# Computational analysis with the HP-25 pocket calculator by Peter Henrici

By Peter Henrici

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G. 38 COROLLARY. L e t A , B be as a b o v e and let z = z2 Z3 ii) I f a = 0 then zl e A f ( E ) , z2 e A [ ( E * ) . e. R e ( ~ ) ¢ 0 . 77~[zl] za =0, "~z~A + - ~ E = 0 and z~B = o. v) If ( E , A, B ) is strongly stabilizable and Tt positive definite ~hen a A - fil3 has no eigenvalues on the imaginary axis. 34. Z3 0. Hence B ' z 2 = 0 and "ffz~A = --flz2E. 17. 39 COROLLARY. ,4',B' be as above and let z = (o~,~) z2 E C2'~+m\{0}, a,/3 C C, Z3 # (o,o), ~ueh that ( ~ A ' - ~B')~ Z3 ii) I f a = 0 -- 0.

46 THEOREM. Let E be nonsingular, R positive det~nite, ( E , A , B ) strongly stabilizable and (E,A, C*(Q - S R - I S * ) C ) strongly detectable. 43). 48) Exk+, = [A - (R + B * X B ) - I ( A ' X B + S*C)]xk 46 lim xk = O. e. g. Kwakernaak/Sivan [K 28] or Lee [L 16]. I In the case that E is singular, previous work is by Bender/Laub [B 11], Silverman [S 8]. 49) has a unique solution. Assume again that e r e - f l A is in Kronecker canonical form and the other matrices are as in the continuous case.

C ) . Then, there exists a unique Q E / 4 & . ( c ) such that T = Q*S = *]TII T12 ] where Tll E C,~,~ 0 T~*J' is upper triangular and has positive diagonal elements. g. Bunse-Cerstner [B 27] or Byers [B 39]. 17 THEOREM [GENERALIZEDSYMPLECTICSCtIUR-FORM]. Let a T - flS be a regular symplectic pencil that has no eigeavalues of modu/us 1. Then, there exists Q E U2n(C) and Z E/482n(C) such that where Tll, S n , T ~ , S~2 E C ~'n are upper triangular and ~ T - flS is a symplectic pencil Moreover, T, S can be chosen such that a T n - fiSh has only eigenva/ues inside the unit cirde.