  The Blahut-Arimoto (BA) algorithm has played a fundamental role in the
numerical computation of rate-distortion (RD) functions. This algorithm
possesses a desirable monotonic convergence property by alternatively
minimizing its Lagrangian with a fixed multiplier. In this paper, we propose a
novel modification of the BA algorithm, wherein the multiplier is updated
through a one-dimensional root-finding step using a monotonic univariate
function, efficiently implemented by Newton's method in each iteration.
Consequently, the modified algorithm directly computes the RD function for a
given target distortion, without exploring the entire RD curve as in the
original BA algorithm. Moreover, this modification presents a versatile
framework, applicable to a wide range of problems, including the computation of
distortion-rate (DR) functions. Theoretical analysis shows that the outputs of
the modified algorithms still converge to the solutions of the RD and DR
functions with rate $O(1/n)$, where $n$ is the number of iterations.
Additionally, these algorithms provide $\varepsilon$-approximation solutions
with $O\left(\frac{MN\log N}{\varepsilon}(1+\log |\log \varepsilon|)\right)$
arithmetic operations, where $M,N$ are the sizes of source and reproduced
alphabets respectively. Numerical experiments demonstrate that the modified
algorithms exhibit significant acceleration compared with the original BA
algorithms and showcase commendable performance across classical source
distributions such as discretized Gaussian, Laplacian and uniform sources.
