ON COMPUTATIONAL SCHEME OF LEAST SQUARES METHOD
The known Householder reflection method allows transforming arbitrary vector in strict accordance with the law of optical reflection. The matrix description of this method sets the Householder transformation which allows, in particular, reducing the arbitrary matrix of full rank to the trapezoidal and upper triangular form. Necessary information on the Householder reflection and its most important properties are provided in the article. On the basis of the specified procedure the simple and efficient algorithm of the solution of the standard least squares problem arising, in particular, in geodetic practice is constructed. Cumulative action of arising Householder matrices is similar to QR decomposition of a matrix, but at the same time there is no need to carry out practically such decomposition. It is shown that the system of normal equations arising in least squares method is equivalent to the triangular system obtained by the reflection method. The given numerical example confirms it. At the same time receiving the triangular system requires much less operations. Also process of constructing a varience-covariance matrix of least squares estimates becomes simpler as the inversion of the upper triangular matrix of the method of reflections is considerably less demanding work, than the one of the square matrix of normal equations. The geometrical interpetation of the reflected observation vector is given. In fact, it represents set of two subvectors - the adjusted observation vector and the residual vector in a new frame. The conclusion of the considerable advantages of the modified scheme from the point of view of both the comparative volume of calculations, and stability, especially when solving problem with a large number of required parameters is made.