∫Adaptive(f,a,b,N,ε)

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Part 1: Romberg-Simpson(n local) single integration

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x)  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 :

3 appVersion 0.98.6179 ≡ Timing 7 min ≡ 21 mat

sanity check the sextuplets Mathcad 11 Adaptiv 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Simpson-Romberg... 5.40752818465635 ≡ Smath Simpson 7½ D 5.4075281 ≡ Simplex 10½ D .... 5.40752818462 ≡ 51 line

x f x sqrt exp :

t0 1 time :

transformed domain [0,1] <= Simpson [a,b] f 022000 Int 5.40752818465635

1 time t0 - 10.2 s *

Part 2: Advanced adaptive single integration

Adaptive integration . . .

Exmples [1, 2, 3] were chosen for their known value.

Upper limit [b] may be discretized for full graph.

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∫Adaptive(f,a,b,N,ε) Program algo style is a bit slower than open style, more convenient. Adapted from Carlos code [20180110]

∫Adaptive(f,a,b,N,ε)

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 :

Example_1: Ordinary difficult region

x f x sqrt exp : x d 10 x ≤ (x 0.0125 ≤ (& cases : 21 sys

Ordinary x f x sqrt exp :

x f x d * 1 -

'N' iterations
precision N 1000 : ε 1015 -^: 21 sys

User inputs

lower limit
upper limit a 0 : b 2 : 21 sys

t0 1 time :

Technical acccuracy .....   5.40752818465627 f a b N ε ∫Adaptive 5.40752818465644

1 time t0 - 25 s *

Example_2: Singular end point

x c 1 0.001 x ≤ (x 2 ≤ (& cases :

TOL 1013 -^:

TOL 1013 -^≤

x f 1 x - ln - x / :

lower limit
upper limit a TOL : b 1 TOL - : 21 sys

Zeta(2) => 1.644934066848226

x f x c * 1 10 1 1 - 2 2 mat x sqrt exp x c * 3 1 sys

t0 1 time :

True value Zeta(2)  [π²/6].......... 1.644934066848226 f a b 200 TOL ∫Adaptive 1.64493406684641

1 time t0 - 23 s *

∫Adaptive(f,a,b,N,ε)

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 c 1106 -^x ≤ (x 1 ≤ (& cases :

x f 1 x - ln - x / :

TOL 1013 -^:

TOL 1013 -^≤

lower limit
upper limit a TOL : b 1 TOL - : 21 sys

t0 1 time :

f a b 100 0.0000001 ∫Adaptive eval 1.644934

1 time t0 - 1 s *

t0 1 time :

S x a b 0.01 range : j 1 x rows range Y j el f a x j el 100 0.0001 ∫Adaptive eval : for x Y augment 31 line :

Zeta(2) => 1.644934066848226

x f x c * 1 10 1 1 - 2 2 mat S 1 1.645 + 40 black 1 5 mat 4 1 sys

1 time t0 - 5 s *

S S rows row 1 1.64502 12 mat

Example_3: infinite limit

infinite limit x f x 0.5 1 -^x - exp * :

'N' iterations
_ precision _iter 1000 : TOL 1015 -^: 21 sys

x f

lower limit 
upper limit a 1012 -^: b 1 : 21 sys

t0 1 time :

sanity Wolfram ... 1.49364826562485 f a b iter TOL ∫Adaptive 1.49364626562488

1 time t0 - min 1.9

π sqrt 1 erf * 1.49364826562485

plot(data,char,size,clr)

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 :

Romberg Simplex 1D kernel

k gencm1 N 2 k matrix : i 1 k range N 1 i el i 1 - : N 2 i el k i - : 21 line for N 31 line :

nr nc v genm M nr nc matrix : i 1 nr range j 1 nc range M i j el v : for for M 31 line :

11 gencm1 012345678910109876543210211 mat

330 genm 00000000033 mat

Carlos added,
enables recursive N 2 k matrix ≡ M nr nc matrix ≡ 21 sys

M idx M transpose 1 col rows :

5 gencm1 idx 5

Carlos  'submat' M 1 n 1 + (range 2 w range el M 1 n 1 + (2 w submatrix ≡

330 genm idx 3

without submatrix(...) n k gencm n 1 ≡ k gencm1 M n 1 + 10 genm : i 0 k range N n 1 - k 1 + (i - gencm : w N idx 1 - : O 1 w 1 + i genm : N O N stack : M M N augment : 51 line for w M idx 0 - : M 1 n 1 + (range 2 w range el 41 line if 11 line :

15 gencm 012344321025 mat

Only true for n=2 and any 'k'

24 gencm 11 mat 000001111222334012340123012010432103210210100315 mat 11 mat

weights(n,s)is universal for any dimension n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - (i^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

Simplex integration algorithm f n s ∫ v 0 : Pt n s pts : w n s 1 - weights : i 1 w rows range P Pt i el : v v w i el P j col f j 1 P idx sum * + : 21 line for v 51 line :

n s pts d 2 s * 1 + : i 0 s range M n s 1 + (i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for 1D

