Convergence of Algebraic multigrid methods for symmetric positive definite matrices with weak diagonal dominance

For symmetric positive definite matrices with weak diagonal dominance, the algebraic sense of smooth errors is discussed and convergence of algebraic multigrid (AMG) methods is proved. It is found that the results of AMG for symmetric positive definite M-matrices with weak diagonal dominance also hold without the M-matrix assumption. The key result is the new formula x^[email protected]?ia"i"[email protected]?j [email protected]"[email protected]"j|a"i"j|+|a"j"i|x^2"[email protected][email protected]?j [email protected]"[email protected]"j|a"i"j|x"[email protected]"[email protected]"ia"i"j|a"i"j|x"j^2.