<content> Legendre wr 63 ... 64 </content>

da06b488-ff54-4851-817e-5e775a8bbd4e 68 4 5 false false 0 0 decimal

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

n 7 :

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

k 1 n range vT 1 el 1 : vT 2 el x : vT k 1 + el k x T : 31 line for

Expand Symbolic ... isolate

vT 1 x 1 - 2 x 2^* + x 3 - 4 x 2^* + (* 18 x 4^* + 8 x 2^* - x 54 x 2^* 5 - 4 x 2^* + (* + (* 116 x 4^* 32 x 2^* - (* + 18 x 2^* - (- x 7 - 8 x 2^* 72 x 2^* 7 - 4 x 2^* + (* + (* + (* 81 mat

vT 1 x 1 - 2 x 2^* + x 3 - 4 x 2^* + (* 18 x 2^* 1 - x 2^+ (* + x 54 x 2^* 5 - 4 x 2^* + (* + (* 12 x 2^* 9 - 8 x 2^* 32 x 2^* - (* + (* + (- x 7 - 8 x 2^* 72 x 2^* 7 - 4 x 2^* + (* + (* + (* 132 x 2^* 1 - x 2^54 x 2^* 2 - x 2^+ (* + (* + (* + 91 mat :

in-situ command line roots n 7 :

Ω 1 el vT n el :

μ 1 n 1 - (range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

Once vT is isolated,
you are in-situ wrt 'n'
within length of \[REGION[c8a2d66593694af488a70292aa75b926]]\

'vT'

r r eval :

R i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

R n 1 - x T 2 1 sys

r 0.2588 - 0.2588 0.7071 - 0.7071 0.9659 - 0.9659 61 mat

r sort 0.9659 - 0.7071 - 0.2588 - 0.2588 0.7071 0.9659 61 mat

n 1 - r T vectorize 2.5 1013 -^* - 2.5 1013 -^* - 1.3 1013 -^* - 1.3 1013 -^* - 1.4 1014 -^* 1.4 1014 -^* 61 mat

Hermite Polynomials

k n r Ω Clear 1

Степень полинома Pangkat polinomial Polynomial order n 4 :

Hermite polynomials x n H 1 - (k^n! (* (k! (n 2 k * - (! (* / 2 x * (n 2 k * -^* (k 0 n 2 / Floor sum :

x n H x 3 H x 2 H 10 x 1 H ( * 4 1 sys

Expand x n H 434 x 4^* + 12 x 2^* - (*

Isolate Ω 1 el 434 x 4^* + 12 x 2^* - (* :

μ 1 n range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

r r eval :

Hr i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

Hr x n H 2 1 sys

r n H vectorize 3.1 1012 -^* 3.1 1012 -^* 3.3 1012 -^* 3.3 1012 -^* 41 mat

Hr 0.5246 - 0 . 0.5246 0 . 1.6507 - 0 . 1.6507 0.43 mat

+ accurate root(s)

a 0.5 : b 0.55 : xn b a + (2 / : 31 line

x f 434 x 4^* + 12 x 2^* - (* :

ε 1013 -^:

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

.. sanity ..

xn 0.52464762327529

r 2 el 0.524647623275216

xn f 1.8 1015 -^*

iterates n 42

X x f x 0.5 roots : 0.524647623275291

X f 2.7 1014 -^* -

Lagrange Iterated Product

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 :

z 2 - 1 - 012316 mat transpose :

data z z 0 * augment :

support data . 14 black plot :

x L x z j el - (j 1 z rows product :

support x L 2 1 sys

data 2 - 01 - 00010203062 mat

Expand x L 2 x + (1 x + (* x * 1 - x + (* 2 - x + (* 3 - x + (*

Isolate Ω 1 el 2 x + (1 x + (* x * 1 - x + (* 2 - x + (* 3 - x + (* :

μ 1 z rows range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

r r eval :

r sort 2 - 1 - 11015 -^* - 12361 mat

k n r Ω Clear 1

Legendre coefficients & roots

Legendre last 'n' polynomial n x L 12 n^/ 1 - (k^2 n * 2 k * - (! (* k! (n k - (! (* n 2 k * - (! (* / x n 2 k * - (^* k 0 n 2 / Floor sum * :

n 7 :

last 'n' polynomial => copy/paste in Ω1 n x L x 35 - 3 x 2^* 10511 x 2^* 21 - 13 x 2^* + (* + (* + (* 16 /

