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"back to A"if:1} if2+1):n} = AS"S")# "S" starting from resul"P" S$i1 == 1, S$i2 == nthe scalingScal (d, A1, A2length(d), dim(A12), == c(n,n## A.diag(1/d, n) \%*\% A1efficientlyA1 * (rep(d, each = n) / d)2,## CS.re2d * rep1,}(d = S, A1 =A2zBBB" : B[oth]PB$i1, PB$##*bothandA.pBi2=Bd[c(if(B$1:-1), 1dBB## returnlist(P = P, S = S, B = B, Sz.eq.Bz = S$z,)) } m4rbind(c(-1,-2, 0, 0(,10,11213) os(str = strOvec.len = 12)) str(b4.Tst(m4.)) with(b4., P, B)) # TRUE (everywhere?) ## better (?) example (matrix(c(0,-1,0,-2,10,), 4,4a)) (eq <- a, nowba$# ditto a non-empty ``less-d'' --- m4outer(2^(0:7), m4[.tri(m40 #-->triangular ==>have manys it; so)the c(4,2:1,3); 4[p,p] m4 dm4much! i1 = i2 = 1 ! ##- Complexs zba + 3i * m4zba4) z*(1 + 1i)) exprs =ba$ Re(z))Imdm4 4 * 3}) op) #t #### "demo": Makse () defini easilyto useRs --> w themests in ..//exact-ex.R ## ~ ### For nilpotentces A, exp(A) is polynomial in A Mathworld givesof g 3 x 3 le nilA3x,y,z## Purpose: six 3x3 A (with A^n = 0n >= 3/ 0 x z \A = [ 0 0 y ]\0 /and its exponential-Author: M Maechler, Date: 19 Jan 2008(nx))yz meric(x),yA = c0,x,0,z,y), 2,2)expc(1,0,0), c(x,1z + x*y/2, y, 1 relative error typically --ified here relErrtarget, currentmean(abs -) /)) facMatn, R_FUN, ev =(n), M = rMat)) Construct randomx of which we "know" expm(xbecause we set x := M %*ev) solve(M)rguments: n:dimenof :numbertorn)ev-n vector of eigenvaluMn x n. 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