对称矩阵行压缩的下标变换定理及在测绘中的应用

Index Transforms of Row-compressing Symmetric Matrix and Its Application in Geomatics

证明了整数环误差引理,进一步证明了对称阵行压缩的下标逆变换公式,这将其解压缩算法的时间复杂度从O(n2)降低到O(1).分析了对称阵的行压缩方式下的2种解压缩算法的相对运行效率,分析表明,当采用下标逆变换算法从压缩的对称矩阵中查询元素时,其查询效率比二重循环算法高得多.将这些公式和算法用来管理测绘工作中的大规模对称矩阵,如ITRF2000-ALASKA站群的协方差矩阵,既可进一步节约计算机存储空间和网络资源,还可提高其数据查询效率.

An Error Lemma of the Integer Ring was proved. The inverse index transform formulae of a Row-compressed symmetric matrix were derived and proved with the help of the Lemma, which reduces the time complexity from O （ n^2） to O （1）. The relative running efficiency of the two decompressions of a Row- Compressed symmetric matrix was analyzed, which shows that the inverse index transform algorithm is much more efficient than the bi-layer loops algorithm when inquiring an element from a compressed symmetric matrix. Using these formulae and algorithms to manage a large scale symmetric matrix in Geomatics, such as the covariance matrix of positions and the velocities of ITRF2000-ALASKA stations, can not only save storage space and network resources, but also improve the efficiency of the data query.