Gallery

Sessions from a transaction log

Per-customer sessions with state, computed one event at a time; transaction velocity and delinquency sit beside it.

The special functions under the distributions

The normal, incomplete gamma and incomplete beta functions that the distributions are computed from.

Least squares that keeps its digits

Ordinary least squares by QR, the code vipyr checks against NIST's certified regressions.

Generalized linear models

Iteratively reweighted least squares for GLMs, compared with R's glm.

A small GPT trained on one paragraph

nanoGPT's layout written in Viv, trained in the browser tab and stepped through call by call.

Options on a binomial tree

Cox-Ross-Rubinstein trees for European and American options, against the Black-Scholes formula.

The Heston model by Fourier cosine series

European prices under stochastic volatility by the COS method, and the smile they imply.

An American put by Crank-Nicolson

The early-exercise boundary from a finite-difference solve, checked against a binomial tree.

The Black-Scholes equation

The price $V(S,t)$ of an option on a stock at price $S$. It is a heat equation run backward from a known final condition — at expiry $T$ a call is worth its payoff, $\max (S-K,0)$ — with diffusion $\frac{1}{2}{\sigma }^2{S}^2$ that grows with $S$, a drift, and a decay at the interest rate. Counting time to expiry, $\tau =T-t$, turns it into an ordinary forward march.

Water by restricted Hartree-Fock, ported from PySCF

The port of PySCF's restricted Hartree-Fock that vipyr calls, run on water in the 6-31G basis.

The Boys function

The integral behind every Gaussian-basis electron repulsion integral, by series, recursion and asymptotics.

The bond length of H2 by geometry optimization

Result

H₂ Rₑ = 1.346 a₀

Checked against

HF/STO-3G equilibrium

The geometry of water by optimization

Result

r 1.870 a₀, ∠ 100.0°

Checked against

HF/STO-3G water

Hartree-Fock for the hydrogen molecule

Result

−1.11671 Eₕ

Checked against

Szabo & Ostlund

Hartree-Fock for a chain of hydrogen atoms

Result

H₈ −4.06498, H₂₀ −4.79091

Checked against

C++ interpreter, exact

Hartree-Fock with a McMurchie-Davidson integral engine

Result

−1.11671 Eₕ

Checked against

general integral engine

Hartree-Fock and MP2 for small molecules in STO-3G

Result

12 molecules + MP2

Checked against

published HF/STO-3G

Hartree-Fock for water in STO-3G

Result

−74.94208 Eₕ

Checked against

Crawford −74.9420787, ~1 µEₕ

Hartree-Fock for water in 6-31G*

Result

−76.0192 Eₕ

Checked against

6-31G*, 19 basis functions

The band structure of silicon

A plane-wave DFT code computes silicon's Kohn-Sham bands along L, Gamma and X, in the browser tab.

Atoms from hydrogen to uranium

The Viv port of dftatom: radial Kohn-Sham equations for one atom, compared with NIST's LDA tables.

Oxygen-16 from a Skyrme energy functional

Skyrme Hartree-Fock with SLy5 on a radial mesh: the binding energy, the charge radius and the densities.

Density mixers for a self-consistent field

Linear, Broyden and Pulay mixing, the library the Skyrme code iterates calcium-40 to self-consistency with.

The nuclear shell model in the sd shell

A configuration-interaction shell model with dense and Lanczos eigensolvers, for neon-20 and magnesium-24.

Nuclear level density by counting states

Result

p(40) = 37338

Checked against

exact count vs Hardy–Ramanujan

The maximum mass of a free-neutron star

Result

0.7087 M☉ maximum

Checked against

Oppenheimer–Volkoff 1939

Landau damping from the plasma dispersion function

The roots of the kinetic dielectric function give the damping rate and frequency of a Langmuir wave at each wavelength.

