library(expm) (sI <- sessionInfo()) packageDescription("Matrix") packageDescription("expm") source(system.file("test-tools.R", package= "expm"), keep.source=FALSE) ## Note that these results are achieved with the default ## settings order=8, method="Pade" -- accuracy could ## presumably be improved still further by some tuning ## of these settings. ### Latest ATLAS (for BDR on F 36; R-devel Oct.2023) has much worse precision: ##==> use much larger tolerance in such cases: ## Simplified (needs R 3.4.0 and newer, from robustbase/inst/xtraR/platform-sessionInfo.R ) : BLAS <- extSoftVersion()[["BLAS"]] Lapack <- La_library() is.BLAS.Lapack <- identical(BLAS, Lapack) ## A cheap check (that works on KH's debian-gcc setup, 2019-05): if(!length(BLAS.is.openBLAS <- grepl("openblas", BLAS, ignore.case=TRUE))) BLAS.is.openBLAS <- NA if(!length(Lapack.is.openBLAS <- grepl("openblas", Lapack, ignore.case=TRUE))) Lapack.is.openBLAS <- NA (maybeATLAS <- is.BLAS.Lapack && !BLAS.is.openBLAS) ## ---------------------------- ## Test case 1 from Ward (1977) ## ---------------------------- T1 <- rbind(c(4, 2, 0), c(1, 4, 1), c(1, 1, 4)) (m1 <- expm(T1, method="Pade")) (m1O <- expm(T1, method="PadeO"))# very slightly different (m1T <- expm(T1, method="Taylor")) (m1TO <- expm(T1, method="TaylorO")) ## True Eigenvalue Decomposition of T1 s2 <- sqrt(2) eV1 <- matrix(c(s2,s2,s2, -2,1,1, 2,-1,-1) / sqrt(6), 3,3) L1 <- diag(lm1 <- c(6, 3, 3)) stopifnot( all.equal(eV1 %*% L1, T1 %*% eV1, tolerance=1e-15) ) ## However, eV1 is not orthogonal, but of rank 2 if(FALSE) { ## require("Rmpfr")) { ## 200 bit precision version of that S2 <- sqrt(mpfr(2,200)) E1 <- c(S2,S2,S2, -2,1,1, 2,-1,-1) / sqrt(mpfr(6,200)) dim(E1) <- c(3,3) print(E1 %*% L1) print(E1) } ## "true" result m1.t <- matrix(c(147.866622446369, 127.781085523181, 127.781085523182, 183.765138646367, 183.765138646366, 163.679601723179, 71.797032399996, 91.8825693231832, 111.968106246371), 3,3) stopifnot(all.equal(m1.t, m1, check.attributes=FALSE, tolerance = 1e-13), all.equal(m1.t, m1O, check.attributes=FALSE, tolerance = 1e-13), all.equal(m1.t,m1T, check.attributes=FALSE, tolerance = 1e-13), all.equal(m1.t,m1TO, check.attributes=FALSE, tolerance = 1e-13), all.equal(m1.t, expm(T1,"Ward77"), tolerance = 1e-13), all.equal(m1.t, expm(T1,"R_Pade"), tolerance = 1e-13), all.equal(m1.t, expm(T1,"R_Ward77"), tolerance = 1e-13)) ## -- these agree with ward (1977, p608) ## m1.2 <- try( expm(T1, "R_Eigen") ) ## 32-bit: gives an error from solve; 64-bit "ok" if(!inherits(m1.2, "try-error")) { if(FALSE)## with libatlas R_Eigen is "sehr eigen" stopifnot(all.equal(m1.t, m1.2, check.attributes=FALSE)) ## but it's less accurate: print( all.equal(m1.t, m1.2, check.attributes=FALSE, tolerance= 1e-12)) ##-> rel.diff = 6.44e-10 / 6.2023e-10 ##__ BUT 0.1228099 ##__ with libatlas (ubuntu 12.04 libatlas-base-dev Version: 3.8.4-3build1) } ## ## ---------------------------- ## Test case 2 from Ward (1977) ## ---------------------------- T2 <- t(matrix(c( 29.87942128909879, .7815750847907159, -2.289519314033932, .7815750847907159, 25.72656945571064, 8.680737820540137, -2.289519314033932, 8.680737820540137, 34.39400925519054), 3, 3)) (m2 <- expm(T2, method="Pade")) ## [,1] [,2] [,3] ##[1,] 5496313853692357 -18231880972009844 -30475770808580828 ##[2,] -18231880972009852 60605228702227024 101291842930256144 ##[3,] -30475770808580840 101291842930256144 169294411240859072 ## -- which agrees with Ward (1977) to 13 significant figures (m2O <- expm(T2, method="PadeO")) (m2T <- expm(T2,method="Taylor")) (m2TO <- expm(T2,method="TaylorO")) m2.t <- matrix(c(5496313853692216, -18231880972008932, -30475770808579672, -18231880972008928, 60605228702222480, 101291842930249776, -30475770808579672, 101291842930249808, 169294411240850528), 3, 3) ## -- in this case a very similar