Abstract: A recent event in the area of computer architecture is the development and production of "vector" machines. These machines are capable of manipulating large arrays of data in an efficient manner. Most existing vector machines, however, are restricted to performing operations on arrays stored in contiguous memory locations. Consequently, algorithms which assess individual data items rather than contiguous arrays of data do not execute efficiently on vector machines. When designing algorithms for a vector machine, therefore, it is important to consider both the machine's capabilities and the machine's restrictions. There is a close connection between the problem of calculating the eigenvalues of a matrix and the problem of calculating the roots of a polynomial. Many algorithms designed to solve one problem, therefore, may be used to solve the other. In an attempt to develop vector algorithms for calculating eigenvalues, the relationship between these two problems was examined. During the course of this examination, a simultaneous root-finding algorithm based on Laguerre's method was developed. This method was found to be more efficient than any other known simultaneous method. An algorithm based on Danilewski's method for calculating eigenvalues was developed and shown to be more efficient under certain circumstances than the widely used QR algorithm. The efficiency of a particular algorithm on a vector machine is determined to a large extent by the rate at which vector operands can be accessed from memory. A number of problems in linear algebra were examined to determine the most frequently occurring patterns of memory access. A system of specifying vectors which could be constructed using these patterns was devised, and hardware to generate the appropriate memory addresses was designed.