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Oakfield Operator Calculus Function Reference Site

Mathematics Examples

CLI Examples

Section titled “CLI Examples”

1) Calculate harmonic series to the billionth term

Section titled “1) Calculate harmonic series to the billionth term”

Uses the digamma function relation

Hn=ψ(n+1)+γH_n = \psi(n + 1) + \gamma

where ψ\psi is the digamma function and γ≈0.5772156649015329\gamma \approx 0.5772156649015329 is the Euler-Mascheroni constant.

local sim = require("libsimcore")
local ctx = sim.sim_create()
local steps = 1000000000
local dt = 0.02


sim.log("Starting simulation with %d steps and dt = %.3f", steps, dt)


local t0 = os.clock()


for n = 1, steps do
    local Hn = sim.digamma(n + 1) + 0.5772156649015329


    if n % 1000000000 == 0 then
        sim.log("H_%d ≈ %.15f", n, Hn)
    end
end


local elapsed = os.clock() - t0


sim.log("Simulation completed in %.15f seconds", elapsed)
sim.sim_shutdown(ctx)


return ctx

Example output:

Terminal window
oak@field % ./bin/sim_cli --script harmonic-series.lua
[INFO] Starting simulation with 1000000000 steps and dt = 0.020
[INFO] H_1000000000 ≈ 21.300481502347946
[INFO] Simulation completed in 28.414504999993369 seconds

2) Jackson q-number ladder

Section titled “2) Jackson q-number ladder”

Walks the Jackson qq-number sequence

[n]q=1−qn1−q[n]_q = \frac{1 - q^{n}}{1 - q}

side-by-side for a fast-decaying deformation (q=0.60q=0.60) and a near-classical case (q=0.95q=0.95).

local sim = require("libsimcore")
local ctx = sim.sim_create()


local q_fast = 0.60
local q_slow = 0.95
local n_max = 12


sim.log("Jackson q-number ladder up to n=%d (q=%.2f vs q=%.2f)", n_max, q_fast, q_slow)


for n = 1, n_max do
    local fast = sim.qnumber(n, q_fast)
    local slow = sim.qnumber(n, q_slow)
    sim.log("n=%02d  [n]_%.2f=%.8f  [n]_%.2f=%.8f", n, q_fast, fast, q_slow, slow)
end


sim.sim_shutdown(ctx)


return ctx

Example output:

Terminal window
oak@field % ./bin/sim_cli --script q-number-ladder.lua
[INFO] Jackson q-number ladder up to n=12 (q=0.60 vs q=0.95)
[INFO] n=01  [n]_0.60=1.00000000  [n]_0.95=1.00000000
[INFO] n=02  [n]_0.60=1.60000000  [n]_0.95=1.95000000
[INFO] n=03  [n]_0.60=1.96000000  [n]_0.95=2.85250000
[INFO] n=04  [n]_0.60=2.17600000  [n]_0.95=3.70987500
[INFO] n=05  [n]_0.60=2.30560000  [n]_0.95=4.52438125
[INFO] n=06  [n]_0.60=2.38336000  [n]_0.95=5.29816219
[INFO] n=07  [n]_0.60=2.43001600  [n]_0.95=6.03325408
[INFO] n=08  [n]_0.60=2.45800960  [n]_0.95=6.73159137
[INFO] n=09  [n]_0.60=2.47480576  [n]_0.95=7.39501181
[INFO] n=10  [n]_0.60=2.48488346  [n]_0.95=8.02526122
[INFO] n=11  [n]_0.60=2.49093007  [n]_0.95=8.62399815
[INFO] n=12  [n]_0.60=2.49455804  [n]_0.95=9.19279825

3) q-Hurwitz zeta spectrum

Section titled “3) q-Hurwitz zeta spectrum”

Samples ζq(s,a)\zeta_q(s, a) across increasing qq to show how the deformation pushes the series upward for a fixed (s,a)(s, a) pair.

local sim = require("libsimcore")
local ctx = sim.sim_create()


local s = 2.50
local a = 0.75
local q_values = {0.10, 0.30, 0.50, 0.70, 0.90}


sim.log("q-Hurwitz zeta sweep for s=%.2f, a=%.2f", s, a)


for _, q in ipairs(q_values) do
    local z = sim.qzeta(s, a, q)
    sim.log("q=%.2f  zeta_q=%.10f", q, z)
end


sim.sim_shutdown(ctx)


return ctx

Example output:

Terminal window
oak@field % ./bin/sim_cli --script qzeta-spectrum.lua
[INFO] q-Hurwitz zeta sweep for s=2.50, a=0.75
[INFO] q=0.10  zeta_q=0.0959811147
[INFO] q=0.30  zeta_q=0.4157875357
[INFO] q=0.50  zeta_q=0.8654166800
[INFO] q=0.70  zeta_q=1.4323668003
[INFO] q=0.90  zeta_q=2.1111821796

4) q-hyperexponential staircase

Section titled “4) q-hyperexponential staircase”

Accumulates the qq-deformed hyperexponential sum

ϕq=∑k=0K−1λλ+εqk\phi_q = \sum_{k=0}^{K-1} \frac{\lambda}{\lambda + \varepsilon q^{k}}

and its derivative ∂λϕq\partial_{\lambda}\phi_q for a few KK values to illustrate how additional terms reshape both the value and slope.

