Viv posts

Finite differences and spectral methods

2026-10-14T12:00:00Z

Two ways to differentiate a function you only know at grid points. A central difference uses the two neighbors; a spectral derivative uses every point. On a smooth periodic function the spectral one reaches rounding error with a few dozen points. Put a jump in the function and, away from the jump, the central difference is the accurate one.

LQR stabilizes the cart-pole at half the gain

2026-10-14T12:00:00Z

Stable for every gain scaling from 1/2 to 1000; stability is lost only below 0.463.

Each digit of Monte Carlo accuracy costs 100× the paths

2026-10-14T12:00:00Z

The error falls like N^(-1/2), with a log-log slope between -0.6 and -0.4 from 1,000 to 1,000,000 paths.

Put-call parity holds to rounding error

2026-10-14T12:00:00Z

Over 3,000 quotes parity holds to 10^-15 of the spot, and early exercise of an American call never pays.

Ripple-carry delay grows like n, carry-lookahead like log n

2026-10-14T12:00:00Z

At 64 bits the critical path is 132 gate delays for ripple carry and 13 for Kogge-Stone, which uses about twice the gates.

Finite differences stay second order away from a jump

2026-10-14T12:00:00Z

Away from the jumps the central difference error falls by 3.9 to 4.1 per doubling of N; the spectral error stays above 0.7.

Conjugate gradients take √κ iterations

2026-10-14T12:00:00Z

Iterations grow like sqrt(κ) (slope 1.05); bad scaling stalls CG past 10,000 iterations, and Jacobi brings it back to 2 times the unscaled count.

The Taylor series of eˣ fails at x = −20

2026-10-14T12:00:00Z

Terms reach 4.3 × 10^7 for an answer of 2.1 × 10^-9, so the sum is off by more than 10%; 1/e^20 is good to 10^-14.

The best forward-difference step is about 10⁻⁸

2026-10-14T12:00:00Z

The forward difference is best near h = 10^-8, with error between 10^-9 and 10^-7; at h = 10^-12 rounding pushes both errors above 10^-6.

Givens QR of a Hessenberg matrix costs O(n²)

2026-10-14T12:00:00Z

About 3n^2 operations against Householder's 4/3n^3; at n = 400 Householder does more than 100 times the work.

Each factor of ten in the condition number costs a digit

2026-10-14T12:00:00Z

Correct digits stay within 1 below and 3 above 16 - log_10κ; at n = 12 fewer than 3 remain.

Leapfrog holds a planet's energy for 100,000 orbits

2026-10-14T12:00:00Z

Leapfrog's energy error is the same after 100,000 periods as after 10,000, while RK4's grows 10 times.

Newton's method doubles the correct digits each step

2026-10-14T12:00:00Z

Full double precision for sqrt(2) in 5 steps, where bisection needs 52; the error ratio e_(k+1)/e_k^2 is near 0.354.

1 − cos x loses its digits as x shrinks

2026-10-14T12:00:00Z

At x = 10^-4 the direct form has lost 7 of 16 digits, while 2sin^2(x/2) keeps at least 15 down to x = 10^-7.

Householder and Givens keep Q orthogonal at any condition number

2026-10-14T12:00:00Z

|| Q^T Q - I || stays below 10 nu from κ = 10 to 10^12, while Gram-Schmidt grows like uκ or uκ^2.

The quadratic formula loses the small root

2026-10-14T12:00:00Z

For x^2 - 10^8x + 1 = 0 the textbook small root is off by 25%, while c/q gets it to 10^-15.

Chebyshev nodes tame Runge's function

2026-10-14T12:00:00Z

At degree 20 equally spaced nodes err by 60 and Chebyshev nodes by 0.018; at degree 40 Chebyshev is below 10^-3.

Spectral derivatives beat every power of N

2026-10-14T12:00:00Z

The spectral error is below 10^-12 by N = 32; the central difference falls like h^2 and is above 10^-4 at N = 128.

On a stiff problem an explicit solver takes hundreds of times the steps

2026-10-14T12:00:00Z

On Robertson's problem Dormand-Prince takes 34,537 steps, Radau IIA 78 and BDF 144, and all three end within 10^-6 of the reference.

Pairwise summation keeps the rounding error of a sum small

2026-10-14T12:00:00Z

On 2^20 uniform numbers pairwise summation errs by under 5u∑|t|, left to right by about 50u∑|t|.

The trapezoid rule is spectrally accurate on periodic functions

2026-10-14T12:00:00Z

16 points give ∫ e^(cos t) dt to 10^-14, while e^x on [0, 1] is off by 3.3 × 10^-4.

