% File src/library/base/man/backsolve.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2025 R Core Team
% Distributed under GPL 2 or later

\name{backsolve}
\alias{backsolve}
\alias{forwardsolve}
\title{Solve an Upper or Lower Triangular System}
\description{
  Solves a triangular system of linear equations.
}
\usage{
   backsolve(r, x, k = ncol(r), upper.tri = TRUE,
             transpose = FALSE)
forwardsolve(l, x, k = ncol(l), upper.tri = FALSE,
             transpose = FALSE)
}
\arguments{
  \item{r, l}{an upper (or lower) triangular matrix giving the
    coefficients for the system to be solved.  Values below (above)
    the diagonal are ignored.}
  \item{x}{a matrix whose columns give the right-hand sides for
    the equations.}
  \item{k}{the number of columns of \code{r} and rows of \code{x} to use.}
  \item{upper.tri}{logical; if \code{TRUE} (default), the \emph{upper}
    \emph{tri}angular part of \code{r} is used.  Otherwise, the lower one.}
  \item{transpose}{logical; if \code{TRUE}, solve \eqn{r' * y = x} for
    \eqn{y}, i.e., \code{t(r) \%*\% y == x}.}
}
\details{
   Solves a system of linear equations where the coefficient matrix is
   upper (or \sQuote{right}, \sQuote{R}) or lower (\sQuote{left},
   \sQuote{L}) triangular.

  \code{x <- backsolve   (R, b)} solves \eqn{R x = b}, and\cr
  \code{x <- forwardsolve(L, b)} solves \eqn{L x = b}, respectively.

  The \code{r}/\code{l} must have at least \code{k} rows and columns,
  and \code{x} must have at least \code{k} rows.

  This is a wrapper for the level-3 BLAS routine \code{dtrsm}.
}
\value{
  The solution of the triangular system.  The result will be a vector if
  \code{x} is a vector and a matrix if \code{x} is a matrix.
}
\references{
  \bibshow{R:Becker+Chambers+Wilks:1988,
    R:Dongarra+Bunch+Moler:1979}
}
\seealso{
  \code{\link{chol}},
  \code{\link{qr}},
  \code{\link{solve}}.
}
\examples{
## upper triangular matrix 'r':
r <- rbind(c(1,2,3),
           c(0,1,1),
           c(0,0,2))
( y <- backsolve(r, x <- c(8,4,2)) ) # -1 3 1
r \%*\% y # == x = (8,4,2)
backsolve(r, x, transpose = TRUE) # 8 -12 -5
}
\keyword{algebra}
\keyword{array}
