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Givens transformations provide a direct method for solving linear least-squares estimation problems without forming the normal equations. This approach has been shown to be particularly advantageous in recursive situations because of characteristics related to data storage requirements, numerical stability and computational efficiency. The following discussion will concentrate on the problem of updating least-squares parameter and error estimates using Givens transformations. Special attention will be given to photogrammetric and geodetic applications.

Roads are one of the most important infrastructures in any country. One problem on road based transportation networks is accident. Current methods to identify of high potential segments of roads for accidents are based on statistical approaches that need statistical data of accident occurrences over an extended period of time so this cannot be applied to newly-built roads. In this research a new approach for road hazardous segment identification (RHSI) is introduced using Geospatial Information System (GIS) and fuzzy reasoning. In this research among all factors that usually play critical roles in the occurrence of traffic accidents, environmental factors and roadway design are considered. Using incomplete data the consideration of uncertainty is herein investigated using fuzzy reasoning. This method is performed in part of Iran's transit roads (Kohin-Loshan) for less expensive means of analyzing the risks and road safety in Iran. Comparing the results of this approach with existing statistical methods shows advantages when data are uncertain and incomplete, specially for recently built transportation roadways where statistical data are limited. Results show in some instances accident locations are somewhat displaced from the segments of highest risk and in few sites hazardous segments are not determined using traditional statistical methods

On the sphere, global Fourier transforms are non Abelian and usually called Spherical Harmonic Transforms (SHTs). Discrete SHTs are defined for various grids of data but most applications have requirements in terms of preferred grids and polar considerations. Chebychev quadrature has proven most appropriate in discrete analysis and synthesis to very high degrees and orders. Multiresolution analysis and synthesis that involve convolutions, dilations and decimations are efficiently carried out using SHTs. The high-resolution global datasets becoming available from satellite systems require very high degree and order SHTs for proper representation of the fields. The implied computational efforts in terms of efficiency and reliability are very challenging. The efforts made to compute SHTs and their inverses to degrees and orders 3600 and higher are discussed with special emphasis on numerical stability and information preservation. Parallel and grid computations are imperative for a number of geodetic, geophysical and related applications where near kilometre resolution is required. Parallel computations have been investigated and preliminary results confirm the expectations in terms of efficiency. Further work is continuing on optimizing the computations.