The one-pass variance comes out negative on four numbers

2026-10-14T12:00:00Z

For 10^9 + 4, +7, +13, +16 (variance 30) the one-pass formula gives -171; Welford's update gives 30.

DC power flow gets worse as the grid loads up

2026-10-14T12:00:00Z

On case118 the DC branch-flow error is 8% at half load, 13% at base load and 52% at three times base load.

Sparse Newton power flow time grows like n^1.2 to n^1.4

2026-10-14T12:00:00Z

Up to 2,383 buses Newton converges in at most 6 iterations, and its solve time grows with the bus count at a fitted log-log slope between 1.0 and 1.6.

LDA atoms miss the virial theorem by under 5%

2026-10-14T12:00:00Z

-V/T exceeds 2 by 4.9% for hydrogen down to 0.19% for argon, with energies within 5 × 10^-7 hartree of NIST.

Strong charges still screen at the Debye length

2026-10-14T12:00:00Z

The decay length is λ_D to 0.1% at Q = 0.1 and at Q = 20; the strong sphere's surface potential is 32% of Debye-Hückel's.

Binder cumulants find the Ising critical point within 1%

2026-10-14T12:00:00Z

The crossings are at 2.263 and 2.269 against Onsager's T_c = 2.2692; the susceptibility grows with slope 1.76, near 7/4.

Spin-orbit coupling makes 28 a magic number

2026-10-14T12:00:00Z

In calcium-40 the gap at 28 neutrons is 4.4 MeV; with W_0 = 0 it closes, while the gaps at 8 and 20 widen past 10 MeV.

Nuclear spin-orbit splitting rivals the shell gap

2026-10-14T12:00:00Z

In oxygen-16 the 1p splitting is 6.2 MeV, 40% of the shell gap; in hydrogen fine structure is 4.4 × 10^-6 of the gap.

Logistic regression on separated data never converges

2026-10-14T12:00:00Z

The slope grows by 2.0 per IRLS iteration, from 4.6 to 44.7, while the deviance falls by a factor of about e each step.

QR keeps the digits the normal equations lose

2026-10-14T12:00:00Z

On nine NIST datasets QR keeps at least 7.5 digits and leads the normal equations by at least 2; on Filip Cholesky keeps none.

Twenty tests on pure noise find something 64.7% of the time

2026-10-14T12:00:00Z

At least one of 20 z-tests is significant in 64.7% of 10,000 studies, as 1 - 0.95^20 = 0.6415 predicts; Bonferroni brings it to 5%.

A random walk spreads as 2Dt and turns Gaussian

2026-09-25T12:00:00Z

4000 walkers each take 200 random steps, drawn evenly from an interval. Their mean squared distance from the start grows in a straight line, 2Dt, with the diffusion constant D = 0.5 set by the step size; a line through the measurements gives D = 0.5004 ± 0.0098. After one step the walkers are spread flat, and by step 200 they are spread as the Gaussian that solves the diffusion equation: the flatness measure called excess kurtosis falls as -6/(5n), from -1.2 to zero within the noise.

BFGS against gradient descent on the Rosenbrock valley

2026-09-25T12:00:00Z

Both methods start at (-1.2, 1) on Rosenbrock's function and use the same backtracking line search; they differ only in the direction they search along. BFGS is within 10^-9 of the minimum at (1, 1) after 35 iterations and has a zero gradient to 10^-12 after 36. Gradient descent takes 22,962 iterations to get within 10^-9, and near the minimum it gains a factor of only 0.99919 per step, close to (κ - 1)/(κ + 1) for the Hessian's condition number κ = 2508.

Black-Scholes by Monte Carlo, against the closed form

2026-09-25T12:00:00Z

The price of a one-year call option on a $100 stock with strike $100, 5% interest and 20% volatility is $10.4506 by the Black-Scholes formula. Simulating the stock price 131072 times and averaging the discounted payoff gives $10.371, 1.95 standard errors away. The error shrinks as 1/sqrt(N): its measured slope on a log-log plot is -0.488, against -0.5, and the standard error 14.72/sqrt(N) comes from an exact formula for the payoff's spread.

Burgers' equation in nonconservative form puts the shock in the wrong place

2026-09-25T12:00:00Z

For smooth solutions, u_t + (u^2/2)_x = 0 and u_t + u u_x = 0 are the same equation. For a shock from u_L = 1 to u_R = 0.25 they are not. A scheme built on the first form, Godunov's, moves the shock at 0.625, the Rankine-Hugoniot speed (u_L + u_R)/2, to rounding. Upwinding the second form moves it at 0.5132, on 400, 800 and 1600 cells alike, so refining the grid does not fix it. With u_R = 0 the nonconservative shock stays at x = 0, with speed 0.

