## 1 Introduction  
Intermetallic compounds are widely exploited and utilized due to their distinguished properties among the common metal in mechanical, electrical, magnetic and chemical properties [1; 2; 3]. For instance, they show high tensile strength, good ductility, high corrosion resistance and thermal stability, which are attributed to the divergences in the chemical bonding of metal atoms [4; 5; 6; 7; 8; 9; 10]. In addition, intermetallic compounds can fill the existing gap between metal and non-metal materials due to their specific properties [11]. In order to make better use of intermetallic compounds as well as exerting their value, it is of importance for us to study their properties. The first principles calculations with ab initio plane-wave pseudopotential density functional theory (PWPDFT) and generalized gradient approximation (GGA) is, in general, considered to be the most effective and convenient method to study the properties of metal materials, since it does not require any experimental parameters but needs some basic physical constants to obtain the basic properties of the ground state of the system. The stability of the crystal structure is the most important properties of intermetallic compounds. First principles calculation with density functional theory (DFT) has been successfully used to evaluate the stability of intermetallic compounds and predict their relevant properties [12; 13; 14]. Moreover, first principles calculations are also a very useful tool to calculate the equation of state (EOS) and thermodynamic properties in materials studies [15; 16; 17]. Co, Zr and Ti are transition metals and the three most basic elements making up alloy materials which under high temperature and high pressure used in business and industry. In the past ten years, more and more attention has been paid to Co-Zr and Co-Ti alloys. In the Co-Zr system, the presence of six compounds (i.e., Co\({}_{11}\)Zr\({}_{2}\), Co\({}_{23}\)Zr\({}_{6}\), Co\({}_{2}\)Zr, CoZr, CoZr\({}_{2}\), CoZr\({}_{3}\)) was confirmed and their phase diagrams and thermodynamic evaluation were reported [18; 19]. Likewise, there are five kinds of compound (TiO\({}_{3}\), h-TiO\({}_{2}\), c-TiO\({}_{2}\), TiO and Ti\({}_{2}\)Co) in Co-Ti system. [PERSON] [20] studied the microstructure and magnetic properties of alloys in the Co-Zr system with an experimental technique. [PERSON] et al. [21] used CALPHAD (calculation of phase diagrams) to evaluate the thermodynamics of the Co-Zr system. [PERSON] et al. [22] computed the extension properties of CoZr and CoTi with the elastic plastic self-consistent (EPSC) polycrystal model. [PERSON] et al. [23] made simple analysis about the elastic properties of Co2 Ti with first principles calculation. [PERSON] et al. [24] studied the structural, elastic and thermodynamic properties of Co\({}_{3}\)Ti by first principle. Meanwhile, the structural, electronic, elastic and thermal properties of CoZr and CoTi were calculated and compared by [PERSON] et al. [25]. [PERSON] et al. [26] studied the structure, elastic and thermal properties of CoX (X=Zr, Ti, Hf) by the methods of ab initio full potential linear augmented plane wave (FP-LAPW) and generalized gradient approximation (GGA). [PERSON] et al. [27] explored the kinetic energy and solute vacancy binding properties of Co\({}_{3}\)Ti and Co\({}_{2}\)Zr via first principles calculations. [PERSON] et al. [28] investigated the synthesis, characterization and hydrogen storage properties of ternary alloy, Co-Zr-Ti, by the methods of XRD, SEM and EDS. Alloy materials work at high temperatures, and some can even perform under high pressures in the aerospace and nuclear industries. The elastic properties and structural anisotropy under high pressure are very important to these materials for further applications. The changes in elastic characteristics under high pressure directly affect the mechanical stability of materials, which is the crucial factor for applications. It is also necessary to produce high quality and light alloys.