## 1 Introduction
Traditionally, there are two methods for ill-conditioned problems i.e., ridge estimation [7] and truncated singular value decomposition (TSVD) [8]. While improving the ill-condition of the matrix, ridge estimation excessively introduces bias, which reduces the reliability of the solution results. While restoring the main properties of the solution, some precision information will be lost in TSVD. Thus, these two methods are not suitable for the identification of integer ambiguity in fast positioning. On the basis of analyzing the structural characteristics of the integer ambiguity identification matrix, [PERSON] [9] proposed a construction method of regularization matrix based on the principle of Tikhonov regularization [10], in which the ill-condition of the integer ambiguity identification matrix is weakened by using a new regularization matrix. Then, a more accurate integer ambiguity floating-point solution and its corresponding mean square error matrix are obtained. To calculate the integer solution of integer ambiguity, there are mainly three methods: Fast Ambiguity Resolution Approach (FARA) [11], Cholesky [12] and Least-square Ambiguity Decorrelation Adjustment (LAMBDA) [13], With the modernization of GNSS, the use of multi-frequency data for positioning has also been put into practice [14]. LAMBDA can reduce the correlation between integer ambiguities and the search range of integer ambiguities through integer transformation. It is a widely used integer ambiguity search algorithm with fast search speed and good effect. However, LAMBDA searches the solution from an initial value in the solution space, which has low fault tolerance and cannot directly correct the search results. Intelligent evolution algorithms and swarm intelligence algorithms have also achieved good results in solving the integer ambiguity [15] for their global search characteristics. [PERSON] [16] combined integer Gaussian transform and particle swarm optimization algorithm to search the integer ambiguity. However, optimization algorithms have the problem of falling into local optimal solutions. Grey Wolf Optimization (GWO) algorithm is a new kind of swarm intelligence optimization algorithm proposed in 2014 [17; 18], which has the characteristics of simplicity, less parameter setting, strong global search ability, etc. It has been proven to be superior to particle swarm optimization algorithms in solution accuracy and convergence speed [19]. After the integer ambiguity of the solution is obtained, an effective confirmation method is required to determine whether the integer ambiguity is correct. The early method to confirm the integer ambiguity is to judge whether there is a significant difference between the optimal solution and the suboptimal solution of integer ambiguity through hypothesis testing, but the threshold value is often set according to experience [20]. The currently developed integer ambiguity confirmation method based on integer aperture estimation theory [21] cannot be applied to fast positioning due to the huge amount of calculation. According to the continuity and slow change of Total Electron Content (TEC), [PERSON] [22] used the TEC double difference of adjacent epochs to test the integer ambiguity. For the failed epoch, the ambiguity is searched again by removing one or more satellites with low altitude angles, but this method is limited by the number of satellites. Therefore, the traditional LAMBDA cannot directly correct the integer ambiguity, and the intelligent optimization methods also cannot guarantee the solution is globally optimal. And even if the global optimal solution is obtained under the designed objective function, it may not be the correct value. Thus, an efficient search strategy and reasonable integer ambiguity test means are necessary for the rapid positioning of GNSS.