Let \(X = (x_ 1,\dots,x_ k)\), where \(x_ 1,\dots,x_ k\) are \(n\)- dimensional vectors (independent variables). Also available is an associated \(n\)-dimensional vector \(y\) (dependent variable). One of the main aims of linear regression is to predict the values of the dependent variable using a linear combination of the independent variables. Unfortunately, one problem arises when \(X' X\) is singular. One of the methods overcoming this problem is the so called partial least squares regression which is the focus of the present paper. A new algorithm for partial least squares regression is introduced that is shown to be equivalent to, and more efficient than, the current algorithm. In addition, the new algorithm works explicitly in terms of the factor loading and is therefore particularly useful when interpretation of the factors is required.