tfQMRgpu: a GPU-accelerated linear solver with block-sparse complex result matrix

Abstract We present , a GPU-accelerated iterative linear solver based on the transpose-free quasi-minimal residual (tfQMR) method. Designed for large-scale electronic structure calculations, particularly in the context of Korringa–Kohn–Rostoker density functional theory, efficiently handles block-sparse complex matrices arising from multiple scattering theory. The solver exploits GPU parallelism to accelerate convergence while leveraging memory-efficient sparse storage formats. By unifying the solution of multiple right-hand side (RHS) block vectors, significantly improves throughput, demonstrating up to a $$3.5\times$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3.5</mml:mn> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> speedup on modern GPUs. Additionally, we introduce a flexible implementation framework that supports both explicit matrix-based and matrix-free operator formulations, such as high-order finite-difference stencils for real-space grid-based Green function calculations. Benchmarks on various NVIDIA GPUs demonstrate the solver’s efficiency, in some cases achieving over 56% of peak floating-point performance for block-sparse matrix multiplications. is open-source, providing interfaces for C, C++, Fortran, Julia, and Python, making it a versatile tool for high-performance computing applications that can benefit from the unification of RHS problems.