Debye screening from the nonlinear Poisson-Boltzmann equation

Result

self-consistent screening

Checked against

Debye length recovered

A tokamak equilibrium from the Grad-Shafranov equation

Result

relaxed equilibrium

Checked against

tokamak flux surfaces

Drift-wave turbulence in the Hasegawa-Wakatani model

Result

enstrophy + energy balance

Checked against

drift-wave turbulence

Particle trapping after Landau damping

Result

O'Neil trapping regime

Checked against

nonlinear saturation

The Spitzer-Härm conductivity of a plasma

Result

within 2% of NRL

Checked against

Spitzer–Härm 1953

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

Result

growth rate vs theory

Checked against

particle-in-cell

Landau damping in a Vlasov-Poisson solver

Result

γ matches the dispersion root

Checked against

Landau damping

The Ising model by Monte Carlo

Metropolis sampling of the two-dimensional Ising model on either side of its critical temperature.

A Lennard-Jones fluid at constant temperature

Molecular dynamics with a thermostat, ported from Allen and Tildesley's Fortran, against the fluid's equation of state.

A Lennard-Jones liquid at constant energy

Velocity Verlet molecular dynamics, ported from Allen and Tildesley: energy conservation and diffusion.

Molecular dynamics of a bead-spring polymer

Result

energy conserved ~0.5%

Checked against

R_g 11.5 → 6.0

Stiff and nonstiff ODE solvers

Dormand-Prince, Radau and BDF behind one solve call, for Robertson's stiff kinetics and batches of epidemics.

The heat equation

A rod with its ends held at zero and a half-sine of heat along it. Diffusion damps a mode $\sin (m\pi x)$ at the rate $\kappa (m\pi {)}^2$, so the exact solution is $u(x,t)=e^{-\kappa {\pi }^{2}t}\sin (\pi x)$ — one number to get right, at every point.

The heat equation in two dimensions

In two dimensions an implicit step couples every point to its four neighbors, and the matrix is no longer tridiagonal. Peaceman and Rachford's way around it: split the step in half, and be implicit in one direction at a time.

The advection equation

The simplest wave equation: whatever shape you start with slides along at speed $c$, unchanged. On a ring of length 1 it is back where it began after a time $1/c$, which makes the exact answer the initial condition itself.

The wave equation

A string fixed at both ends, plucked into its fundamental: $u=\sin (\pi x)\cos (\pi ct)$. Centered differences in both variables give the leapfrog scheme, $u^{n+1}=2u^n-u^{n-1}+{\nu }^2(u_{i+1}-2u_i+u_{i-1})$, with $\nu =c\Delta t/\Delta x$.

Burgers' equation with a shock and a rarefaction

Take the viscosity away and smooth data stops being enough: characteristics cross, the derivative blows up in finite time, and the solution continues as a shock. A finite difference of a discontinuity means nothing, so the method has to be built on the integral form instead — what flows into a cell minus what flows out.

The Euler equations of gas dynamics

Mass, momentum and energy of a compressible gas, with $p=(\gamma -1)(E-\frac{1}{2}\rho u^2)$. Sod's problem is a membrane at $x=\frac{1}{2}$ with dense, high-pressure gas on the left; at $t=0$ it bursts. Three waves come out: a rarefaction fan moving left, a contact discontinuity (density jumps, pressure and velocity do not) and a shock, both moving right.

Laplace's equation

A square plate: one edge held at 1, the other three at 0. What is the temperature at the center?

Poisson's equation

The equation of electrostatics, steady heat flow and pressure in an incompressible fluid. There is no time to march through: every unknown depends on every other, and the work is solving a large sparse linear system.

The modes of a drum

Separate variables in the wave equation and what is left is an eigenproblem: the shapes a membrane can vibrate in, and the frequencies that go with them. For a unit square fixed at its edge the answer is known, ${\lambda }_{mn}={\pi }^2(m^2+n^2)$ with $u=\sin (m\pi x)\sin (n\pi y)$.

The time-dependent Schrödinger equation

Two operators, each trivial in its own basis. The potential multiplies $\psi (x)$ by a phase; the kinetic energy multiplies $\hat{\psi }(k)$ by a phase. Neither can be applied exactly to the other's representation — but an FFT moves between them, so alternate: half a potential step, a full kinetic step, half a potential step. That is Strang splitting. Every factor is a pure phase, so the norm is conserved to rounding whatever the time step, and the error is $O(\Delta t^2)$, from the two operators failing to commute.

