base (version 3.6.2)

# backsolve: 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

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`.

## 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.

## 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`.

## 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`.

## Examples

Run this code
```# 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
# }
```

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