If we approximate J to be very small (this approximation is valid as we want to minimize J and now if after completing this method, we calculate J, it would be very small),
[[File:Equation46.png|frame|center]]
Defining So, generally, QuEST uses Newton Raphson method starting with [[File:Equation44.png]] calculated above, and writing over multiple iterations reaches at a more accurate value for λ.<br \>Notice how this is different from the equation in terms qmethod. QuEST doesn’t directly solve for the eigen values. It starts with a reasonable guess and then moves on to check out if there are values close to this guess that can do better.<br \>Now to solve for the quaternion, both scalar and vector parts are divided q_4 =cosϕ/2 (q_4 is the scalar part; ϕis the angle of Rodrigue’s parameter '''p''' simplifies rotation). Note, this only changes the magnitude of the quaternion and the final answer can be corrected for later by normalising it. So, the equationvector part now is
[[File:Equation47.png|frame|center]]
where a is the vector along axis of rotation and Φ is the angle of rotation. Eigenvalue The initial matrix equation, which was to be solved is now written turns out asfollows
[[File:Equation48.png|frame|center]]
Which can be solved p is being operated on by gauss an invertible matrix. And Gauss eliminationcould be employed to find its inverse and hence solve for p. <br \>Once the '''p''' has been obtainedFinally, the quaternion can be found out by notingis normalised to get an unit quaternion.
[[File:Equation49.png|frame|center]]
Hence In general, while choosing between algorithm one has to make a tradeoff between getting a more accurate answer and implementing a computationally intensive code. QuEST adopts less accurate approach than qmethod at the estimated quaternion is obtained without solving eigenvalue equationcost of reducing computation. <br \>
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If you are done reading this page, you can go back to [[Attitude Determination and Control Subsystem]]