R Under development (unstable) (2024-12-05 r87423 ucrt) -- "Unsuffered Consequences"
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> library(cluster)
> 
> tools::assertWarning(eh <- ellipsoidhull(cbind(x=1:4, y = 1:4)), verbose=TRUE) #singular
Error in Fortran routine computing the spanning ellipsoid,
 probably collinear data
Asserted warning: algorithm possibly not converged in 5000 iterations
> eh ## center ok, shape "0 volume" --> Warning
'ellipsoid' in 2 dimensions:
 center = ( 2.5 2.5 ); squared ave.radius d^2 =  0 
 and shape matrix =
     x    y
x 1.25 1.25
y 1.25 1.25
  hence, area  =  0 

** Warning: ** the algorithm did not terminate reliably!
   most probably because of collinear data 
> stopifnot(volume(eh) == 0)
> 
> set.seed(157)
> for(n in 4:10) { ## n=2 and 3 still differ -- platform dependently!
+     cat("n = ",n,"\n")
+     x2 <- rnorm(n)
+     print(ellipsoidhull(cbind(1:n, x2)))
+     print(ellipsoidhull(cbind(1:n, x2, 4*x2 + rnorm(n))))
+ }
n =  4 
'ellipsoid' in 2 dimensions:
 center = ( 2.66215 0.82086 ); squared ave.radius d^2 =  2 
 and shape matrix =
                x2
   1.55901 0.91804
x2 0.91804 0.67732
  hence, area  =  2.9008 
'ellipsoid' in 3 dimensions:
 center = ( 2.50000 0.74629 2.95583 ); squared ave.radius d^2 =  3 
 and shape matrix =
                x2       
   1.25000 0.72591 1.8427
x2 0.72591 0.52562 1.5159
   1.84268 1.51588 4.7918
  hence, volume  =  3.1969 
n =  5 
'ellipsoid' in 2 dimensions:
 center = ( 3.0726 1.2307 ); squared ave.radius d^2 =  2 
 and shape matrix =
                x2
   2.21414 0.45527
x2 0.45527 2.39853
  hence, area  =  14.194 
'ellipsoid' in 3 dimensions:
 center = ( 2.7989 1.1654 4.6782 ); squared ave.radius d^2 =  3 
 and shape matrix =
                x2        
   1.92664 0.40109  1.4317
x2 0.40109 1.76625  6.9793
   1.43170 6.97928 28.0530
  hence, volume  =  26.631 
n =  6 
'ellipsoid' in 2 dimensions:
 center = ( 3.04367 0.97016 ); squared ave.radius d^2 =  2 
 and shape matrix =
                x2
   4.39182 0.30833
x2 0.30833 0.59967
  hence, area  =  10.011 
'ellipsoid' in 3 dimensions:
 center = ( 3.3190 0.7678 3.2037 ); squared ave.radius d^2 =  3 
 and shape matrix =
                    x2        
    2.786928 -0.044373 -1.1467
x2 -0.044373  0.559495  1.5496
   -1.146728  1.549620  5.5025
  hence, volume  =  24.804 
n =  7 
'ellipsoid' in 2 dimensions:
 center = (  3.98294 -0.16567 ); squared ave.radius d^2 =  2 
 and shape matrix =
                  x2
    4.62064 -0.83135
x2 -0.83135  0.37030
  hence, area  =  6.3453 
'ellipsoid' in 3 dimensions:
 center = (  4.24890 -0.25918 -0.76499 ); squared ave.radius d^2 =  3 
 and shape matrix =
                 x2        
    4.6494 -0.93240 -4.0758
x2 -0.9324  0.39866  1.9725
   -4.0758  1.97253 10.4366
  hence, volume  =  16.152 
n =  8 
'ellipsoid' in 2 dimensions:
 center = (  3.6699 -0.4532 ); squared ave.radius d^2 =  2 
 and shape matrix =
                x2
    9.4327 -2.5269
x2 -2.5269  3.7270
  hence, area  =  33.702 
'ellipsoid' in 3 dimensions:
 center = (  4.22030 -0.37953 -1.53922 ); squared ave.radius d^2 =  3 
 and shape matrix =
                x2        
    7.5211 -1.4804 -6.6587
x2 -1.4804  2.6972 11.8198
   -6.6587 11.8198 52.6243
  hence, volume  =  84.024 
n =  9 
'ellipsoid' in 2 dimensions:
 center = (  5.324396 -0.037779 ); squared ave.radius d^2 =  2 
 and shape matrix =
