2.4 Solution of the normal equation

From equation (2.1/1) , the normal equation (2.1/2) or (2.1/3) is established. If the vector of unknown parameters x from equation (2.1/3) was computed according to

(2.4/1)

then the complete matrix would have to be computed because all its elements are non-zero. This is impossible for complex equation systems. Therefore the equations are always resolved by factorisation. BINGO applies the Cholesky algorithm, where N is split into

(2.4/2)

Introduce into (2.1/3)

(2.4/3)

and substitute

(2.4/4)

Then from (2.4/3) we have

(2.4/5)

After factorisation according to (2.4/2), g is determined from (2.4/5) and x from (2.4/4).

This equation can only be solved when the normal equation matrix is regular or positive definite, i.e. it has the full rank of r=n-u.

As the unknowns of block adjustments generally are approximately known, the adjustment has to be done in several iterations, where for each iteration i+1 the adjustment results of iteration i are used as approximations. When the corrections to the unknowns are small enough, no further iteration is necessary.