R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> library( "miscTools" )
>
> set.seed( 123 )
>
> # not symmetric
> m1 <- matrix( rnorm( 9 ), ncol = 3 )
> print( m1 )
[,1] [,2] [,3]
[1,] -0.5604756 0.07050839 0.4609162
[2,] -0.2301775 0.12928774 -1.2650612
[3,] 1.5587083 1.71506499 -0.6868529
> try( semidefiniteness( m1 ) )
Error in isSemidefinite.matrix(m = m, ...) :
argument 'm' must be a symmetric matrix
> try( semidefiniteness( m1, method = "eigen" ) )
Error in isSemidefinite.matrix(m = m, ...) :
argument 'm' must be a symmetric matrix
> try( semidefiniteness( m1, positive = FALSE ) )
Error in isSemidefinite.matrix(m = m, ...) :
argument 'm' must be a symmetric matrix
> try( semidefiniteness( m1, positive = FALSE, method = "eigen" ) )
Error in isSemidefinite.matrix(m = m, ...) :
argument 'm' must be a symmetric matrix
>
> # positive semidefinite
> m2 <- crossprod( m1 )
> print( m2 )
[,1] [,2] [,3]
[1,] 2.796686 2.604009 -1.037747
[2,] 2.604009 2.963135 -1.309056
[3,] -1.037747 -1.309056 2.284591
> semidefiniteness( m2 )
[1] TRUE
> semidefiniteness( m2, method = "eigen" )
[1] TRUE
> semidefiniteness( m2, positive = FALSE )
[1] FALSE
> semidefiniteness( m2, positive = FALSE, method = "eigen" )
[1] FALSE
> # negative semidefinite
> semidefiniteness( -m2 )
[1] FALSE
> semidefiniteness( -m2, method = "eigen" )
[1] FALSE
> semidefiniteness( -m2, positive = FALSE )
[1] TRUE
> semidefiniteness( -m2, positive = FALSE, method = "eigen" )
[1] TRUE
>
> # positive semidefinite, singular
> m3 <- cbind( m2, - rowSums( m2 ) )
> m3 <- rbind( m3, - colSums( m3 ) )
> print( m3 )
[,1] [,2] [,3] [,4]
[1,] 2.796686 2.604009 -1.03774694 -4.36294799
[2,] 2.604009 2.963135 -1.30905572 -4.25808763
[3,] -1.037747 -1.309056 2.28459052 0.06221214
[4,] -4.362948 -4.258088 0.06221214 8.55882348
> semidefiniteness( m3 )
[1] TRUE
> semidefiniteness( m3, method = "eigen" )
[1] TRUE
> semidefiniteness( m3, positive = FALSE )
[1] FALSE
> semidefiniteness( m3, positive = FALSE, method = "eigen" )
[1] FALSE
>
> # positive semidefinite, singular, and large numbers
> m4 <- m3 * 1e6
> print( m4 )
[,1] [,2] [,3] [,4]
[1,] 2796686 2604009 -1037746.94 -4362947.99
[2,] 2604009 2963135 -1309055.72 -4258087.63
[3,] -1037747 -1309056 2284590.52 62212.14
[4,] -4362948 -4258088 62212.14 8558823.48
> # rcond(m4)
> # det(m4)
> semidefiniteness( m4 )
[1] TRUE
> semidefiniteness( m4, method = "eigen" )
[1] TRUE
> semidefiniteness( m4, positive = FALSE )
[1] FALSE
> semidefiniteness( m4, positive = FALSE, method = "eigen" )
[1] FALSE
>
> # negative semidefinite, diagonal
> m5 <- diag( -1, 4, 4 )
> print( m5 )
[,1] [,2] [,3] [,4]
[1,] -1 0 0 0
[2,] 0 -1 0 0
[3,] 0 0 -1 0
[4,] 0 0 0 -1
> semidefiniteness( m5 )
[1] FALSE
> semidefiniteness( m5, method = "eigen" )
[1] FALSE
> semidefiniteness( m5, positive = FALSE )
[1] TRUE
> semidefiniteness( m5, positive = FALSE, method = "eigen" )
[1] TRUE
>
> # negative semidefinite, singular
> m6 <- matrix( -1, 4, 4 )
> print( m6 )
[,1] [,2] [,3] [,4]
[1,] -1 -1 -1 -1
[2,] -1 -1 -1 -1
[3,] -1 -1 -1 -1
[4,] -1 -1 -1 -1
> semidefiniteness( m6 )
[1] FALSE
> semidefiniteness( m6, method = "eigen" )
[1] FALSE
> semidefiniteness( m6, positive = FALSE )
[1] TRUE
> semidefiniteness( m6, positive = FALSE, method = "eigen" )
[1] TRUE
>
> # negative semidefinite, diagonal
> m7 <- diag( c( -1, -3 ) )
> print( m7 )
[,1] [,2]
[1,] -1 0
[2,] 0 -3
> semidefiniteness( m7 )
[1] FALSE
> semidefiniteness( m7, method = "eigen" )
[1] FALSE
> semidefiniteness( m7, positive = FALSE )
[1] TRUE
> semidefiniteness( m7, positive = FALSE, method = "eigen" )
[1] TRUE
>
> # positive semidefinite
> m8 <- symMatrix( c( 2, -1, 0, 2, -1, 2 ) )
> print( m8 )
[,1] [,2] [,3]
[1,] 2 -1 0
[2,] -1 2 -1
[3,] 0 -1 2
> semidefiniteness( m8 )
[1] TRUE
> semidefiniteness( m8, method = "eigen" )
[1] TRUE
> semidefiniteness( m8, positive = FALSE )
[1] FALSE
> semidefiniteness( m8, positive = FALSE, method = "eigen" )
[1] FALSE
>
> # indefinite
> m9 <- symMatrix( rnorm( 6 ) )