The Korteweg-de Vries equation

Shallow-water waves, and the equation solitons were discovered in. Nonlinearity steepens a hump; third-order dispersion spreads it; and for exactly one family of shapes the two cancel: $u=\frac{c}{2}{\mathrm{sech}}^2\left(\frac{\sqrt{c}}{2}(x-ct-x_0)\right)$. Taller solitons are narrower and faster — the speed $c$ is twice the height.

The nonlinear Schrödinger equation

Light in a fiber, a Bose-Einstein condensate, deep-water wave envelopes. Dispersion spreads a pulse; the focusing nonlinearity squeezes it. When they balance the pulse is a soliton, and the simplest is $\psi =\mathrm{sech}(x)e^{it/2}$: a shape that does not change at all, with a phase that turns at a steady rate.

Burgers' equation, viscous

The smallest equation with both of a fluid's habits in it: $u{u}_x$ steepens a wave, and $\nu u_{xx}$ smooths it. Where they balance the wave stops changing shape and simply travels: $u=c-a\tanh \left(\frac{a(x-ct-x_0)}{2\nu }\right)$, a smoothed step from $c+a$ down to $c-a$, of width about $4\nu /a$, moving at the mean of the two states.

The Fisher-KPP equation

A population that grows logistically and spreads by diffusion: a gene through a species, a flame, an invading plant. Behind the front $u=1$, ahead of it $u=0$, and the front advances at a constant speed.

The Poisson equation by finite elements

Result

error ratio → 4

Checked against

O(h²), the P1 rate

The same finite-element solver written with records

Result

identical to the above

Checked against

records are zero-cost

The Poisson equation solved spectrally

Result

error 1.7e-16

Checked against

exact solution, spectral, to rounding

Randomized trace estimation and low-rank approximation

Hutch++ for the trace of a matrix seen only through products, and randomly pivoted Cholesky for a kernel matrix.

Thurber's electron mobility fitted to NIST's certified values

MINPACK's Levenberg-Marquardt, ported to Viv, fits a rational model to NIST's Thurber data.

Which digits of a result are real

Cancellation, Kahan summation and a Hilbert solve, the programs vipyr runs under the digits lens.

The exact derivative as a compiler transform

Result

f′(2) = 9.926737

Checked against

AD as an IR transform

Forward-mode differentiation with dual numbers

Result

f′(2) = 9.926737

Checked against

dual numbers, exact

Reverse-mode differentiation on a tape

Result

∇g matches forward

Checked against

one backward pass

Identifying a function through the Meijer G-function

Result

log ✓ rational ✓ arctan ✗

Checked against

identification by snapping

A small computer-algebra system on flat arrays

Result

11 asserts pass

Checked against

symbolic diff, simplify to fixpoint

Finding a symmetry with a neural network

Result

exact after the substitution

Checked against

the oracle finds x−y

Recovering a formula from data

Result

4 / 4 recovered exactly

Checked against

AI-Feynman inner loop

Balancing a cart-pole with LQR

The Riccati equation gives the feedback gains, and the simulated pole comes back upright from 25 degrees.

Lyapunov equations for the cart-pole's feedback gains

The Lyapunov solver behind vipyr's stability map of the cart-pole, on the Schur form.

A four-bit adder from NAND gates

Gates built from NAND, a full adder from gates, and four of them in a row, checked on every input.

Fibonacci on a RISC-V processor

A C program compiled, assembled and run on an RV32IM simulator, each step written in Viv.

AC power flow on MATPOWER cases

Newton-Raphson and fast-decoupled power flow with a sparse LU, from the 9-bus case to 9241 buses.

An interconnection study

A new generator added to the IEEE 118-bus system, with every single-line outage checked.

A PV curve by continuation power flow

The voltage at a bus as the transfer into it grows, traced around the nose of the curve.