Excitation energies of water from the Casida equation, with and without de-excitations

2026-09-25T12:00:00Z

Water in the 6-31G basis, from its Hartree-Fock ground state to all 40 singlet and 40 triplet excitation energies, two ways. Time-dependent Hartree-Fock (the random-phase approximation) keeps the coupling B between excitations and de-excitations; the Tamm-Dancoff approximation drops it and diagonalizes A alone. The lowest singlet is at 0.344074 Ha (9.363 eV) in TDHF and 0.346161 Ha in TDA, and every one of the 160 energies matches PySCF's to 10^-8 Ha.

The double scroll of Chua's circuit

2026-09-25T12:00:00Z

Two capacitors, an inductor, a resistor and one diode whose current is a three-piece straight line. The voltages never settle and never repeat: they wind around two centers, hopping between them at irregular times. Nearby starts separate like e^(λ t) with λ = 0.33, which is what a separate scipy calculation gives within the scatter expected of a run this long.

Crank-Nicolson is second order in time, backward Euler first

2026-09-25T12:00:00Z

The heat equation u_t = u_xx from u(x, 0) = x(1 - x), stepped to t = 0.1 with six time steps from 0.02 down to 0.000625. Each halving cuts the Crank-Nicolson error by a factor of 4.0 to 4.7 (measured order 2.09 to 2.23, then 2.000) and the backward Euler error by 1.9 to 2.0 (order 0.94 to 0.996). The errors are measured against the exact solution of the space-discretized equation, a discrete sine series. Against the exact Fourier series of the PDE, the Crank-Nicolson error stops falling at the space error, 1.95 × 10^-6.

Electron mobility fitted to NIST's certified digits

2026-09-25T12:00:00Z

A seven-parameter rational model fitted to NIST's semiconductor electron-mobility data by lmder, the Levenberg-Marquardt routine of MINPACK, translated to Viv step by step. Every trial step matches a gfortran build of the original Fortran to the last bit, the answer matches scipy.optimize.leastsq, and the parameters land within 0.001 standard deviations of NIST's certified values.

Forward and backward Euler on either side of h = 2/λ

2026-09-25T12:00:00Z

The exact solution of this equation is cos t, which a step of 0.1 would follow to a few digits. With λ = 1000, forward Euler is within 4 × 10^-7 of it at t = 2 with h = 0.0019, and off by 2.7 × 10^33 at h = 0.0021. A disturbance is multiplied by 1 - hλ each step, and the measured factor matches |1 - hλ| to 2 × 10^-14 at 36 step sizes. Backward Euler's factor is 1/(1 + hλ), below 1 at every step size, and at h = 0.1 it is still within 2 × 10^-5 of cos 2.

Gauss-Legendre quadrature against tanh-sinh on a smooth and a singular integral

2026-09-25T12:00:00Z

Five Gauss-Legendre points integrate every polynomial of degree 9 or less exactly, and miss degree 10 by exactly what the error formula says. On e^x seven points reach rounding. On 1/sqrt(1 - x^2), which is infinite at both ends of the interval, 128 points still miss by 1.4 × 10^-2, and the error only halves when the points double. The tanh-sinh rule gets that integral to rounding with 37 points.

Grad-Shafranov by finite differences, against Solov'ev's closed form

2026-09-25T12:00:00Z

A tokamak equilibrium solved on a grid and graded against an exact one. Solov'ev's ψ = (R^2-1)^2/8 + R^2Z^2/(2κ^2) solves the Grad-Shafranov equation with constant pressure gradient; the five-point solver, relaxed by SOR until the residual is below 10^-11, misses it by 1.68×10^-4, 4.21×10^-5 and 1.06×10^-5 on grids of spacing 0.1, 0.05 and 0.025, an observed order of 1.995 and then 1.994. Those errors are scipy's direct solve of the same equations, to 10^-10.

Graphene's Dirac cones

2026-09-25T12:00:00Z

An electron hopping between neighboring carbon atoms of a honeycomb has two energy bands, and they touch at the corners of the Brillouin zone. Near a corner the energy rises in a straight line in every direction, a cone, with slope v_F = 3ta/2ℏ. A centered finite difference at the corner gives 1.5 to eight digits in every direction.

RHF, UHF and full CI for H₂ in STO-3G along the bond

2026-09-25T12:00:00Z

Pull the two atoms of H₂ apart in the smallest basis there is, and compute its energy three ways. Restricted Hartree-Fock is off by 0.021 Ha at the equilibrium bond length and heads for the wrong limit, 0.39 Ha too high. Unrestricted Hartree-Fock splits from it at R = 2.180 bohr and reaches two hydrogen atoms. Full CI is exact in this basis. Every energy matches PySCF to 10^-9 Ha.

