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### Solving Linear Least Squares Problems

Table 5.11 summarizes performance results obtained for the ScaLAPACK routine PSGELS /PDGELS that solves full-rank linear least squares problems. Solving such problems of the form , where x and b are vectors and A is a rectangular matrix having full rank is traditionally achieved via the computation of the QR factorization of the matrix A. In ScaLAPACK, the QR factorization is based on the use of elementary Householder matrices of the general form
where v is a column vector and is a scalar. This leads to an algorithm with excellent vector performance, especially if coded to use Level 2 PBLAS.

The key to developing a distributed block form of this algorithm is to represent a product of K elementary Householder matrices of order N as a block form of a Householder matrix. This can be done in various ways. ScaLAPACK uses the form [108]
where V is an N-by-K matrix whose columns are the individual vectors associated with the Householder matrices and T is an upper triangular matrix of order K. Extra work is required to compute the elements of T, but this is compensated for by the greater speed of applying the block form.

Table 5.11: Speed in Mflop/s of PSGELS/PDGELS for square matrices of order N