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USTC 2025 Spring Semester Course Notes for Optimization Algorithms.

USTC 2025 Spring Semester Course Notes for Optimization Algorithms.

USTC 2025 Spring Semester Course Notes for Numerical Methods for Nonlinear Equations, including basic numerical methods e.g. finite volume method with Godunov’s flux, high order Godunov’s flux, Lax-Friedrichs flux, TVD/TVB limiter, and WENO method etc.

USTC 2025 Spring Semester Course Notes for Numerical Methods for Nonlinear Equations, including the basic theory of nonlinear equations, e.g. weak solution, shock waves, rarefaction waves, contact discontinuities, Riemann problem, Rankine-Hugoniot condition, entropy solution, etc.

A note on a collection of papers on spectral approximation of convolution operators.

1. Xu, K., Austin, A. P., & Wei, K. (2017). A fast algorithm for the convolution of functions with compact support using Fourier extensions. SIAM Journal on Scientific Computing.
2. Xu, K., & Loureiro, A. F. (2018). Spectral approximation of convolution operators. SIAM Journal on Scientific Computing.
3. Loureiro, A. F., & Xu, K. (2019).Volterra-type convolution of classical polynomials. Mathematics of Computation.
4. Liu, X., Deng, K., & Xu, K. (2024). Spectral approximation of convolution operators of Fredholm type. SIAM Journal on Scientific Computing.

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for a particular eigenvalue algorithm, the QR algorithm.

With the help of the FFT, we can compute the forward and backward Chebyshev transform in O(N log(N)) operations.

Gauss-type quadratures are a class of quadrature rules that approximate the integral of a function by a weighted sum of the function values at certain points.

Brief introduction to fractional derivatives and integrals.

To the Lighthouse is a novel by Virginia Woolf. The novel centres on the Ramsay family and their visits to the Isle of Skye in Scotland between 1910 and 1920.