Compute Hessenberg matrix
stm_hessenberg.Rdstm_hessenberg() computes the Hessenberg form of a square matrix,
optionally also returning the orthogonal matrix \( Q \) used in the
transformation. This is useful in numerical linear algebra, particularly
for simplifying matrix operations such as eigenvalue calculations.
Arguments
- mat
A numeric square matrix to be transformed into its Hessenberg form.
- calc_q
Optional. A logical indicating whether to return the orthogonal matrix \( Q \) in addition to the Hessenberg matrix. Default is
FALSE, meaning only the Hessenberg matrix is returned.- digits
Optional. The number of digits to round the result to. If
NULL, no rounding is performed.
Value
If calc_q = FALSE, a matrix representing the Hessenberg form
of mat. If calc_q = TRUE, a list containing two
elements: H (the Hessenberg matrix) and Q (the
orthogonal matrix \( Q \)).
Details
The function uses the pracma::hessenberg() function to compute the
Hessenberg form of a square matrix. Optionally, it also returns the
orthogonal matrix used in the transformation, which is useful for certain
numerical methods.
Examples
A <- matrix(
c(4, 1, 2, 2, 3, 1, 1, 0, 2), nrow = 3, byrow = TRUE
)
print(A)
#> [,1] [,2] [,3]
#> [1,] 4 1 2
#> [2,] 2 3 1
#> [3,] 1 0 2
stm_hessenberg(A)
#> [,1] [,2] [,3]
#> [1,] 4.000000 -1.788854 1.341641
#> [2,] -2.236068 3.200000 -0.400000
#> [3,] 0.000000 0.600000 1.800000
stm_hessenberg(A, calc_q = TRUE)
#> $h
#> [,1] [,2] [,3]
#> [1,] 4.000000 -1.788854 1.341641
#> [2,] -2.236068 3.200000 -0.400000
#> [3,] 0.000000 0.600000 1.800000
#>
#> $q
#> [,1] [,2] [,3]
#> [1,] 1 0.0000000 0.0000000
#> [2,] 0 -0.8944272 -0.4472136
#> [3,] 0 -0.4472136 0.8944272
#>
stm_hessenberg(A, calc_q = TRUE, digits = 3)
#> $h
#> [,1] [,2] [,3]
#> [1,] 4.000 -1.789 1.342
#> [2,] -2.236 3.200 -0.400
#> [3,] 0.000 0.600 1.800
#>
#> $q
#> [,1] [,2] [,3]
#> [1,] 1 0.000 0.000
#> [2,] 0 -0.894 -0.447
#> [3,] 0 -0.447 0.894
#>