backsolve {bdsmatrix} R Documentation

## Solve an Upper or Lower Triangular System

### Description

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.

### Usage

```  backsolve(r, ...)
## S4 method for signature 'gchol'
backsolve(r, x, k=ncol(r), upper.tri=TRUE, ...)
## S4 method for signature 'gchol.bdsmatrix'
backsolve(r, x, k=ncol(r), upper.tri=TRUE, ...)
```

### Arguments

 `r` a matrix or matrix-like object `x` a vector or 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. `...` further arguments passed to other methods

### Details

The generalized Cholesky decompostion of a symmetric matrix A is A= LD t(L) where D is diagonal, L is lower triangular, and t(L) is the transpose of L. These functions solve either L sqrt(D) x=b (when `upper.tri=FALSE`) or sqrt(D) t(L) x=b.

### 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.

Note that `forwardsolve(L, b)` is just a wrapper for `backsolve(L, b, upper.tri=FALSE)`.

### Methods

Use `showMethods(backsolve)` to see all the defined methods; the two created by the bdsmatrix library are described here:

bdsmatrix

`signature=(r= "gchol")` for a generalized cholesky decomposition

bdsmatrix

`signature=(r= "gchol.bdsmatrix")` for the generalize cholesky decomposition of a bdsmatrix object

### Note

The `bdsmatrix` package promotes the base R `backsolve` function to a generic. To see the full documentation for the default method, view `backsolve` from the `base` package.

`forwardsolve`, `gchol`