Nearest Symmetric Positive-definite Matrix

Find nearest (in Frobenius norm) symmetric positive-definite matrix to A.

nearest_spd(A)

Arguments

A

square numeric matrix.

Details

"The nearest symmetric positive semidefinite matrix in the Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2, where H is the symmetric polar factor of B=(A + A')/2."
N. J. Highham

Value

Returns a matrix of the same size.

References

Nicholas J. Higham (1988). Computing a nearest symmetric positive semidefinite matrix. Linear Algebra and its Applications. Vol. 103, pp.103-118.

See also

randortho, procrustes

Examples

A <- matrix(1:9, 3, 3)
B <- nearest_spd(A); B
#>          [,1]     [,2]     [,3]
#> [1,] 2.034900 3.202344 4.369788
#> [2,] 3.202344 5.039562 6.876781
#> [3,] 4.369788 6.876781 9.383774
#          [,1]     [,2]     [,3]
# [1,] 2.034900 3.202344 4.369788
# [2,] 3.202344 5.039562 6.876781
# [3,] 4.369788 6.876781 9.383774
norm(B - A, type = 'F')
#> [1] 3.758517
# [1] 3.758517