Square Root Method

The square root method is an algorithm which solves the Matrix Equation

 (1)

for , with a Symmetric Matrix and g a given Vector. Convert to a Triangular Matrix such that
 (2)

where is the Matrix Transpose. Then
 (3) (4)

so
 (5)

giving the equations
 (6)

These give
 (7)

giving from A. Now solve for k in terms of the s and g,
 (8)

which gives
 (9)

Finally, find from the s and k,
 (10)

giving the desired solution,
 (11)