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In this work we present new kernels for the generation and application of block-Jacobi precon-ditioners that accelerate the iterative solution of sparse linear systems on graphics processing units (GPUs). Our approach departs from the conventional LU factorization and decomposes the diagonal blocks of the matrix using the Gauss-Huard method. When enhanced with column pivoting, this method is as stable as LU with partial/row pivoting. Due to extensive use of GPU registers and integration of implicit pivoting, our variable size batched Gauss-Huard implementation outperforms the batched version of LU factorization. In addition, the application kernel combines the conventional two-stage triangular solve procedure, consisting of a backward solve followed by a forward solve, into a single stage that performs both operations simultaneously. [ABSTRACT FROM AUTHOR]