Ω 1 el x 35 - 3 x 2^* 10511 x 2^* 21 - 13 x 2^* + (* + (* + (* 16 / :

μ 1 n range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

◄ from Smath Handbook

r r eval :

R i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

R n x L 2 1 sys

R 00 . 0.4058 - 0 . 0.4058 0 . 0.7415 - 0 . 0.7415 0 . 0.9491 - 0 . 0.9491 0.73 mat

x y axes x y / - :

x plot n x L :

x plot R axes 3 1 sys

r sort 0.9491 - 0.7415 - 0.4058 - 0 0.4058 0.7415 0.9491 71 mat

x plot R axes 31 sys

Explode Legendre . . .

n Clear 1

Legendre last 'n' polynomial n x L 12 n^/ 1 - (k^2 n * 2 k * - (! (* k! (n k - (! (* n 2 k * - (! (* / x n 2 k * - (^* k 0 n 2 / Floor sum * :

n 08 range :

n x L vectorize 1 x 1 - 3 x 2^* + 2 / x 3 - 5 x 2^* + (* 2 / 335 x 4^* + 30 x 2^* - 8 / x 157 x 2^* 10 - 9 x 2^* + (* + (* 8 / 5163 x 4^* + (* - 231 x 6^* + 105 x 2^* + 16 / x 35 - 3 x 2^* 10511 x 2^* 21 - 13 x 2^* + (* + (* + (* 16 / 571287 x 8^* + 1386 x 4^* + (* - 12012 x 6^* + 1260 x 2^* + 128 / - 91 mat

Sparse Polynomial(s) ...

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 f 2 - 7 x 2^* - 2 x 3^* - x 4^+ :

r Clear 1

vector polyroots c 2 - 07 - 2 - 115 mat transpose :

r c polyroots :

x Φ x r j el - (j 1 r rows product :

x c 1 r 3 el x ≤ (x r 4 el ≤ (& cases :

Xroots r 3 el 0 augment r 4 el 0 augment stack :

Xr Xroots . 14 black plot :

r 0.0352 0.5185 i * + 0.0352 0.5185 i * - 1.9225 - 3.8522 41 mat

x f x c * x Φ Xr 3 1 sys

Xroots 1.9225 - 0 3.8522 022 mat

accurate roots(s) ...

canonical Lgrange x P 42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+ :

X1 x P x 1 - roots : 0.9429 -

X2 x P x 4 roots : 4.0624

Croots X1 0 augment X2 0 augment stack :

Xr Croots . 14 black plot :

x P Xr 2 1 sys

X1 P 0.000000000000021 -

X2 P 0.000000000000568

Circle roots(s) ...

n P x r Clear 1

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 :

r 1 :

n 12 :

x circle r 2^x 2^- sqrt ± :

roots x n^r - 0 ≡ x solve maple :

P roots Re roots Im augment r * :

roots 5 el arg ° 120.000000000000000

rootsP P . 12 black plot :

x circle rootsP r 0 + 100 green 1 5 mat x asin 6 1 sys

P 01 - 011 - 010 0.5 - 0.866 0.5 0.866 - 0.866 - 0.5 - 0.866 0.5 0.5 - 0.866 - 0.5 0.866 0.866 - 0.5 0.866 0.5 - 122 mat

x circle rootsP r 0 + 100 green 15 mat x asin 61 sys

n 12

Legendre Quadrature . . .

n r Ω Clear 1

Legendre last 'n' polynomial n x L 12 n^/ 1 - (k^2 n * 2 k * - (! (* k! (n k - (! (* n 2 k * - (! (* / x n 2 k * - (^* k 0 n 2 / Floor sum * :

can not calculate n > 11 n 11 :

n x L x 693 - 13 x 2^* 1155 x 2^6930 - 17 x 2^* 99019 x 2^* 55 - 21 x 2^* + (* + (* + (* + (* + (* 256 /

Ω 1 el x 693 - 13 x 2^* 1155 x 2^6930 - 17 x 2^* 99019 x 2^* 55 - 21 x 2^* + (* + (* + (* + (* + (* 256 / :

μ 1 n range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

r r eval :

