I am afraid that these suggestions may not work. There are more choices than Win32 and Win64, including several flavours of BLAS/Lapack which probably are involved if you evaluate eigenvalues, and also differences in hardware, compilers and phase of the moon. If there are several equal eigenvalues, any solution of axes is arbitrary and it can be made stable for testing only by chance. If you have M equal eigenvalues, you should try to find a test that the M-dimensional (sub)space is approximately correct irrespective of random orientation of axes in this subspace.
Cheers, Jari Oksanen On 18 May 2018, at 00:06 am, Kevin Coombes <kevin.r.coom...@gmail.com<mailto:kevin.r.coom...@gmail.com>> wrote: Yes; but I have been running around all day without time to sit down and try them. The suggestions make sense, and I'm looking forward to implementing them. On Thu, May 17, 2018, 3:55 PM Ben Bolker <bbol...@gmail.com<mailto:bbol...@gmail.com>> wrote: There have been various comments in this thread (by me, and I think Duncan Murdoch) about how you can identify the platform you're running on (some combination of .Platform and/or R.Version()) and use it to write conditional statements so that your tests will only be compared with reference values that were generated on the same platform ... did those get through? Did they make sense? On Thu, May 17, 2018 at 3:30 PM, Kevin Coombes <kevin.r.coom...@gmail.com<mailto:kevin.r.coom...@gmail.com>> wrote: Yes; I'm pretty sure that it is exactly the repeated eigenvalues that are the issue. The matrices I am using are all nonsingular, and the various algorithms have no problem computing the eigenvalues correctly (up to numerical errors that I can bound and thus account for on tests by rounding appropriately). But an eigenvalue of multiplicity M has an M-dimensional eigenspace with no preferred basis. So, any M-dimensional (unitary) change of basis is permitted. That's what give rise to the lack of reproducibility across architectures. The choice of basis appears to use different heuristics on 32-bit windows than on 64-bit Windows or Linux machines. As a result, I can't include the tests I'd like as part of a CRAN submission. On Thu, May 17, 2018, 2:29 PM William Dunlap <wdun...@tibco.com<mailto:wdun...@tibco.com>> wrote: Your explanation needs to be a bit more general in the case of identical eigenvalues - each distinct eigenvalue has an associated subspace, whose dimension is the number repeats of that eigenvalue and the eigenvectors for that eigenvalue are an orthonormal basis for that subspace. (With no repeated eigenvalues this gives your 'unique up to sign'.) E.g., for the following 5x5 matrix with two eigenvalues of 1 and two of 0 x <- tcrossprod( cbind(c(1,0,0,0,1),c(0,1,0,0,1),c(0,0,1,0,1)) ) x [,1] [,2] [,3] [,4] [,5] [1,] 1 0 0 0 1 [2,] 0 1 0 0 1 [3,] 0 0 1 0 1 [4,] 0 0 0 0 0 [5,] 1 1 1 0 3 the following give valid but different (by more than sign) eigen vectors e1 <- structure(list(values = c(4, 1, 0.999999999999999, 0, -2.22044607159862e-16 ), vectors = structure(c(-0.288675134594813, -0.288675134594813, -0.288675134594813, 0, -0.866025403784439, 0, 0.707106781186547, -0.707106781186547, 0, 0, 0.816496580927726, -0.408248290463863, -0.408248290463863, 0, -6.10622663543836e-16, 0, 0, 0, -1, 0, -0.5, -0.5, -0.5, 0, 0.5), .Dim = c(5L, 5L))), .Names = c("values", "vectors"), class = "eigen") e2 <- structure(list(values = c(4, 1, 1, 0, -2.29037708937563e-16), vectors = structure(c(0.288675134594813, 0.288675134594813, 0.288675134594813, 0, 0.866025403784438, -0.784437556312061, 0.588415847923579, 0.196021708388481, 0, 4.46410900710223e-17, 0.22654886208902, 0.566068420404321, -0.79261728249334, 0, -1.11244069540181e-16, 0, 0, 0, -1, 0, -0.5, -0.5, -0.5, 0, 0.5), .Dim = c(5L, 5L))), .Names = c("values", "vectors" ), class = "eigen") I.e., all.equal(crossprod(e1$vectors), diag(5), tol=0) [1] "Mean relative difference: 1.407255e-15" all.equal(crossprod(e2$vectors), diag(5), tol=0) [1] "Mean relative difference: 3.856478e-15" all.equal(e1$vectors %*% diag(e1$values) %*% t(e1$vectors), x, tol=0) [1] "Mean relative difference: 1.110223e-15" all.equal(e2$vectors %*% diag(e2$values) %*% t(e2$vectors), x, tol=0) [1] "Mean relative difference: 9.069735e-16" e1$vectors [,1] [,2] [,3] [,4] [,5] [1,] -0.2886751 0.0000000 8.164966e-01 0 -0.5 [2,] -0.2886751 0.7071068 -4.082483e-01 0 -0.5 [3,] -0.2886751 -0.7071068 -4.082483e-01 0 -0.5 [4,] 0.0000000 0.0000000 0.000000e+00 -1 0.0 [5,] -0.8660254 0.0000000 -6.106227e-16 0 0.5 e2$vectors [,1] [,2] [,3] [,4] [,5] [1,] 0.2886751 -7.844376e-01 2.265489e-01 0 -0.5 [2,] 0.2886751 5.884158e-01 5.660684e-01 0 -0.5 [3,] 0.2886751 1.960217e-01 -7.926173e-01 0 -0.5 [4,] 0.0000000 0.000000e+00 0.000000e+00 -1 0.0 [5,] 0.8660254 4.464109e-17 -1.112441e-16 0 0.5 Bill Dunlap TIBCO Software wdunlap tibco.com<http://tibco.com> On Thu, May 17, 2018 at 10:14 AM, Martin Maechler < maech...@stat.math.ethz.ch<mailto:maech...@stat.math.ethz.ch>> wrote: Duncan Murdoch .... on Thu, 17 May 2018 12:13:01 -0400 writes: On 17/05/2018 11:53 AM, Martin Maechler wrote: Kevin Coombes ... on Thu, 17 May 2018 11:21:23 -0400 writes: [..................] [3] Should the documentation (man page) for "eigen" or "mvrnorm" include a warning that the results can change from machine to machine (or between things like 32-bit and 64-bit R on the same machine) because of difference in linear algebra modules? (Possibly including the statement that "set.seed" won't save you.) The problem is that most (young?) people do not read help pages anymore. help(eigen) has contained the following text for years, and in spite of your good analysis of the problem you seem to not have noticed the last semi-paragraph: Value: The spectral decomposition of ‘x’ is returned as a list with components values: a vector containing the p eigenvalues of ‘x’, sorted in _decreasing_ order, according to ‘Mod(values)’ in the asymmetric case when they might be complex (even for real matrices). For real asymmetric matrices the vector will be complex only if complex conjugate pairs of eigenvalues are detected. vectors: either a p * p matrix whose columns contain the eigenvectors of ‘x’, or ‘NULL’ if ‘only.values’ is ‘TRUE’. The vectors are normalized to unit length. Recall that the eigenvectors are only defined up to a constant: even when the length is specified they are still only defined up to a scalar of modulus one (the sign for real matrices). It's not a warning but a "recall that" .. maybe because the author already assumed that only thorough users would read that and for them it would be a recall of something they'd have learned *and* not entirely forgotten since ;-) I don't think you're really being fair here: the text in ?eigen doesn't make clear that eigenvector values are not reproducible even within the same version of R, and there's nothing in ?mvrnorm to suggest it doesn't give reproducible results. Ok, I'm sorry ... I definitely did not want to be unfair. I've always thought the remark in eigen was sufficient, but I'm probably wrong and we should add text explaining that it practically means that eigenvectors are only defined up to sign switches (in the real case) and hence results depend on the underlying {Lapack + BLAS} libraries and therefore are platform dependent. Even further, we could consider (optionally, by default FALSE) using defining a deterministic scheme for postprocessing the current output of eigen such that at least for the good cases where all eigenspaces are 1-dimensional, the postprocessing would result in reproducible signs, by e.g., ensuring the first non-zero entry of each eigenvector to be positive. MASS::mvrnorm() and mvtnorm::rmvnorm() both use "eigen", whereas mvtnorm::rmvnorm() *does* have method = "chol" which AFAIK does not suffer from such problems. OTOH, the help page of MASS::mvrnorm() mentions the Cholesky alternative but prefers eigen for better stability (without saying more). In spite of that, my personal recommendation would be to use mvtnorm::rmvnorm(.., method = "chol") { or the 2-3 lines of R code to the same thing without an extra package, just using rnorm(), chol() and simple matrix operations } because in simulations I'd expect the var-cov matrix Sigma to be far enough away from singular for chol() to be stable. 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