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We present a design space exploration for synthesizing optimized, high-throughput implementations of multiple multi-dimensional tridiagonal system solvers on FPGAs. Re-evaluating the characteristics of algorithms for the direct solution of tridiagonal systems, we develop a new tridiagonal solver library aimed at implementing high-performance computing applications on Xilinx FPGA hardware. Key new features of the library are (1) the unification of standard state-of-the-art techniques for implementing implicit numerical solvers with a number of novel high-gain optimizations such as vectorization and batching, motivated by multi-dimensional systems in real-world applications, (2) data-flow techniques that provide application specific optimizations for both 2D and 3D problems, including integration of explicit loops commonplace in real workloads, and (3) the development of an analytic model to explore the design space, and obtain rapid performance estimates. The new library provide an order of magnitude better performance for solving large batches of systems compared to Xilinx's current tridiagonal solver library. Two representative applications are implemented using the new solver on a Xilinx Alveo U280 FPGA, demonstrating over 85% predictive model accuracy. These are compared with a current state-of-the-art GPU library for solving multi-dimensional tridiagonal systems on an Nvidia V100 GPU, analyzing time to solution, bandwidth, and energy consumption. Results show the FPGAs achieving competitive or better runtime performance for a range of multi-dimensional problems compared to the V100 GPU. Additionally, the significant energy savings offered by FPGA implementations, over 30% for the most complex application, are quantified. We discuss the algorithmic trade-offs required to obtain good performance on FPGAs, giving insights into the feasibility and profitability of FPGA implementations.