The Hasegawa-Wakatani drift-wave model

2026-09-25T12:00:00Z

The Hasegawa-Wakatani equations are the simplest model of drift-wave turbulence at the edge of a tokamak that contains the instability driving it. Five rungs build their linear theory: the drift-wave frequency, the growth rate as the eigenvalue of a 2×2 matrix (0.098684 at C = 1, as numpy's eigensolver gives), the return to the drift wave as C → ∞ at the predicted rates 1/C and 1/C^2, a zonal flow that keeps k^2/(1+k^2) of its start as it relaxes, and the one-line change in the modified model that leaves it untouched.

Jacobi's eigenvalue method

2026-09-25T12:00:00Z

Jacobi's method diagonalizes a symmetric matrix by plane rotations, each of which zeroes one off-diagonal pair. On an 8 by 8 matrix the off-diagonal norm goes 1.33, 0.453, 0.0299, 3.2 × 10^-5, 8 × 10^-13 over four sweeps, and after the fifth it is under 10^-15, rounding error. The diagonal is then the spectrum, and it matches numpy.linalg.eigh to 10^-14.

Jacobi, SOR and conjugate gradients on the Poisson equation

2026-09-25T12:00:00Z

The five-point Poisson equation on a 15 by 15 grid, solved to a relative residual of 10^-8 four ways. Jacobi takes 948 sweeps, Gauss-Seidel 475, SOR with the optimal factor 60, and conjugate gradients 25 iterations. Each count follows from one number the theory predicts: Jacobi's error shrinks by cos π h per sweep once the smoothest mode dominates, Gauss-Seidel's by its square, and SOR and CG depend on sqrt(κ) ∝ 1/h. Halve h and Jacobi takes four times as many sweeps; SOR and CG take twice as many.

Three floating-point failures in summation, cancellation and RANDU

2026-09-25T12:00:00Z

Adding 1/k for k up to a million left to right ends 414 units in the last place (ulps) from the correctly rounded sum; Kahan's compensated sum is correctly rounded at all 31 checkpoints from 1 to 10^6. Evaluating (x - 1)^7 from its expanded coefficients near x = 1 gives rounding noise: 89 sign changes among 401 points, where the polynomial has one, and 8.5 times the true value at x = 1.005. The factored form is within 1.4ε everywhere. The random number generator RANDU puts every triple of successive draws on one of 15 planes.

Kinetic Monte Carlo against the master equation

2026-09-25T12:00:00Z

A molecule hops between three states, A to B to C and back, at fixed rates. The master equation gives the exact probability of each state at every time; kinetic Monte Carlo instead follows 10000 molecules one random jump at a time. At 25 times the fraction of molecules in each state matches the exact probability within four standard errors, the intermediate B rises to 0.460 and falls back, and the mean time to first reach C, exactly 1.75, is measured as 1.721 ± 0.014.

Lanczos finds the largest eigenvalue three times

2026-09-25T12:00:00Z

Sixty steps of Lanczos on a 200 by 200 symmetric matrix whose eigenvalues are known exactly. Both ends of the spectrum, ± 4, come out right to 10^-14. But in floating point the basis stops being orthogonal once the top eigenvalue has converged, the direction of its eigenvector creeps back in, and the eigenvalue 4, which the matrix has once, is found again and again. Orthogonalizing each new vector against all the old ones stops it.

Landau damping from the root of the dispersion relation

2026-09-25T12:00:00Z

A Langmuir wave in a collisionless plasma still decays: electrons moving near its phase velocity take energy from it. Newton's method on the kinetic dispersion relation, with the plasma dispersion function from lib/zfunc, puts the least-damped root at kλ_D = 0.5 at ω = 1.415662 - 0.153359i (in units of ω_pe), matching a 30-digit mpmath root to 10^-12. The usual small-k damping formula is 1.3% off there and 59% off at kλ_D = 0.3.

Make Euler stable on a stiff equation

2026-09-25T12:00:00Z

The solution of y' = -50y decays to zero, but forward Euler with step h = 0.05 multiplies y by 1 - 50h = -1.5 every step, and after 12 steps it is 129.7 instead of 10^-13. Rewrite decay_step and system_step as backward Euler, which takes the slope at the end of the step instead of the start, so that no step size makes them grow.

Make Gram-Schmidt orthogonal to rounding

2026-09-25T12:00:00Z

Ten nearly parallel vectors, each e_0 plus a tiny 10^-8 in its own direction. Classical Gram-Schmidt turns them into ten "orthonormal" vectors of which two are at 60 degrees: their dot product is 0.5. Rewrite orthonormalize so the vectors it returns are orthogonal to 10^-15 and equal the exact ones to 10^-12.

