Iterative solution of large systems of normal equations arising from the fitting of surfaces to irregularely spaced data

1977 ◽  
Vol 21 (2) ◽  
pp. B21-B36
Author(s):  
P. F. Czeglédy
Keyword(s):  
Iterative Solution ◽  
Normal Equations ◽  
Large Systems

ACCUMULATION OF THE CHOLESKY SQUARE ROOT IN HELMERT BLOCKING

1986 ◽  
Vol 40 (3) ◽  
pp. 297-314
Author(s):  
D. R. Junkins ◽  
R. R. Steeves
Keyword(s):  
Efficient Solution ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Computational Method ◽  
Normal Equation ◽  
Square Root ◽  
Unknown Parameters ◽  
Normal Equations ◽  
Systems Of Linear Equations ◽  
Large Systems

The Helmert blocking method is being used in the present effort to readjust North American geodetic networks. Combining this method with the Cholesky computational method enables the efficient solution of very large systems of linear equations. A by-product of this approach is a “partial” Cholesky square root for each Helmert block. This paper demonstrates that the Cholesky square root for the entire system of normal equations can be constructed from partial Cholesky square root blocks that are produced during the Helmert block adjustment, even though various reorderings of the unknown parameters are necessary throughout the computations. The entire Cholesky square root can be used to compute the inverse of the normal equation coefficient matrix, which is needed for post-adjustment statistical analyses.

scholarly journals On Iterative Solution of the Extended Normal Equations

2020 ◽  
Vol 41 (4) ◽  
pp. 1571-1589
Author(s):  
Henri Calandra ◽  
Serge Gratton ◽  
Elisa Riccietti ◽  
Xavier Vasseur
Keyword(s):  
Iterative Solution ◽  
Normal Equations

Iterative solution of systems of linear differential equations

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 259-307 ◽  
Author(s):  
Ulla Miekkala ◽  
Olavi Nevanlinna
Keyword(s):  
Parallel Processing ◽  
Iterative Methods ◽  
Initial Value Problems ◽  
Theoretical Basis ◽  
Iterative Solution ◽  
Attractive Alternative ◽  
Initial Value ◽  
History Of ◽  
Large Systems ◽  
Linear Differential

Parallel processing has made iterative methods an attractive alternative for solving large systems of initial value problems. Iterative methods for initial value problems have a history of more than a century, and in the works of Picard (1893) and Lindelöf (1894) they were given a firm theoretical basis. In particular, the superlinear convergence on finite intervals is included in Lindelöf (1894).

Structure of the system of normal equations in the construction and equalization of analytical phototriangulation by the ligament method using the coplanarity condition

2020 ◽  
Vol 64 (5) ◽  
pp. 507-514
Author(s):  
Bezmenov V.M. ◽  
Keyword(s):  
Normal Equations

Рассматривается построение и уравнивание аналитической пространственной фототриангуляции, ос- нованной на совместном использованием условия коллинеарности и условия компланарности векторов. Получена структура системы нормальных уравнений. Предлагаемое решение позволяет выполнять по- строение фототриангуляции с одновременным определением (уточнением) элементов внешнего ори- ентирования и внутреннего ориентирования съёмочной камеры. Для назначения начальных значений весов уравнений, составленных с использованием условия компланарности, для некоторой точки ис- следуемого объекта (местности) предлагается использовать ошибки определения пространственных координат точки, рассчитанные методом прямой фотограмметрической засечки для произвольного случая съёмки.

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