# backsolve

##### Solve an Upper or Lower Triangular System

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

- 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**tri*angular 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.

##### See Also

##### Examples

`library(base)`

```
# NOT RUN {
## 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
# }
```

*Documentation reproduced from package base, version 3.6.0, License: Part of R 3.6.0*