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% File src/library/base/man/backsolve.Rd
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% Part of the R package, https://www.R-project.org
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% Copyright 1995-2025 R Core Team
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% Distributed under GPL 2 or later
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\name{backsolve}
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\alias{backsolve}
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\alias{forwardsolve}
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\title{Solve an Upper or Lower Triangular System}
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\description{
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  Solves a triangular system of linear equations.
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}
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\usage{
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   backsolve(r, x, k = ncol(r), upper.tri = TRUE,
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             transpose = FALSE)
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forwardsolve(l, x, k = ncol(l), upper.tri = FALSE,
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             transpose = FALSE)
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}
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\arguments{
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  \item{r, l}{an upper (or lower) triangular matrix giving the
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    coefficients for the system to be solved.  Values below (above)
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    the diagonal are ignored.}
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  \item{x}{a matrix whose columns give the right-hand sides for
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    the equations.}
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  \item{k}{the number of columns of \code{r} and rows of \code{x} to use.}
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  \item{upper.tri}{logical; if \code{TRUE} (default), the \emph{upper}
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    \emph{tri}angular part of \code{r} is used.  Otherwise, the lower one.}
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  \item{transpose}{logical; if \code{TRUE}, solve \eqn{r' * y = x} for
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    \eqn{y}, i.e., \code{t(r) \%*\% y == x}.}
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}
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\details{
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   Solves a system of linear equations where the coefficient matrix is
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   upper (or \sQuote{right}, \sQuote{R}) or lower (\sQuote{left},
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   \sQuote{L}) triangular.
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  \code{x <- backsolve   (R, b)} solves \eqn{R x = b}, and\cr
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  \code{x <- forwardsolve(L, b)} solves \eqn{L x = b}, respectively.
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  The \code{r}/\code{l} must have at least \code{k} rows and columns,
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  and \code{x} must have at least \code{k} rows.
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  This is a wrapper for the level-3 BLAS routine \code{dtrsm}.
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}
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\value{
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  The solution of the triangular system.  The result will be a vector if
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  \code{x} is a vector and a matrix if \code{x} is a matrix.
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}
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\references{
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  \bibshow{R:Becker+Chambers+Wilks:1988,
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    R:Dongarra+Bunch+Moler:1979}
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}
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\seealso{
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  \code{\link{chol}},
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  \code{\link{qr}},
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  \code{\link{solve}}.
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}
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\examples{
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## upper triangular matrix 'r':
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r <- rbind(c(1,2,3),
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           c(0,1,1),
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           c(0,0,2))
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( y <- backsolve(r, x <- c(8,4,2)) ) # -1 3 1
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r \%*\% y # == x = (8,4,2)
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backsolve(r, x, transpose = TRUE) # 8 -12 -5
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}
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\keyword{algebra}
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\keyword{array}