  In this study, a novel preconditioner based on the absolute-value block
$\alpha$-circulant matrix approximation is developed, specifically designed for
nonsymmetric dense block lower triangular Toeplitz (BLTT) systems that emerge
from the numerical discretization of evolutionary equations. Our preconditioner
is constructed by taking an absolute-value of a block $\alpha$-circulant matrix
approximation to the BLTT matrix. To apply our preconditioner, the original
BLTT linear system is converted into a symmetric form by applying a
time-reversing permutation transformation. Then, with our preconditioner, the
preconditioned minimal residual method (MINRES) solver is employed to solve the
symmetrized linear system. With properly chosen $\alpha$, the eigenvalues of
the preconditioned matrix are proven to be clustered around $\pm1$ without any
significant outliers. With the clustered spectrum, we show that the
preconditioned MINRES solver for the preconditioned system has a convergence
rate independent of system size. To the best of our knowledge, this is the
first preconditioned MINRES method with size-independent convergence rate for
the dense BLTT system. The efficacy of the proposed preconditioner is
corroborated by our numerical experiments, which reveal that it attains optimal
convergence.
