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

27aabd1e-20d9-4546-b780-3d6a5d7e2b04 23 4 5 decimal

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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 :

f a b n ∫kernel

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, Y^X 
Integration accuracy from user command line 'n' ... NUMERIC.
The integration variate is in Romberg [0,1] within 'n' ... 'n' independent of work sheet option. 
Thus, on long range of the integration variate ∫kernel(f,a,b,n) improves the convergence. 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 :

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Frechet extrema value -II
fit to not given data set.
Mathcad fails plotting !!! x Frechet α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

x sub α 2 el x α 4 el - (/ :

x c 10 x ≤ (x end ≤ (& cases :

end 12 :

Fréchet: extreme value-II x f α 1 el x sub α 5 el^* x sub α 3 el^- exp * α 6 el + :

4 appVersion 0.98.6179.21440 ≡

agnitude > 10^307
"result above max"
... NOT TRUE !!! x f x 012 int #

x canvas f 0 x 12 ∫kernel :

x f x c * x canvas x c * 2 1 sys

12 canvas 29808

f 01212 ∫kernel 29808

Saddly enough . . .

Mathcad 11 fails to plot Frechet !

Forcibly => integrate !

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x sub α 2 el x α 4 el - (/ :

Fréchet: extreme value-II
Recast the system ... x f α 1 el x sub α 5 el^* x sub α 3 el^- exp * α 6 el + :

x Ω_1 α 5 el x sub ln * exp :

x Ω_2 α 3 el x sub ln * exp - exp :

x Fr α 1 el x Ω_1 * x Ω_2 * α 6 el + :

Smath integrator
fails miserably !
Sanity what you thread ! x Fr x 012 int 97644165264443 i * +

x f x c * x Fr 2 1 sys

x f x Frechet :

native Frechet f 012120 ∫kernel 29808

x f x Fr :

recast Frechet f 012120 ∫kernel 29808

x y D x f :

init 0 :

sol init 0 end 120 x y D dn_GearsBDF :

sol sol rows row 122980412 mat

t0 1 time :

x f x c * sol 2 1 sys

1 time t0 - 1 s *

x Smath x Frechet :

Symbolic
OPTIMIZ NONE x Smath 1523043253 - x 1730274 / - / (25388310 / -^* 43253 - x 1730274 / - / (27354710 / -^- exp * 400 +

4 Smath 6000

x f x Smath :

native Frechet f 012120 ∫kernel 29808

x Frechet x c * x Smath 2 1 sys

magnitude > 10^307
"result above max"
... NOT TRUE !!! x Smath x 012 int #

∫Adaptive(f,a,b,N,ε) Carlos

f a b N ε ∫Adaptive sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 41 line eval :

ADAPTIVE RECURSIVE ... Carlos

Adaptive recursive quadrature based on the Simpson Rule for the calculation of integrals

f a b N ε Io Recursive N 0 ≡ Io Number of Iterations exceeded : c a b + (2 / : I1 c a - 6 / a f 4 a c + 2 / f * + c f + (* : I2 b c - 6 / c f 4 c b + 2 / f * + b f + (* : Io I1 - I2 - abs ε > Io f a c N 1 - ε 2 / I1 Recursive f c b N 1 - ε 2 / I2 Recursive + : Io I1 I2 + : if 41 line if Io 21 line :

f a b N ε ∫AdapRecursive Io b a - 6 / a f 4 a b + 2 / f * + b f + (* : f a b N ε Io Recursive eval 21 line eval :

x f x Smath :

N 5 :

TOL 30 :

search best 'N'recursive

adapt TOL wrt project

x ∫Frechet f 0 x N TOL ∫AdapRecursive :

timing canvas t0 1 time :

canvas = 160 pixels solutions

x f x c * x ∫Frechet x c * 2 1 sys

1 time t0 - 31 s *

data X 0121 range : i 1 X rows range F i el X i el ∫Frechet : for X F augment 31 line :

data transpose 0123456789101112047018595726114941711121432243882633927642285562924529807213 mat

A more convivial Frechet !

