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Type 'q()' to quit R. > require(DPQmpfr) Loading required package: DPQmpfr > require(Rmpfr) Loading required package: Rmpfr Loading required package: gmp Attaching package: 'gmp' The following objects are masked from 'package:base': %*%, apply, crossprod, matrix, tcrossprod C code of R package 'Rmpfr': GMP using 64 bits per limb Attaching package: 'Rmpfr' The following object is masked from 'package:gmp': outer The following objects are masked from 'package:stats': dbinom, dgamma, dnbinom, dnorm, dpois, dt, pnorm The following objects are masked from 'package:base': cbind, pmax, pmin, rbind > > ## very accurate set of *large* abscissa values > x <- mpfr(2,1024)^ seq(3, 100, by=1/2) > xN <- asNumeric(x) > > ## Looking at pnormU_S53() = upper bound to pnorm(), for large x, > ## the term log(4 / (3 + sqrt(1 + 8/x^2))) early underflows to 0: > eps_S53 <- function(x, method = c("direct", "log1pExct", "deltaFull", + "delta1", "delta2", "delta3")) { + method <- match.arg(method) + delta <- function(x) { xm2 <- x^-2; (.5*(1 - sqrt(1 + 8*xm2)) + xm2) / (1 + xm2) } + switch(method, + "direct" = log(4 / (3 + sqrt(1 + 8*x^-2))), + "log1pExct" = log1p(delta(x)), + "deltaFull" = delta(x), + "delta1" = -1/(x^2+1) , ## 1 term for delta(x) + "delta2" = -(1 - (2/x)^2) / (x^2+1) , ## 2 terms for delta(x) + "delta3" = {tx2 <- (2/x)^2 ; -(1 - tx2*(1 - tx2)) / (x^2+1) }, ## 3 terms + stop("invalid 'method': ", method)) + } > > > (S53meth <- eval(formals(eps_S53)$method)) [1] "direct" "log1pExct" "deltaFull" "delta1" "delta2" "delta3" > ## "direct" "log1pExct" "deltaFull" "delta1" "delta2" "delta3" > > epsAllS53 <- function(x, meth = eval(formals(eps_S53)$method)) { + stopifnot(is.character(meth), is.numeric(x) || is(x,"mpfr")) + Rl <- lapply(meth, function(m) eps_S53(x, method=m)) + new("mpfrMatrix", unlist(Rl), + Dim = c(length(x), length(meth)), + Dimnames = list(NULL, meth)) + } > > EA <- epsAllS53(x) > asNumeric(EA) # looks +- ok direct log1pExct deltaFull delta1 delta2 [1,] 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-1.401298e-45 -1.401298e-45 -1.401298e-45 -1.401298e-45 [145,] -7.006492e-46 -7.006492e-46 -7.006492e-46 -7.006492e-46 -7.006492e-46 [146,] -3.503246e-46 -3.503246e-46 -3.503246e-46 -3.503246e-46 -3.503246e-46 [147,] -1.751623e-46 -1.751623e-46 -1.751623e-46 -1.751623e-46 -1.751623e-46 [148,] -8.758115e-47 -8.758115e-47 -8.758115e-47 -8.758115e-47 -8.758115e-47 [149,] -4.379058e-47 -4.379058e-47 -4.379058e-47 -4.379058e-47 -4.379058e-47 [150,] -2.189529e-47 -2.189529e-47 -2.189529e-47 -2.189529e-47 -2.189529e-47 [151,] -1.094764e-47 -1.094764e-47 -1.094764e-47 -1.094764e-47 -1.094764e-47 [152,] -5.473822e-48 -5.473822e-48 -5.473822e-48 -5.473822e-48 -5.473822e-48 [153,] -2.736911e-48 -2.736911e-48 -2.736911e-48 -2.736911e-48 -2.736911e-48 [154,] -1.368456e-48 -1.368456e-48 -1.368456e-48 -1.368456e-48 -1.368456e-48 [155,] -6.842278e-49 -6.842278e-49 -6.842278e-49 -6.842278e-49 -6.842278e-49 [156,] -3.421139e-49 -3.421139e-49 -3.421139e-49 -3.421139e-49 -3.421139e-49 [157,] -1.710569e-49 -1.710569e-49 -1.710569e-49 -1.710569e-49 -1.710569e-49 [158,] -8.552847e-50 -8.552847e-50 -8.552847e-50 -8.552847e-50 -8.552847e-50 [159,] -4.276424e-50 -4.276424e-50 -4.276424e-50 -4.276424e-50 -4.276424e-50 [160,] -2.138212e-50 -2.138212e-50 -2.138212e-50 -2.138212e-50 -2.138212e-50 [161,] -1.069106e-50 -1.069106e-50 -1.069106e-50 -1.069106e-50 -1.069106e-50 [162,] -5.345529e-51 -5.345529e-51 -5.345529e-51 -5.345529e-51 -5.345529e-51 [163,] -2.672765e-51 -2.672765e-51 -2.672765e-51 -2.672765e-51 -2.672765e-51 [164,] -1.336382e-51 -1.336382e-51 -1.336382e-51 -1.336382e-51 -1.336382e-51 [165,] -6.681912e-52 -6.681912e-52 -6.681912e-52 -6.681912e-52 -6.681912e-52 [166,] -3.340956e-52 -3.340956e-52 -3.340956e-52 -3.340956e-52 -3.340956e-52 [167,] -1.670478e-52 -1.670478e-52 -1.670478e-52 -1.670478e-52 -1.670478e-52 [168,] -8.352390e-53 -8.352390e-53 -8.352390e-53 -8.352390e-53 -8.352390e-53 [169,] -4.176195e-53 -4.176195e-53 -4.176195e-53 -4.176195e-53 -4.176195e-53 [170,] -2.088097e-53 -2.088097e-53 -2.088097e-53 -2.088097e-53 -2.088097e-53 [171,] -1.044049e-53 -1.044049e-53 -1.044049e-53 -1.044049e-53 -1.044049e-53 [172,] -5.220244e-54 -5.220244e-54 -5.220244e-54 -5.220244e-54 -5.220244e-54 [173,] -2.610122e-54 -2.610122e-54 -2.610122e-54 -2.610122e-54 -2.610122e-54 [174,] -1.305061e-54 -1.305061e-54 -1.305061e-54 -1.305061e-54 -1.305061e-54 [175,] -6.525304e-55 -6.525304e-55 -6.525304e-55 -6.525304e-55 -6.525304e-55 [176,] -3.262652e-55 -3.262652e-55 -3.262652e-55 -3.262652e-55 -3.262652e-55 [177,] -1.631326e-55 -1.631326e-55 -1.631326e-55 -1.631326e-55 -1.631326e-55 [178,] -8.156631e-56 -8.156631e-56 -8.156631e-56 -8.156631e-56 -8.156631e-56 [179,] -4.078315e-56 -4.078315e-56 -4.078315e-56 -4.078315e-56 -4.078315e-56 [180,] -2.039158e-56 -2.039158e-56 -2.039158e-56 -2.039158e-56 -2.039158e-56 [181,] -1.019579e-56 -1.019579e-56 -1.019579e-56 -1.019579e-56 -1.019579e-56 [182,] -5.097894e-57 -5.097894e-57 -5.097894e-57 -5.097894e-57 -5.097894e-57 [183,] -2.548947e-57 -2.548947e-57 -2.548947e-57 -2.548947e-57 -2.548947e-57 [184,] -1.274474e-57 -1.274474e-57 -1.274474e-57 -1.274474e-57 -1.274474e-57 [185,] -6.372368e-58 -6.372368e-58 -6.372368e-58 -6.372368e-58 -6.372368e-58 [186,] -3.186184e-58 -3.186184e-58 -3.186184e-58 -3.186184e-58 -3.186184e-58 [187,] -1.593092e-58 -1.593092e-58 -1.593092e-58 -1.593092e-58 -1.593092e-58 [188,] -7.965460e-59 -7.965460e-59 -7.965460e-59 -7.965460e-59 -7.965460e-59 [189,] -3.982730e-59 -3.982730e-59 -3.982730e-59 -3.982730e-59 -3.982730e-59 [190,] -1.991365e-59 -1.991365e-59 -1.991365e-59 -1.991365e-59 -1.991365e-59 [191,] -9.956824e-60 -9.956824e-60 -9.956824e-60 -9.956824e-60 -9.956824e-60 [192,] -4.978412e-60 -4.978412e-60 -4.978412e-60 -4.978412e-60 -4.978412e-60 [193,] -2.489206e-60 -2.489206e-60 -2.489206e-60 -2.489206e-60 -2.489206e-60 [194,] -1.244603e-60 -1.244603e-60 -1.244603e-60 -1.244603e-60 -1.244603e-60 [195,] -6.223015e-61 -6.223015e-61 -6.223015e-61 -6.223015e-61 -6.223015e-61 delta3 [1,] -1.448317e-02 [2,] -7.517260e-03 [3,] -3.831203e-03 [4,] -1.934208e-03 [5,] -9.718137e-04 [6,] -4.870916e-04 [7,] -2.438429e-04 [8,] -1.219958e-04 [9,] -6.101653e-05 [10,] -3.051292e-05 [11,] -1.525762e-05 [12,] -7.629104e-06 [13,] -3.814625e-06 [14,] -1.907330e-06 [15,] -9.536698e-07 [16,] -4.768360e-07 [17,] -2.384183e-07 [18,] -1.192092e-07 [19,] -5.960463e-08 [20,] -2.980232e-08 [21,] -1.490116e-08 [22,] -7.450580e-09 [23,] -3.725290e-09 [24,] -1.862645e-09 [25,] -9.313226e-10 [26,] -4.656613e-10 [27,] -2.328306e-10 [28,] -1.164153e-10 [29,] -5.820766e-11 [30,] -2.910383e-11 [31,] -1.455192e-11 [32,] -7.275958e-12 [33,] -3.637979e-12 [34,] -1.818989e-12 [35,] -9.094947e-13 [36,] -4.547474e-13 [37,] -2.273737e-13 [38,] -1.136868e-13 [39,] -5.684342e-14 [40,] -2.842171e-14 [41,] -1.421085e-14 [42,] -7.105427e-15 [43,] -3.552714e-15 [44,] -1.776357e-15 [45,] -8.881784e-16 [46,] -4.440892e-16 [47,] -2.220446e-16 [48,] -1.110223e-16 [49,] -5.551115e-17 [50,] -2.775558e-17 [51,] -1.387779e-17 [52,] -6.938894e-18 [53,] -3.469447e-18 [54,] -1.734723e-18 [55,] -8.673617e-19 [56,] -4.336809e-19 [57,] -2.168404e-19 [58,] -1.084202e-19 [59,] -5.421011e-20 [60,] -2.710505e-20 [61,] -1.355253e-20 [62,] -6.776264e-21 [63,] -3.388132e-21 [64,] -1.694066e-21 [65,] -8.470329e-22 [66,] -4.235165e-22 [67,] -2.117582e-22 [68,] -1.058791e-22 [69,] -5.293956e-23 [70,] -2.646978e-23 [71,] -1.323489e-23 [72,] -6.617445e-24 [73,] -3.308722e-24 [74,] -1.654361e-24 [75,] -8.271806e-25 [76,] -4.135903e-25 [77,] -2.067952e-25 [78,] -1.033976e-25 [79,] -5.169879e-26 [80,] -2.584939e-26 [81,] -1.292470e-26 [82,] -6.462349e-27 [83,] -3.231174e-27 [84,] -1.615587e-27 [85,] -8.077936e-28 [86,] -4.038968e-28 [87,] -2.019484e-28 [88,] -1.009742e-28 [89,] -5.048710e-29 [90,] -2.524355e-29 [91,] -1.262177e-29 [92,] -6.310887e-30 [93,] -3.155444e-30 [94,] -1.577722e-30 [95,] -7.888609e-31 [96,] -3.944305e-31 [97,] -1.972152e-31 [98,] -9.860761e-32 [99,] -4.930381e-32 [100,] -2.465190e-32 [101,] -1.232595e-32 [102,] -6.162976e-33 [103,] -3.081488e-33 [104,] -1.540744e-33 [105,] -7.703720e-34 [106,] -3.851860e-34 [107,] -1.925930e-34 [108,] -9.629650e-35 [109,] -4.814825e-35 [110,] -2.407412e-35 [111,] -1.203706e-35 [112,] -6.018531e-36 [113,] -3.009266e-36 [114,] -1.504633e-36 [115,] -7.523164e-37 [116,] -3.761582e-37 [117,] -1.880791e-37 [118,] -9.403955e-38 [119,] -4.701977e-38 [120,] -2.350989e-38 [121,] -1.175494e-38 [122,] -5.877472e-39 [123,] -2.938736e-39 [124,] -1.469368e-39 [125,] -7.346840e-40 [126,] -3.673420e-40 [127,] -1.836710e-40 [128,] -9.183550e-41 [129,] -4.591775e-41 [130,] -2.295887e-41 [131,] -1.147944e-41 [132,] -5.739719e-42 [133,] -2.869859e-42 [134,] -1.434930e-42 [135,] -7.174648e-43 [136,] -3.587324e-43 [137,] -1.793662e-43 [138,] -8.968310e-44 [139,] -4.484155e-44 [140,] -2.242078e-44 [141,] -1.121039e-44 [142,] -5.605194e-45 [143,] -2.802597e-45 [144,] -1.401298e-45 [145,] -7.006492e-46 [146,] -3.503246e-46 [147,] -1.751623e-46 [148,] -8.758115e-47 [149,] -4.379058e-47 [150,] -2.189529e-47 [151,] -1.094764e-47 [152,] -5.473822e-48 [153,] -2.736911e-48 [154,] -1.368456e-48 [155,] -6.842278e-49 [156,] -3.421139e-49 [157,] -1.710569e-49 [158,] -8.552847e-50 [159,] -4.276424e-50 [160,] -2.138212e-50 [161,] -1.069106e-50 [162,] -5.345529e-51 [163,] -2.672765e-51 [164,] -1.336382e-51 [165,] -6.681912e-52 [166,] -3.340956e-52 [167,] -1.670478e-52 [168,] -8.352390e-53 [169,] -4.176195e-53 [170,] -2.088097e-53 [171,] -1.044049e-53 [172,] -5.220244e-54 [173,] -2.610122e-54 [174,] -1.305061e-54 [175,] -6.525304e-55 [176,] -3.262652e-55 [177,] -1.631326e-55 [178,] -8.156631e-56 [179,] -4.078315e-56 [180,] -2.039158e-56 [181,] -1.019579e-56 [182,] -5.097894e-57 [183,] -2.548947e-57 [184,] -1.274474e-57 [185,] -6.372368e-58 [186,] -3.186184e-58 [187,] -1.593092e-58 [188,] -7.965460e-59 [189,] -3.982730e-59 [190,] -1.991365e-59 [191,] -9.956824e-60 [192,] -4.978412e-60 [193,] -2.489206e-60 [194,] -1.244603e-60 [195,] -6.223015e-61 > > ## testing cbind() i.e., the Rmpfr-method a bit here ! > chkDn <- function(r) stopifnot(identical( + dimnames(r), + list(NULL, c("x", "log2x", eval(formals(eps_S53)$method))))) > chkDn2 <- function(r) stopifnot(identical( + colnames(r)[-(1:2)], eval(formals(eps_S53)$method))) > chkDn ( r1 <- cbind(x, log2x = log2(x), EA)) > chkDn2( r2 <- selectMethod("cbind","mpfr")(x, log2x = log2(x), EA) ) > ## are the same if names are *made* to match: > stopifnot(identical(r1, local({t <- r2; dimnames(t) <- dimnames(r1); t}))) > chkDn2( r3 <- selectMethod("cbind","mpfr")(x, log2(x), epsAllS53(x)) ) > chkDn2( r4 <- cbind(x, log2(x), epsAllS53(x)) ) > stopifnot(identical(r1, local({t <- r4; dimnames(t) <- dimnames(r1); t}))) > > U_S53mat <- function(x) cbind(x, log2x = log2(x), epsAllS53(x)) > chkDn(Rx <- U_S53mat(x)) > dimnames(Rx) # .. all empty, i.e., list(NULL, NULL) ... even though epsAll..() gives nice colnames [[1]] NULL [[2]] [1] "x" "log2x" "direct" "log1pExct" "deltaFull" "delta1" [7] "delta2" "delta3" > ## > if(FALSE) ## no longer needed ! + ## A version with correct colnames : + U_S53mat <- function(x) { + eA <- epsAllS53(x) + r <- cbind(x, log2(x), eA) + colnames(r) <- c("x", "log2.x", colnames(eA)) + r + } > > stopifnot(Rx[,-(1:2)] < 0) > > op <- options(width = 100) > print(asNumeric(Rx), digits=5) ; options(op) x log2x direct log1pExct deltaFull delta1 delta2 delta3 [1,] 8.0000e+00 3.0 -1.5051e-02 -1.4585e-02 -1.4479e-02 -1.5385e-02 -1.4423e-02 -1.4483e-02 [2,] 1.1314e+01 3.5 -7.6647e-03 -7.5454e-03 -7.5170e-03 -7.7519e-03 -7.5097e-03 -7.5173e-03 [3,] 1.6000e+01 4.0 -3.8687e-03 -3.8385e-03 -3.8312e-03 -3.8911e-03 -3.8303e-03 -3.8312e-03 [4,] 2.2627e+01 4.5 -1.9437e-03 -1.9361e-03 -1.9342e-03 -1.9493e-03 -1.9341e-03 -1.9342e-03 [5,] 3.2000e+01 5.0 -9.7419e-04 -9.7229e-04 -9.7181e-04 -9.7561e-04 -9.7180e-04 -9.7181e-04 [6,] 4.5255e+01 5.5 -4.8769e-04 -4.8721e-04 -4.8709e-04 -4.8804e-04 -4.8709e-04 -4.8709e-04 [7,] 6.4000e+01 6.0 -2.4399e-04 -2.4387e-04 -2.4384e-04 -2.4408e-04 -2.4384e-04 -2.4384e-04 [8,] 9.0510e+01 6.5 -1.2203e-04 -1.2200e-04 -1.2200e-04 -1.2206e-04 -1.2200e-04 -1.2200e-04 [9,] 1.2800e+02 7.0 -6.1026e-05 -6.1018e-05 -6.1017e-05 -6.1031e-05 -6.1017e-05 -6.1017e-05 [10,] 1.8102e+02 