Tests for the Linear Least Squares (LLS) routines

The LLS test program tests the PxGELS driver routine for computing solutions to over- and underdetermined, full-rank systems of linear equations $A X = B$ ($A$ is $m$-by-$n$). For each test matrix type, we generate three matrices: One which is scaled near underflow, a matrix with moderate norm, and one which is scaled near overflow.

The PxGELS driver computes the least-squares solutions (when $m \ge n$) and the minimum-norm solution (when $m < n$) for an $m$-by-$n$ matrix $A$ of full rank. To test PxGELS, we generate a diagonally dominant matrix $A$, and for $C = A$ and $C = A^H$, we

• generate a consistent right-hand side $B$ such that $X$ is in the range space of $C$, compute a matrix $X$ using PxGELS, and compute the ratio
  
  $$\vert\vert AX - B\vert\vert/(\max(m,n) \vert\vert A\vert\vert \vert\vert X\vert\vert \epsilon)$$
  
  

• If $C$ has more rows than columns (i.e. we are solving a least-squares problem), form $R = AX - B$, and check whether $R$ is orthogonal to the column space of $A$ by computing
  
  $$\vert\vert R^H C \vert\vert/(\max(m,n,nrhs)\vert\vert A\vert\vert \vert\vert B\vert\vert \epsilon)$$
  
  

• If $C$ has more columns than rows (i.e. we are solving an overdetermined system), check whether the solution $X$ is in the row space of $C$ by scaling both $X$ and $C$ to have norm one, and forming the QR factorization of $D = [A,X]$ if $C = A^H$, and the LQ factorization of $D = [A^H,X]^H$ if $C = A$. Letting $E = D(n:n+nrhs,n+1,n+nrhs)$ in the first case, and $E = D(m+1:m+nrhs,m+1:m+nrhs)$ in the latter, we compute
  
  $$\max \vert d_{ij}\vert / (\max(m,n,nrhs) \epsilon)$$