swh:1:snp:d1587d616651317fdcebcbb237dce82c32266449
Tip revision: dfe6577bb164f53feaedf2da9530457197edfb1a authored by Georgi N. Boshnakov on 20 October 2022, 11:25:10 UTC
version 4021.93
version 4021.93
Tip revision: dfe6577
matrix-triang.Rd
\name{triang}
\alias{triang}
\alias{Triang}
\title{Upper and lower triangular matrices}
\description{
Extracs the upper or lower tridiagonal part from a matrix.
}
\usage{
triang(x)
Triang(x)
}
\arguments{
\item{x}{
a numeric matrix.
}
}
\details{
The functions \code{triang} and \code{Triang} allow to transform a
square matrix to a lower or upper triangular form.
A triangular matrix is either an upper triangular matrix or lower
triangular matrix. For the first case all matrix elements \code{a[i,j]}
of matrix \code{A} are zero for \code{i>j}, whereas in the second case
we have just the opposite situation. A lower triangular matrix is
sometimes also called left triangular. In fact, triangular matrices
are so useful that much computational linear algebra begins with
factoring or decomposing a general matrix or matrices into triangular
form. Some matrix factorization methods are the Cholesky factorization
and the LU-factorization. Even including the factorization step,
enough later operations are typically avoided to yield an overall
time savings. Triangular matrices have the following properties: the
inverse of a triangular matrix is a triangular matrix, the product of
two triangular matrices is a triangular matrix, the determinant of a
triangular matrix is the product of the diagonal elements, the
eigenvalues of a triangular matrix are the diagonal elements.
}
\references{
Higham, N.J., (2002);
\emph{Accuracy and Stability of Numerical Algorithms},
2nd ed., SIAM.
Golub, van Loan, (1996);
\emph{Matrix Computations},
3rd edition. Johns Hopkins University Press.
}
\examples{
## Create Pascal Matrix:
P = pascal(3)
P
## Create lower triangle matrix
L = triang(P)
L
}
\keyword{math}