Computing an eigenvector of a tridiagonal when the eigenvalue is known

One of the oldest methods for computing an eigenvector of a matrix is based or, the solution of a set of homogeneous equations. This approach call be traced back to the times of Cauchy (1829). The method of Holzer (1921) in vibration analysis is a special instance of this method for tridiagonal matrix pencils. We analyse and develop art effective algorithm for tridiagonal matrices by removing the principal difficulty encountered by Wilkinson (1958). This leads to the twisted factorisation of Henrici (1963) and also to a measure of singularity which was known to Babuska (1972) in a different context. The eigen vectors can be computed concurrently and hence the method is suitable for modern parallel architectures.