MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.Comment: 26 pages, 6 figure

China's grave demographic challenges in coming decades

This paper systematically analyzes the uncertainties of major demographic indicators from China's 2000 census, such as fertility, gender ratio at birth, and age structure, and through a probability demographic forecast gives an assessment of the situation facing the country. Research outcomes suggest that great differences exist in the estimate of China's fertility, gender ratio at birth and low-age child population. These differences directly affect China's current and future demographic uncertainties, and have implications for policy and future research. The demographic uncertainties caused by current conditions are of great value to decision-makers and the public alike

Porous Graphitic Carbon Nitride Nanosheets by Pre-polymerization for Enhanced Hydrogen Evolution from Water Splitting under Solar Light

A facile and green method was developed to fabricate porous graphitic carbon nitride (g-C3N4) nanosheets by simple pre-polymerizing melamine. Porous structures were formed in polymerized g-C3N4 at 350∘C for 2h, which greatly enhanced the specifi  surface area and pore volume, resulting in superior photocatalytic evolution. The hydrogen evolution rate was 11.2 higher than that of bulk g-C3N4 under visible light. The porous structure not only provided abundant active catalytic sites and cross-plane diffusion channels to facilitate the charge and mass transportation, but also promoted the charge separation in the photocatalytic reaction. This g-C3N4 is suitable for mass-production to generate hydrogen from water splitting. Keywords: graphitic carbon nitride, photocatalytic, porous structures, prepolymerization, hydrogen evolution from water splittin