degree of accuracy -- even Taylor is ok stopifnot(all.equal(m2.t, m2, check.attributes=FALSE, tolerance = 1e-12), all.equal(m2.t, m2O,check.attributes=FALSE, tolerance = 1e-12), all.equal(m2.t,m2T, check.attributes=FALSE, tolerance = 1e-12), all.equal(m2.t,m2TO,check.attributes=FALSE, tolerance = 1e-12), all.equal(m2.t, expm(T2,"Ward77"), tolerance = 1e-12), all.equal(m2.t, expm(T2,"R_Ward77"), tolerance = 1e-12), all.equal(m2.t, expm(T2,"R_Pade"), tolerance = 1e-12), TRUE) ## ---------------------------- ## Test case 3 from Ward (1977) ## ---------------------------- T3 <- t(matrix(c( -131, 19, 18, -390, 56, 54, -387, 57, 52), 3, 3)) (m3 <- expm(T3, method="Pade")) ## [,1] [,2] [,3] ##[1,] -1.5096441587713636 0.36787943910439874 0.13533528117301735 ##[2,] -5.6325707997970271 1.47151775847745725 0.40600584351567010 ##[3,] -4.9349383260294299 1.10363831731417195 0.54134112675653534 ## -- agrees to 10dp with Ward (1977), p608. (m3O <- expm(T3, method="PadeO")) (m3T <- expm(T3,method="Taylor")) (m3TO <- expm(T3,method="TaylorO")) m3.t <- matrix(c(-1.50964415879218, -5.6325707998812, -4.934938326092, 0.367879439109187, 1.47151775849686, 1.10363831732856, 0.135335281175235, 0.406005843524598, 0.541341126763207), 3,3) stopifnot(all.equal(m3.t, m3, check.attributes=FALSE, tolerance = 3e-11), # ^^^^^ # 1.2455e-11 for libatlas (above) all.equal(m3.t, m3T, check.attributes=FALSE, tolerance = 1e-11), all.equal(m3.t, m3O, check.attributes=FALSE, tolerance = 8e-11),# M1: 1.39e-11 all.equal(m3.t, m3TO, check.attributes=FALSE, tolerance = 1e-11), all.equal(m3.t, expm(T3,"R_Eigen"), tolerance = 1e-11), all.equal(m3.t, expm(T3,"Ward77"), tolerance = 1e-11), all.equal(m3.t, expm(T3,"R_Ward"), tolerance = 1e-11), all.equal(m3.t, expm(T3,"R_Pade"), tolerance = 1e-11), TRUE) ## -- in this case, a similar level of agreement with Ward (1977). ## ---------------------------- ## Test case 4 from Ward (1977) ## ---------------------------- T4 <- array(c(0, 0, 0, 0, 0, 0, 0, 0, 0, 1e-10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0), dim = c(10, 10)) (m4 <- expm(T4, method="Pade")) (m4O <- expm(T4, method="PadeO")) (m4T <- expm(T4,method="Taylor")) (m4TO <- expm(T4,method="TaylorO")) ## ATLAS on BDR's gannet (Fedora 26; gcc-13; R-devel 2023-10-24) tol1 <- if(maybeATLAS) 4e-7 else 5e-15 # (m4, m4O) gave "Mean relative difference: 1.242879e-07" stopifnot(all.equal(m4 [,10], 1/gamma(10:1), tolerance=1e-14), all.equal(m4O [,10], 1/gamma(10:1), tolerance=1e-14), all.equal(m4T [,10], 1/gamma(10:1), tolerance=1e-14), all.equal(m4TO[,10], 1/gamma(10:1), tolerance=1e-14), all.equal(m4, m4O, check.attributes=FALSE, tolerance=tol1), all.equal(m4, m4T, check.attributes=FALSE, tolerance=tol1), all.equal(m4, m4TO,check.attributes=FALSE, tolerance=tol1), all.equal(m4, expm(T4,"Ward77"), check.attributes=FALSE, tolerance = 1e-14), all.equal(m4, expm(T4,"R_Ward"), check.attributes=FALSE, tolerance = 1e-14), all.equal(m4, expm(T4,"R_Pade"), check.attributes=FALSE, tolerance = 1e-14), max(abs(m4 - expm(T4,"R_Eigen"))) < 1e-7) ## here expm(., EV ) is accurate only to 7 d.p., whereas ## expm(.,Pade) is correct to at least 14 d.p. ### Test case with diagonalizable matrix with multiple Eigen values: A4 <- rbind(c(-1, 3, -1), c(-3, 5, -1), c(-3, 3, 1)) Ea4 <- eigen(A4) stopifnot(all.equal(Ea4$values, (lam4 <- c(2,2,1)))) ## However, the eigen values don't show the nice property: V4 <- Ea4$vectors crossprod(V4) ## i.e., they are *not* orthogonal ## but still diagonalize: stopifnot(all.equal(A4, V4 %*% diag(x=lam4) %*% solve(V4))) ## whereas this diagonalizes *and* looks nice W4 <- rbind(c(1, 1, -1), c(1, 1, 0), c(1, 0, 3)) (sW4 <- solve(W4)) assert.EQ(diag(x = c(1,2,2)), solve(W4) %*% A4 %*% W4, giveRE=TRUE) assert.EQ(A4, logm(expm(A4)), tol = 3e-13, giveRE=TRUE) ## seen 5.5e-14 with R's own matprod