Arnoldi Iteration
Arnoldi iteration generates an orthonormal basis of the Krylov space and a Hessenberg matrix.
arnoldi(A, q, m)
A |
a square n-by-n matrix. |
q |
a vector of length n. |
m |
an integer. |
arnoldi(A, q, m) carries out m iterations of the
Arnoldi iteration with n-by-n matrix A and starting vector
q (which need not have unit 2-norm). For m < n it
produces an n-by-(m+1) matrix Q with orthonormal columns
and an (m+1)-by-m upper Hessenberg matrix H such that
A*Q[,1:m] = Q[,1:m]*H[1:m,1:m] + H[m+1,m]*Q[,m+1]*t(E_m),
where E_m is the m-th column of the m-by-m identity matrix.
Returns a list with two elements:
Q A matrix of orthonormal columns that generate the Krylov
space A, A q, A^2 q, ....
H A Hessenberg matrix such that A = Q * H * t(Q).
Nicholas J. Higham (2008). Functions of Matrices: Theory and Computation, SIAM, Philadelphia.
A <- matrix(c(-149, -50, -154,
537, 180, 546,
-27, -9, -25), nrow = 3, byrow = TRUE)
a <- arnoldi(A, c(1,0,0))
a
## $Q
## [,1] [,2] [,3]
## [1,] 1 0.0000000 0.0000000
## [2,] 0 0.9987384 -0.0502159
## [3,] 0 -0.0502159 -0.9987384
##
## $H
## [,1] [,2] [,3]
## [1,] -149.0000 -42.20367124 156.316506
## [2,] 537.6783 152.55114875 -554.927153
## [3,] 0.0000 0.07284727 2.448851
a$Q %*% a$H %*% t(a$Q)
## [,1] [,2] [,3]
## [1,] -149 -50 -154
## [2,] 537 180 546
## [3,] -27 -9 -25Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.