  With the recent emergence of mixed precision hardware, there has been a
renewed interest in its use for solving numerical linear algebra problems fast
and accurately. The solution of total least squares problems, i.e., solving
$\min_{E,r} \| [E, r]\|_F$ subject to $(A+E)x=b+r$, arises in numerous
applications. Solving this problem requires finding the smallest singular value
and corresponding right singular vector of $[A,b]$, which is challenging when
$A$ is large and sparse. An efficient algorithm for this case due to Bj\"{o}rck
et al. [SIAM J. Matrix Anal. Appl. 22(2), 2000], called RQI-PCGTLS, is based on
Rayleigh quotient iteration coupled with the preconditioned conjugate gradient
method.
  We develop a mixed precision variant of this algorithm, RQI-PCGTLS-MP, in
which up to three different precisions can be used. We assume that the lowest
precision is used in the computation of the preconditioner, and give
theoretical constraints on how this precision must be chosen to ensure
stability. In contrast to standard least squares, for total least squares, the
resulting constraint depends not only on the matrix $A$, but also on the
right-hand side $b$. We perform a number of numerical experiments on model
total least squares problems used in the literature, which demonstrate that our
algorithm can attain the same accuracy as RQI-PCGTLS albeit with a potential
convergence delay due to the use of low precision. Performance modeling shows
that the mixed precision approach can achieve up to a $4\times$ speedup
depending on the size of the matrix and the number of Rayleigh quotient
iterations performed.
