echelon: Reduced row echelon form

Description

Transform the equalities in a system of linear (in)equations or Reduced Row Echelon form (RRE)

Usage

echelon(A, b, neq = nrow(A), nleq = 0, eps = 1e-08)

Value

A list with the following components:

A: the A matrix with equalities transformed to RRE form.
b: the constant vector corresponding to A
neq: the number of equalities in the resulting system.
nleq: the number of inequalities of the form a.x <= b. This will only be passed to the output.

Arguments

A: [numeric] matrix
b: [numeric] vector
neq: [numeric] The first neq rows of A, b are treated as equations.
nleq: [numeric] The nleq rows after neq are treated as inequations of the form a.x<=b. All remaining rows are treated as strict inequations of the form a.x<b.
eps: [numeric] Values of magnitude less than eps are considered zero (for the purpose of handling machine rounding errors).

Details

The parameters A, b and neq describe a system of the form Ax<=b, where the first neq rows are equalities. The equalities are transformed to RRE form.

A system of equations is in reduced row echelon form when

All zero rows are below the nonzero rows
For every row, the leading coefficient (first nonzero from the left) is always right of the leading coefficient of the row above it.
The leading coefficient equals 1, and is the only nonzero coefficient in its column.

Examples

Run this code

echelon(
 A = matrix(c(
    1,3,1,
    2,7,3,
    1,5,3,
    1,2,0), byrow=TRUE, nrow=4)
 , b = c(4,-9,1,8)
 , neq=4
)

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