\Header{backsolve}{Solve an Upper or Lower Triangular System}
\alias{forwardsolve}{backsolve}
\keyword{algebra}{backsolve}
\keyword{array}{backsolve}
\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{Value}
The solution of the triangular system.  The result will be a
a vector if \texttt{x} is a vector and a matrix if \texttt{x} is a matrix.\end{Value}
\begin{References}
Dongarra, J. J., J. R. Bunch, C. B. Moler and G. W. Stewart (1978).
\emph{LINPACK Users Guide}.  Philadelphia: SIAM Publications.\end{References}
\begin{SeeAlso}
\texttt{\Link{chol}},
\texttt{\Link{qr}},
\texttt{\Link{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)
( y2 <- backsolve(r, x, transpose = TRUE)) # 8 -12 -5
all(t(r) %*% y2 == x)# exactly on Linux (Pentium)
all(y  == backsolve(t(r), x, upper = FALSE, transpose = TRUE))
all(y2 == backsolve(t(r), x, upper = FALSE,  transpose = FALSE))
\end{ExampleCode}
\end{Examples}

