plot(data,a,b,c)

a4ae6c4e-c716-4aad-b62d-fd7bb18879d8 40 4 5 decimal

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

plot(data,a,b,c)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Hermite Polynomials ... roots 6 decimals ±

Hermite polynomial form ... expand in vector function n x H 1 n 0 ≡ 2 x * n 1 ≡ 1 - (k^n! (* k! (n 2 k * - (! (* / 2 x * (n 2 k * -^* k 0 n 2 / Floor sum cases 11 line :

j 05 range :

j x H vectorize 12 x * 21 - 2 x 2^* + (* 4 x * 3 - 2 x 2^* + (* 434 x 2^* 3 - x 2^+ (* + (* 8 x * 154 x 2^* 5 - x 2^+ (* + (* 61 mat

0 x H 1 x H 2 x H 3 x H 4 x H 5 x H 6 1 sys

m 5 :

m x H Hermite roots [20] for Wr

Hermite roots [20] r 0.245340708300893 - 0.245340708300893 0.737473728545395 - 0.737473728545395 1.23407621539532 - 1.23407621539532 1.73853771211659 - 1.73853771211659 2.25497400208926 - 2.25497400208926 2.78880605842768 - 2.78880605842768 3.34785456738297 - 3.34785456738297 3.94476404011613 - 3.94476404011613 4.60368244955075 - 4.60368244955075 5.38748089001125 - 5.38748089001125 201 mat :

m r rows : 20

i 1 r rows range W i el 2 m 1 +^m! (* π sqrt * m 1 + r i el H 2^/ : for

W W eval : Wei W r 2^exp * vectorize eval : Wr r W Wei augment : 31 sys

sanity Abramowitz [20]

..... roots ..... W ....... Wei ... Wr 0.2453407083009 - 0.4622436696006 0.4909215006667 0.2453407083009 0.4622436696006 0.4909215006667 0.7374737285454 - 0.2866755053628 0.493843385272 0.7374737285454 0.2866755053628 0.493843385272 1.2340762153953 - 0.10901720602 0.4999208713362 1.2340762153953 0.10901720602 0.4999208713362 1.7385377121166 - 0.0248105208874 0.5096790271158 1.7385377121166 0.0248105208874 0.5096790271158 2.2549740020893 - 0.0032437733422 0.5240803509482 2.2549740020893 0.0032437733422 0.5240803509482 2.7888060584277 - 0.000228338636 0.5448517423528 2.7888060584277 0.000228338636 0.5448517423528 3.347854567383 - 0.0000078025565 0.5752624427386 3.347854567383 0.0000078025565 0.5752624427386 3.9447640401161 - 0.0000001086069 0.6222786957531 3.9447640401161 0.0000001086069 0.6222786957531 4.6036824495507 - 0.0000000004399 0.7043328023144 4.6036824495507 0.0000000004399 0.7043328023144 5.3874808900112 - 0.0000000000002 0.8988363621001 5.3874808900112 0.0000000000002 0.8988363621001 203 mat

e x 2^-^x f * x ∞ - ∞ int W i el r i el f * i 0 n sum ≡

x g x ∞ - ∞ int W i el e r i el 2^^* r i el g * i 0 n sum ≡

plot(data,a,b,c)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Hermite polynomial form ... expand in vector function n x H 1 n 0 ≡ 2 x * n 1 ≡ 1 - (k^n! (* k! (n 2 k * - (! (* / 2 x * (n 2 k * -^* k 0 n 2 / Floor sum cases 11 line :

Hermite explorer ... Export

j 9 : n 0 j range H n 1 + el n x H : for 21 line

select/collect H 3 el 21 - 2 x 2^* + (*

eval SYMBOLIC view all x H H :

n 6 :

x N n x H x expand maple : x N x N op maple : x 1 : v x N reverse : 12 mat 31 line

i 1 v rows range n 2 mod 0 ≡ Her 2 i * 1 - el v i el : Her 2 i * el v i el : if for

'n' even

'n' odd

r Her polyroots : r r eval : 21 sys

Plot roots of Lobatto roots r r 0 * augment : roots roots . 14 black plot : 21 line

roots n x H 2 1 sys

Her 120 - 07200480 - 06471 mat

r 0.436077411928 - 0.436077411928 1.335849074014 - 1.335849074014 2.350604973674 - 2.350604973674 61 mat

