Toward the kernel of the vector epsilon algorithm
John A. Steele, Allan T. Dolovich · International Journal for Numerical Methods in Engineering · 2000
The vector epsilon algorithm (VEA) is a non-linear sequence-to-sequence transformation which has been in use for over 35 years to determine the limits (antilimits) of convergent (divergent) vector sequences. Recently, it has been used in a variety of engineering applications to accelerate iterative solution processes, including iterative finite element techniques. The VEA has been shown to give the limiting value of many sequences. However, an expression describing the kernel of the VEA, the set of all sequences {vn} which the VEA extrapolates successfully to the sequence's limit (antilimit) vector v, remains elusive. Here, this question is addressed with a simple proof giving the kernel of the first-order VEA with some comments about the kernel for higher orders. We prove that the first-order VEA assumes that each term of the related sequence {vn-v} is rotated by a fixed angle and scaled in length by a constant factor with respect to the preceding term. Copyright © 2000 John Wiley & Sons, Ltd.