# Nullspace: Nullspace method for LSE problem. In LSE: Constrained Least Squares and Generalized QR Factorization

## Description

Null Space method allows to give an analytic solution for equality constrained least squares problem (LSE). Requires pracma library.

## Usage

 `1` ```Nullspace(A,C,b,d) ```

## Arguments

 `A` Design matrix, m rows and n columns. `C` Constraint matrix, p rows and n columns. `b` Response vector for A, Ax=b, m rows and 1 column. `d` Response vector for C, Cx=d, p rows and 1 column.

## Details

Null Space method gives a numerical vector as the solution of a least squares problem (Ax=b), using an unconstrained problem equivalent to the LSE proposed, this method an be applied when impose some restrictions (additional information, extramuestral information or a priori information) that lead to another linear equality system (Cx=d). See significance constraint (x=0) or inclusion restriction (x+y=1), etc.

## Value

Numerical vector for a LSE problem.

## Author(s)

Sergio Andrés Cabrera Miranda Statician sergio05acm@gmail.com

## References

Lawson, C. L., & Hanson, R. J. (1974). Linear least squares with linear inequality constraints. Solving least squares problems, 158-173.

Van Benthem, M. H., Keenan, M. R., & Haaland, D. M. (2002). Application of equality constraints on variables during alternating least squares procedures. Journal of Chemometrics: A Journal of the Chemometrics Society, 16(12), 613-622.

## Examples

 ```1 2 3 4 5 6``` ```A = matrix(runif(50,-1,1),10,5) C = matrix(runif(20,-1,1),4,5) b = matrix(runif(10,-1,1),10,1) d = matrix(runif(4,-1,1),4,1) Nullspace(A,C,b,d) ```

LSE documentation built on Feb. 2, 2022, 5:07 p.m.