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Ew 7Wq \`'|B}N`C$FZ'"(0 T[4 Ei6p=PLhA/nkцVi|@نքB4R>vY?C:ڈד5ˋHGB@t"3RekA_I-?zACJ.(uw QxI^Z=x@A Y73yY0{ofle3eV P138}߉hs E;ߐp-q?F{aq?0sh}Hl_CI\½/~8G|H)_R|uGrxoCt#~vs$~ "&&Ѧz>:ϋDcڒrFcT7KJr}&cNW4.;=ƥqiKIf[lv j1Wٱݮ #Hre.o-R^S( gRWdz?#}V֜j'{&&V;61gLҧŏUg:--#}V+pO牔x8A] ltiR7$' *_]pG{UØk\1ZJTdN G,Vטu CM  &matStig <- matrix(c( 1.725, -0.765, -15.00, 0, 0, 0, 0, 0, -0.795, 0.765, 0, 0, 0, 0, 0, 0, 0, 0, 0.3949, -0.3949, 0, 0, 343.4851, 0, 0, 0, 0, 0.1456, 0, 0, 0, 0, 0, 0, 0, 0, -1.725, 0.795, 0, 0, 0, 0, 0, 0, 0.765,-0.765, 0, 0, 0, 0, 0, 0, 15.00, 0, -0.3949, 0, 0, 0, 0, 0, 0, 0, 0.3949, -0.1456), nrow=8, byrow=TRUE) balanceTst <- function(A) { ## Purpose: Consistency checking of balance() {was "dgebal()"} ## ---------------------------------------------------------------------- ## Arguments: a square matrix ## ---------------------------------------------------------------------- ## Author: Martin Maechler, 20 Feb 2008 and on n <- dim(A)[1] ## do *the* three calls and look at result P <- balance(A, "P") doPerm <- function(A, pp, i1, i2) { stopifnot(length(pp) == n, dim(A) == c(n,n), 1 <= i1, i1 <= i2, i2 <= n) A. <- A if(i2 < n) { ## The upper part for(i in n:(i2+1)) { # 'p2' in *reverse* order ## swap i <-> pp[i] both rows and columns tt <- A.[,i]; A.[,i] <- A.[,pp[i]]; A.[,pp[i]] <- tt tt <- A.[i,]; A.[i,] <- A.[pp[i],]; A.[pp[i],] <- tt } } if(i1 > 1) { ## The lower part for(i in 1:(i1-1)) { # 'p1' in *forward* order tt <- A.[,i]; A.[,i] <- A.[,pp[i]]; A.[,pp[i]] <- tt tt <- A.[i,]; A.[i,] <- A.[pp[i],]; A.[pp[i],] <- tt } } A. } checkPerm <- function(P, orig.A) { didPerm <- ((leftP <- (i1 <- P$i1) != 1L) | (rightP <- (i2 <- P$i2) != n)) if(didPerm) { ## *had* permutation -- now check my idea about it pp <- as.integer(P$scale) ## Permute A to become P$z : A. <- doPerm(orig.A, pp = pp, i1=i1, i2=i2) stopifnot(isTRUE(all.equal(A., P$z, tolerance = 1e-15))) ## Now the reverse: Use pp[] and permute A. "back to A": if(leftP) { ## The lower part for(i in (i1-1):1) { # 'p1' in *reverse* order tt <- A.[,i];pp[i]]; <- tti,i,} if(P) { ##upper par2+1):n2forward## swap i <-> both rows and columnsstopifnot(isTRUE(all.equal(A., orig.A, tolerance = 1e-15)} checkPerm(P = A) S <- balance(P$z, "S")# "S" starting from resul"P" S$i1 == 1, S$i2 == n## NowscalingScal (d, A1, A2) { length(d), dim(A12), == c(n,n)) ## A.scaled <- diag(1/d, n) \%*\% A1efficiently:A1 * (rep(d, each = n) / d)2,## CS.re2d * rep1,}(d = S$, A1 = A2zBA, "BB" : B[oth]PB$i1, PB$i2)n*both* permutaandA.permoA, pp = as.integer(B), i1=i2=Bd[c(if(B$i1 > 1) 1:-1), 2 < n) 1dBB## returnlist(P = P, S = S, B = B, Sz.eq.Bz = S$z,)) } m4. <- rbind(c(-1,-2, 0, 0(,10,11213) op <-s(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$# ditto a non-empty ``less-d'' --- m4outer(2^(0:7), m4[lower.tri(m40 #-->triangular ==>have manys e it; so)the c(4,2:1,3); 4[p,p] m4 dm4much! i1 = i2 = 1 ! ##- Complexs zba + 3i * m4zba4) z*(1 + 1i)) exprs =ba$ 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 nilA3(x,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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