# backsolve: Solve an Upper or Lower Triangular System

## Description

Solves a triangular system of linear equations.

## Usage

 ```1 2 3 4``` ``` backsolve(r, x, k = ncol(r), upper.tri = TRUE, transpose = FALSE) forwardsolve(l, x, k = ncol(l), upper.tri = FALSE, transpose = FALSE) ```

## Arguments

 `r, l` an upper (or lower) triangular matrix giving the coefficients for the system to be solved. Values below (above) the diagonal are ignored. `x` a matrix whose columns give the right-hand sides for the equations. `k` The number of columns of `r` and rows of `x` to use. `upper.tri` logical; if `TRUE` (default), the upper triangular part of `r` is used. Otherwise, the lower one. `transpose` logical; if `TRUE`, solve r' * y = x for y, i.e., `t(r) %*% y == x`.

## Details

Solves a system of linear equations where the coefficient matrix is upper (or ‘right’, ‘R’) or lower (‘left’, ‘L’) triangular.

`x <- backsolve (R, b)` solves R x = b, and
`x <- forwardsolve(L, b)` solves L x = b, respectively.

The `r`/`l` must have at least `k` rows and columns, and `x` must have at least `k` rows.

This is a wrapper for the level-3 BLAS routine `dtrsm`.

## Value

The solution of the triangular system. The result will be a vector if `x` is a vector and a matrix if `x` is a matrix.

## References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978) LINPACK Users Guide. Philadelphia: SIAM Publications.

`chol`, `qr`, `solve`.
 ```1 2 3 4 5 6 7``` ```## 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 ```