Prefered ▼ roots r r 0 * augment : Ωr roots . 12 black plot : 21 sys

R i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

Ωr n x L 2 1 sys

Wr i 1 n range W i el 21 r i el (2^- (* n 1 + (2^n 1 + r i el L (2^* / eval : for W r augment 21 line :

Wr 0.273 0 0.263 0.27 - 0.263 0.27 0.233 0.519 - 0.233 0.519 0.186 0.73 - 0.186 0.73 0.126 0.887 - 0.126 0.887 0.056 0.978 - 0.056 0.978 112 mat

https://pomax.github.io/bezierinfo/legendre-gauss.html

<content> Legendre wr 63 ... 64 </content>

wr 0.0494723666239310 0.0000000000000000 0.0494118330399182 0.0494521871161596 - 0.0494118330399182 0.0494521871161596 0.0492303804237476 0.0987833564469453 - 0.0492303804237476 0.0987833564469453 0.0489284528205120 0.1478727863578720 - 0.0489284528205120 0.1478727863578720 0.0485067890978838 0.1966003467915067 - 0.0485067890978838 0.1966003467915067 0.0479664211379951 0.2448467932459534 - 0.0479664211379951 0.2448467932459534 0.0473086713122689 0.2924940585862514 - 0.0473086713122689 0.2924940585862514 0.0465351492453837 0.3394255419745844 - 0.0465351492453837 0.3394255419745844 0.0456477478762926 0.3855263942122479 - 0.0456477478762926 0.3855263942122479 0.0446486388259414 0.4306837987951116 - 0.0446486388259414 0.4306837987951116 0.0435402670830276 0.4747872479948044 - 0.0435402670830276 0.4747872479948044 0.0423253450208158 0.5177288132900333 - 0.0423253450208158 0.5177288132900333 0.0410068457596664 0.5594034094862850 - 0.0410068457596664 0.5594034094862850 0.0395879958915441 0.5997090518776252 - 0.0395879958915441 0.5997090518776252 0.0380722675843496 0.6385471058213654 - 0.0380722675843496 0.6385471058213654 0.0364633700854573 0.6758225281149861 - 0.0364633700854573 0.6758225281149861 0.0347652406453559 0.7114440995848458 - 0.0347652406453559 0.7114440995848458 0.0329820348837793 0.7453246483178474 - 0.0329820348837793 0.7453246483178474 0.0311181166222198 0.7773812629903724 - 0.0311181166222198 0.7773812629903724 0.0291780472082805 0.8075354957734567 - 0.0291780472082805 0.8075354957734567 0.0271665743590979 0.8357135543195029 - 0.0271665743590979 0.8357135543195029 0.0250886205533450 0.8618464823641238 - 0.0250886205533450 0.8618464823641238 0.0229492710048899 0.8858703285078534 - 0.0229492710048899 0.8858703285078534 0.0207537612580391 0.9077263027785316 - 0.0207537612580391 0.9077263027785316 0.0185074644601613 0.9273609206218432 - 0.0185074644601613 0.9273609206218432 0.0162158784103383 0.9447261340410098 - 0.0162158784103383 0.9447261340410098 0.0138846126161156 0.9597794497589419 - 0.0138846126161156 0.9597794497589419 0.0115193760768800 0.9724840346975701 - 0.0115193760768800 0.9724840346975701 0.0091259686763267 0.9828088105937273 - 0.0091259686763267 0.9828088105937273 0.0067102917659601 0.9907285468921895 - 0.0067102917659601 0.9907285468921895 0.0042785083468638 0.9962240127779701 - 0.0042785083468638 0.9962240127779701 0.0018398745955771 0.9992829840291237 - 0.0018398745955771 0.9992829840291237 632 mat :