                x2
   10.1098 -1.3708
x2 -1.3708  2.1341
  hence, area  =  27.885 
'ellipsoid' in 3 dimensions:
 center = (  5.44700 -0.12504 -1.13538 ); squared ave.radius d^2 =  3 
 and shape matrix =
                x2        
    7.0364 -1.2424 -5.5741
x2 -1.2424  1.7652  7.3654
   -5.5741  7.3654 31.5558
  hence, volume  =  64.161 
n =  10 
'ellipsoid' in 2 dimensions:
 center = ( 4.85439 0.28401 ); squared ave.radius d^2 =  2 
 and shape matrix =
               x2
   13.932 0.64900
x2  0.649 0.95132
  hence, area  =  22.508 
'ellipsoid' in 3 dimensions:
 center = ( 5.12537 0.25024 0.86441 ); squared ave.radius d^2 =  3 
 and shape matrix =
                x2       
   9.29343 0.56973 1.4143
x2 0.56973 0.76519 1.8941
   1.41427 1.89409 6.3803
  hence, volume  =  73.753 
> 
> set.seed(1)
> x <- rt(100, df = 4)
> y <- 100 + 5 * x + rnorm(100)
> ellipsoidhull(cbind(x,y))
'ellipsoid' in 2 dimensions:
 center = ( -1.3874 93.0589 ); squared ave.radius d^2 =  2 
 and shape matrix =
        x      y
x  32.924 160.54
y 160.543 785.88
  hence, area  =  62.993 
> z <- 10  - 8 * x + y + rnorm(100)
> (e3 <- ellipsoidhull(cbind(x,y,z)))
'ellipsoid' in 3 dimensions:
 center = (  -0.71678  96.09950 111.61029 ); squared ave.radius d^2 =  3 
 and shape matrix =
        x       y        z
x  26.005  126.41  -80.284
y 126.410  616.94 -387.459
z -80.284 -387.46  254.006
  hence, volume  =  301.25 
> d3o <- cbind(x,y + rt(100,3), 2 * x^2 + rt(100, 2))
> (e. <- ellipsoidhull(d3o, ret.sq = TRUE))
'ellipsoid' in 3 dimensions:
 center = (   0.32491 101.68998  39.48045 ); squared ave.radius d^2 =  3 
 and shape matrix =
       x                 
x 19.655  94.364   48.739
  94.364 490.860  181.022
  48.739 181.022 1551.980
  hence, volume  =  21856 
> stopifnot(all.equal(e.$sqdist,
+ 		    with(e., mahalanobis(d3o, center=loc, cov=cov)),
+ 		    tol = 1e-13))
> d5 <- cbind(d3o, 2*abs(y)^1.5 + rt(100,3), 3*x - sqrt(abs(y)))
> (e5 <- ellipsoidhull(d5, ret.sq = TRUE))
'ellipsoid' in 5 dimensions:
 center = (   -0.32451   98.54780   37.33619 1973.88383  -10.81891 ); squared ave.radius d^2 =  5 
 and shape matrix =
          x                                       
x   17.8372    87.0277    8.3389   2607.9   49.117
    87.0277   446.9453   -2.0502  12745.4  239.470
     8.3389    -2.0502 1192.8439   2447.8   24.458
  2607.9264 12745.3826 2447.8006 384472.1 7179.239
    49.1172   239.4703   24.4582   7179.2  135.260
  hence, volume  =  191410 
> tail(sort(e5$sqdist)) ## 4 values 5.00039 ... 5.0099
[1] 4.999915 5.000005 5.000010 5.000088 5.001444 5.009849
> 
> (e5.1e77 <- ellipsoidhull(1e77*d5))
'ellipsoid' in 5 dimensions:
 center = ( -3.2451e+76  9.8548e+78  3.7336e+78  1.9739e+80 -1.0819e+78 ); squared ave.radius d^2 =  5 
 and shape matrix =
            x                                                  
x 1.7837e+155  8.7028e+155  8.3389e+154 2.6079e+157 4.9117e+155
  8.7028e+155  4.4695e+156 -2.0502e+154 1.2745e+158 2.3947e+156
  8.3389e+154 -2.0502e+154  1.1928e+157 2.4478e+157 2.4458e+155
  2.6079e+157  1.2745e+158  2.4478e+157 3.8447e+159 7.1792e+157
  4.9117e+155  2.3947e+156  2.4458e+155 7.1792e+157 1.3526e+156
  hence, volume  =  exp(898.66) 
> stopifnot(# proof correct scaling c^5
+     all.equal(volume(e5.1e77, log=TRUE) - volume(e5, log=TRUE),
+               ncol(d5) *  77* log(10))
+ )
> 
> proc.time()
   user  system elapsed 
   0.25    0.09    0.34