Make Numerov find the decaying solution

2026-09-25T12:00:00Z

Numerov's method integrates y'' = y outward from y(0) = 1, y(h) = e^(-h), aiming at the decaying solution e^(-x). By x = 9 it is off by a factor of 30, and by x = 18 by 2 × 10^9. Rewrite decaying so that it follows e^(-x) to Numerov's own accuracy, within 2.7 × 10^-5, on both intervals.

Make power iteration converge on a tied pair

2026-09-25T12:00:00Z

Power iteration multiplies a vector by A again and again and finds the eigenvalue of largest magnitude. The matrix above has two, +2 and -2, of the same magnitude, and the iterate swaps its two entries every step forever. Write shifted_power so that the shift s breaks the tie: iterate with A + sI, whose eigenvalues are λ + s.

Make the heat-equation step stable past r = 1/2

2026-09-25T12:00:00Z

The explicit (FTCS) step for u_t = u_xx multiplies the grid's most jagged mode by about 1 - 4r each step, and for r = 0.6 that is -1.4: a checkerboard that should fade grows to 215 in 16 steps. Rewrite heat_step as backward Euler, which needs a tridiagonal solve per step and damps every mode for every r.

Metropolis sampling finds Onsager's magnetization

2026-09-25T12:00:00Z

A 16 by 16 grid of spins, each flipped at random by the Metropolis rule, settles at T = 1.8 to an average magnetization of 0.9567 ± 0.0013. The exact infinite-lattice answer, from Onsager and Yang, is 0.95686. The critical temperature 2/ln(1 + sqrt(2)) = 2.26919 is where that formula reaches zero.

Nuclear level density, by counting the states

2026-09-25T12:00:00Z

Put 100 fermions in equally spaced levels and count, one by one, the states with excitation energy E. The counts are the partition numbers 1, 1, 2, 3, 5, 7, 11, …, exactly, up to p(100) = 190569292. They grow like Bethe's e^(2sqrt(aE)); at E = 100 the count is 4.4% below the leading Hardy-Ramanujan term, and the next term of the series, -0.4433/sqrt(E), accounts for all but 0.06% of that.

Neutron pairing in a neutron-star crust

2026-09-25T12:00:00Z

One cell of the neutron-star inner crust in one dimension: a slab of nuclear matter in a gas of free neutrons, with a contact pairing force. Solving the Bogoliubov-de Gennes equations self-consistently gives a pairing gap of 0.9151 MeV in the gas and 0.1518 MeV at the middle of the slab, as a separate numpy solver finds. The same code on a uniform ring reproduces the BCS gap equation's 0.961089 MeV to 10^-8.

Silicon's band structure, from the crystal up

2026-09-25T12:00:00Z

A density-functional calculation of silicon, one cell at a time: the crystal, a plane-wave basis, the pseudopotential, the Hamiltonian, the self-consistent loop and the bands from L through Gamma to X. It runs at a 5 Ha cutoff on four k-points so it fits in a browser tab; its tests check each stage, from the basis sizes to the band energies, against an independent numpy implementation at the same settings.

Skyrme Hartree-Fock of oxygen-16

2026-09-25T12:00:00Z

The ground state of ^16O from the SLy5 Skyrme functional, solved on a radial mesh at two spacings and extrapolated. The binding energy comes out 8.0250 MeV per nucleon and the charge radius 2.7993 fm; skyrme_rpa, the published code of Colò et al. (2013), gives 8.0261 MeV and 2.7991 fm in its own test run with the same force, and the six single-particle energies agree with it to 3 keV.

A gyro-orbit and the E×B drift by the Boris push

2026-09-25T12:00:00Z

The Boris push moves a charged particle through electric and magnetic fields in almost every particle-in-cell code. In a magnetic field alone it keeps the speed to rounding and puts every position on one circle; the gyroperiod comes out 6.28449 instead of 2π, longer by (ω_c Δ t)^2/12. With an electric field added, a particle started at mathbf E × mathbf B / B^2 moves at exactly that velocity forever, and any other start averages to it.

The error of a linear solve follows the condition number

2026-09-25T12:00:00Z

Gaussian elimination with partial pivoting solves Hilbert systems whose exact solution is all ones. From n = 3 to 12 the condition number κ grows from 524 to 1.7 × 10^16, and the error grows with it, staying between 0.005 and 0.18 times κε, while the residual stays under ε. In least squares, fitting a degree-11 polynomial to 40 points, Householder QR is off by 3.6 × 10^-9 (about κε), and the normal equations, which square κ, by 0.18.

