Efficient Algorithms for Computing the Condition Number of a Tridiagonal Matrix

Let A be a tridiagonal matrix of order n. We show that it is possible to compute and hence condo (A), in O(n) operations. Several algorithms which perform this task are given and their numerical properties are investigated. If A is also positive definite then I[A-[[o can be computed as the norm of the solution to a positive definite tridiagonal linear system whose coeffcient matrix is closely related to A. We show how this computation can be carried out in parallel with the solution of a linear system Ax b. In particular we describe some simple modifications to the LINPACK routine SPTSL which enable this routine to compute condt (A), efficiently, in addition to solving Ax b. Key words, matrix condition number, tridiagonal matrix, positive definite matrix, LINPACK