# Power Method Algorithm and Flowchart

Power method is used to find the largest Eigen value and the corresponding Eigen vector for a given system of linear simultaneous equations. In this post, I have included simple algorithm and flowchart for Power method to calculate eigen value and eigen vector. Eigen value problems are generally encountered in dynamics problems and structural stability analysis.

To students, Eigen value problems may seem to be theoretical and raw. But in reality, these sort of problems arise in numerous engineering and scientific analysis. Eigen values are required for analyzing the frequencies associated with various phenomena such as:

• vibrations of beams
• systematic vibrations of of an annular membrane
• the vibrations of a system of springs and masses
• the oscillations of a triple pendulum
• the torsional oscillations of a multi-cylinder engine
• the torsional oscillations of a uniform cantilever structure, and more.

For the Power Method algorithm and flowchart, we will consider a given matrix A for which we have to find the column vector X and a constant L (read as Lamda) such that:

[A] [X] = L [X] which gives
{ [A] – L [I] } [X] = [0]

Now, for a system with 3 linear algebraic equations, we get
– a set of {X1, X2, X3} column vector for L1
– a set of {X1, X2, X3} column vector for L2
– a set of {X1, X2, X3} column vector for L3

## Power Method Algorithm:

1. Start
2. Define matrix X
3. Calculate Y = AX
4. Find the largest element in magnitude of matrix Y and assign it to K.
5. Calculate fresh value X = (1/K) * Y
6. If [Kn – K(n-1)] > delta, goto step 3.
7. Stop

## Power Method Flowchart:

The procedure for Power method and these algorithm and flowchart to calculate the Eigen value and the corresponding Eigen vector is based on the method of “Characteristic polynomial”. For finding the dominant Eigen value of a given matrix, the iterative procedure implemented is known as “Rayleigh’s power method”.

With the power method algorithm and flowchart presented here, you can write source code for Power method in any high level programming language. The algorithm and flowchart are not exactly similarly presented with respect to one another. But, being based on iterative method, they are quite convenient to implement, and the method overall is well-suited for machine computations as well.