A star of free neutrons can weigh at most 0.71 suns

2026-09-25T12:00:00Z

Hold a ball of neutrons up by their Fermi pressure alone, with gravity from general relativity, and solve for the star at each central density. The mass rises, peaks at 0.7102 solar masses with a radius of 9.16 km, and falls: no heavier star of free neutrons exists. A scipy twin written in different variables gets 0.71018, and Narain, Schaffner-Bielich and Mishustin (2006) quote 0.710.

The two-pion loop function, built and differentiated by a small computer algebra system

2026-09-25T12:00:00Z

The loop function L(q) sets the momentum dependence of the two-pion-exchange nuclear force at next-to-leading order. The program writes it as an expression tape, evaluates it, and differentiates the tape symbolically into a second tape. Both are checked against a 30-digit Feynman-parameter integral that never uses the closed form: L(200 MeV) = 1.14642288565 and dL/dq = 1.24075106×10^-3/MeV there, to 10^-14 and 2×10^-17/MeV.

Tunneling through one and two barriers by transfer matrices

2026-09-25T12:00:00Z

An electron meets a barrier it does not have the energy to climb. Chop the potential into flat pieces, and the wave on each side of every edge is related by a 2 by 2 matrix; the product of those matrices gives the chance of getting through, T, and of bouncing back, R. They add to 1 at every energy, to within 10^-13. Two barriers 4 apart, each of which alone lets through less than 1% at E = 0.2108, together let through 100% at that energy.

The bounce period of an electron trapped in a plasma wave

2026-09-25T12:00:00Z

In the frame of a large-amplitude plasma wave, an electron near the wave's phase velocity moves like a pendulum. Released 0.1 radian from the bottom of a trough, it takes 6.287115 time units to bounce back, which is 4K(sin^2 0.05) from the complete elliptic integral, and 2π(1 + θ_0^2/16) to six digits. Electrons with speed below 2ω_B/k at the trough stay in it; faster ones pass over the crests.

The two-stream instability in a particle-in-cell code

2026-09-25T12:00:00Z

Two cold electron beams passing through each other are unstable. A particle-in-cell code with 2000 particles measures the growth rate of the fastest mode as 0.35366 ω_p; cold-beam linear theory gives 1/(2sqrt(2)) = 0.35355. The phase-space film shows the beams rolling up into vortices of trapped electrons, which is where the growth stops.

Upwind advection smears the pulse by its modified equation

2026-09-25T12:00:00Z

First-order upwind carries a Gaussian pulse once around a periodic domain at Courant number ν = 0.5. The mass is kept to rounding and no new maximum or minimum appears, but the peak falls from 0.9975 to 0.5767. The scheme's modified equation adds a diffusion term D u_xx with D = ch(1 - ν)/2, which predicts the fall to 1/sqrt(3) = 0.5774 and predicts the pulse's variance growth, 2DT, exactly.

The tension in each of two ropes holding up a box

2026-09-24T12:00:00Z

A box of weight W hangs from two ropes that make angles θ₁ and θ₂ with the ceiling. Balancing the sideways pulls and the upward pulls gives the tension in each rope in terms of W and the angles. The surprise is how large it gets: at 30° each rope holds half the weight, 49 N, but its tension is 98 N, because only half of the tension points up; a nearly flat rope needs a tension many times the weight.

Doubling time breaks at tiny rates and at negative ones

2026-09-24T12:00:00Z

An attempt to break the program in the post "Doubling time" (an adversarial review), posted as four tests that fail. At a rate of 10^-12 a year it is off by a relative 9 × 10^-5; at 10^-16 it says the money never doubles; at -5% it says the money doubled 13.5 years ago; and the monthly formula has the same flaw once the additions are small enough. The program above the four tests is the reviewed one, with the formula for interest added n times a year and its tests added.

Doubling time, fixed for tiny and negative rates

2026-09-24T12:00:00Z

A fix for the four problems the review of "Doubling time" found. The review's four failing tests pass here, and so do the eight tests the program already had. The logarithm of 1 + r is computed without the rounding that lost the small rates, and a rate that is not positive gets an infinite doubling time.

Hartree-Fock for water

2026-09-24T12:00:00Z

Water in the STO-3G basis, from the basis to the restricted Hartree-Fock energy, using lib/hf, a library whose tests compare it against PySCF. The electron repulsion integrals are computed two independent ways, McMurchie-Davidson and Head-Gordon-Pople, which must agree; the energy matches PySCF 2.5.0 within 10^-9 hartree.

