An Iterative Algorithm for Maximal and Minimal Solutions of a Class Matrix Equations

In the paper, the peak solutions of a class equation is studied, the peak solutions are the maximal and minimal solutions. There is an iterative algorithm given for the solutions of the class equation. First, the existence of the peak solutions of the class equations is obtained. Second, when the peak solutions exist, an iterative algorithm is established to converge to the peak solutions of the class equation. By an upper bound and a lower bound of the solutions of the equation solution as the initial matrix, the iterative algorithm of the paper converges to the peak solutions of the class equation. The convergence problem of the algorithm is proved by the mathematical induction in the paper. The above results are verified by the examples.


Introduction
The class equation H1 X A X A I − += theory has been increasingly applied in engineering techniques such as control theory, trapezoidal networks, dynamic programming, etc. [1].In recent years, the solutions of the class matrix equations have also received high attention [1][2], and more and more scholars have begun to focus on numerical solutions of the equations, which are being followed in algebra.The class matrix equations have profound theoretical significance and broad application backgrounds in research fields.The theory for the existence of positive definite solutions and numerical solutions were studied for the nonlinear matrix equations [3].This paper derives an upper and lower bound for the solution of the class equation, an algorithm was established, the solution sequence obtained by the algorithm, and the solution sequence are convergence to the Maximum and minimum solution of the equation, the Maximum and minimum solutions are positive definite solutions.The following is a discussion on the numerical calculation of the Maximum and minimum solution for Wherein, A is an n order invertible square matrix, A H represents the conjugate transposition of A , and I is an n order unit matrix.

Definition 1The *
X is the solution of the matrix equation ( 1), and it is the positive definite.If there is any solution X of the (1), satisfies * XX  , then * X is called the Maximum positive definite solution of the matrix equation (1).

Definition 2
The * X is the solution of the matrix equation ( 1), and it is the positive definite.If there is any solution X of the matrix equation (1), satisfies * XX  , then * X is called the minimum positive definite solution of the matrix equation (1).
For any two n order matrices C ,D , CD  denoted CD − as a positive definite matrix.CO  denoted C as a positive definite matrix, wherein O is an n order zero matrix.
Lemma 2 [4] If n order matrix C O , D O  and satisfy CD  , then Lemma 3 If A is an n order invertible matrix, XO  , then

Main Conclusion
Theorem 1 If the n order matrix XO  , and it is a maximal solution of (1), the iteration format is established as following then k X () converges to According to the principles of mathematical induction, it can be concluded that . Then by the principles of mathematical induction proving . (3) Step 1: From Theorem 1 and Lemma 2, 1 XI −  can be inferred, according to Lemma 3 From the equation ( 2), it can be concluded that Step 2: Assumption k, the inequality holds, the inequality (3) holds, Proving the inequality From the hypothesis and Lemma 2, it is According to the equation ( 2) ( 1) 1 ( 1) Combining the (4), when k+1, the inequality (3) can be concluded.By the principles of mathematical induction, the equation (3) holds.
According to the convergence of monotonic matrix sequences [5], when k →, the limit of () k X exists, k X () converges to ( ) Any n dimensional nonzero column vector  , the definition of matrix sequence convergence [6] knows In the equation ( 2 .So then X * is the maximum solution.According to the conclusion of Theorem 2, an iterative algorithm 1for finding the maximum solution of the (1) can be established.Given the initial matrix (0) X I = , by the iterative format (2) (that is Theorem 2 to calculate the matrix sequence ,, , whose limit is the maximum solution of the equation (1).The specific steps are as the algorithm1: (1) Given convergence criteria ε and 1  .The formula ( 5) is equivalent to the formula (6).
Theorem5 assuming the equation ( 1) has a minimal solution and establishes an iterative format then then k X () converges to ( ) From Lemma 1 and the properties of contractual transformation, we obtain . According to the principles of mathematical induction, it can be concluded that Step 1: From Theorem 1 and Lemma 2, .
Step 2: Assumption k holds, proof k+1 also holds.From hypothesis and Lemma 1, it is Given the initial matrix (0) H X AA = , following the iterative format (8) to calculate the matrix sequence XX (0) (1) ,, , whose limit is the minimal solution of the formula (1).The specific steps are as algorithm2: (1) Given convergence criteria ε and 1 k = , calculating the initial matrix , the maximal and minimal solution of the formula (1) can be obtained by the algorithm1and the algorithm 2 proposed in this paper, The calculation numerical values are in Table 1and Table 2 (convergence criterion 1 0 7 .e = − ).

XA
satisfies (1), and XO  , then H Because XO  , by Lemma 1 and the properties of contract transformation, H1 Theorem 1, If XO  ,and X satisfies (1), the necessary condition is H

First
, by mathematical induction to prove.When replace k + 2 with k + 1and return (2) Theorem 3 If n order e matrix B O ,C O  and satisfy I B C , When A is an n matrix, and A  0 , there is prove that the formula (5) is equivalent to the formula (6).The formula XI − − multiplied A and right multiplied H A , both ends of the above equation by on the left and it can be concluded that and the equation (10), when k+1, the conclusion is obtained．From the principles of mathematical induction, inequality (9) holds.According to the convergence [4] of monotonic matrix sequence (9), when k →,The limit of

Table 1 .
Iteration times, calculation time, and actual error.

Table 2 .
Iteration times, calculation time, and actual error.