Product triangular systems with shift

Systems of the form $(R^{(1)}\cdots R^{(p)}-\lambda I)x=b$, where each R(i) is an n-by-n upper triangular matrix, can be solved in O(pn3) flops if the matrix of coefficients is explicitly formed. We develop a new method for this system that circumvents the explicit product and requires only O(pn2) flops to execute. The error bounds for the new algorithm are essentially the same as the error bounds for the explicit method. The new algorithm extends readily to the situation when R(1) is upper quasi-triangular.