[[File:Equation74.png|frame|center]]
The error function J can be simplified to a summation involving <math>w_k </math> (the weights, which are constants set by us) and the gain function g as defined below
[[File:Equation75.png|frame|center]]
It is easy to see that minimising J is equivalent to maximising g. <br \>
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 and writing the equation in terms of Rodrigue’s parameter '''p''' simplifies the equationSo, generally, QuEST uses Newton Raphson method starting with [[File:Equation47Equation44.png|frame|center]]calculated above, and over multiple iterations reaches at a more accurate value for λ.<br \>where Notice how this is different from the 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 <math>q_4 =cos \frac{\phi}{2}</math> (<math>q_4</math> is the scalar part; <math> \phi</math> is the vector along axis angle of rotation ). Note, this only changes the magnitude of the quaternion and Φ the final answer can be corrected for later by normalising it. So, the vector part now is <math>\boldsymbol{p} = \frac{\bar{\boldsymbol{q}}}{q_4} = a*tan\frac{\phi}{2}</math>,where <math>a</math> is the angle vector along axis 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 elimination. <br \>Once the could be employed to find its inverse and hence solve for '''p''' has been obtained. <br \>Finally, 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]]
== References ==