\HeaderA{backsolve}{Solve an Upper or Lower Triangular System}{backsolve}
\aliasA{forwardsolve}{backsolve}{forwardsolve}
\keyword{algebra}{backsolve}
\keyword{array}{backsolve}
\begin{Description}\relax
Solves a system of linear equations where the coefficient matrix is
upper or lower triangular.
\end{Description}
\begin{Usage}
\begin{verbatim}
   backsolve(r, x, k=ncol(r), upper.tri=TRUE, transpose=FALSE)
forwardsolve(l, x, k=ncol(l), upper.tri=FALSE, transpose=FALSE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{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[\code{x}] a matrix whose columns give the right-hand sides for
the equations.
\item[\code{k}] The number of columns of \code{r} and rows of \code{x} to use.
\item[\code{upper.tri}] logical; if \code{TRUE} (default), the \emph{upper}
\emph{tri}angular part of \code{r} is used.  Otherwise, the lower one.
\item[\code{transpose}] logical; if \code{TRUE}, solve \eqn{r' * y = x}{} for
\eqn{y}{}, i.e., \code{t(r) \%*\% y == x}.
\end{ldescription}
\end{Arguments}
\begin{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.
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Dongarra, J. J., Bunch,J. R.,  Moler, C. B. and  Stewart, G. W. (1978)
\emph{LINPACK Users Guide.}  Philadelphia: SIAM Publications.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{chol}{chol}},
\code{\LinkA{qr}{qr}},
\code{\LinkA{solve}{solve}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## 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
\end{ExampleCode}
\end{Examples}

