Numerics & scientific computing

Step an ODE forward by its current slope, y += h·f(t, y); first-order, one evaluation per step, the baseline every other integrator is measured against.

The classic four-stage, fourth-order ODE step; error drops 16× when the step halves, with no derivatives or past steps needed.

An embedded 5(4) Runge-Kutta pair estimates each step's error for free and resizes the step to meet a tolerance; the default non-stiff solver.

Implicit steps solve a small nonlinear system so fast decaying modes stay stable at any step size; BDF methods are the stiff-ODE workhorse.

Half-kick, drift, half-kick: a second-order, time-reversible, symplectic integrator with one force evaluation per step whose energy error stays bounded for millions of steps.

Update velocity from forces first, then position with the new velocity: explicit Euler cost, but symplectic and stable; the default game-physics integrator.

Replace derivatives with weighted differences of grid values; simple stencils plus a stability condition (CFL) turn PDEs like heat, advection and waves into array loops.

Writes the PDE in weak form, approximates it with piecewise polynomials on a mesh and assembles a sparse system; handles any geometry.

Represent the solution by a global Fourier or Chebyshev series; derivatives become multiplications by ik in coefficient space, giving exponential convergence for smooth problems.

Approximate an integral by a weighted sum of n function values at roots of orthogonal polynomials; exact for polynomials up to degree 2n-1.

Factor A = PLU (any square matrix, partial pivoting) or A = LLᵀ (SPD) once in O(n³), then solve each right-hand side in O(n²).

Compressed sparse row stores only nonzeros as row pointers, column indices and values; SpMV streams them once, so it is memory-bandwidth bound.

Krylov iteration for symmetric positive definite Ax = b using one matrix-vector product and two dot products per step; converges in about √κ iterations.

Smooths high-frequency error on the fine grid and solves the smooth remainder on coarser grids; O(n) elliptic solves at any resolution.

Solve sparse Ax = b exactly by reordering, symbolic analysis, numeric LLᵀ/LU and triangular solves; fill-in, not flops per entry, decides the cost.

Permute a sparse symmetric matrix before Cholesky/LU so the factor has few new nonzeros; minimum degree, nested dissection and RCM trade quality, speed and parallelism.

Sparse approximate LU or Cholesky factors (ILU, IC) that precondition Krylov solvers: fewer iterations than Jacobi, less memory than a direct solve.

Computes the DFT in O(n log n) by recursively splitting into even and odd halves; the basis of fast convolution and spectral methods.

Linearize at the guess and jump to the tangent root; quadratic convergence near a simple root, made robust by brackets (1D) or line search.

How IEEE 754 rounding error accumulates and how to contain it: compensated (Kahan/Neumaier) and pairwise summation, error-free transformations with FMA, ULP comparisons.

Quasi-Newton minimizer that approximates the inverse Hessian from the last m gradient differences via the two-loop recursion; O(m·n) memory and work per step.

Small, fast, statistically strong non-cryptographic generators: PCG32 and xoshiro256**, with seeding, parallel streams and unbiased ranges.

Estimate an expectation by averaging random samples; error falls as σ/√N in any dimension, and variance reduction (antithetic, control variates, importance sampling) shrinks σ.