| % File src/library/base/man/backsolve.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 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{ |
| 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. |
| } |
| \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} |