Power method pseudo-code

The power method finds the dominant eigenvalue (largest in magnitude) and its eigenvector. Repeated multiplication A·x makes the vector turn toward the dominant eigenvector, and the scale factor approaches the eigenvalue.

Pseudo-code

Algorithm: Power method
Input : n, matrix A[n][n], initial vector x[n] (non-zero,
        e.g. all 1), tolerance e, max iterations N
Output: dominant eigenvalue lambda, eigenvector x

1.  Read n, A, x, e, N
2.  Set lambda_old = 0, k = 1
3.  Repeat steps 4 to 11 while k <= N
4.      y = A * x            // y[i] = sum_j A[i][j]*x[j]
5.      lambda = element of y with largest absolute value
6.      If lambda = 0 then
            print "Zero eigenvalue; try another x"; stop
7.      For i = 1 to n do
            x[i] = y[i] / lambda     // scale
        End for
8.      If |lambda - lambda_old| < e then
9.          print "Dominant eigenvalue =", lambda
10.         print "Eigenvector =", x
            stop
        End if
11.     lambda_old = lambda
        k = k + 1
12. End repeat
13. Print "Not converged in N iterations"
14. Stop

Notes

• Convergence rate depends on the ratio |λ2/λ1| of the two largest eigenvalues; the smaller the ratio, the faster.
• The starting vector must have a component along the dominant eigenvector.
• Scaling by the largest component keeps numbers from overflowing.