#codingexercise

Int GetCountDuplicatesCount (int [] A)

{

if (A == null) return 0;

return A.CountDuplicatesCount();

}


#codingexercise

Int GetCountDuplicatesAvg (int [] A)

{

if (A == null) return 0;

return A.CountDuplicatesAvg();

}

We continue to discuss CG method. We were talking about starting and stopping the iterations. We next discuss preconditioning. Preconditioning is a technique for improving the condition number of a matrix. Let us say M is a symmetric, positive definite matrix that approximates A, but is easier to invert. Then applying M inverse to both sides of the linear equation we solve with Steepest Descent or CG method,  we can solve it more quickly because the eigenvalues of Minverse applied to matrix A on the left hand side of the linear equation are better clustered than those of A. However, the problem is M-inverse applied to A doesn't always come out to be positive symmetric matrix. We could avoid this difficulty by taking a pair of matrices E and E-Transpose whose product is M because for every positive symmetric M there exists such. Now the M-inverse applied to A and Einverse applied to A applied to E-Transponse-inverse have the same eigenvalues. By transforming, the latter is now symmetric and positive definite This process of using CG to solve this system is called Transformed Preconditioned Conjugate Gradient Method.