I need to know what is the order of complexity and also how to calculate it.
Thanks in advance.
The most reliable way is to make SVD decomposition (for example using MATLAB) and than ratio of first and the last singular value (the last one which we recognize as non equal to zero) will be the condition numer.
The SVD is the most time consuming but the best approach.
I think that the question means the following: what is the computational complexity of a procedure that finds the condition number of the matrix. Therefore, one of the answers is - computational complexity of the SVD decomposition. (if implemented regularly - O(n*m^2+n^3), if your matrix is m x n.
But usually (from my practice) you don't need to know the exact condition number (technically, if it is large it is impossible or very hard to calculate it accurately as the difference between the largest and smallest singular values is huge). Maybe you may want to clarify your needs and some equivalent approximate metric for the matrix will be sufficient.
But I don't know anything better than SVD to find condition number exactly.
Thank you all for your help, but I think Dr. Breuer is right. My question was not clear enough! Please accept my apology.
Let me explain it, I am trying to solve a linear equation Ax = B.
As Matrix A is not a square matrix, I use x= (ATA)-1ATB
Different experiments result different matrix A and B. matrix A is always accurate but, Matrix B is obtained from some practical experiments therefore, is not accurate.
I know the accuracy of the answer is dependent on the condition number of matrix (ATA), and I am using "Cond()" command in Matlab to calculate it. and then I want to use this number to find the most accurate answer.
Now I want to Know what is the computational complexity of doing this procedure.How many operations is used to calculate it? (matrix A is M by 2 and matrix B is M by 1).
I hope this is clear enough.
Thanks in advance.
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