base (version 3.3.2)

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.

See Also

chol, qr, 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