TMtx.LQRSolve Method (TMtx, TMtx, TMtx, TMtxOperation)

Solve overdetermined or underdetermined system of real linear equations.

Pascal

function LQRSolve(const B: TMtx; const X: TMtx; const R: TMtx = nil; Op: TMtxOperation = opNone): TMtx; overload;

Description

Solve overdetermined or underdetermined real linear systems involving a Rows-by-Cols matrix or its transpose, using a QR or LQ factorization of the calling matrix. It is assumed that the calling matrix has full rank. The following options are provided:

1.If Op = opNone and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

minimize || B - A*X ||.

2.If Op = opNone and m < n: find the minimum norm solution of an underdetermined system

A * X = B.

3.If Op = opTrans and m >= n: find the minimum norm solution of an undetermined system

A**T * X = B.

4.If Op = opTran and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

minimize || B - A**T * X ||

5.If Op = opHerm and m >= n: find the minimum norm solution of an undetermined system

A**H * X = B.

6.If Op = opHerm and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

minimize || B - A**H * X ||.

Notes

If the parameters are of TVec type (vectors), the routine requires less pre and post processing.