The Differential Theory And Applications Of Non-analytic Quaternion Function

Quaternion was a non exchanged numerical system.It was put forward by Hamilton who was a physicist.In the 19^th century,Cauchy and Riemann made great contribution on the theoretical principle of complex analysis and provided a complete system of function theory on the complex domain.This system play an important role in Two-dimensional Physics,Engineering Mechanics,Signal Processing,Optimization Problems and other fields.Therefore extending this theory into quaternion for further research is of great significance.Research on quaternion function's differential theory has wide and important background.Quaternion's product is not satisfied with commutativity.It makes regular conclusions and characters invalid,which leads theoretical research more challenging.This paper summarized existing research results.Currently many domestic and foreign scholars research on quaternion function's analytic properties,but their conclusions are limited.This paper focused on non analytic quaternion function's differential theory.There are two definitions about quaternion derivative:left derivative and right derivative.This paper mainly discussed left derivative and proposed algorithms and properties based on left derivative.The conclusions of right derivative are similar to left derivative.Quaternion is non exchanged,which makes regular product and composite derivation invalid.This paper research from particularity to universality.Firstly researching real valued function's derivation rule and proving two quaternion functions'product and composite derivation.Then using some examples to prove derivation rules'validity,proposing quaternion function's mean value theorem and Taylor expansion and proving them in detail.Lastly applying the obtained product and composite derivation rule in quaternion neural network learning algorithm.Least Square,Gauss-Newton and Levenberg-Marquardt algorithms are deduced.Wide linear LS problems are solved using quaternion matrix real-valued characteristic.Then Gauss-Newton's and LM convergence are proved.In addition,Two different ways are used in solving quaternion matrix equation.Results of the two ways are the same.Indicate the correctness.