ALL YOU NEED IS LOVE

国境の長いトンネルを抜けると雪国であった。夜の底が白くなった。信号所に汽車が止まった。

一切都不会过去。 — 契诃夫

Many classic iterative methods are based on subspace projection, i.e., krylov subspace methods for solving sparse linear systems, including the Conjugate Gradient (CG) method, the Generalized Minimal Residual (GMRES) method, the BiConjugate Gradient Stabilized (BiCGSTAB) method, and the Preconditioned Conjugate Gradient (PCG) method.

Scientific computing has traditionally required the highest performance, yet domain experts have largely moved to slower dynamic languages for daily work. We believe there are many good reasons to prefer dynamic languages for these applications, and we do not expect their use to diminish … The Julia programming language fills this role: it is a flexible dynamic language, appropriate for scientific and numerical computing, with performance comparable to traditional statically-typed languages. — Julia Documentation

‘The tau-method is both an algorithm and a philosophy. It was invented by Cornelius Lanczos in the same paper (1938) that gave the world the pseudospectral method.’ - Chebyshev and Fourier Spectral Methods, John P. Boyd

Basic definition, important characteristics and some useful properties of some orthogonal polynomials.

Mikael Mortensen. A Generic and Strict Banded Spectral Petrov-Galerkin Method for Differential Equations with Polynomial Coefficients. SIAM J. Sci. Comput. 45 (2023).

A series of seminal papers by Jie Shen:
1.Direct Solvers for the Second and Fourth Order Equations Using Legendre Polynomials;
2.Direct Solvers for the Second and Fourth Order Equations Using Chebyshev Polynomials;
3.Polar and Cylindrical Geometries;
4.Spherical Geometries.

Sheehan Olver, Alex Townsend. A Fast and Well-Conditioned Spectral Method, SIAM Rev. 55 (2013)

Schwarz method, Optimal Schwarz method, Krylov Subspace methdod.