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This paper first describes some of the various types of algorithms encountered in scientific and engineering computations. Examples include (1) deterministic versus probabilistic applications, (2) modeling media that range from continuum to particles to molecules to atoms to electrons to quarks, (3) solving a differential equation (DE), (4) interpolation versus least-squares approximations, (5) optimization problems, (6) search algorithms, (7) integer problems, and (8) general-purpose algorithms such as the Fast Fourier Transform and a linear solver. Because most of the Technology Insertion 2005 (TI-05) benchmark applications solve a DE, the paper then explains the algorithmic consequences of each type of DE (ordinary and elliptic, hyperbolic, and parabolic partial DE) and then relates this information to specific Department of Defense applications, most of which are in the TI-05 application benchmark suite (TI05ABS). Various types of grids are then described, as most of the programs in the TI05ABS are grid-based. These grids are then related to the TI05ABS programs with accompanying descriptions of the resulting algorithmic complexities. Finally, a major aspect of algorithm complexity is the use of linear solvers. This paper presents a detailed analysis of each TI05ABS application regarding its characteristics with specific focus on linear solvers.