7.5 -3.0515e-05 -3.0513e-05 -3.0513e-05 -3.0517e-05 -3.0513e-05 -3.0513e-05 [11,] 2.5600e+02 8.0 -1.5258e-05 -1.5258e-05 -1.5258e-05 -1.5259e-05 -1.5258e-05 -1.5258e-05 [12,] 3.6204e+02 8.5 -7.6292e-06 -7.6291e-06 -7.6291e-06 -7.6293e-06 -7.6291e-06 -7.6291e-06 [13,] 5.1200e+02 9.0 -3.8147e-06 -3.8146e-06 -3.8146e-06 -3.8147e-06 -3.8146e-06 -3.8146e-06 [14,] 7.2408e+02 9.5 -1.9073e-06 -1.9073e-06 -1.9073e-06 -1.9073e-06 -1.9073e-06 -1.9073e-06 [15,] 1.0240e+03 10.0 -9.5367e-07 -9.5367e-07 -9.5367e-07 -9.5367e-07 -9.5367e-07 -9.5367e-07 [16,] 1.4482e+03 10.5 -4.7684e-07 -4.7684e-07 -4.7684e-07 -4.7684e-07 -4.7684e-07 -4.7684e-07 [17,] 2.0480e+03 11.0 -2.3842e-07 -2.3842e-07 -2.3842e-07 -2.3842e-07 -2.3842e-07 -2.3842e-07 [18,] 2.8963e+03 11.5 -1.1921e-07 -1.1921e-07 -1.1921e-07 -1.1921e-07 -1.1921e-07 -1.1921e-07 [19,] 4.0960e+03 12.0 -5.9605e-08 -5.9605e-08 -5.9605e-08 -5.9605e-08 -5.9605e-08 -5.9605e-08 [20,] 5.7926e+03 12.5 -2.9802e-08 -2.9802e-08 -2.9802e-08 -2.9802e-08 -2.9802e-08 -2.9802e-08 [21,] 8.1920e+03 13.0 -1.4901e-08 -1.4901e-08 -1.4901e-08 -1.4901e-08 -1.4901e-08 -1.4901e-08 [22,] 1.1585e+04 13.5 -7.4506e-09 -7.4506e-09 -7.4506e-09 -7.4506e-09 -7.4506e-09 -7.4506e-09 [23,] 1.6384e+04 14.0 -3.7253e-09 -3.7253e-09 -3.7253e-09 -3.7253e-09 -3.7253e-09 -3.7253e-09 [24,] 2.3170e+04 14.5 -1.8626e-09 -1.8626e-09 -1.8626e-09 -1.8626e-09 -1.8626e-09 -1.8626e-09 [25,] 3.2768e+04 15.0 -9.3132e-10 -9.3132e-10 -9.3132e-10 -9.3132e-10 -9.3132e-10 -9.3132e-10 [26,] 4.6341e+04 15.5 -4.6566e-10 -4.6566e-10 -4.6566e-10 -4.6566e-10 -4.6566e-10 -4.6566e-10 [27,] 6.5536e+04 16.0 -2.3283e-10 -2.3283e-10 -2.3283e-10 -2.3283e-10 -2.3283e-10 -2.3283e-10 [28,] 9.2682e+04 16.5 -1.1642e-10 -1.1642e-10 -1.1642e-10 -1.1642e-10 -1.1642e-10 -1.1642e-10 [29,] 1.3107e+05 17.0 -5.8208e-11 -5.8208e-11 -5.8208e-11 -5.8208e-11 -5.8208e-11 -5.8208e-11 [30,] 1.8536e+05 17.5 -2.9104e-11 -2.9104e-11 -2.9104e-11 -2.9104e-11 -2.9104e-11 -2.9104e-11 [31,] 2.6214e+05 18.0 -1.4552e-11 -1.4552e-11 -1.4552e-11 -1.4552e-11 -1.4552e-11 -1.4552e-11 [32,] 3.7073e+05 18.5 -7.2760e-12 -7.2760e-12 -7.2760e-12 -7.2760e-12 -7.2760e-12 -7.2760e-12 [33,] 5.2429e+05 19.0 -3.6380e-12 -3.6380e-12 -3.6380e-12 -3.6380e-12 -3.6380e-12 -3.6380e-12 [34,] 7.4146e+05 19.5 -1.8190e-12 -1.8190e-12 -1.8190e-12 -1.8190e-12 -1.8190e-12 -1.8190e-12 [35,] 1.0486e+06 20.0 -9.0949e-13 -9.0949e-13 -9.0949e-13 -9.0949e-13 -9.0949e-13 -9.0949e-13 [36,] 1.4829e+06 20.5 -4.5475e-13 -4.5475e-13 -4.5475e-13 -4.5475e-13 -4.5475e-13 -4.5475e-13 [37,] 2.0972e+06 21.0 -2.2737e-13 -2.2737e-13 -2.2737e-13 -2.2737e-13 -2.2737e-13 -2.2737e-13 [38,] 2.9658e+06 21.5 -1.1369e-13 -1.1369e-13 -1.1369e-13 -1.1369e-13 -1.1369e-13 -1.1369e-13 [39,] 4.1943e+06 22.0 -5.6843e-14 -5.6843e-14 -5.6843e-14 -5.6843e-14 -5.6843e-14 -5.6843e-14 [40,] 5.9316e+06 22.5 -2.8422e-14 -2.8422e-14 -2.8422e-14 -2.8422e-14 -2.8422e-14 -2.8422e-14 [41,] 8.3886e+06 23.0 -1.4211e-14 -1.4211e-14 -1.4211e-14 -1.4211e-14 -1.4211e-14 -1.4211e-14 [42,] 1.1863e+07 23.5 -7.1054e-15 -7.1054e-15 -7.1054e-15 -7.1054e-15 -7.1054e-15 -7.1054e-15 [43,] 1.6777e+07 24.0 -3.5527e-15 -3.5527e-15 -3.5527e-15 -3.5527e-15 -3.5527e-15 -3.5527e-15 [44,] 2.3727e+07 24.5 -1.7764e-15 -1.7764e-15 -1.7764e-15 -1.7764e-15 -1.7764e-15 -1.7764e-15 [45,] 3.3554e+07 25.0 -8.8818e-16 -8.8818e-16 -8.8818e-16 -8.8818e-16 -8.8818e-16 -8.8818e-16 [46,] 4.7453e+07 25.5 -4.4409e-16 -4.4409e-16 -4.4409e-16 -4.4409e-16 -4.4409e-16 -4.4409e-16 [47,] 6.7109e+07 26.0 -2.2204e-16 -2.2204e-16 -2.2204e-16 -2.2204e-16 -2.2204e-16 -2.2204e-16 [48,] 9.4906e+07 26.5 -1.1102e-16 -1.1102e-16 -1.1102e-16 -1.1102e-16 -1.1102e-16 -1.1102e-16 [49,] 1.3422e+08 27.0 -5.5511e-17 -5.5511e-17 -5.5511e-17 -5.5511e-17 -5.5511e-17 -5.5511e-17 [50,] 1.8981e+08 27.5 -2.7756e-17 -2.7756e-17 -2.7756e-17 -2.7756e-17 -2.7756e-17 -2.7756e-17 [51,] 2.6844e+08 28.0 -1.3878e-17 -1.3878e-17 -1.3878e-17 -1.3878e-17 -1.3878e-17 -1.3878e-17 [52,] 3.7963e+08 28.5 -6.9389e-18 -6.9389e-18 -6.9389e-18 -6.9389e-18 -6.9389e-18 -6.9389e-18 [53,] 5.3687e+08 29.0 -3.4694e-18 -3.4694e-18 -3.4694e-18 -3.4694e-18 -3.4694e-18 -3.4694e-18 [54,] 7.5925e+08 29.5 -1.7347e-18 -1.7347e-18 -1.7347e-18 -1.7347e-18 -1.7347e-18 -1.7347e-18 [55,] 1.0737e+09 30.0 -8.6736e-19 -8.6736e-19 -8.6736e-19 -8.6736e-19 -8.6736e-19 -8.6736e-19 [56,] 1.5185e+09 30.5 -4.3368e-19 -4.3368e-19 -4.3368e-19 -4.3368e-19 -4.3368e-19 -4.3368e-19 [57,] 2.1475e+09 31.0 -2.1684e-19 -2.1684e-19 -2.1684e-19 -2.1684e-19 -2.1684e-19 -2.1684e-19 [58,] 3.0370e+09 31.5 -1.0842e-19 -1.0842e-19 -1.0842e-19 -1.0842e-19 -1.0842e-19 -1.0842e-19 [59,] 4.2950e+09 32.0 -5.4210e-20 -5.4210e-20 -5.4210e-20 -5.4210e-20 -5.4210e-20 -5.4210e-20 [60,] 6.0740e+09 32.5 -2.7105e-20 -2.7105e-20 -2.7105e-20 -2.7105e-20 -2.7105e-20 -2.7105e-20 [61,] 8.5899e+09 33.0 -1.3553e-20 -1.3553e-20 -1.3553e-20 -1.3553e-20 -1.3553e-20 -1.3553e-20 [62,] 1.2148e+10 33.5 -6.7763e-21 -6.7763e-21 -6.7763e-21 -6.7763e-21 -6.7763e-21 -6.7763e-21 [63,] 1.7180e+10 34.0 -3.3881e-21 -3.3881e-21 -3.3881e-21 -3.3881e-21 -3.3881e-21 -3.3881e-21 [64,] 2.4296e+10 34.5 -1.6941e-21 -1.6941e-21 -1.6941e-21 -1.6941e-21 -1.6941e-21 -1.6941e-21 [65,] 3.4360e+10 35.0 -8.4703e-22 -8.4703e-22 -8.4703e-22 -8.4703e-22 -8.4703e-22 -8.4703e-22 [66,] 4.8592e+10 35.5 -4.2352e-22 -4.2352e-22 -4.2352e-22 -4.2352e-22 -4.2352e-22 -4.2352e-22 [67,] 6.8719e+10 36.0 -2.1176e-22 -2.1176e-22 -2.1176e-22 -2.1176e-22 -2.1176e-22 -2.1176e-22 [68,] 9.7184e+10 36.5 -1.0588e-22 -1.0588e-22 -1.0588e-22 -1.0588e-22 -1.0588e-22 -1.0588e-22 [69,] 1.3744e+11 37.0 -5.2940e-23 -5.2940e-23 -5.2940e-23 -5.2940e-23 -5.2940e-23 -5.2940e-23 [70,] 1.9437e+11 37.5 -2.6470e-23 -2.6470e-23 -2.6470e-23 -2.6470e-23 -2.6470e-23 -2.6470e-23 [71,] 2.7488e+11 38.0 -1.3235e-23 -1.3235e-23 -1.3235e-23 -1.3235e-23 -1.3235e-23 -1.3235e-23 [72,] 3.8874e+11 38.5 -6.6174e-24 -6.6174e-24 -6.6174e-24 -6.6174e-24 -6.6174e-24 -6.6174e-24 [73,] 5.4976e+11 39.0 -3.3087e-24 -3.3087e-24 -3.3087e-24 -3.3087e-24 -3.3087e-24 -3.3087e-24 [74,] 7.7747e+11 39.5 -1.6544e-24 -1.6544e-24 -1.6544e-24 -1.6544e-24 -1.6544e-24 -1.6544e-24 [75,] 1.0995e+12 40.0 -8.2718e-25 -8.2718e-25 -8.2718e-25 -8.2718e-25 -8.2718e-25 -8.2718e-25 [76,] 1.5549e+12 40.5 -4.1359e-25 -4.1359e-25 -4.1359e-25 -4.1359e-25 -4.1359e-25 -4.1359e-25 [77,] 2.1990e+12 41.0 -2.0680e-25 -2.0680e-25 -2.0680e-25 -2.0680e-25 -2.0680e-25 -2.0680e-25 [78,] 3.1099e+12 41.5 -1.0340e-25 -1.0340e-25 -1.0340e-25 -1.0340e-25 -1.0340e-25 -1.0340e-25 [79,] 4.3980e+12 42.0 -5.1699e-26 -5.1699e-26 -5.1699e-26 -5.1699e-26 -5.1699e-26 -5.1699e-26 [80,] 6.2198e+12 42.5 -2.5849e-26 -2.5849e-26 -2.5849e-26 -2.5849e-26 -2.5849e-26 -2.5849e-26 [81,] 8.7961e+12 43.0 -1.2925e-26 -1.2925e-26 -1.2925e-26 -1.2925e-26 -1.2925e-26 -1.2925e-26 [82,] 1.2440e+13 43.5 -6.4623e-27 -6.4623e-27 -6.4623e-27 -6.4623e-27 -6.4623e-27 -6.4623e-27 [83,] 1.7592e+13 44.0 -3.2312e-27 -3.2312e-27 -3.2312e-27 -3.2312e-27 -3.2312e-27 -3.2312e-27 [84,] 2.4879e+13 44.5 -1.6156e-27 -1.6156e-27 -1.6156e-27 -1.6156e-27 -1.6156e-27 -1.6156e-27 [85,] 3.5184e+13 45.0 -8.0779e-28 -8.0779e-28 -8.0779e-28 -8.0779e-28 -8.0779e-28 -8.0779e-28 [86,] 4.9758e+13 45.5 -4.0390e-28 -4.0390e-28 -4.0390e-28 -4.0390e-28 -4.0390e-28 -4.0390e-28 [87,] 7.0369e+13 46.0 -2.0195e-28 -2.0195e-28 -2.0195e-28 -2.0195e-28 -2.0195e-28 -2.0195e-28 [88,] 9.9516e+13 46.5 -1.0097e-28 -1.0097e-28 -1.0097e-28 -1.0097e-28 -1.0097e-28 -1.0097e-28 [89,] 1.4074e+14 47.0 -5.0487e-29 -5.0487e-29 -5.0487e-29 -5.0487e-29 -5.0487e-29 -5.0487e-29 [90,] 1.9903e+14 47.5 -2.5244e-29 -2.5244e-29 -2.5244e-29 -2.5244e-29 -2.5244e-29 -2.5244e-29 [91,] 2.8147e+14 48.0 -1.2622e-29 -1.2622e-29 -1.2622e-29 -1.2622e-29 -1.2622e-29 -1.2622e-29 [92,] 3.9807e+14 48.5 -6.3109e-30 -6.3109e-30 -6.3109e-30 -6.3109e-30 -6.3109e-30 -6.3109e-30 [93,] 5.6295e+14 49.0 -3.1554e-30 -3.1554e-30 -3.1554e-30 -3.1554e-30 -3.1554e-30 -3.1554e-30 [94,] 7.9613e+14 49.5 -1.5777e-30 -1.5777e-30 -1.5777e-30 -1.5777e-30 -1.5777e-30 -1.5777e-30 [95,] 1.1259e+15 50.0 -7.8886e-31 -7.8886e-31 -7.8886e-31 -7.8886e-31 -7.8886e-31 -7.8886e-31 [96,] 1.5923e+15 50.5 -3.9443e-31 -3.9443e-31 -3.9443e-31 -3.9443e-31 -3.9443e-31 -3.9443e-31 [97,] 2.2518e+15 51.0 -1.9722e-31 -1.9722e-31 -1.9722e-31 -1.9722e-31 -1.9722e-31 -1.9722e-31 [98,] 3.1845e+15 51.5 -9.8608e-32 -9.8608e-32 -9.8608e-32 -9.8608e-32 -9.8608e-32 -9.8608e-32 [99,] 4.5036e+15 52.0 -4.9304e-32 -4.9304e-32 -4.9304e-32 -4.9304e-32 -4.9304e-32 -4.9304e-32 [100,] 6.3691e+15 52.5 -2.4652e-32 -2.4652e-32 -2.4652e-32 -2.4652e-32 -2.4652e-32 -2.4652e-32 [101,] 9.0072e+15 53.0 -1.2326e-32 -1.2326e-32 -1.2326e-32 -1.2326e-32 -1.2326e-32 -1.2326e-32 [102,] 1.2738e+16 53.5 -6.1630e-33 -6.1630e-33 -6.1630e-33 -6.1630e-33 -6.1630e-33 -6.1630e-33 [103,] 1.8014e+16 54.0 -3.0815e-33 -3.0815e-33 -3.0815e-33 -3.0815e-33 -3.0815e-33 -3.0815e-33 [104,] 2.5476e+16 54.5 -1.5407e-33 -1.5407e-33 -1.5407e-33 -1.5407e-33 -1.5407e-33 -1.5407e-33 [105,] 3.6029e+16 55.0 -7.7037e-34 -7.7037e-34 -7.7037e-34 -7.7037e-34 -7.7037e-34 -7.7037e-34 [106,] 5.0952e+16 55.5 -3.8519e-34 -3.8519e-34 -3.8519e-34 -3.8519e-34 -3.8519e-34 -3.8519e-34 [107,] 7.2058e+16 56.0 -1.9259e-34 -1.9259e-34 -1.9259e-34 -1.9259e-34 -1.9259e-34 -1.9259e-34 [108,] 1.0190e+17 56.5 -9.6296e-35 -9.6296e-35 -9.6296e-35 -9.6296e-35 -9.6296e-35 -9.6296e-35 [109,] 1.4412e+17 57.0 -4.8148e-35 -4.8148e-35 -4.8148e-35 -4.8148e-35 -4.8148e-35 -4.8148e-35 [110,] 2.0381e+17 57.5 -2.4074e-35 -2.4074e-35 -2.4074e-35 -2.4074e-35 -2.4074e-35 -2.4074e-35 [111,] 2.8823e+17 58.0 -1.2037e-35 -1.2037e-35 -1.2037e-35 -1.2037e-35 -1.2037e-35 -1.2037e-35 [112,] 4.0762e+17 58.5 -6.0185e-36 -6.0185e-36 -6.0185e-36 -6.0185e-36 -6.0185e-36 -6.0185e-36 [113,] 5.7646e+17 59.0 -3.0093e-36 -3.0093e-36 -3.0093e-36 -3.0093e-36 -3.0093e-36 -3.0093e-36 [114,] 8.1524e+17 59.5 -1.5046e-36 -1.5046e-36 -1.5046e-36 -1.5046e-36 -1.5046e-36 -1.5046e-36 [115,] 1.1529e+18 60.0 -7.5232e-37 -7.5232e-37 -7.5232e-37 -7.5232e-37 -7.5232e-37 -7.5232e-37 [116,] 1.6305e+18 60.5 -3.7616e-37 -3.7616e-37 -3.7616e-37 -3.7616e-37 -3.7616e-37 -3.7616e-37 [117,] 2.3058e+18 61.0 -1.8808e-37 -1.8808e-37 -1.8808e-37 -1.8808e-37 -1.8808e-37 -1.8808e-37 [118,] 3.2610e+18 61.5 -9.4040e-38 -9.4040e-38 -9.4040e-38 -9.4040e-38 -9.4040e-38 -9.4040e-38 [119,] 4.6117e+18 62.0 -4.7020e-38 -4.7020e-38 -4.7020e-38 -4.7020e-38 -4.7020e-38 -4.7020e-38 [120,] 6.5219e+18 62.5 -2.3510e-38 -2.3510e-38 -2.3510e-38 -2.3510e-38 -2.3510e-38 -2.3510e-38 [121,] 9.2234e+18 63.0 -1.1755e-38 -1.1755e-38 -1.1755e-38 -1.1755e-38 -1.1755e-38 -1.1755e-38 [122,] 1.3044e+19 63.5 -5.8775e-39 -5.8775e-39 -5.8775e-39 -5.8775e-39 -5.8775e-39 -5.8775e-39 [123,] 1.8447e+19 64.0 -2.9387e-39 -2.9387e-39 -2.9387e-39 -2.9387e-39 -2.9387e-39 -2.9387e-39 [124,] 2.6088e+19 64.5 -1.4694e-39 -1.4694e-39 -1.4694e-39 -1.4694e-39 -1.4694e-39 -1.4694e-39 [125,] 3.6893e+19 65.0 -7.3468e-40 -7.3468e-40 -7.3468e-40 -7.3468e-40 -7.3468e-40 -7.3468e-40 [126,] 5.2175e+19 65.5 -3.6734e-40 -3.6734e-40 -3.6734e-40 -3.6734e-40 -3.6734e-40 -3.6734e-40 [127,] 7.3787e+19 66.0 -1.8367e-40 -1.8367e-40 -1.8367e-40 -1.8367e-40 -1.8367e-40 -1.8367e-40 [128,] 1.0435e+20 66.5 -9.1835e-41 -9.1835e-41 -9.1835e-41 -9.1835e-41 -9.1835e-41 -9.1835e-41 [129,] 1.4757e+20 67.0 -4.5918e-41 -4.5918e-41 -4.5918e-41 -4.5918e-41 -4.5918e-41 -4.5918e-41 [130,] 2.0870e+20 67.5 -2.2959e-41 -2.2959e-41 -2.2959e-41 -2.2959e-41 -2.2959e-41 -2.2959e-41 [131,] 2.9515e+20 68.0 -1.1479e-41 -1.1479e-41 -1.1479e-41 -1.1479e-41 -1.1479e-41 -1.1479e-41 [132,] 4.1740e+20 68.5 -5.7397e-42 -5.7397e-42 -5.7397e-42 -5.7397e-42 -5.7397e-42 -5.7397e-42 [133,] 5.9030e+20 69.0 -2.8699e-42 -2.8699e-42 -2.8699e-42 -2.8699e-42 -2.8699e-42 -2.8699e-42 [134,] 8.3480e+20 69.5 -1.4349e-42 -1.4349e-42 -1.4349e-42 -1.4349e-42 -1.4349e-42 -1.4349e-42 [135,] 1.1806e+21 70.0 -7.1746e-43 -7.1746e-43 -7.1746e-43 -7.1746e-43 -7.1746e-43 -7.1746e-43 [136,] 1.6696e+21 70.5 -3.5873e-43 -3.5873e-43 -3.5873e-43 -3.5873e-43 -3.5873e-43 -3.5873e-43 [137,] 2.3612e+21 71.0 -1.7937e-43 -1.7937e-43 -1.7937e-43 -1.7937e-43 -1.7937e-43 -1.7937e-43 [138,] 3.3392e+21 71.5 -8.9683e-44 -8.9683e-44 -8.9683e-44 -8.9683e-44 -8.9683e-44 -8.9683e-44 [139,] 4.7224e+21 72.0 -4.4842e-44 -4.4842e-44 -4.4842e-44 -4.4842e-44 -4.4842e-44 -4.4842e-44 [140,] 6.6784e+21 72.5 -2.2421e-44 -2.2421e-44 -2.2421e-44 -2.2421e-44 -2.2421e-44 -2.2421e-44 [141,] 9.4447e+21 73.0 -1.1210e-44 -1.1210e-44 -1.1210e-44 -1.1210e-44 -1.1210e-44 -1.1210e-44 [142,] 1.3357e+22 73.5 -5.6052e-45 -5.6052e-45 -5.6052e-45 -5.6052e-45 -5.6052e-45 -5.6052e-45 [143,] 1.8889e+22 74.0 -2.8026e-45 -2.8026e-45 -2.8026e-45 -2.8026e-45 -2.8026e-45 -2.8026e-45 [144,] 2.6714e+22 74.5 -1.4013e-45 -1.4013e-45 -1.4013e-45 -1.4013e-45 -1.4013e-45 -1.4013e-45 [145,] 3.7779e+22 75.0 -7.0065e-46 -7.0065e-46 -7.0065e-46 -7.0065e-46 -7.0065e-46 -7.0065e-46 [146,] 5.3427e+22 75.5 -3.5032e-46 -3.5032e-46 -3.5032e-46 -3.5032e-46 -3.5032e-46 -3.5032e-46 [147,] 7.5558e+22 76.0 -1.7516e-46 -1.7516e-46 -1.7516e-46 -1.7516e-46 -1.7516e-46 -1.7516e-46 [148,] 1.0685e+23 76.5 -8.7581e-47 -8.7581e-47 -8.7581e-47 -8.7581e-47 -8.7581e-47 -8.7581e-47 [149,] 1.5112e+23 77.0 -4.3791e-47 -4.3791e-47 -4.3791e-47 -4.3791e-47 -4.3791e-47 -4.3791e-47 [150,] 2.1371e+23 77.5 -2.1895e-47 -2.1895e-47 -2.1895e-47 -2.1895e-47 -2.1895e-47 -2.1895e-47 [151,] 3.0223e+23 78.0 -1.0948e-47 -1.0948e-47 -1.0948e-47 -1.0948e-47 -1.0948e-47 -1.0948e-47 [152,] 4.2742e+23 78.5 -5.4738e-48 -5.4738e-48 -5.4738e-48 -5.4738e-48 -5.4738e-48 -5.4738e-48 [153,] 6.0446e+23 79.0 -2.7369e-48 -2.7369e-48 -2.7369e-48 -2.7369e-48 -2.7369e-48 -2.7369e-48 [154,] 8.5484e+23 79.5 -1.3685e-48 -1.3685e-48 -1.3685e-48 -1.3685e-48 -1.3685e-48 -1.3685e-48 [155,] 1.2089e+24 80.0 -6.8423e-49 -6.8423e-49 -6.8423e-49 -6.8423e-49 -6.8423e-49 -6.8423e-49 [156,] 1.7097e+24 80.5 -3.4211e-49 -3.4211e-49 -3.4211e-49 -3.4211e-49 -3.4211e-49 -3.4211e-49 [157,] 2.4179e+24 81.0 -1.7106e-49 -1.7106e-49 -1.7106e-49 -1.7106e-49 -1.7106e-49 -1.7106e-49 [158,] 3.4194e+24 81.5 -8.5528e-50 -8.5528e-50 -8.5528e-50 -8.5528e-50 -8.5528e-50 -8.5528e-50 [159,] 4.8357e+24 82.0 -4.2764e-50 -4.2764e-50 -4.2764e-50 -4.2764e-50 -4.2764e-50 -4.2764e-50 [160,] 6.8387e+24 82.5 -2.1382e-50 -2.1382e-50 -2.1382e-50 -2.1382e-50 -2.1382e-50 -2.1382e-50 [161,] 9.6714e+24 83.0 -1.0691e-50 -1.0691e-50 -1.0691e-50 -1.0691e-50 -1.0691e-50 -1.0691e-50 [162,] 1.3677e+25 83.5 -5.3455e-51 -5.3455e-51 -5.3455e-51 -5.3455e-51 -5.3455e-51 -5.3455e-51 [163,] 1.9343e+25 84.0 -2.6728e-51 -2.6728e-51 -2.6728e-51 -2.6728e-51 -2.6728e-51 -2.6728e-51 [164,] 2.7355e+25 84.5 -1.3364e-51 -1.3364e-51 -1.3364e-51 -1.3364e-51 -1.3364e-51 -1.3364e-51 [165,] 3.8686e+25 85.0 -6.6819e-52 -6.6819e-52 -6.6819e-52 -6.6819e-52 -6.6819e-52 -6.6819e-52 [166,] 5.4710e+25 85.5 -3.3410e-52 -3.3410e-52 -3.3410e-52 -3.3410e-52 -3.3410e-52 -3.3410e-52 [167,] 7.7371e+25 86.0 -1.6705e-52 -1.6705e-52 -1.6705e-52 -1.6705e-52 -1.6705e-52 -1.6705e-52 [168,] 1.0942e+26 86.5 -8.3524e-53 -8.3524e-53 -8.3524e-53 -8.3524e-53 -8.3524e-53 -8.3524e-53 [169,] 1.5474e+26 87.0 -4.1762e-53 -4.1762e-53 -4.1762e-53 -4.1762e-53 -4.1762e-53 -4.1762e-53 [170,] 2.1884e+26 87.5 -2.0881e-53 -2.0881e-53 -2.0881e-53 -2.0881e-53 -2.0881e-53 -2.0881e-53 [171,] 3.0949e+26 88.0 -1.0440e-53 -1.0440e-53 -1.0440e-53 -1.0440e-53 -1.0440e-53 -1.0440e-53 [172,] 4.3768e+26 88.5 -5.2202e-54 -5.2202e-54 -5.2202e-54 -5.2202e-54 -5.2202e-54 -5.2202e-54 [173,] 6.1897e+26 89.0 -2.6101e-54 -2.6101e-54 -2.6101e-54 -2.6101e-54 -2.6101e-54 -2.6101e-54 [174,] 8.7536e+26 89.5 -1.3051e-54 -1.3051e-54 -1.3051e-54 -1.3051e-54 -1.3051e-54 -1.3051e-54 [175,] 1.2379e+27 90.0 -6.5253e-55 -6.5253e-55 -6.5253e-55 -6.5253e-55 -6.5253e-55 -6.5253e-55 [176,] 1.7507e+27 90.5 -3.2627e-55 -3.2627e-55 -3.2627e-55 -3.2627e-55 -3.2627e-55 -3.2627e-55 [177,] 2.4759e+27 91.0 -1.6313e-55 -1.6313e-55 -1.6313e-55 -1.6313e-55 -1.6313e-55 -1.6313e-55 [178,] 3.5014e+27 91.5 -8.1566e-56 -8.1566e-56 -8.1566e-56 -8.1566e-56 -8.1566e-56 -8.1566e-56 [179,] 4.9518e+27 92.0 -4.0783e-56 -4.0783e-56 -4.0783e-56 -4.0783e-56 -4.0783e-56 -4.0783e-56 [180,] 7.0028e+27 92.5 -2.0392e-56 -2.0392e-56 -2.0392e-56 -2.0392e-56 -2.0392e-56 -2.0392e-56 [181,] 9.9035e+27 93.0 -1.0196e-56 -1.0196e-56 -1.0196e-56 -1.0196e-56 -1.0196e-56 -1.0196e-56 [182,] 1.4006e+28 93.5 -5.0979e-57 -5.0979e-57 -5.0979e-57 -5.0979e-57 -5.0979e-57 -5.0979e-57 [183,] 1.9807e+28 94.0 -2.5489e-57 -2.5489e-57 -2.5489e-57 -2.5489e-57 -2.5489e-57 -2.5489e-57 [184,] 2.8011e+28 94.5 -1.2745e-57 -1.2745e-57 -1.2745e-57 -1.2745e-57 -1.2745e-57 -1.2745e-57 [185,] 3.9614e+28 95.0 -6.3724e-58 -6.3724e-58 -6.3724e-58 -6.3724e-58 -6.3724e-58 -6.3724e-58 [186,] 5.6023e+28 95.5 -3.1862e-58 -3.1862e-58 -3.1862e-58 -3.1862e-58 -3.1862e-58 -3.1862e-58 [187,] 7.9228e+28 96.0 -1.5931e-58 -1.5931e-58 -1.5931e-58 -1.5931e-58 -1.5931e-58 -1.5931e-58 [188,] 1.1205e+29 96.5 -7.9655e-59 -7.9655e-59 -7.9655e-59 -7.9655e-59 -7.9655e-59 -7.9655e-59 [189,] 1.5846e+29 97.0 -3.9827e-59 -3.9827e-59 -3.9827e-59 -3.9827e-59 -3.9827e-59 -3.9827e-59 [190,] 2.2409e+29 97.5 -1.9914e-59 -1.9914e-59 -1.9914e-59 -1.9914e-59 -1.9914e-59 -1.9914e-59 [191,] 3.1691e+29 98.0 -9.9568e-60 -9.9568e-60 -9.9568e-60 -9.9568e-60 -9.9568e-60 -9.9568e-60 [192,] 4.4818e+29 98.5 -4.9784e-60 -4.9784e-60 -4.9784e-60 -4.9784e-60 -4.9784e-60 -4.9784e-60 [193,] 6.3383e+29 99.0 -2.4892e-60 -2.4892e-60 -2.4892e-60 -2.4892e-60 -2.4892e-60 -2.4892e-60 [194,] 8.9636e+29 99.5 -1.2446e-60 -1.2446e-60 -1.2446e-60 -1.2446e-60 -1.2446e-60 -1.2446e-60 [195,] 1.2677e+30 100.0 -6.2230e-61 -6.2230e-61 -6.2230e-61 -6.2230e-61 -6.2230e-61 -6.2230e-61 > > matplot(Rx[,"x"], - Rx[,-(1:2)], type = "b", log="xy") > > ## not useful really > opa <- par(ask=TRUE) > for(j in 3:ncol(Rx)) { + D <- Rx[,-c(1:2,j)] - Rx[,j] + neg <- mean(D < 0) > 1/2 + matplot(Rx[,"x"], if(neg) -D else D, + main = sprintf("difference %s(Rx[,*] - Rx[, \"%s\"])", + if(neg) " -" else "", colnames(Rx)[j]), + type = "b", log="xy") + } Warning messages: 1: In xy.coords(x, y, xlabel, ylabel, log = log, recycle = TRUE) : 195 y values <= 0 omitted from logarithmic plot 2: In xy.coords(x, y, xlabel, ylabel, log = log, recycle = TRUE) : 390 y values <= 0 omitted from logarithmic plot 3: In xy.coords(x, y, xlabel, ylabel, log) : 195 y values <= 0 omitted from logarithmic plot 4: In xy.coords(x, y, xlabel, ylabel, log = log, recycle = TRUE) : 195 y values <= 0 omitted from logarithmic plot 5: In xy.coords(x, y, xlabel, ylabel, log = log, recycle = TRUE) : 390 y values <= 0 omitted