wr 0.0486909570091397 0.0243502926634244 - 0.0486909570091397 0.0243502926634244 0.0485754674415034 0.0729931217877990 - 0.0485754674415034 0.0729931217877990 0.0483447622348030 0.1214628192961206 - 0.0483447622348030 0.1214628192961206 0.0479993885964583 0.1696444204239928 - 0.0479993885964583 0.1696444204239928 0.0475401657148303 0.2174236437400071 - 0.0475401657148303 0.2174236437400071 0.0469681828162100 0.2646871622087674 - 0.0469681828162100 0.2646871622087674 0.0462847965813144 0.3113228719902110 - 0.0462847965813144 0.3113228719902110 0.0454916279274181 0.3572201583376681 - 0.0454916279274181 0.3572201583376681 0.0445905581637566 0.4022701579639916 - 0.0445905581637566 0.4022701579639916 0.0435837245293235 0.4463660172534641 - 0.0435837245293235 0.4463660172534641 0.0424735151236536 0.4894031457070530 - 0.0424735151236536 0.4894031457070530 0.0412625632426235 0.5312794640198946 - 0.0412625632426235 0.5312794640198946 0.0399537411327203 0.5718956462026340 - 0.0399537411327203 0.5718956462026340 0.0385501531786156 0.6111553551723933 - 0.0385501531786156 0.6111553551723933 0.0370551285402400 0.6489654712546573 - 0.0370551285402400 0.6489654712546573 0.0354722132568824 0.6852363130542333 - 0.0354722132568824 0.6852363130542333 0.0338051618371416 0.7198818501716109 - 0.0338051618371416 0.7198818501716109 0.0320579283548516 0.7528199072605319 - 0.0320579283548516 0.7528199072605319 0.0302346570724025 0.7839723589433414 - 0.0302346570724025 0.7839723589433414 0.0283396726142595 0.8132653151227975 - 0.0283396726142595 0.8132653151227975 0.0263774697150547 0.8406292962525803 - 0.0263774697150547 0.8406292962525803 0.0243527025687109 0.8659993981540928 - 0.0243527025687109 0.8659993981540928 0.0222701738083833 0.8893154459951141 - 0.0222701738083833 0.8893154459951141 0.0201348231535302 0.9105221370785028 - 0.0201348231535302 0.9105221370785028 0.0179517157756973 0.9295691721319396 - 0.0179517157756973 0.9295691721319396 0.0157260304760247 0.9464113748584028 - 0.0157260304760247 0.9464113748584028 0.0134630478967186 0.9610087996520538 - 0.0134630478967186 0.9610087996520538 0.0111681394601311 0.9733268277899110 - 0.0111681394601311 0.9733268277899110 0.0088467598263639 0.9833362538846260 - 0.0088467598263639 0.9833362538846260 0.0065044579689784 0.9910133714767443 - 0.0065044579689784 0.9910133714767443 0.0041470332605625 0.9963401167719553 - 0.0041470332605625 0.9963401167719553 0.0017832807216964 0.9993050417357722 - 0.0017832807216964 0.9993050417357722 642 mat :

This page was compiled by Mike "Pomax" Kamermans, on June 5th, 2011,
as a resource for the primer on Bezier curves, specifically the
section that deals with arc length computation. You are free to do
what you like with this data. Go make the world a better place than
it was when you found it.

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

weight, wi w wr 1 col :

abscissea, xi u wr 2 col :

m wr rows : 64

x f 1 x / :

a 1 - : b 2 : 21 line

a b Quad i 1 m range y i el b a - 2 / (u i el * b a + 2 / (+ eval : for b a - (2 / w i el y i el f * (i 1 m sum eval * 21 line :

x f 1 - 2 1 - 2 - 2 2 mat 2 2 2 2 - 2 2 mat 3 1 sys

ε 1015 -^:

a b Quad 0.916522

12 Quad 0.693147180559945

2 ln 0.693147180559945

ε 2 Quad 9.487781807410380

ε Clear 1

<content> ∫Adaptive(f,a,b,N,ε) </content>

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 :

ε 1015 -^:

Infinite  limit x f 1 x / :

'N' iterations N 1000 : 11 sys

lower limitupper limit a ε : b 2 : 21 sys

Disastrous Smath Simpsonacc 10000 fuckt it ! x f x a b int 3.3333 1010^*

Ad Absurdum x ln a b at 35.231923575471

t0 1 time :

WoiframAlpha is 4 decimals f a b N ε ∫Adaptive 35.231923575472

1 time t0 - min 1.4

<content> ∫Adaptive(f,a,b,N,ε) Carlos </content>

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 :

x 0.5 1 - ^ x - exp * π sqrt x sqrt erf * 2 1 sys

x f x 0.5 1 -^x - exp * eval : N 1000 : ε 1015 -^: a 1015 -^: b e : 51 sys

f a b N ε ∫Adaptive 1.73750201148017

Ad Absurdum π sqrt x sqrt erf vectorize * a b at 1.7375020114801

x f x a b int 1434.3831

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 :

