<p> ∫kernel(f,a,b,n) ... Simpson/Romberg Kernel [Jean] </p>

a3ae3110-2a4c-4330-be2f-56217124feae 11 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Premise algo companion . . . ∫kernel

]]> <p> ∫kernel(f,a,b,n) ... Simpson/Romberg Kernel [Jean] </p>

Basic Romberg-Simpson ... integration acc from options ... NUMERIC f a b Romberg w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

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 
Integration accuracy from user command line 'n' ... NUMERIC f a b n ∫kernel 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 / : 15 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 :

x f x ln :

f a b n ∫kernel

lo 1012 -^:

t x Ω x ln :

acc 1000 enough 
for that example
10000  ~ exact lo t part t u Ω u lo t int :

x ∫ t x Ω x int maple : x 1 - x ln + (*

t u Ω u ln

x ∫ x 1 - x ln + (* :

x λ 1 lo x ≤ (x 4 ≤ (& cases :

f lo x 100 ∫kernel x λ * x ∫ 2 1 sys

Native ∫ vs kernel

n 1000 :

native  ∫  100 4 ∫ lo 4 part - 0.1413

x Δ x ∫ f lo x n ∫kernel - :

4 Δ 0.000000924454 -

lo 1012 -^: hi 4 : n 100 : 13 mat

Premise ∫ 0 : t lo hi lo 0.1 + range : i 1 t rows range ∫ i el f lo t i el n ∫kernel : for t ∫ augment 41 line :

Premise x ∫ 2 1 sys

data 00 0.025 0.025 0.05 0.05 0.075 0.0749 0.1 0.0998 0.125 0.1247 0.15 0.1494 0.175 0.1741 0.2 0.1987 0.225 0.2231 0.25 0.2474 0.275 0.2715 0.3 0.2955 0.325 0.3193 0.35 0.3429 0.375 0.3663 0.4 0.3894 0.425 0.4123 0.45 0.435 0.475 0.4573 0.5 0.4794 0.525 0.5012 0.55 0.5227 0.575 0.5438 0.6 0.5646 0.625 0.5851 0.65 0.6052 0.675 0.6249 0.7 0.6442 0.725 0.6631 0.75 0.6816 0.775 0.6997 0.8 0.7174 0.825 0.7345 0.85 0.7513 0.875 0.7675 0.9 0.7833 0.925 0.7986 0.95 0.8134 0.975 0.8277 1 0.8415 1.025 0.8547 1.05 0.8674 1.075 0.8796 1.1 0.8912 1.125 0.9023 1.15 0.9128 1.175 0.9227 1.2 0.932 1.225 0.9408 1.25 0.949 1.275 0.9566 1.3 0.9636 1.325 0.9699 1.35 0.9757 1.375 0.9809 1.4 0.9854 1.425 0.9894 1.45 0.9927 1.475 0.9954 1.5 0.9975 1.525 0.999 1.55 0.9998 1.575 1 1.6 0.9996 1.625 0.9985 1.65 0.9969 1.675 0.9946 1.7 0.9917 1.725 0.9881 1.75 0.984 1.775 0.9792 1.8 0.9738 1.825 0.9679 1.85 0.9613 1.875 0.9541 1.9 0.9463 1.925 0.9379 1.95 0.929 1.975 0.9194 2 0.9093 2.025 0.8986 2.05 0.8874 2.075 0.8756 2.1 0.8632 2.125 0.8503 2.15 0.8369 2.175 0.823 2.2 0.8085 2.225 0.7935 2.25 0.7781 2.275 0.7621 2.3 0.7457 2.325 0.7288 2.35 0.7115 2.375 0.6937 2.4 0.6755 2.425 0.6568 2.45 0.6378 2.475 0.6183 2.5 0.5985 2.525 0.5783 2.55 0.5577 2.575 0.5368 2.6 0.5155 2.625 0.4939 2.65 0.472 2.675 0.4498 2.7 0.4274 2.725 0.4046 2.75 0.3817 2.775 0.3584 2.8 0.335 2.825 0.3113 2.85 0.2875 2.875 0.2634 2.9 0.2392 2.925 0.2149 2.95 0.1904 2.975 0.1658 3 0.1411 3.025 0.1163 3.05 0.0915 3.075 0.0665 3.1 0.0416 3.125 0.0166 1262 mat :

Data set from foreign source

data rows 126

t0 1 time :

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 * (* + : 21 line for vx Y augment 61 line :

Finite differences Cumulative Integral

data Σ data x sin 3 1 sys

1 time t0 - 0.52 s *

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 + el Y i el - X i 1 + el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X 1 deriv * augment 101 line :

data Deriv x cos 2 1 sys data Deriv Σ x sin 2 1 sys