> print( m9 )
[,1] [,2] [,3]
[1,] -0.4456620 1.2240818 0.3598138
[2,] 1.2240818 0.4007715 0.1106827
[3,] 0.3598138 0.1106827 -0.5558411
> semidefiniteness( m9 )
[1] FALSE
> semidefiniteness( m9, method = "eigen" )
[1] FALSE
> semidefiniteness( m9, positive = FALSE )
[1] FALSE
> semidefiniteness( m9, positive = FALSE, method = "eigen" )
[1] FALSE
>
> # positive and negative semidefinite
> m10 <- matrix( 0, 3, 3 )
> print( m10 )
[,1] [,2] [,3]
[1,] 0 0 0
[2,] 0 0 0
[3,] 0 0 0
> semidefiniteness( m10 )
[1] TRUE
> semidefiniteness( m10, method = "eigen" )
[1] TRUE
> semidefiniteness( m10, positive = FALSE )
[1] TRUE
> semidefiniteness( m10, positive = FALSE, method = "eigen" )
[1] TRUE
>
> # indefinite
> m11 <- symMatrix( 1:6 )
> print( m11 )
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 2 4 5
[3,] 3 5 6
> semidefiniteness( m11 )
[1] FALSE
> semidefiniteness( m11, method = "eigen" )
[1] FALSE
> semidefiniteness( m11, positive = FALSE )
[1] FALSE
> semidefiniteness( m11, positive = FALSE, method = "eigen" )
[1] FALSE
>
> # indefinite, singular
> m12 <- cbind( m9, - rowSums( m9 ) )
> m12 <- rbind( m12, - colSums( m12 ) )
> print( m12 )
[,1] [,2] [,3] [,4]
[1,] -0.4456620 1.2240818 0.35981383 -1.13823365
[2,] 1.2240818 0.4007715 0.11068272 -1.73553596
[3,] 0.3598138 0.1106827 -0.55584113 0.08534459
[4,] -1.1382337 -1.7355360 0.08534459 2.78842503
> semidefiniteness( m12 )
[1] FALSE
> semidefiniteness( m12, method = "eigen" )
[1] FALSE
> semidefiniteness( m12, positive = FALSE )
[1] FALSE
> semidefiniteness( m12, positive = FALSE, method = "eigen" )
[1] FALSE
>
> # indefinite, singular, small numbers
> m13 <- m12 * 1e-6
> print( m13 )
[,1] [,2] [,3] [,4]
[1,] -4.456620e-07 1.224082e-06 3.598138e-07 -1.138234e-06
[2,] 1.224082e-06 4.007715e-07 1.106827e-07 -1.735536e-06
[3,] 3.598138e-07 1.106827e-07 -5.558411e-07 8.534459e-08
[4,] -1.138234e-06 -1.735536e-06 8.534459e-08 2.788425e-06
> semidefiniteness( m13 )
[1] FALSE
> semidefiniteness( m13, method = "eigen" )
[1] FALSE
> semidefiniteness( m13, positive = FALSE )
[1] FALSE
> semidefiniteness( m13, positive = FALSE, method = "eigen" )
[1] FALSE
>
> # 'large' matrix
> m14 <- symMatrix( 1:( 13 * (13+1) / 2 ) )
> semidefiniteness( m14 )
[1] FALSE
> semidefiniteness( m14, method = "det" )
Warning in isSemidefinite.matrix(m = m, ...) :
using method 'det' can take a very long time for matrices with more than 12 rows and columns; it is suggested to use method 'eigen' for larger matrices
[1] FALSE
> semidefiniteness( m14, method = "eigen" )
[1] FALSE
>
> # list, one element not a matrix
> ml1 <- list( m2, c( m1 ), m3, m4 )
> try( semidefiniteness( ml1 ) )
Error in isSemidefinite.list(m = m, ...) :
all components of the list specified by argument 'm' must be matrices
>
> # list of matrices, one non-symmetric
> ml2 <- list( m2, m1, m3, m4 )
> try( semidefiniteness( ml2 ) )
Error in isSemidefinite.matrix(m[[t]], ...) :
argument 'm' must be a symmetric matrix
>
> # list of matrices, one 'large' matrix
> ml3 <- list( m2, m14, m3, m4 )
> semidefiniteness( ml3 )
[1] TRUE FALSE TRUE TRUE
> semidefiniteness( ml3, method = "det" )
Warning in isSemidefinite.matrix(m[[t]], ...) :
using method 'det' can take a very long time for matrices with more than 12 rows and columns; it is suggested to use method 'eigen' for larger matrices
[1] TRUE FALSE TRUE TRUE
> semidefiniteness( ml3, method = "eigen" )
[1] TRUE FALSE TRUE TRUE
> semidefiniteness( ml3, positive = FALSE )
[1] FALSE FALSE FALSE FALSE
> semidefiniteness( ml3, positive = FALSE, method = "det" )
Warning in isSemidefinite.matrix(m[[t]], ...) :
using method 'det' can take a very long time for matrices with more than 12 rows and columns; it is suggested to use method 'eigen' for larger matrices
[1] FALSE FALSE FALSE FALSE
> semidefiniteness( ml3, positive = FALSE, method = "eigen" )
[1] FALSE FALSE FALSE FALSE
>
> proc.time()
user system elapsed
0.268 0.020 0.312
Generated by dwww version 1.15 on Wed Jun 26 00:51:44 CEST 2024.