# backsolve

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

Solves a triangular system of linear equations.

Keywords
algebra, array
##### 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.

##### 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.
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 # }