from logarithmic plot > par(opa) > ## rather: > str(epsMn <- rowMeans(Rx[,-(1:2)])) Class 'mpfr' [package "Rmpfr"] of length 195 and precision 1024 -0.0147342696055 -0.00758431492287 ... > > matplot(Rx[,"x"], Rx[,-(1:2)] - epsMn, type = "b", log="x") > matplot(Rx[,"x"], abs(Rx[,-(1:2)] - epsMn), type = "b", log="xy") # don't see really > > print(asNumeric(cbind(Rx[, 1:2], Rx[,-(1:2)] - epsMn)), digits=4) x log2x direct log1pExct deltaFull delta1 delta2 [1,] 8.000e+00 3.0 -3.169e-04 1.496e-04 2.554e-04 -6.503e-04 3.112e-04 [2,] 1.131e+01 3.5 -8.034e-05 3.894e-05 6.734e-05 -1.676e-04 7.463e-05 [3,] 1.600e+01 4.0 -2.022e-05 9.948e-06 1.731e-05 -4.256e-05 1.824e-05 [4,] 2.263e+01 4.5 -5.070e-06 2.515e-06 4.388e-06 -1.072e-05 4.505e-06 [5,] 3.200e+01 5.0 -1.270e-06 6.322e-07 1.105e-06 -2.691e-06 1.120e-06 [6,] 4.525e+01 5.5 -3.176e-07 1.585e-07 2.772e-07 -6.742e-07 2.790e-07 [7,] 6.400e+01 6.0 -7.944e-08 3.968e-08 6.941e-08 -1.687e-07 6.965e-08 [8,] 9.051e+01 6.5 -1.986e-08 9.927e-09 1.737e-08 -4.220e-08 1.740e-08 [9,] 1.280e+02 7.0 -4.967e-09 2.483e-09 4.344e-09 -1.055e-08 4.348e-09 [10,] 1.810e+02 7.5 -1.242e-09 6.208e-10 1.086e-09 -2.638e-09 1.087e-09 [11,] 2.560e+02 8.0 -3.104e-10 1.552e-10 2.716e-10 -6.596e-10 2.717e-10 [12,] 3.620e+02 8.5 -7.761e-11 3.880e-11 6.791e-11 -1.649e-10 6.791e-11 [13,] 5.120e+02 9.0 -1.940e-11 9.701e-12 1.698e-11 -4.123e-11 1.698e-11 [14,] 7.241e+02 9.5 -4.851e-12 2.425e-12 4.244e-12 -1.031e-11 4.244e-12 [15,] 1.024e+03 10.0 -1.213e-12 6.063e-13 1.061e-12 -2.577e-12 1.061e-12 [16,] 1.448e+03 10.5 -3.032e-13 1.516e-13 2.653e-13 -6.442e-13 2.653e-13 [17,] 2.048e+03 11.0 -7.579e-14 3.790e-14 6.632e-14 -1.611e-13 6.632e-14 [18,] 2.896e+03 11.5 -1.895e-14 9.474e-15 1.658e-14 -4.026e-14 1.658e-14 [19,] 4.096e+03 12.0 -4.737e-15 2.368e-15 4.145e-15 -1.007e-14 4.145e-15 [20,] 5.793e+03 12.5 -1.184e-15 5.921e-16 1.036e-15 -2.517e-15 1.036e-15 [21,] 8.192e+03 13.0 -2.961e-16 1.480e-16 2.591e-16 -6.291e-16 2.591e-16 [22,] 1.159e+04 13.5 -7.401e-17 3.701e-17 6.476e-17 -1.573e-16 6.476e-17 [23,] 1.638e+04 14.0 -1.850e-17 9.252e-18 1.619e-17 -3.932e-17 1.619e-17 [24,] 2.317e+04 14.5 -4.626e-18 2.313e-18 4.048e-18 -9.830e-18 4.048e-18 [25,] 3.277e+04 15.0 -1.156e-18 5.782e-19 1.012e-18 -2.458e-18 1.012e-18 [26,] 4.634e+04 15.5 -2.891e-19 1.446e-19 2.530e-19 -6.144e-19 2.530e-19 [27,] 6.554e+04 16.0 -7.228e-20 3.614e-20 6.325e-20 -1.536e-19 6.325e-20 [28,] 9.268e+04 16.5 -1.807e-20 9.035e-21 1.581e-20 -3.840e-20 1.581e-20 [29,] 1.311e+05 17.0 -4.518e-21 2.259e-21 3.953e-21 -9.600e-21 3.953e-21 [30,] 1.854e+05 17.5 -1.129e-21 5.647e-22 9.882e-22 -2.400e-21 9.882e-22 [31,] 2.621e+05 18.0 -2.823e-22 1.412e-22 2.471e-22 -6.000e-22 2.471e-22 [32,] 3.707e+05 18.5 -7.059e-23 3.529e-23 6.176e-23 -1.500e-22 6.176e-23 [33,] 5.243e+05 19.0 -1.765e-23 8.823e-24 1.544e-23 -3.750e-23 1.544e-23 [34,] 7.415e+05 19.5 -4.412e-24 2.206e-24 3.860e-24 -9.375e-24 3.860e-24 [35,] 1.049e+06 20.0 -1.103e-24 5.515e-25 9.650e-25 -2.344e-24 9.650e-25 [36,] 1.483e+06 20.5 -2.757e-25 1.379e-25 2.413e-25 -5.859e-25 2.413e-25 [37,] 2.097e+06 21.0 -6.893e-26 3.447e-26 6.032e-26 -1.465e-25 6.032e-26 [38,] 2.966e+06 21.5 -1.723e-26 8.616e-27 1.508e-26 -3.662e-26 1.508e-26 [39,] 4.194e+06 22.0 -4.308e-27 2.154e-27 3.770e-27 -9.155e-27 3.770e-27 [40,] 5.932e+06 22.5 -1.077e-27 5.385e-28 9.424e-28 -2.289e-27 9.424e-28 [41,] 8.389e+06 23.0 -2.693e-28 1.346e-28 2.356e-28 -5.722e-28 2.356e-28 [42,] 1.186e+07 23.5 -6.732e-29 3.366e-29 5.890e-29 -1.430e-28 5.890e-29 [43,] 1.678e+07 24.0 -1.683e-29 8.415e-30 1.473e-29 -3.576e-29 1.473e-29 [44,] 2.373e+07 24.5 -4.207e-30 2.104e-30 3.681e-30 -8.940e-30 3.681e-30 [45,] 3.355e+07 25.0 -1.052e-30 5.259e-31 9.203e-31 -2.235e-30 9.203e-31 [46,] 4.745e+07 25.5 -2.630e-31 1.315e-31 2.301e-31 -5.588e-31 2.301e-31 [47,] 6.711e+07 26.0 -6.574e-32 3.287e-32 5.752e-32 -1.397e-31 5.752e-32 [48,] 9.491e+07 26.5 -1.643e-32 8.217e-33 1.438e-32 -3.492e-32 1.438e-32 [49,] 1.342e+08 27.0 -4.109e-33 2.054e-33 3.595e-33 -8.731e-33 3.595e-33 [50,] 1.898e+08 27.5 -1.027e-33 5.136e-34 8.988e-34 -2.183e-33 8.988e-34 [51,] 2.684e+08 28.0 -2.568e-34 1.284e-34 2.247e-34 -5.457e-34 2.247e-34 [52,] 3.796e+08 28.5 -6.420e-35 3.210e-35 5.617e-35 -1.364e-34 5.617e-35 [53,] 5.369e+08 29.0 -1.605e-35 8.025e-36 1.404e-35 -3.411e-35 1.404e-35 [54,] 7.593e+08 29.5 -4.012e-36 2.006e-36 3.511e-36 -8.526e-36 3.511e-36 [55,] 1.074e+09 30.0 -1.003e-36 5.015e-37 8.777e-37 -2.132e-36 8.777e-37 [56,] 1.519e+09 30.5 -2.508e-37 1.254e-37 2.194e-37 -5.329e-37 2.194e-37 [57,] 2.147e+09 31.0 -6.269e-38 3.135e-38 5.486e-38 -1.332e-37 5.486e-38 [58,] 3.037e+09 31.5 -1.567e-38 7.837e-39 1.371e-38 -3.331e-38 1.371e-38 [59,] 4.295e+09 32.0 -3.918e-39 1.959e-39 3.429e-39 -8.326e-39 3.429e-39 [60,] 6.074e+09 32.5 -9.796e-40 4.898e-40 8.571e-40 -2.082e-39 8.571e-40 [61,] 8.590e+09 33.0 -2.449e-40 1.224e-40 2.143e-40 -5.204e-40 2.143e-40 [62,] 1.215e+10 33.5 -6.122e-41 3.061e-41 5.357e-41 -1.301e-40 5.357e-41 [63,] 1.718e+10 34.0 -1.531e-41 7.653e-42 1.339e-41 -3.253e-41 1.339e-41 [64,] 2.430e+10 34.5 -3.826e-42 1.913e-42 3.348e-42 -8.131e-42 3.348e-42 [65,] 3.436e+10 35.0 -9.566e-43 4.783e-43 8.370e-43 -2.033e-42 8.370e-43 [66,] 4.859e+10 35.5 -2.392e-43 1.196e-43 2.093e-43 -5.082e-43 2.093e-43 [67,] 6.872e+10 36.0 -5.979e-44 2.989e-44 5.232e-44 -1.271e-43 5.232e-44 [68,] 9.718e+10 36.5 -1.495e-44 7.474e-45 1.308e-44 -3.176e-44 1.308e-44 [69,] 1.374e+11 37.0 -3.737e-45 1.868e-45 3.270e-45 -7.941e-45 3.270e-45 [70,] 1.944e+11 37.5 -9.342e-46 4.671e-46 8.174e-46 -1.985e-45 8.174e-46 [71,] 2.749e+11 38.0 -2.335e-46 1.168e-46 2.044e-46 -4.963e-46 2.044e-46 [72,] 3.887e+11 38.5 -5.839e-47 2.919e-47 5.109e-47 -1.241e-46 5.109e-47 [73,] 5.498e+11 39.0 -1.460e-47 7.298e-48 1.277e-47 -3.102e-47 1.277e-47 [74,] 7.775e+11 39.5 -3.649e-48 1.825e-48 3.193e-48 -7.755e-48 3.193e-48 [75,] 1.100e+12 40.0 -9.123e-49 4.562e-49 7.983e-49 -1.939e-48 7.983e-49 [76,] 1.555e+12 40.5 -2.281e-49 1.140e-49 1.996e-49 -4.847e-49 1.996e-49 [77,] 2.199e+12 41.0 -5.702e-50 2.851e-50 4.989e-50 -1.212e-49 4.989e-50 [78,] 3.110e+12 41.5 -1.425e-50 7.127e-51 1.247e-50 -3.029e-50 1.247e-50 [79,] 4.398e+12 42.0 -3.564e-51 