Summary: Two new matrix iterative methods are presented to solve the matrix equation

$AXB=C$, the minimum residual problem

${min}_{X\in \mathcal{S}}\parallel AXB-C\parallel $ and the matrix nearness problem

${min}_{X\in {S}_{E}}\parallel X-{X}^{*}\parallel $, where

$\mathcal{S}$ is the set of constraint matrices, such as symmetric, symmetric

$R$-symmetric and

$(R,S)$-symmetric, and

${S}_{E}$ is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in [

*Y.-B. Deng* et al., Numer. Linear Algebra Appl. 13, No. 10, 801–823 (2006;

Zbl 1174.65382);

*G.-X. Huang* et al., J. Comput. Appl. Math. 212, No. 2, 231–244 (2008;

Zbl 1146.65036); the author, Appl. Math. Comput. 170, No. 1, 711–723 (2005;

Zbl 1081.65039); and

*Y. Lei* and

*A. Liao*, Appl. Math. Comput. 188, No. 1, 499–513 (2007;

Zbl 1131.65038)]. Paige’s algorithm is used as the frame method for deriving these matrix iterative methods. Numerical examples are used to illustrate the efficiency of these new methods.