Linear Least Squares (LLS) Problems     Next: Generalized Linear Least Up: Driver Routines Previous: Linear Equations

## Linear Least Squares (LLS) Problems

The linear least squares problem  is: where A is an m-by-n matrix, b is a given m element vector and x is the n element solution vector.

In the most usual case m > = n and rank(A) = n, and in this case the solution to problem (2.1) is unique, and the problem is also referred to as finding a least squares solution to an overdetermined  system of linear equations.

When m < n and rank(A) = m, there are an infinite number of solutions x which exactly satisfy b - Ax = 0. In this case it is often useful to find the unique solution x which minimizes , and the problem is referred to as finding a minimum norm solution  to an underdetermined  system of linear equations.

The driver routine xGELS  solves problem (2.1) on the assumption that rank(A) = min(m , n) -- in other words, A has full rank - finding a least squares solution of an overdetermined  system when m > n, and a minimum norm solution of an underdetermined  system when m < n. xGELS     uses a QR or LQ factorization of A, and also allows A to be replaced by in the statement of the problem (or by if A is complex).

In the general case when we may have rank(A) < min(m , n) -- in other words, A may be rank-deficient - we seek the minimum norm least squares solution  x which minimizes both and .

The driver routines xGELSX      and xGELSS      solve this general formulation of problem 2.1, allowing for the possibility that A is rank-deficient; xGELSX     uses a complete orthogonal factorization of A, while xGELSS     uses the singular value decomposition of A.

The LLS  driver routines are listed in Table 2.3.

All three routines allow several right hand side vectors b and corresponding solutions x to be handled in a single call, storing these vectors as columns of matrices B and X, respectively. Note however that problem 2.1 is solved for each right hand side vector independently; this is not the same as finding a matrix X which minimizes .

```-------------------------------------------------------------------
Single precision    Double precision
Operation                      real     complex    real     complex
-------------------------------------------------------------------
solve LLS using QR or          SGELS    CGELS      DGELS    ZGELS
LQ factorization

solve LLS using complete       SGELSX   CGELSX     DGELSX   ZGELSX
orthogonal factorization

solve LLS using SVD            SGELSS   CGELSS     DGELSS   ZGELSS
-------------------------------------------------------------------
```
Table 2.3: Driver routines for linear least squares problems     Next: Generalized Linear Least Up: Driver Routines Previous: Linear Equations

Tue Nov 29 14:03:33 EST 1994