Radau Polynomials @ accuracy of solve(▪,▪,▪,▪)

Legendre  setup 6179 needs "local assignment" n x L 1 - (k^2 n * 2 k * - (! (x n 2 k * -^* (* 2 n^k! (* n k - (! (n 2 k * - (! (* (* / (k 0 n 2 / Floor sum 11 line :

n x Radau n 1 - x L n x L + x 1 + / :

n 13 :

Plot roots of Radau r n x Radau x 1 - 1 solve : r r eval : roots r r 0 * augment : roots roots . 14 black plot : 41 line

Radau ODE solver was introduced
in Mathcad 11 ... NO code found.
HIRES application ~ = Rkadapt.

n x Radau x r 4 el 1 int 0.0321

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 :

roots n x Radau 2 1 sys

r 1 - r stack (:

i 1 r rows range W i el 1 n 2^/ 1 r i el - (n 1 - r i el L (2^/ * : Wr W r augment : 21 line for

≈  Sanity MCD 11F.B. Hildebrand Wr 0.0118 1 - 0.0719 0.9569 - 0.1251 0.8579 - 0.171 0.7091 - 0.207 0.5192 - 0.2309 0.2992 - 0.2414 0.0619 - 0.2379 0.1789 0.2205 0.4092 0.1904 0.6157 0.1492 0.7863 0.0993 0.9111 0.045 0.9829 132 mat

x f x 1 - 1 int 2 n 2^/ 1 - f * W i el x i el f * i 0 n 1 - sum * ≡

Radau Wr . 12 black plot :

Wr Radau 2 1 sys

be more maths . . . .
F.B.Hildebrand [8.11] p .406

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 :

k n r Ω Clear 1

Legendre  setup 6179 needs "local assignment" n x L 1 - (k^2 n * 2 k * - (! (x n 2 k * -^* (* 2 n^k! (* n k - (! (n 2 k * - (! (* (* / (k 0 n 2 / Floor sum 11 line :

n x Radau n 1 - x L n x L + x 1 + / :

n 6 :

n x Radau 5 - x 2157 x 2^* 10 - 9 x 2^* + (* + (* 21 x * 5 x 2^15 - 11 x 2^* + (* + (* + (* + 161 x + (* /

Ω 1 el 5 - x 2157 x 2^* 10 - 9 x 2^* + (* + (* 21 x * 5 x 2^15 - 11 x 2^* + (* + (* + (* + 161 x + (* / :

μ 1 n 1 - (range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

r r sort :

r 0.8029 - 0.3909 - 0.1241 0.604 0.9204 51 mat

roots r r 0 * augment : Ωrad roots . 12 black plot : 21 sys

Ωrad n x Radau 2 1 sys

Accurate Radau roots
limited . . . n ≤ 6

n 20 :

Plot roots of Radau r n x Radau x 1 - 1.01 solve : r r eval : roots r r 0 * augment : roots roots . 12 black plot : 41 line

r 0.9817 - 0.9391 - 0.8734 - 0.7862 - 0.6796 - 0.5564 - 0.4194 - 0.2722 - 0.1182 - 0.0386 0.1945 0.3456 0.4881 0.6187 0.7339 0.8311 0.9078 0.9621 0.9928 191 mat

r rows 19

roots n x Radau 2 1 sys

r 1 - r stack (:

i 1 r rows range W i el 1 n 2^/ 1 r i el - (n 1 - r i el L (2^/ * : Wr W r augment : 21 line for

solve Radau roots
limited . . . n ≤ 27

an exact polynomial fit ▼

Legendre  setup 6179 needs "local assignment" n x L 1 - (k^2 n * 2 k * - (! (x n 2 k * -^* (* 2 n^k! (* n k - (! (n 2 k * - (! (* (* / (k 0 n 2 / Floor sum 11 line :

n x Radau n 1 - x L n x L + x 1 + / :

n 5 :

