In surveying, equations relating observations with other observations or coordinates are seldom in a linear form. For example, the observation equation for the observation of distance between two points A and B is defined by Pythagoras’ equation which relates the differences of the eastings and northings of the points A and B with the distance between the two points.
So the observation equations are not normally capable of being expressed in terms of a series of unknowns multiplied by their own numerical coefficients. The solution of a series of equations by matrix methods requires that this is so. For example, the following pair of equations:
can be expressed in matrix terms as:
and solved as:
The same cannot be done with these non-linear equations!
A route to the solution of these equations, but obviously not the only one, is to make use of the first part of a Taylor expansion of the two functions:
The application of Taylor’s theorem leads to:
where x0 and y0 are estimated or provisional values of x and y. f1(x0, y0) is f1(x, y) where x and y take the values x0 and y0 and the notation | means ‘under the condition that . . .’. Therefore
is f1(x, y) differentiated with respect to x, and with x and y then taking the values x0 and y0. (x−x0) is now the correction to the provisional value of x since
and so the equations are now linear in (x −x0) and (y −y0). A simple one-dimensional example may help to illustrate the mathematical statements above.