那时候好像永远是夏天，太阳总是有空出来伴随着我。阳光充足，太亮，使得眼前一阵阵发黑。
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Jacobi Matrices
Jacobi matrices represent a cornerstone binding two wings of the same building: One is built from moments, continued fractions, and polynomials, with the purpose of approximating functions and integrals. The other is built from vectors, vector spaces, operators, and matrices with the purpose of matrix computations such as solving linear algebraic systems and approximating eigenvalues.
– Krylov Subspace Methods - Principles and Analysis -
Optimization Algorithms - Dual Algorithms & ALM & ADMM & BCD
USTC 2025 Spring Semester Course Notes for Optimization Algorithms.
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Optimization Algorithms - Newton's Methods
USTC 2025 Spring Semester Course Notes for Optimization Algorithms.
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Optimization Algorithms - Gradient Methods
USTC 2025 Spring Semester Course Notes for Optimization Algorithms.
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Basic Theory of Optimization Algorithms
USTC 2025 Spring Semester Course Notes for Optimization Algorithms.
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Numerical Methods for Nonlinear Equations
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.
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Theory of Nonlinear Equations
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.
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Spectral Approximation of Convolution Operators
A note on a collection of papers on spectral approximation of convolution operators.
- 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.
- Xu, K., & Loureiro, A. F. (2018). Spectral approximation of convolution operators. SIAM Journal on Scientific Computing.
- Loureiro, A. F., & Xu, K. (2019).Volterra-type convolution of classical polynomials. Mathematics of Computation.
- Liu, X., Deng, K., & Xu, K. (2024). Spectral approximation of convolution operators of Fredholm type. SIAM Journal on Scientific Computing.
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QR decomposition & QR algorithm
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.