Plot roots of Radau r n x Radau x 1 - 1.01 solve : r r eval : roots r r 0 * augment : roots roots . 14 black plot : 41 line

x 0.75 - 0.25 - 0.125 0.375 0.625 161 mat :

n x Radau vectorize 0.26513671875 0.24267578125 - 0.475555419922 0.170379638672 0.391143798828 - 161 mat

roots n x Radau 2 1 sys

XY x 0.26513671875 0.24267578125 - 0.475555419922 0.170379638672 0.391143798828 - 161 mat augment : 0.75 - 0.26513671875 0.25 - 0.24267578125 - 0.125 0.475555419922 0.375 0.170379638672 0.625 0.391143798828 - 1162 mat

x λ 11 - x ≤ (x 1 ≤ (& cases :

<content> plotG, discrete poly </content>

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

Low High step x fnct inc U Low High step range : i 1 U rows range f i el U i el fnct : for U f augment 31 line :

XY 0.75 - 0.26513671875 0.25 - 0.24267578125 - 0.125 0.475555419922 0.375 0.170379638672 0.625 0.391143798828 - 1162 mat :

x φ 1 x x 2^x 3^x 4^51 mat :

X XY 1 col : Y XY 2 col : pts X Y + 14 blue plotG : 31 line

Cholesky companion solver... j 1 x φ length range i 1 X length range M i j el X i el φ j el : for for Φ M transpose M * : d M transpose Y * : 12 mat b Φ 1 -^eval d * : 31 line

b 0.375 1.5 5.25 - 3.5 - 7.875 51 mat

recast the scalar model x fit b x φ * :

n x Radau pts roots 4 1 sys

x 0.375 - :

n x Radau x fit - 21014 -^*

x r Clear 1

Gram-Schmidt Orthogonal Polynomials

N 7 :

i 1 N range f i el x i 1 -^: for

f transpose 1 x x 2^x 3^x 4^x 5^x 6^17 mat

kernel algo: MUST BE pre-initialised o o 1 el f 1 el : i 2 f rows range d 0 : j 1 i 1 - (range a f i el o j el * x 1 - 1 int o j el o j el * x 1 - 1 int / 15 round : d d a o j el * - : 21 line for o i el f i el d + (: 31 line for o 31 line :

recast scalar algo x o o :

Poly x o 1 el x o 2 el x o 3 el x o 4 el x o 5 el x o 6 el 61 sys :

Poly x o 6 el <content> Expand Syymbolic </content>

x o 3 el x 2^- x 687714221422867544566615537974000000000000000 / 15 round - (4101568306834652189666619894700000000000000000 / 15 round * + 13 / 15 round + (-

x G x 2^- x 687714221422867544566615537974000000000000000 / - (4101568306834652189666619894700000000000000000 / * + 13 / + (- :

<content> hatch </content>

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) between functions f(x) and g(x)in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

Plot roots of Gram-Schmidt r x G x 1 - 1.01 solve : r r eval : roots r r 0 * augment : roots roots . 14 black plot : 41 line

x f x G :

x g 0 x * :

Hatch f g r 1 el r 2 el 10 hatch :

x G Hatch roots 3 1 sys

Integral curve x ∫ x G x int maple :

r 0.5774 - 0.5774 21 mat

x G Hatch roots 31 sys

≈ 15 Decimals x ∫ r 1 el r 2 el at 0.256600119639832 -

x ∫ 1 0.125 - 1 0.125 2 2 mat 2 1 sys

Simpson very exact Gaussians x G x r 1 el r 2 el int 0.256600119639831 -

Rational Fraction Auto Procedure

<content> Function Utilities </content>

vx vy char size clr plotG n vx length : plot vx 1 el vy 1 el char size clr augment : i 2 n range plot plot vx i el vy i el char size clr augment stack : for plot 41 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 :

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 :

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 :

Module xd(L,H,N) is a discretiser
for use in-situ [user spec's]:
range [L,H], density [N].

