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[matrix] Annotation of /pkg/man/Schur.Rd
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Annotation of /pkg/man/Schur.Rd

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1 : bates 10 \name{Schur}
2 :     \title{Schur Decomposition of a Matrix}
3 :     \usage{
4 :     Schur(x, vectors, \dots)
5 :     }
6 :     \alias{Schur}
7 :     %\alias{Schur.Matrix}
8 :     \description{
9 :     Computes the Schur decomposition and eigenvalues of a square matrix.
10 :     }
11 :     \arguments{
12 :     \item{x}{
13 :     numeric or complex square Matrix inheriting from class
14 :     \code{"Matrix"}. Missing values (NAs) are not allowed.
15 :     }
16 :     \item{vectors}{logical. When \code{TRUE} (the default), the Schur
17 :     vectors are computed.
18 :     }
19 :     \item{\dots}{further arguments passed to or from other methods.}
20 :     }
21 :     \value{
22 :     An object of class \code{c("schur.Matrix", "decomp")} whose
23 :     attributes include the eigenvalues, the Schur quasi-triangular form
24 :     of the matrix, and the Schur vectors (if requested).
25 :     }
26 :     \details{
27 :     Based on the Lapack functions \code{dgeesx}
28 :     }
29 :     \section{BACKGROUND}{
30 :     If \code{A} is a square matrix, then \code{A = Q T t(Q)}, where
31 :     \code{Q} is orthogonal, and \code{T} is upper quasi-triangular
32 :     (nearly triangular with either 1 by 1 or 2 by 2 blocks on the
33 :     diagonal).
34 :     The eigenvalues of \code{A} are the same as those of \code{T},
35 :     which are easy to compute. The Schur form is used most often for
36 :     computing non-symmetric eigenvalue decompositions, and for computing
37 :     functions of matrices such as matrix exponentials.
38 :     }
39 :     \references{
40 :     Anderson, E., et al. (1994).
41 :     \emph{LAPACK User's Guide,}
42 :     2nd edition, SIAM, Philadelphia.
43 :     }
44 :     \examples{
45 :     #Schur(Hilbert(9)) # Schur factorization (real eigenvalues)
46 :     #A <- Matrix(rnorm( 9*9, sd = 100), nrow = 9)
47 :     #schur.A <- Schur(A)
48 :     #mod.eig <- Mod(schur.A$values) # eigenvalue modulus
49 :     #schur.A
50 :     }
51 :     \keyword{algebra}

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