Computes the least-squares solution using Householder transformations applied in blocked form.

The routine LQRRV computes the QR decomposition of a matrix A using blocked Householder transformations. The underlying code is based on either LINPACK , LAPACK, or ScaLAPACK code depending upon which supporting libraries are used during linking. For a detailed explanation see Using ScaLAPACK, LAPACK, LINPACK, and EISPACK in the Introduction section of this manual. The standard algorithm is based on the storage-efficient WY representation for products of Householder transformations. See Schreiber and Van Loan (1989).

The routine LQRRV determines an orthogonal matrix Q and an upper triangular matrix R such that A = QR. The QR factorization of a matrix A having NRA rows and NCA columns is as follows:

Initialize A1 ← A
For k = 1, min(NRA - 1, NCA)

End k

Thus,

where p = min(NRA − 1, NCA). The matrix Q is not produced directly by LQRRV. The information needed to construct the Householder transformations is saved instead. If the matrix Q is needed explicitly, QT can be determined while the matrix is factored. No pivoting among the columns is done. The primary purpose of LQRRV is to give the user a high-performance QR least-squares solver. It is intended for least-squares problems that are well-posed. For background, see Golub and Van Loan (1989, page 225). During the QR factorization, the most time-consuming step is computing the matrix-vector update Ak ← HkAk−1. The routine LQRRV constructs “block” of NB Householder transformations in which the update is “rich” in matrix multiplication. The product of NB Householder transformations are written in the form

where YNRA×NB is a lower trapezoidal matrix and TNB×NB is upper triangular. The optimal choice of the block size parameter NB varies among computer systems. Users may want to change it from its default value of 1.

The arguments which differ from the standard version of this routine are:

All other arguments are global and are the same as described for the standard version of the routine. In the argument descriptions above, MXLDA, MXLDX, MXCOL, and MXCOLX can be obtained through a call to SCALAPACK_GETDIM (see Utilities) after a call to SCALAPACK_SETUP (see Utilities) has been made. See the ScaLAPACK Example below.

Given a real m × k matrix B it is often necessary to compute the k least-squares solutions of the linear system AX = B, where A is an m × n real matrix. When m > n the system is considered overdetermined. A solution with a zero residual normally does not exist. Instead the minimization problem

is solved k times where xj, bj are the j-th columns of the matrices X, B respectively. When A is of full column rank there exits a unique solution XLS that solves the above minimization problem. By using the routine LQRRV, XLS is computed.

The previous example is repeated here as a distributed computing example. Given a real m × k matrix B it is often necessary to compute the k least-squares solutions of the linear system AX = B, where A is an m × n real matrix. When m > n the system is considered overdetermined. A solution with a zero residual normally does not exist. Instead the minimization problem

is solved k times where xj, bj are the j-th columns of the matrices X, B respectively. When A is of full column rank there exits a unique solution XLS that solves the above minimization problem. By using the routine LQRRV, XLS is computed. SCALAPACK_MAP and SCALAPACK_UNMAP are IMSL utility routines (see Utilities) used to map and unmap arrays to and from the processor grid. They are used here for brevity. DESCINIT is a ScaLAPACK tools routine which initializes the descriptors for the local arrays.