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The Green's function or boundary element method (BEM) is the best available technique for rigorous surface acoustic wave (SAW) device analysis. However, its computational cost usually means that it cannot be applied directly to devices with complex, nonperiodic electrode structures. In this paper, approximate forms for the Green's function are employed. They are based on rigorous representations, they can represent the Green's function to any required degree of accuracy, and they can be applied to any type of substrate and acoustic wave. The use of this type of approximation for practical device analysis is considered, and computational procedures are presented that can exploit the special approximate Green's function structure. It is shown that highly efficient computational algorithms can be constructed, in which the computational effort increases linearly with the number of electrodes in the device. These methods can be applied to any type of device structure, and they do not require any empirically derived parameters. The practical application of the methods is illustrated by examples of longitudinally coupled resonator filter (LCRF) designs implemented using leaky wave cuts of lithium tantalate. Agreement between theory and experiment is excellent, even for devices of this complexity.