LOBPCG finds the lowest states of a particle in a box

2026-09-24T12:00:00Z

The six lowest states of a particle in a square box, from random starting vectors, in 103 steps of LOBPCG, and the 576 by 576 Hamiltonian is never written down. Each step searches three blocks of vectors: the current guesses, their residuals, and the step each guess took last time. Leave out the third block and the same run takes 797 steps.

Muon orthogonalizes the update

2026-09-24T12:00:00Z

Muon is an optimizer that keeps the directions of a gradient matrix and gives every one of them a step of about the same size, however lopsided the gradient is. Five Newton-Schulz steps get close to that without an SVD. On a problem whose inputs differ in size a hundredfold, it ends at least 4.5 times closer to the answer than SGD, momentum or Adam at their best step sizes, and behind momentum in training loss.

Newton's method doubles the correct digits

2026-09-24T12:00:00Z

Newton's method on x^2 = 2, one cell at a time. From x_0 = 1 the error is 1.6 × 10^-12 after four steps and zero after five; bisection needs 39 halvings to promise as much. Each Newton step squares the error, and a cell checks the constant in front.

The rule of 72 against the rule of 70

2026-09-24T12:00:00Z

A copy of the doubling-time post that adds the rule of 72. Above a rate of 4.90% a year it is closer to the exact doubling time than the rule of 70, and below that the rule of 70 is closer; at 8% the rule of 72 is off by 0.07%. The original four tests are kept word for word.

Two methods compute water's 6-31G integrals

2026-09-24T12:00:00Z

McMurchie-Davidson and Head-Gordon-Pople compute the same 4186 electron repulsion integrals for water in the 6-31G basis by different recurrences. They agree to rounding, each matches PySCF, and each one alone gives PySCF's Hartree-Fock energy. With a third implementation in the comparison, the cross-check names the implementation that disagrees and the integral where it does.

A swing-up policy trained offline

2026-09-23T12:00:00Z

An acrobot is two links with one motor, at the elbow; the shoulder turns freely. A linear controller can balance it only near upright, so from hanging down it first has to swing itself up. The swing-up controller here was not derived: it is nine weights found offline by a cross-entropy search over 1600 simulated swings, run as compiled native code, and the page contains only those nine numbers. In your browser the arm swings up in 1.188 seconds, hands over to a balancing controller, and settles to within 0.005 radians of upright.

A transformer trained with Muon

2026-09-23T12:00:00Z

One attention head, 336 parameters, a vocabulary of six. The task is to output the previous token, which a transformer can do only by learning to look one position back. It is trained by Muon, an optimizer that orthogonalizes the momentum with five Newton-Schulz steps before stepping along it. Chance is a loss of 1.792; sixty steps bring it to 0.00058 with every position right. In the attention matrix it ends with, each position puts its weight on the one before it.

Solving the Riccati equation without integrating it

2026-09-23T12:00:00Z

Every LQR controller comes from a quadratic matrix equation, and the tempting way to solve it is to integrate the differential version backwards until it stops moving. On a one-state system that method returns 73.783 where the answer is 414.214, with no warning. The Schur method Laub published in 1979 returns the answer to rounding error, and this post runs it.

Building a four-bit adder out of NAND gates

2026-09-22T12:00:00Z

A NAND gate wires into NOT, AND, OR and XOR. Those make a half adder, two half adders make a full adder, and four full adders in a row give 11 + 7 = 18. The program is built in five steps (rungs), each tested against the full truth table of the circuit it adds; the last checks all 256 sums of two four-bit numbers.

A cart-pole, balanced by a PD controller

2026-09-22T12:00:00Z

The pole starts 15 degrees from upright. A controller that pushes the cart in proportion to the pole's angle and rate, and a little toward the origin, catches it in under two seconds and holds it. At a fifth of the gains it falls.

Compound interest

2026-09-22T12:00:00Z

$1000 at 5% a year becomes $1628.89 in ten years if the interest is added once a year, and $1647.01 if it is added monthly. Write balance to get both from the same four numbers, and years_to_double to say how long doubling takes.

Doubling time

2026-09-22T12:00:00Z

Something growing 7% a year doubles in ln 2 / ln 1.07 = 10.24 years, not 100/7 = 14.3: growth compounds. The logarithm is the tool that turns "how long until" into a division, and the rule of 70 is that division with ln 2 ≈ 0.7 and ln(1 + r) ≈ r.

LQR computes the cart-pole's gains from the Riccati equation

2026-09-22T12:00:00Z

The PD-controlled cart-pole has four gains that were tuned by hand. Linearize the same dynamics, choose how much each state error and each unit of push should cost, and the Riccati equation gives the gains instead. From 25 degrees they land the pole thirteen times closer to upright than the hand-tuned ones, for a third less cost and about the same peak push.

