,and becomes the desired answer only when N is reached. Levinson's method is well documented in standard texts(e.g.,[5]).The useful fact that the method generalizes to the nonsymmetric case seems to be less well known.At some risk of excessive detail,we therefore give a derivation here,due to G.B.Rybicki. In following a recursion from step M to step M+1 we find that our developing solution (M)changes in this way: ∑R-0= i=1,,M (2.8.11) j=1 becomes M RM)+R-+)=i=1....,M+1 (2.8.12) A8/ j=1 By eliminating y we find ∑R（ C(MD)-2(M+) =R-(M+1) i=1,,M (2.8.13) America Press. = or by letting i M +1-i and jM+1-j, Programs ∑R-G)=R- (2.8.14) SCIENTIFIC j=1 where 6 c0=, (2.8.15) COMPUTING (ISBN To put this another way, ,=牌--G写” j=1,,M (2.8.16) Thus,if we can use recursion to find the order M quantities(M)and G()and the single Numerical 10621 orderM+1 quantity,then all of the other)will follow.Fortunately,the 43106 quantity)follows from equation (28.12)withiM+1. (outside Recipes M RM+1-写+)+RoxH)=M+1 (2.8.17) North Software. j=1 For the unknown orderM+1 quantities we can substitute the previous order quantities in G since c2-,= 0-x+ M+1) (2.8.18) IM+1 The result of this operation is = ∑Ru+1-0-1 (M) (2.8.19) ∑1R+1-,G21-,-R2.8 Vandermonde Matrices and Toeplitz Matrices 93 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). a recursive procedure that solves the M-dimensional Toeplitz problem M j=1 Ri−jx(M) j = yi (i = 1,...,M) (2.8.10) in turn for M = 1, 2,... until M = N, the desired result, is finally reached. The vector x(M) j is the result at the Mth stage, and becomes the desired answer only when N is reached. Levinson’s method is well documented in standard texts (e.g., [5]). The useful fact that the method generalizes to the nonsymmetric case seems to be less well known. At some risk of excessive detail, we therefore give a derivation here, due to G.B. Rybicki. In following a recursion from step M to step M + 1 we find that our developing solution x(M) changes in this way: M j=1 Ri−jx(M) j = yi i = 1,...,M (2.8.11) becomes M j=1 Ri−jx(M+1) j + Ri−(M+1)x(M+1) M+1 = yi i = 1,...,M +1 (2.8.12) By eliminating yi we find M j=1 Ri−j x(M) j − x(M+1) j x(M+1) M+1  = Ri−(M+1) i = 1,...,M (2.8.13) or by letting i → M + 1 − i and j → M + 1 − j, M j=1 Rj−iG(M) j = R−i (2.8.14) where G(M) j ≡ x(M) M+1−j − x(M+1) M+1−j x(M+1) M+1 (2.8.15) To put this another way, x(M+1) M+1−j = x(M) M+1−j − x(M+1) M+1 G(M) j j = 1,...,M (2.8.16) Thus, if we can use recursion to find the order M quantities x(M) and G(M) and the single order M + 1 quantity x(M+1) M+1 , then all of the other x(M+1) j will follow. Fortunately, the quantity x(M+1) M+1 follows from equation (2.8.12) with i = M + 1, M j=1 RM+1−jx(M+1) j + R0x(M+1) M+1 = yM+1 (2.8.17) For the unknown order M + 1 quantities x(M+1) j we can substitute the previous order quantities in G since G(M) M+1−j = x(M) j − x(M+1) j x(M+1) M+1 (2.8.18) The result of this operation is x(M+1) M+1 = M j=1 RM+1−jx(M) j − yM+1 M j=1 RM+1−jG(M) M+1−j − R0 (2.8.19)