Conjugate and semiconjugate direction methods with preconditioning projectors

Various conjugate and semiconjugate direction methods, with preconditioning projectors, are discussed. The system of linear algebraic equations (SLAE) is considered for which there is a converging stationary iterative process with a transition matrix. SLAE can be solved by the semiconjugate residual (SCR) method, which is a stable modification of the generalized conjugate residual method. The convergence of the iterations can be accelerated by formally applying conjugate direction algorithms to preconditioned SLAE, as Kaczmarz's alternating block method can be represented in the form of converging stationary iterative process with symmetric positive definite (SPD) matrix B. Several theorems have also been provided to prove the direction methods.