arnoldi: Arnoldi Iteration

In pracma: Practical Numerical Math Functions

Arnoldi Iteration

Description

Arnoldi iteration generates an orthonormal basis of the Krylov space and a Hessenberg matrix.

Usage

arnoldi(A, q, m)

Arguments

A	a square n-by-n matrix.
q	a vector of length n.
m	an integer.

Details

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.

Value

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).

References

Nicholas J. Higham (2008). Functions of Matrices: Theory and Computation, SIAM, Philadelphia.

See Also

hessenberg

Examples

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  -25

pracma documentation built on Nov. 10, 2023, 1:14 a.m.