Parallel operations with band symmetric and positive definite matrices are considered. For effective realization of these computations on vector computers, the principal strategy is to vectorize the code and to exploit the sparsity structure of the matrix properly. Using the diagonal scheme and the unrolling procedure a new kernel subroutine for matrix-vector multiplication is developed. The authors show that by taking into account the above principles, the computing time for the conjugate gradient algorithm can be decreased significantly. In the test examples, band matrices with five non-zero diagonals are used. For a large size of the matrices, the speedup achieved tends to the optimal value 4, which corresponds to the number of processors in the supercomputer system CRAY- XMP/4.