Positive Definiteness of a Matrix, Rank and Condition Number
is.positive.definite tests whether all eigenvalues of a symmetric matrix
are positive.
make.positive.definite computes the nearest positive definite of a
real symmetric matrix, using the algorithm of NJ Higham (1988) <DOI:10.1016/0024-3795(88)90223-6>.
rank.condition estimates the rank and the condition
of a matrix by
computing its singular values D[i] (using svd).
The rank of the matrix is the number of singular values D[i] > tol)
and the condition is the ratio of the largest and the smallest
singular value.
The definition tol= max(dim(m))*max(D)*.Machine$double.eps
is exactly compatible with the conventions used in "Octave" or "Matlab".
Also note that it is not checked whether the input matrix m is real and symmetric.
is.positive.definite(m, tol, method=c("eigen", "chol"))
make.positive.definite(m, tol)
rank.condition(m, tol)m |
a matrix (assumed to be real and symmetric) |
tol |
tolerance for singular values and for absolute eigenvalues
- only those with values larger than
tol are considered non-zero (default:
|
method |
Determines the method to check for positive definiteness:
eigenvalue computation ( |
is.positive.definite returns
a logical value (TRUE or FALSE).
rank.condition returns a list object with the following components:
rank |
Rank of the matrix. |
condition |
Condition number. |
tol |
Tolerance. |
make.positive.definite returns a symmetric positive definite matrix.
Korbinian Strimmer (https://strimmerlab.github.io).
# load corpcor library
library("corpcor")
# Hilbert matrix
hilbert = function(n) { i = 1:n; 1 / outer(i - 1, i, "+") }
# positive definite ?
m = hilbert(8)
is.positive.definite(m)
# numerically ill-conditioned
m = hilbert(15)
rank.condition(m)
# make positive definite
m2 = make.positive.definite(m)
is.positive.definite(m2)
rank.condition(m2)
m2 - mPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.