Trace method

6.9 Trace method

The trace method is a numerical method for solving the partial problem of eigenvalues, which means that for given real or complex matrix, it calculates only one eigenvalue (the one with greatest value of its euclidian norm) and its corresponding eigenvector. The method is iterative and it has three variations. The approximation of the eigenvalue calculated in k-th iteration, eigenvalue_k, is calculated as follows (M is input matrix):

k-th root method: eigenvalue_k = |trace(M^k)|^1/k
quotient method: eigenvalue_k = trace(M^k+1) / trace(M^k)
speeded up k-th root method: eigenvalue_k = |trace(M^{2^k})|^{1/2^k}

The eigenvector is calculated by multiplying M^k matrix of the last iteration with the starting vector (1,1,...1) and normalizing the result.

The methods stop when desired accuracy is achieved or maximal allowed number of iterations is exceeded. Be sure to read the help page for the result panel of the applet for information on how to interpret the error graph with regard to success/failure of the methods.