Hermite polyroots . . . suite ▼

]]> [r,W,Wei] ... 20

Wr 0.2453407083009 - 0.4622436696006 0.4909215006667 0.2453407083009 0.4622436696006 0.4909215006667 0.7374737285454 - 0.2866755053628 0.493843385272 0.7374737285454 0.2866755053628 0.493843385272 1.2340762153953 - 0.10901720602 0.4999208713362 1.2340762153953 0.10901720602 0.4999208713362 1.7385377121166 - 0.0248105208874 0.5096790271158 1.7385377121166 0.0248105208874 0.5096790271158 2.2549740020893 - 0.0032437733422 0.5240803509482 2.2549740020893 0.0032437733422 0.5240803509482 2.7888060584277 - 0.000228338636 0.5448517423528 2.7888060584277 0.000228338636 0.5448517423528 3.347854567383 - 0.0000078025565 0.5752624427386 3.347854567383 0.0000078025565 0.5752624427386 3.9447640401161 - 0.0000001086069 0.6222786957531 3.9447640401161 0.0000001086069 0.6222786957531 4.6036824495507 - 0.0000000004399 0.7043328023144 4.6036824495507 0.0000000004399 0.7043328023144 5.3874808900112 - 0.0000000000002 0.8988363621001 5.3874808900112 0.0000000000002 0.8988363621001 203 mat :

x Φ x 2^- exp x * (:

Sanity Quadrature ...

x Φ

limit ± immaterial
integrand is symmetric
! ... not displaying '0' x Φ x 1 - 1 int 41017 -^* -

e x 2^-^x f * x ∞ - ∞ int W i el r i el f * i 0 n sum ≡

r Wr 1 col : W Wr 2 col : 21 sys

exact sanity
Mathcad 11 W j el r j el Φ * j 12 sum 0.000000000000000

10 Φ 3.7201 1043 -^*

Ordinary Simpson can't integrate ...
the range of integration is much 
too long for the default   ∫impson
The [0 ... 10] is not true either.
∫kernel(f,0,1000,5000)verdicts the dispute x Φ x 010 int x Φ x 01000 int 21 mat 0.500000208956 4.62931 1010 -^* 21 mat

∫kernel(f,a,b,n)) ... Simpson/Romberg Kernel

Basic Romberg-Simpson ... integration acc from options f a b 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^- (* : 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' 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 / : 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 :

x f x 2^- exp x * :

[0 ... ∞] ≡ 0.5 f 010005000 ∫kernel 0.49999999994

x f x 2^- exp x 2^* :

x f

For that example ∫kernel ~ ≡ FD" !

Declared ~ exact ! f 051000 ∫kernel 0.443113

exact sanity
Mathcad 11
but wrong . W j el r j el f * j 120 sum 0.3133

L Δ N inc X 0 : i 1 N range X i 1 + el L Δ i * + eval : X 1 el L : 21 line for X 31 line :

Δ 0 0.001 5000 inc : data Δ Δ f vectorize augment : data data eval : 31 sys

end coord's data data rows row 5 3.5 1010 -^* 12 mat

Riemann rectangles Finite Differences Arc length Finite differences derivative X 0 : Y 0 : 12 mat X data 1 col : Y data 2 col : 12 mat 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 - / + (* eval : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : Dp X deriv augment : Arc integrand .................... xx Dp 1 col : δx Dp 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for Arc xx yy augment : Arc length ....................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - eval : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + eval : 21 line for Σ vx S augment : 191 line

f(x) & derivative

Dp data 2 1 sys

Arc integrand

Arc 1 1 sys

Cumulative arc length ... curiously 'linear'

Σ 5.1306 5 / x * 2 1 sys

Σ Σ rows row 5 5.1309 12 mat

Finite differences cumulative integ'n vx data 1 col : vy data 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - eval : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + eval : 21 line for Σ vx S augment : 51 line

Area Σ Σ rows row (: 5 0.443113 12 mat

Hermite polyroots otherwise . . . adapted . . . Andrey Ivashov

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data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

x Clear 1

Степень полинома Pangkat polinomial Polynomial order for polyroots n 5 :

z n H 1 - (n^e z 2^2 /^* e z 2^2 / -^(z n diff * :

T 1 el z n H : k 1 n range T k 1 + el T k el (z diff k / : for z D T : 21 line 21 line

Коэффициенты полинома Эрмита Koefisien polinomial Hermite Hermite polyroots 
coefficients.
NOT documented 
wrt 'QUADRATURES' v 0 D : 015010 - 0161 mat

HerRoots v polyroots : rH HerRoots eval : roots rH rH 0 * augment : roots roots . 14 black plot : 41 sys

plot roots x n H 21 sys :

plot

rH 0 1.355626179974 - 1.355626179974 2.856970013873 - 2.856970013873 51 mat