local sim = require("libsimcore")
local ctx = sim.sim_create()


local lambda = 1.10
local epsilon = 0.60
local q = 0.82
local steps = {3, 5, 7, 9, 11}


sim.log("q-hyperexponential ladder with lambda=%.2f, epsilon=%.2f, q=%.2f", lambda, epsilon, q)


for _, K in ipairs(steps) do
    local phi = sim.qhyperexp_phi(lambda, epsilon, K, q)
    local dphi = sim.qhyperexp_phi_deriv(lambda, epsilon, K, q)
    sim.log("K=%02d  phi=%.8f  dphi=%.8f", K, phi, dphi)
end


sim.sim_shutdown(ctx)


return ctx

Example output:

Terminal window
oak@field % ./bin/sim_cli --script q-hyperexp-staircase.lua
[INFO] q-hyperexponential ladder with lambda=1.10, epsilon=0.60, q=0.82
[INFO] K=03  phi=2.06966900  dphi=0.58022341
[INFO] K=05  phi=3.64063279  dphi=0.88608059
[INFO] K=07  phi=5.33019012  dphi=1.12418797
[INFO] K=09  phi=7.11017829  dphi=1.30202556
[INFO] K=11  phi=8.95666551  dphi=1.43078130

5) Trigamma curvature ladder

Section titled “5) Trigamma curvature ladder”

Samples the trigamma (first polygamma) ψ1(z)\psi_1(z) across half-steps to highlight how curvature decays as zz grows.

local sim = require("libsimcore")
local ctx = sim.sim_create()


local z_values = {0.5, 1.0, 1.5, 2.0, 3.5, 5.0}


sim.log("Trigamma curvature samples psi1(z) for select z values")


for _, z in ipairs(z_values) do
    local psi1 = sim.trigamma(z)
    sim.log("z=%3.1f  psi1=%.12f", z, psi1)
end


sim.sim_shutdown(ctx)


return ctx

Example output:

Terminal window
oak@field % ./bin/sim_cli --script trigamma-curvature.lua
[INFO] Trigamma curvature samples psi1(z) for select z values
[INFO] z=0.5  psi1=4.934797200545
[INFO] z=1.0  psi1=1.644929066861
[INFO] z=1.5  psi1=0.934797200570
[INFO] z=2.0  psi1=0.644929066886
[INFO] z=3.5  psi1=0.330352756175
[INFO] z=5.0  psi1=0.221317955850

6) q-digamma convergence to classical digamma

Section titled “6) q-digamma convergence to classical digamma”

Shows how the qq-digamma ψq(z)\psi_q(z) approaches the classical digamma ψ(z)\psi(z) as q→1q \to 1 by logging the delta.

local sim = require("libsimcore")
local ctx = sim.sim_create()


local z = 5
local q_values = {0.50, 0.70, 0.90, 0.97, 0.995}


local psi_classic = sim.digamma(z)
sim.log("q-digamma convergence toward digamma at z=%d (psi=%.12f)", z, psi_classic)


for _, q in ipairs(q_values) do
    local psi_q = sim.qdigamma(z, q)
    local delta = psi_q - psi_classic
    sim.log("q=%.3f  psi_q=%.12f  delta=%.12f", q, psi_q, delta)
end


sim.sim_shutdown(ctx)


return ctx

Example output:

Terminal window
oak@field % ./bin/sim_cli --script qdigamma-limit.lua
[INFO] q-digamma convergence toward digamma at z=5 (psi=1.506117668432)
[INFO] q=0.500  psi_q=0.648898044222  delta=-0.857219624210
[INFO] q=0.700  psi_q=0.981376082021  delta=-0.524741586411
[INFO] q=0.900  psi_q=1.330585006503  delta=-0.175532661929
[INFO] q=0.970  psi_q=1.453554858225  delta=-0.052562810207
[INFO] q=0.995  psi_q=1.497365785717  delta=-0.008751882715

7) Hyperexponential gain sweep

Section titled “7) Hyperexponential gain sweep”

Sweeps λ\lambda while holding ε\varepsilon and KK fixed to show how the hyperexponential sum

ϕ=∑k=0K−1λλ+ε+k\phi = \sum_{k=0}^{K-1} \frac{\lambda}{\lambda + \varepsilon + k}

and its slope ∂λϕ\partial_{\lambda}\phi change with signal level.

local sim = require("libsimcore")
local ctx = sim.sim_create()


local epsilon = 0.40
local K = 10
local lambdas = {0.60, 1.00, 1.80, 3.00}


sim.log("Hyperexponential gain sweep (epsilon=%.2f, K=%d)", epsilon, K)
sim.log("lambda  phi(lambda)    dphi/dlambda   slope_ratio")


for _, lambda in ipairs(lambdas) do
    local phi = sim.hyperexp_phi(lambda, epsilon, K)
    local dphi = sim.hyperexp_phi_deriv(lambda, epsilon, K)
    local ratio = dphi / phi
    sim.log("%5.2f  %11.8f  %12.8f  %11.8f", lambda, phi, dphi, ratio)
end


sim.sim_shutdown(ctx)


return ctx

Example output:

Terminal window
oak@field % ./bin/sim_cli --script hyperexp-gain-sweep.lua
[INFO] Hyperexponential gain sweep (epsilon=0.40, K=10)
[INFO] lambda  phi(lambda)    dphi/dlambda   slope_ratio
[INFO]  0.60    1.75738095     1.99910762     1.13754938
[INFO]  1.00    2.45049752     1.51681989     0.61898446
[INFO]  1.80    3.44807892     1.03807316     0.30105841
[INFO]  3.00    4.46372879     0.69572733     0.15586237