Convolutional potential and linear response problems are widely present in the fields of science and engineering, including physics, chemistry, materials science and other disciplines. Their high-precision and rapid calculations are often the key to numerical simulation. Common convolution methods include Coulomb potential, dipole potential and Yukawa potential, etc. The nonlocality of convolution, the singularity of the kernel function and the strong anisotropy of the density function bring challenges in terms of accuracy and efficiency to the rapid calculation of potential. We will report a series of fast algorithms with spectral accuracy O(NlogN), including the non-uniform fast Fourier transform method, Gaussian sum method, kernel truncation method, anisotropic kernel truncation method, far-field approximation method and moment matching method, as well as their applications. Linear response problems are a special type of asymmetric eigenvalue problem, which have wide applications in condensed matter physics and electronic structure calculations. We have designed and developed a set of parallel large-scale solvers by taking advantage of the biorthogonal structure of its eigenspace. Based on the full combination of its zero-invariant subspace structure, it can achieve high-precision and efficient solution of high-dimensional BdG problems.