Support Vector Machine (SVM) is a machine learning method based on statistical learning theory. Because of its superior learning performance. It has become the hot topic of pattern recognition, data mining, machine learning and other large data processing areas. Access to relevant similar articles. Found that the theory of SVM formula,such as the distance function , Lagrange function , quadratic convex optimization function lack detailed exposition. This paper tries to make readers to the typical linear optimal binary classification problem have a complete idea. The results of the classification and identification of rock cuttings are presented.It also wants to make a foundation of nonlinear multi-class pattern decomposition.