Editing Cholesky decomposition (section)

===Linear least squares===
In [[linear least squares (mathematics)|linear least squares]] problem one seeks a solution {{math|1='''x'''}} of an over-determined system {{math|1='''Ax''' = '''l'''}}, such that quadratic norm of the residual vector {{math|1='''Ax-l'''}} is minimum. This may be accomplished by solving by Cholesky decomposition
normal equations <math>\mathbf{Nx}=\mathbf{A}^\mathsf{T}\mathbf{l}</math>, where <math>\mathbf{N}=\mathbf{A}^\mathsf{T}\mathbf{A}</math> is symmetric positive definite. Symmetric equation matrix
may also come from an energy functional, which must be positive from physical considerations; this happens frequently in the numerical solution of [[partial differential equation]]s.

Such method is economic and works well in many applications, however it fails for near singular
{{math|1='''N'''}}. This is best illustrated in pathological case of square <math>\mathbf{A}</math>,
where determinant of {{math|1='''N'''}} is square of that of the original system {{math|1='''Ax''' = '''l'''}}.
Then it is best to apply SVD or QR decomposition. Givens QR has the advantage that similarly
to normal equations there is no need to keep the whole matrix {{math|1='''A'''}} as it is
possible to update Cholesky factor with consecutive rows of {{math|1='''A'''}}.