1.782e-51 3.118e-51 -7.573e-51 3.118e-51 [80,] 6.220e+12 42.5 -8.909e-52 4.455e-52 7.796e-52 -1.893e-51 7.796e-52 [81,] 8.796e+12 43.0 -2.227e-52 1.114e-52 1.949e-52 -4.733e-52 1.949e-52 [82,] 1.244e+13 43.5 -5.568e-53 2.784e-53 4.872e-53 -1.183e-52 4.872e-53 [83,] 1.759e+13 44.0 -1.392e-53 6.960e-54 1.218e-53 -2.958e-53 1.218e-53 [84,] 2.488e+13 44.5 -3.480e-54 1.740e-54 3.045e-54 -7.395e-54 3.045e-54 [85,] 3.518e+13 45.0 -8.700e-55 4.350e-55 7.613e-55 -1.849e-54 7.613e-55 [86,] 4.976e+13 45.5 -2.175e-55 1.088e-55 1.903e-55 -4.622e-55 1.903e-55 [87,] 7.037e+13 46.0 -5.438e-56 2.719e-56 4.758e-56 -1.156e-55 4.758e-56 [88,] 9.952e+13 46.5 -1.359e-56 6.797e-57 1.190e-56 -2.889e-56 1.190e-56 [89,] 1.407e+14 47.0 -3.399e-57 1.699e-57 2.974e-57 -7.222e-57 2.974e-57 [90,] 1.990e+14 47.5 -8.496e-58 4.248e-58 7.434e-58 -1.806e-57 7.434e-58 [91,] 2.815e+14 48.0 -2.124e-58 1.062e-58 1.859e-58 -4.514e-58 1.859e-58 [92,] 3.981e+14 48.5 -5.310e-59 2.655e-59 4.647e-59 -1.128e-58 4.647e-59 [93,] 5.629e+14 49.0 -1.328e-59 6.638e-60 1.162e-59 -2.821e-59 1.162e-59 [94,] 7.961e+14 49.5 -3.319e-60 1.659e-60 2.904e-60 -7.053e-60 2.904e-60 [95,] 1.126e+15 50.0 -8.297e-61 4.149e-61 7.260e-61 -1.763e-60 7.260e-61 [96,] 1.592e+15 50.5 -2.074e-61 1.037e-61 1.815e-61 -4.408e-61 1.815e-61 [97,] 2.252e+15 51.0 -5.186e-62 2.593e-62 4.538e-62 -1.102e-61 4.538e-62 [98,] 3.185e+15 51.5 -1.296e-62 6.482e-63 1.134e-62 -2.755e-62 1.134e-62 [99,] 4.504e+15 52.0 -3.241e-63 1.621e-63 2.836e-63 -6.887e-63 2.836e-63 [100,] 6.369e+15 52.5 -8.103e-64 4.051e-64 7.090e-64 -1.722e-63 7.090e-64 [101,] 9.007e+15 53.0 -2.026e-64 1.013e-64 1.773e-64 -4.305e-64 1.773e-64 [102,] 1.274e+16 53.5 -5.064e-65 2.532e-65 4.431e-65 -1.076e-64 4.431e-65 [103,] 1.801e+16 54.0 -1.266e-65 6.330e-66 1.108e-65 -2.690e-65 1.108e-65 [104,] 2.548e+16 54.5 -3.165e-66 1.583e-66 2.770e-66 -6.726e-66 2.770e-66 [105,] 3.603e+16 55.0 -7.913e-67 3.956e-67 6.924e-67 -1.682e-66 6.924e-67 [106,] 5.095e+16 55.5 -1.978e-67 9.891e-68 1.731e-67 -4.204e-67 1.731e-67 [107,] 7.206e+16 56.0 -4.946e-68 2.473e-68 4.327e-68 -1.051e-67 4.327e-68 [108,] 1.019e+17 56.5 -1.236e-68 6.182e-69 1.082e-68 -2.627e-68 1.082e-68 [109,] 1.441e+17 57.0 -3.091e-69 1.546e-69 2.705e-69 -6.568e-69 2.705e-69 [110,] 2.038e+17 57.5 -7.728e-70 3.864e-70 6.762e-70 -1.642e-69 6.762e-70 [111,] 2.882e+17 58.0 -1.932e-70 9.659e-71 1.690e-70 -4.105e-70 1.690e-70 [112,] 4.076e+17 58.5 -4.830e-71 2.415e-71 4.226e-71 -1.026e-70 4.226e-71 [113,] 5.765e+17 59.0 -1.207e-71 6.037e-72 1.056e-71 -2.566e-71 1.056e-71 [114,] 8.152e+17 59.5 -3.019e-72 1.509e-72 2.641e-72 -6.414e-72 2.641e-72 [115,] 1.153e+18 60.0 -7.546e-73 3.773e-73 6.603e-73 -1.604e-72 6.603e-73 [116,] 1.630e+18 60.5 -1.887e-73 9.433e-74 1.651e-73 -4.009e-73 1.651e-73 [117,] 2.306e+18 61.0 -4.716e-74 2.358e-74 4.127e-74 -1.002e-73 4.127e-74 [118,] 3.261e+18 61.5 -1.179e-74 5.896e-75 1.032e-74 -2.506e-74 1.032e-74 [119,] 4.612e+18 62.0 -2.948e-75 1.474e-75 2.579e-75 -6.264e-75 2.579e-75 [120,] 6.522e+18 62.5 -7.370e-76 3.685e-76 6.448e-76 -1.566e-75 6.448e-76 [121,] 9.223e+18 63.0 -1.842e-76 9.212e-77 1.612e-76 -3.915e-76 1.612e-76 [122,] 1.304e+19 63.5 -4.606e-77 2.303e-77 4.030e-77 -9.788e-77 4.030e-77 [123,] 1.845e+19 64.0 -1.151e-77 5.757e-78 1.008e-77 -2.447e-77 1.008e-77 [124,] 2.609e+19 64.5 -2.879e-78 1.439e-78 2.519e-78 -6.117e-78 2.519e-78 [125,] 3.689e+19 65.0 -7.197e-79 3.598e-79 6.297e-79 -1.529e-78 6.297e-79 [126,] 5.218e+19 65.5 -1.799e-79 8.996e-80 1.574e-79 -3.823e-79 1.574e-79 [127,] 7.379e+19 66.0 -4.498e-80 2.249e-80 3.936e-80 -9.558e-80 3.936e-80 [128,] 1.044e+20 66.5 -1.125e-80 5.623e-81 9.839e-81 -2.390e-80 9.839e-81 [129,] 1.476e+20 67.0 -2.811e-81 1.406e-81 2.460e-81 -5.974e-81 2.460e-81 [130,] 2.087e+20 67.5 -7.028e-82 3.514e-82 6.150e-82 -1.493e-81 6.150e-82 [131,] 2.951e+20 68.0 -1.757e-82 8.785e-83 1.537e-82 -3.734e-82 1.537e-82 [132,] 4.174e+20 68.5 -4.393e-83 2.196e-83 3.844e-83 -9.334e-83 3.844e-83 [133,] 5.903e+20 69.0 -1.098e-83 5.491e-84 9.609e-84 -2.334e-83 9.609e-84 [134,] 8.348e+20 69.5 -2.745e-84 1.373e-84 2.402e-84 -5.834e-84 2.402e-84 [135,] 1.181e+21 70.0 -6.863e-85 3.432e-85 6.005e-85 -1.458e-84 6.005e-85 [136,] 1.670e+21 70.5 -1.716e-85 8.579e-86 1.501e-85 -3.646e-85 1.501e-85 [137,] 2.361e+21 71.0 -4.290e-86 2.145e-86 3.753e-86 -9.115e-86 3.753e-86 [138,] 3.339e+21 71.5 -1.072e-86 5.362e-87 9.384e-87 -2.279e-86 9.384e-87 [139,] 4.722e+21 72.0 -2.681e-87 1.341e-87 2.346e-87 -5.697e-87 2.346e-87 [140,] 6.678e+21 72.5 -6.703e-88 3.351e-88 5.865e-88 -1.424e-87 5.865e-88 [141,] 9.445e+21 73.0 -1.676e-88 8.378e-89 1.466e-88 -3.561e-88 1.466e-88 [142,] 1.336e+22 73.5 -4.189e-89 2.095e-89 3.665e-89 -8.902e-89 3.665e-89 [143,] 1.889e+22 74.0 -1.047e-89 5.236e-90 9.164e-90 -2.225e-89 9.164e-90 [144,] 2.671e+22 74.5 -2.618e-90 1.309e-90 2.291e-90 -5.564e-90 2.291e-90 [145,] 3.778e+22 75.0 -6.545e-91 3.273e-91 5.727e-91 -1.391e-90 5.727e-91 [146,] 5.343e+22 75.5 -1.636e-91 8.182e-92 1.432e-91 -3.477e-91 1.432e-91 [147,] 7.556e+22 76.0 -4.091e-92 2.045e-92 3.580e-92 -8.693e-92 3.580e-92 [148,] 1.069e+23 76.5 -1.023e-92 5.114e-93 8.949e-93 -2.173e-92 8.949e-93 [149,] 1.511e+23 77.0 -2.557e-93 1.278e-93 2.237e-93 -5.433e-93 2.237e-93 [150,] 2.137e+23 77.5 -6.392e-94 3.196e-94 5.593e-94 -1.358e-93 5.593e-94 [151,] 3.022e+23 78.0 -1.598e-94 7.990e-95 1.398e-94 -3.396e-94 1.398e-94 [152,] 4.274e+23 78.5 -3.995e-95 1.998e-95 3.496e-95 -8.489e-95 3.496e-95 [153,] 6.045e+23 79.0 -9.988e-96 4.994e-96 8.739e-96 -2.122e-95 8.739e-96 [154,] 8.548e+23 79.5 -2.497e-96 1.248e-96 2.185e-96 -5.306e-96 2.185e-96 [155,] 1.209e+24 80.0 -6.242e-97 3.121e-97 5.462e-97 -1.326e-96 5.462e-97 [156,] 1.710e+24 80.5 -1.561e-97 7.803e-98 1.365e-97 -3.316e-97 1.365e-97 [157,] 2.418e+24 81.0 -3.901e-98 1.951e-98 3.414e-98 -8.290e-98 3.414e-98 [158,] 3.419e+24 81.5 -9.753e-99 4.877e-99 8.534e-99 -2.073e-98 8.534e-99 [159,] 4.836e+24 82.0 -2.438e-99 1.219e-99 2.134e-99 -5.182e-99 2.134e-99 [160,] 6.839e+24 82.5 -6.096e-100 3.048e-100 5.334e-100 -1.295e-99 5.334e-100 [161,] 9.671e+24 83.0 -1.524e-100 7.620e-101 1.333e-100 -3.238e-100 1.333e-100 [162,] 1.368e+25 83.5 -3.810e-101 1.905e-101 3.334e-101 -8.096e-101 3.334e-101 [163,] 1.934e+25 84.0 -9.525e-102 4.762e-102 8.334e-102 -2.024e-101 8.334e-102 [164,] 2.735e+25 84.5 -2.381e-102 1.191e-102 2.084e-102 -5.060e-102 2.084e-102 [165,] 3.869e+25 85.0 -5.953e-103 2.977e-103 5.209e-103 -1.265e-102 5.209e-103 [166,] 5.471e+25 85.5 -1.488e-103 7.441e-104 1.302e-103 -3.163e-103 1.302e-103 [167,] 7.737e+25 86.0 -3.721e-104 1.860e-104 3.256e-104 -7.906e-104 3.256e-104 [168,] 1.094e+26 86.5 -9.302e-105 4.651e-105 8.139e-105 -1.977e-104 8.139e-105 [169,] 1.547e+26 87.0 -2.325e-105 1.163e-105 2.035e-105 -4.942e-105 2.035e-105 [170,] 2.188e+26 87.5 -5.814e-106 2.907e-106 5.087e-106 -1.235e-105 5.087e-106 [171,] 3.095e+26 88.0 -1.453e-106 7.267e-107 1.272e-106 -3.088e-106 1.272e-106 [172,] 4.377e+26 88.5 -3.633e-107 1.817e-107 3.179e-107 -7.721e-107 3.179e-107 [173,] 6.190e+26 89.0 -9.084e-108 4.542e-108 7.948e-108 -1.930e-107 7.948e-108 [174,] 8.754e+26 89.5 -2.271e-108 1.135e-108 1.987e-108 -4.826e-108 1.987e-108 [175,] 1.238e+27 90.0 -5.677e-109 2.839e-109 4.968e-109 -1.206e-108 4.968e-109 [176,] 1.751e+27 90.5 -1.419e-109 7.097e-110 1.242e-109 -3.016e-109 1.242e-109 [177,] 2.476e+27 91.0 -3.548e-110 1.774e-110 3.105e-110 -7.540e-110 3.105e-110 [178,] 3.501e+27 91.5 -8.871e-111 4.435e-111 7.762e-111 -1.885e-110 7.762e-111 [179,] 4.952e+27 92.0 -2.218e-111 1.109e-111 1.940e-111 -4.713e-111 1.940e-111 [180,] 7.003e+27 92.5 -5.544e-112 2.772e-112 4.851e-112 -1.178e-111 4.851e-112 [181,] 9.904e+27 93.0 -1.386e-112 6.930e-113 1.213e-112 -2.945e-112 1.213e-112 [182,] 1.401e+28 93.5 -3.465e-113 1.733e-113 3.032e-113 -7.363e-113 3.032e-113 [183,] 1.981e+28 94.0 -8.663e-114 4.331e-114 7.580e-114 -1.841e-113 7.580e-114 [184,] 2.801e+28 94.5 -2.166e-114 1.083e-114 1.895e-114 -4.602e-114 1.895e-114 [185,] 3.961e+28 95.0 -5.414e-115 2.707e-115 4.737e-115 -1.151e-114 4.737e-115 [186,] 5.602e+28 95.5 -1.354e-115 6.768e-116 1.184e-115 -2.876e-115 1.184e-115 [187,] 7.923e+28 96.0 -3.384e-116 1.692e-116 2.961e-116 -7.191e-116 2.961e-116 [188,] 1.120e+29 96.5 -8.460e-117 4.230e-117 7.402e-117 -1.798e-116 7.402e-117 [189,] 1.585e+29 97.0 -2.115e-117 1.057e-117 1.851e-117 -4.494e-117 1.851e-117 [190,] 2.241e+29 97.5 -5.287e-118 2.644e-118 4.626e-118 -1.124e-117 4.626e-118 [191,] 3.169e+29 98.0 -1.322e-118 6.609e-119 1.157e-118 -2.809e-118 1.157e-118 [192,] 4.482e+29 98.5 -3.305e-119 1.652e-119 2.892e-119 -7.022e-119 2.892e-119 [193,] 6.338e+29 99.0 -8.262e-120 4.131e-120 7.229e-120 -1.756e-119 7.229e-120 [194,] 8.964e+29 99.5 -2.065e-120 1.033e-120 1.807e-120 -4.389e-120 1.807e-120 [195,] 1.268e+30 100.0 -5.163e-121 2.582e-121 4.518e-121 -1.097e-120 4.518e-121 delta3 [1,] 2.511e-04 [2,] 6.705e-05 [3,] 1.729e-05 [4,] 4.386e-06 [5,] 1.105e-06 [6,] 2.772e-07 [7,] 6.941e-08 [8,] 1.737e-08 [9,] 4.344e-09 [10,] 1.086e-09 [11,] 2.716e-10 [12,] 6.791e-11 [13,] 1.698e-11 [14,] 4.244e-12 [15,] 1.061e-12 [16,] 2.653e-13 [17,] 6.632e-14 [18,] 1.658e-14 [19,] 4.145e-15 [20,] 1.036e-15 [21,] 2.591e-16 [22,] 6.476e-17 [23,] 1.619e-17 [24,] 4.048e-18 [25,] 1.012e-18 [26,] 2.530e-19 [27,] 6.325e-20 [28,] 1.581e-20 [29,] 3.953e-21 [30,] 9.882e-22 [31,] 2.471e-22 [32,] 6.176e-23 [33,] 1.544e-23 [34,] 3.860e-24 [35,] 9.650e-25 [36,] 2.413e-25 [37,] 6.032e-26 [38,] 1.508e-26 [39,] 3.770e-27 [40,] 9.424e-28 [41,] 2.356e-28 [42,] 5.890e-29 [43,] 1.473e-29 [44,] 3.681e-30 [45,] 9.203e-31 [46,] 2.301e-31 [47,] 5.752e-32 [48,] 1.438e-32 [49,] 3.595e-33 [50,] 8.988e-34 [51,] 2.247e-34 [52,] 5.617e-35 [53,] 1.404e-35 [54,] 3.511e-36 [55,] 8.777e-37 [56,] 2.194e-37 [57,] 5.486e-38 [58,] 1.371e-38 [59,] 3.429e-39 [60,] 8.571e-40 [61,] 2.143e-40 [62,] 5.357e-41 [63,] 1.339e-41 [64,] 3.348e-42 [65,] 8.370e-43 [66,] 2.093e-43 [67,] 5.232e-44 [68,] 1.308e-44 [69,] 3.270e-45 [70,] 8.174e-46 [71,] 2.044e-46 [72,] 5.109e-47 [73,] 1.277e-47 [74,] 3.193e-48 [75,] 7.983e-49 [76,] 1.996e-49 [77,] 4.989e-50 [78,] 1.247e-50 [79,] 3.118e-51 [80,] 7.796e-52 [81,] 1.949e-52 [82,] 4.872e-53 [83,] 1.218e-53 [84,] 3.045e-54 [85,] 7.613e-55 [86,] 1.903e-55 [87,] 4.758e-56 [88,] 1.190e-56 [89,] 2.974e-57 [90,] 7.434e-58 [91,] 1.859e-58 [92,] 4.647e-59 [93,] 1.162e-59 [94,] 2.904e-60 [95,] 7.260e-61 [96,] 1.815e-61 [97,] 4.538e-62 [98,] 1.134e-62 [99,] 2.836e-63 [100,] 7.090e-64 [101,] 1.773e-64 [102,] 4.431e-65 [103,] 1.108e-65 [104,] 2.770e-66 [105,] 6.924e-67 [106,] 1.731e-67 [107,] 4.327e-68 [108,] 1.082e-68 [109,] 2.705e-69 [110,] 6.762e-70 [111,] 1.690e-70 [112,] 4.226e-71 [113,] 1.056e-71 [114,] 2.641e-72 [115,] 6.603e-73 [116,] 1.651e-73 [117,] 4.127e-74 [118,] 1.032e-74 [119,] 2.579e-75 [120,] 6.448e-76 [121,] 1.612e-76 [122,] 4.030e-77 [123,] 1.008e-77 [124,] 2.519e-78 [125,] 6.297e-79 [126,] 1.574e-79 [127,] 3.936e-80 [128,] 9.839e-81 [129,] 2.460e-81 [130,] 6.150e-82 [131,] 1.537e-82 [132,] 3.844e-83 [133,] 9.609e-84 [134,] 2.402e-84 [135,] 6.005e-85 [136,] 1.501e-85 [137,] 3.753e-86 [138,] 9.384e-87 [139,] 2.346e-87 [140,] 5.865e-88 [141,] 1.466e-88 [142,] 3.665e-89 [143,] 9.164e-90 [144,] 2.291e-90 [145,] 5.727e-91 [146,] 1.432e-91 [147,] 3.580e-92 [148,] 8.949e-93 [149,] 2.237e-93 [150,] 5.593e-94 [151,] 1.398e-94 [152,] 3.496e-95 [153,] 8.739e-96 [154,] 2.185e-96 [155,] 5.462e-97 [156,] 1.365e-97 [157,] 3.414e-98 [158,] 8.534e-99 [159,] 2.134e-99 [160,] 5.334e-100 [161,] 1.333e-100 [162,] 3.334e-101 [163,] 8.334e-102 [164,] 2.084e-102 [165,] 5.209e-103 [166,] 1.302e-103 [167,] 3.256e-104 [168,] 8.139e-105 [169,] 2.035e-105 [170,] 5.087e-106 [171,] 1.272e-106 [172,] 3.179e-107 [173,] 7.948e-108 [174,] 1.987e-108 [175,] 4.968e-109 [176,] 1.242e-109 [177,] 3.105e-110 [178,] 7.762e-111 [179,] 1.940e-111 [180,] 4.851e-112 [181,] 1.213e-112 [182,] 3.032e-113 [183,] 7.580e-114 [184,] 1.895e-114 [185,] 4.737e-115 [186,] 1.184e-115 [187,] 2.961e-116 [188,] 7.402e-117 [189,] 1.851e-117 [190,] 4.626e-118 [191,] 1.157e-118 [192,] 2.892e-119 [193,] 7.229e-120 [194,] 1.807e-120 [195,] 4.518e-121 > > > > > if(FALSE) ## not used here anymore : + Mx <- cbind(x, log2x = log2(x), hSqrt = .5*(1-sqrt(1 + 8/x^2)), T1 = -2*x^-2, D = -2*x^-2 - .5*(1-sqrt(1 + 8/x^2)), D2= -2*x^-2*(1 - 2*x^-2) - .5*(1-sqrt(1 + 8/x^2))) > > > > > proc.time() user system elapsed 2.09 0.18 2.25