]]> Minimize Conjugate-Gradient

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 :

init iter Minimize XY f φ β LS X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx-> raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line : β 1 el init : i 1 : i iter ≤ β i 1 + el XY f φ β i el LS : i i 1 + : 21 line while β β iter el : 41 line β 31 line :

Vector Gradient ... Optimiz Symbolic
seemingly a scalar kernel  not 
recognized in program loop x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

corr Pearson correlation χ X Mean : μ Y Mean : n XY rows : 13 mat X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el β f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el β f eval μ - (2^(i 1 n sum * sqrt / / 31 line :

correlation between [X,Y data]

correlation ... [X,fitted model]

Absolute  residuals observer
fetches β from calculated δ j 1 XY rows range Δ j el Y j el X j el β f - eval : for X Δ augment 21 line :

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 XY rows sum :

The Fréchet (“Fr” for short) distribution is one of the important distributions in extreme value theory and it has applications ranging from accelerated life testing through floods, earthquakes, horse racing, rainfall, queues in supermarkets, wind speeds and sea waves.

XY 1 0.4897 2 1.3079 3 2.84 4 3.0364 5 1.7709 6 0.5491 7 0.4606 8 0.3345 9 0.2073 10 0.0012 11 0.0024 112 mat :

X XY 1 col : Y XY 2 col : 12 mat Data XY . 10 black plotG : 21 line

x β f β 1 el β 3 el x β 4 el - / (β 2 el 1 +^* β 3 el x β 4 el - / (β 2 el^- exp * :

x β φ x β f 4 φ :

init 10573 - 14 mat transpose :

iter 3 :

algo solving for β β init iter Minimize :

β 8.5418 40.7318 52.0514 48.7382 - 41 mat

assign/isolate .. β β :

SSD 0.472

corr 1.069

Absolute residuals Residuals i 1 XY rows range Δ i el Y i el X i el β f - eval : for X Δ augment 21 line :

FIT range, populate U 011 0.05 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 41 line :

Residuals 0.1 BarSize 0 2 1 sys

β 8.5418 40.7318 52.0514 48.7382 - 41 mat :

XY 0.1 BarSize FIT 2 1 sys

Cumulative discrete integration
wants β's isolated near the module Σ u 011 0.1 range : i 1 u rows range Cumul i el x β f x u 1 el u i el int eval : for u Cumul augment 31 line :

Σ Σ :

Σ FIT 2 1 sys

cumul ∫f(z) @ point Σ Σ rows row 11 10.9 12 mat

The bug(s) and the fix . . .

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x q 43253 - x 1730274 / - / (27354710 / -^- exp :

x g 43253 - x 1730274 / - / (25388310 / -^:

x f 15230 x q * x g * 400 + :

1 x f *

integrator unfit integrate. x f x 012 int #

Σ U 012 0.1 range : i 1 U rows range Cumul i el x f x U 1 el U i el int eval : for Σ U Cumul augment (: 31 line :

... NOT TRUE !!! Σ #

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

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 :

f a b n ∫kernel

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, Y^X 
Integration accuracy from user command line 'n' ... NUMERIC.
The integration variate is in Romberg [0,1] within 'n' ... 'n' independent of work sheet option. 
Thus, on long range of the integration variate ∫kernel(f,a,b,n) improves the convergence. 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 :

Σ U 012 0.1 range : i 1 U rows range Cumul i el f 0 U i el 10 ∫kernel : for Σ U Cumul augment (: 31 line :

Integrator error message(s)

result above max
'x' not defined
...............
spline(x)... doctored
∫kernel(,,,,)doctored x f 15230 x q * x g * 400 + (≡ x f 15230 x q * x g * (≡ x spline f a b n ∫kernel 41 sys

x f x c * Σ 2 1 sys

Σ Σ rows row 122980712 mat

Doctor Frechet integration from the onset as given u 012 0.1 range : data u u Frechet vectorize eval augment : x spline data 1 col data 2 col x cinterp : 31 sys

x spline x 012 int 29808

> 10^307
NOT TRUE
simply !!! x f x 012 int #