## 1 Introduction  
Mobile positioning, allowing us to obtain the location information, is becoming increasingly important. In general, the position of the mobile device can be determined by a set of base stations (BSs), satellites, or wireless sensors [1, 2]. For example, Global Positioning System (GPS) can provide an accurate position to the user from signals received from the satellites [3]. However, GPS users usually require a particular handset device to calculate their positions when they are fully or partially equipped with a GPS receiver. In addition, GPS-embedded handsets are not functional in buildings or shadowed environments, where direct line-of-sight (LOS) propagation is not achievable. Without the aid of GPS, network-based positioning schemes become a solution, which use time or/and angle measurements from the set of BSs to determine the mobile station (MS) location [2, 4]. Wireless sensor networks (WSN) are another solution, which is usually applied to indoor measurements, but the locations of the sensor nodes should be known [1]. Initially, geometric dilution of precision (GDOP) is set as a criterion to select better satellites to meet the desired positioning precision in GPS. In general, a smaller GDOP indicates the positioning accuracy is better. Therefore, the subset with the smallest GDOP is selected for the optimal subset and can be used for positioning [5, 6, 7]. Nowadays, GDOP can be used to find the best subset of measurement units. For example, if there are \(N\) (\(N\geq 4\)) usable measurement units, an optimal subset of four measurement units can be found by GDOP. Hence, there will be \(C\) (\(N\), 4) possible GDOP values to be calculated. The GDOP calculation using the matrix inversion method is not suitable here because it requires significant computational operations. When more measurements are used, the complexity of the matrix inversion method increases rapidly [8]. When sufficient measurements are available, the determined optimal measurements with the minimum GDOP can improve the positioning accuracy. Satellite selection techniques can be used with a limited number of channels on the GPS receivers and for application in real time navigation. Due to the limited resources associated with mobile devices and the vast number of measurements required, selecting a subset with the most appropriate measurement units rapidly and reasonably is very critical. The conventional method for calculating GDOP is to use matrix inversion, requiring an enormous amount of computation. Recently, there have been extensive publications discussing how to obtain the GDOP value, or the approximate one, without executing matrix inversion. Some of those papers propose various methods to obtain approximate GDOP [9, 10, 11, 12]. In the conventional calculation of GDOP, the variances of range errors are assumed to be identical and independent distributed [13]. However, this assumption contradicts the practical environment, because the error variance from each measurement does not have equivalent amplitude [14]. For example, some GPS satellites are relatively old and have less accuracy. If we assume the same error variance, it will cause positioning errors. A satellite signal is related with the user range accuracy value, carrier-to-noise ratio, elevation angle, and the date of ephemeris, it is unreasonable to be assumed that all these signals have the same variance. Thus, the weighted GDOP (WGDOP) is proposed to overcome this problem [15]. WGDOP is performed by measuring the elevation and the signal-to-noise ratio (SNR) of the receiver [16], so that the location is estimated. A WGDOP minimum algorithm for the combined GPS-Galileo navigation receiver was proposed in [17]. In the past two decades, there has been extensive research on calculating WGDOP to yield superior GPS positioningaccuracy [18, 19, 20, 21, 22, 23, 24, 25]. Matrix inversion is a widely-used method for calculating WGDOP in most literatures.