<p>Int(f,a,b,n) ... Simpson/Romberg Kernel </p>

e728f743-5b05-4f6e-8d29-85149396e315 29 4 5 decimal

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Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline f a b n Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : α 0 : β 1 : h β α - n / : 14 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

Linear integration of discontinuous functions

x f 0 x π 2 / ≤ 0 x π ≥ (x 3 π * 2 / ≤ (& x sin cases :

x f

u f u 04 π * int 0.084

Fnct x 03 π * (0.05 range : i 1 x rows range u i el x i el : Fnct i el u i el f eval : 21 line for u Fnct augment 31 line :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + eval : 21 line for vx Y augment 61 line :

Fnct 1 1 sys

Discontinuous functions must be specified completely. The Smath numerical integrator supports "if/otherwise" as well the finite differences integrator.

Finite differences Cumulative Integral

Fnct Σ Fnct 2 1 sys

Inaccurate from native Simpson
set option integral acc 10000 f 0410000 Int 0.999908

u f u 04 int 1.004

δ 2 :

t δ UnitStep 1 t δ - sign + (2 / :

Δ δ 1 - δ 222 mat :

Delay δ 0.5 - δ 12 black 15 mat :

UnitStep ... sign = signum

x δ UnitStep Δ Delay 3 1 sys

x linearInt t δ UnitStep t 0 x int :

Unit u 0.01 4 0.025 range : i 1 u rows range u i el u i el : Ustep i el u i el δ UnitStep eval : 21 line for u Ustep augment 31 line :

Σ v 1015 -^4 0.025 range : i 1 v rows range F i el v i el linearInt : for v F augment 31 line :

Unit Σ 2 1 sys x linearInt