Vector gradient ... Optimiz Symbolic 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 :

This section automates the fit . . .
slight recast from Jeff rev.3 In some ways,
iworks same as Mathcad GenfitMatrix, except4
fthe vector of support points that must equal 'k'

XY 8.0957 - 210.002 - 7.7963 - 195.802 - 7.4807 - 182.962 - 3.7187 - 78.476 - 1.9063 - 38.862 - 00 1.3739 26.87 5.2677 100 6.4886 122.37 8.3743 156.634 12.5517 231.9681 14.7427 271.442 17.4928 321.108 17.8462 327.502 19.456 356.66 152 mat :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

...n is total number of points in the data set n 15

Num orderDenom order p.n 4 : p.d 3 : 21 sys

k 1 p.n + p.d + : 8

x β f β 1 el β j 1 + el x j^* j 1 p.n sum + 1 β j 1 + p.n + el x j^* j 1 p.d sum + / 11 line :

f(x,β) ... is Normalized leading 1 denom
very stable numerically Padé style.
βinit & β from collaboration + Minimize

recast the system x β φ x β f k φ :

support points ≡ k support 134579121518 mat :

iter 5 :

βinit ct 1 : j support X.g ct el XY j 1 el : Y.g ct el XY j 2 el : 12 mat ct ct 1 + : 21 line for M.g k Ones : i 1 p.n range M.g M.g X.g i^vectorize augment : for i 1 p.d range M.g M.g X.g i^Y.g * vectorize - augment : for M.g 1 -^Y.g * 61 line :

β β 1 el βinit : iterate ConjugateGradient i 1 : Δβ'.max 1 : i iter ≤ (Δβ'.max 106 -^> (& β i 1 + el XY f φ β i el Minimize : i i 1 + : n 1 k range Δβ' n el β i el n el β i 1 - el n el - β i 1 - el n el / 1 try abs : for Δβ'.max Δβ' max : 41 line while β β i el i Δβ'.max stack : 61 line :

β β :

x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + β 5 el x 4^* + 1 β 6 el x * + β 7 el x 2^* + β 8 el x 3^* + / :

1. iterates2. Minimize N.iter β k 1 + el : Δβ' β k 2 + el : 21 mat 6021 mat

β β 1 k 11 submatrix :

βinit 0.0023 19.8516 2.9582 0.1117 0.0014 0.1611 0.0066 6.3444 105 -^* 81 mat

β 0.0081 19.8477 3.0839 0.1146 0.0008 0.1675 0.0068 2.9472 105 -^* 81 mat

FIT u 8.1 - 21 8.05 - range : i 1 u rows range FIT i el u i el β f eval : for u FIT augment 31 line :

Absolute residuals [mutual conformity] δ i 1 XY rows range Δ i el Y i el X i el β f - 3 round eval : for X Y Δ augment 21 line :

Δelta X δ 3 col augment : stem Δelta Error : 21 line

data X Y . 12 black plotG :

Rational fraction to Type 'J' T/C, NBS Monograph 125

data FIT 2 1 sys

δ 8.0957 - 210.002 - 0.003 - 7.7963 - 195.802 - 0.012 7.4807 - 182.962 - 0.01 - 3.7187 - 78.476 - 0.012 1.9063 - 38.862 - 0.01 - 00 0.008 - 1.3739 26.87 0.001 5.2677 100 0.013 6.4886 122.37 0.008 8.3743 156.634 0.006 - 12.5517 231.9681 0.019 - 14.7427 271.442 0.002 - 17.4928 321.108 0.016 17.8462 327.502 0.015 19.456 356.66 0.018 - 153 mat

Golden-Ratio search [min/max]

<content> Golden ratio min/max </content>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x 4^2 x 3^* - 7 x 2^* - 2 - eval :

ε 1015 -^:

By setting f(x):=eval(,)by default, it will cope with all "_not defined"
As well, you may have to use local definition.
Some functions will ONLY accept higher 'ε' ... 10^{-14} ... 10^{-13}

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 41 line :

X f 1.5 - 1 - y1 y2 ≤ ε Extrema f 0.5 - 0.5 y1 y2 ≥ ε Extrema f 2.5 3 y1 y2 ≤ ε Extrema 31 mat :

on failure , increase TOL TOL 1012 -^:

r 2.5 - 1.5 - TOL D 3.5 4 TOL D 21 mat : 1.922518473318 - 3.852153430325 21 mat

x f X 1 el X 1 el f . 14 red X 2 el X 2 el f . 14 red X 3 el X 3 el f . 14 red 3 5 mat r 1 el 0 . 14 black 1 5 mat r 2 el 0 . 14 black 1 5 mat 4 1 sys

X 1.26556446000827 - 0.000000028160548 - 2.76556437660309 31 mat

r f vectorize 21013 -^* - 51013 -^* - 21 mat

X f vectorize 6.5923 - 2 - 39.3452 - 31 mat

jmG: 20230820