A Lennard-Jones liquid by molecular dynamics in two dimensions

2026-09-22T12:00:00Z

100 atoms in a periodic box, velocity Verlet, no thermostat. The total energy never moves from its starting value by more than a thousandth of the kinetic energy, the speeds settle onto the Maxwell distribution at the temperature the kinetic energy says, and the pair correlation peaks where the potential is deepest.

Solve the quadratic

2026-09-22T12:00:00Z

A ball is thrown straight up at 4 m/s from 1 m above the ground. When does it land? Write roots(a, b, c), the two solutions of ax^2 + bx + c = 0, smaller first.

Building the derivative from its definition

2026-09-22T12:00:00Z

The slope of a secant, then a secant centered on the point, then two of those combined: three estimates of the derivative of x^3, each with an error the algebra predicts exactly, and the tests check the algebra. The last rung tries the same three on e^x, where nothing is exact and the orders show up as ratios: halve h and the three errors fall by factors of 2, 4 and 16.

The geometric series adds up

2026-09-22T12:00:00Z

1 + 1/2 + 1/4 + cdots never reaches 2, and the partial sums say exactly how far short: S_n = 2 - 2^(1-n). The general fact is S_n = a (1 - r^n)/(1 - r), and when |r| < 1 the r^n goes to zero and the whole series is a/(1 - r). The program adds the terms one by one and checks the formula at every n.

The quadratic formula, applied

2026-09-22T12:00:00Z

The ball thrown up at 4 m/s from 1 m lands after 1 s. The quadratic formula gives that, but for a < 0 its minus sign gives the larger root, so the solution writes the roots as h ± w with w ≥ 0.

The quadratic formula, derived

2026-09-22T12:00:00Z

A garden has a perimeter of 10 m and an area of 6 m². How long are its sides? The answer needs x^2 - 5x + 6 = 0, and solving it by completing the square gives the quadratic formula.

Two lines cross

2026-09-22T12:00:00Z

Two straight lines, ax + by = c and dx + ey = f, meet at one point unless they are parallel. Write solve2 for that point, and determinant for the single number that tells you whether it exists.

Forward Euler loses the planet

2026-09-21T12:00:00Z

Ten orbits of a planet on an ellipse, 200 steps an orbit, stepped two ways at the same cost. Forward Euler gains energy until the planet is unbound. Leapfrog's energy error stays under one percent, and it keeps the angular momentum to rounding.

The Gibbs overshoot never shrinks

2026-09-21T12:00:00Z

Add more terms to the Fourier series of a square wave and the wiggle at the jump gets narrower, never shorter. It overshoots by 8.95% of the jump however many terms you take, and the number comes from an integral.

The logistic map at r = 4 has Lyapunov exponent ln 2

2026-09-21T12:00:00Z

Iterate x → r x(1 - x). At r = 3.2 it settles on two values you can write down. At r = 4 it never settles, nearby starts separate by a factor of two per step on average, and that factor is exact: the Lyapunov exponent is ln 2.

Which root does Newton's method find?

2026-09-21T12:00:00Z

Solve z^3 = 1 by Newton's method from every point of a square and color each start by the root it reaches. Near a root the answer is obvious. Along the lines between them it is a fractal: every boundary point touches all three basins.

How far a finite-difference Hamiltonian's levels sit below n + 1/2

2026-09-21T12:00:00Z

Put H = -1/2 ∂_x^2 + 1/2 x^2 on a grid and diagonalize: the levels come out at n + 1/2, a little low. How low is predictable. First-order perturbation theory gives -(h^2)/32(2n^2 + 2n + 1), and for the six lowest levels the measured shift matches it to within half a percent (the test allows 4%).

Pi from escape counts at the cusp of the Mandelbrot set

2026-09-21T12:00:00Z

Start just right of the cusp, at c = 1/4 + ε, and count the steps of z → z^2 + c until z > 2. Multiply the count by sqrt(ε). It tends to π.

The Runge phenomenon, where more points give a worse fit

2026-09-21T12:00:00Z

Interpolate 1/(1 + 25x^2) by a polynomial through equally spaced points and the error grows as you add points. Move the same number of points to cos(jπ/n) and it falls geometrically, at a rate you can work out from where the function's poles are.

The eigenvalues of a random matrix fill Wigner's semicircle

2026-09-21T12:00:00Z

Fill a symmetric 200 × 200 matrix with independent random numbers, divide by sqrt(N), and look at its eigenvalues. They do not scatter. They fill the semicircle 1/(2π)sqrt(4 - x^2), whatever distribution the entries came from.