111 pts 0.955 0.864 0.773 0.682 0.591 0.5 0.409 0.318 0.227 0.136 0.045 111 mat 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05 110 mat 0.944 0.833 0.722 0.611 0.5 0.389 0.278 0.167 0.056 19 mat 0.938 0.812 0.688 0.562 0.438 0.312 0.188 0.062 18 mat 0.929 0.786 0.643 0.5 0.357 0.214 0.071 17 mat 0.917 0.75 0.583 0.417 0.25 0.083 16 mat 0.9 0.7 0.5 0.3 0.1 15 mat 0.875 0.625 0.375 0.125 14 mat 0.833 0.5 0.167 13 mat 0.75 0.25 12 mat 0.5 11 mat 111 mat

Recast the simplex domain

Romberg-Simplex 1D integrator  ... domain [a,b] ... f a b s ∫ u b a - (34 / 2 x 1 el * 1 - (* 14 / 2 x 1 el * 1 - (3^* - 12 / + (* a + : x f u f 12 x 1 el * 1 - (2^- (* : simplex 32 / b a - (* f 1 s ∫ * : simplex 41 line :

x f x - exp x sqrt / :

106 -^f 1000

ε 1012 -^: b 0.1 : 12 mat

x c 1 ε x ≤ (x b ≤ (& cases :

x f x c * b 50 b 50 - 2 2 mat 2 1 sys

Reserve Simplex [infinite limit and singular end point]. Explore for best 's' below. Next best => Adaptive 14 D's 2 min.

t0 1 time :

sanity 12 D f 0111 ∫ 1.493648265624

1 time t0 - 0.4 s *

π sqrt 1 erf * 1.49364826562485

Explore f 019 ∫ f 0110 ∫ f 0111 ∫ f 0112 ∫ f 0113 ∫ 51 mat : 1.49364826611909 1.49364826565873 1.49364826562379 1.49364826563812 1.49364826566673 51 mat

Simplex reverses convergence as 's' increases. Once you have determined the direction wrt f(x) use the reported best 's' ... here 's=11'

Search: 's' reverse convergence s k 6 : s k 15 range f 01 s ∫ f 01 s 1 - ∫ ≥ index k el s : k k 1 + : 21 line k if for s k 1 + : 31 line :

s 11

sanity ...  1.49364826562485 ... 12 D f 0111 ∫ 1.49364826562379

π sqrt 1 erf * f 0111 ∫ - 11012 -^*

2D simplex integration ... demo

2D Simplex maths

doctored by Carlos
to enable recursive N 2 k matrix ≡ M nr nc matrix ≡ 21 sys

k gencm1 N 2 k matrix : i 1 k range N 1 i el i 1 - : N 2 i el k i - : 21 line for N 31 line :

Sanity checked 5 gencm1 012344321025 mat

M last M transpose 1 col rows :

nr nc v genm M nr nc matrix : i 1 nr range j 1 nc range M i j el v : for for M 31 line :

330 genm 00000000033 mat

410 genm 000041 mat

M idx M transpose 1 col rows :

without submatrix(...) tip from Carlos n k gencm n 1 ≡ k gencm1 M n 1 + 10 genm : i 0 k range N n 1 - k 1 + (i - gencm : w N idx 1 - : O 1 w 1 + i genm : N O N stack : M M N augment : 51 line for w M idx 0 - : M 1 n 1 + (range 2 w range el 41 line if 11 line :

15 gencm 012344321025 mat

Only true for n=2 and any 'k'

24 gencm 11 mat 000001111222334012340123012010432103210210100315 mat 11 mat

21 gencm 00101010033 mat

22 gencm 00011201201021010036 mat

23 gencm 000011122301230120103210210100310 mat

n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - (i^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

n s pts d 2 s * 1 + : i 1 s range M n s i - gencm : M M 2 n 1 + 1 M idx 0 - submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

s > 7 ... numerical instability n 2 : s 7 : 12 mat

pts n s pts :

Simplex integration algorithm f n s ∫∫ v 0 : Pt n s pts : w n s 1 - weights : i 1 w rows range P Pt i el : v v w i el P j col f j 1 P idx sum * + : 21 line for v 51 line :

t0 1 time :

x f e x 1 el^0.5 x 2 el (3^- (* :

sanity ... 0.299449943475250 f n s ∫∫ 0.29944994347527

1 time t0 - 0.4 s *

3D Simplex Maths

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

k gencm1 N 2 k matrix : i 1 k range N 1 i el i 1 - : N 2 i el k i - : 21 line for N 31 line :

nr nc v genm M nr nc matrix : i 1 nr range j 1 nc range M i j el v : for for M 31 line :

5 gencm1 012344321025 mat

330 genm 00000000033 mat

M idx M transpose 1 col rows :

5 gencm1 idx 5

M 1 n 1 + (range 2 w range el M 1 n 1 + (2 w submatrix ≡

330 genm idx 3

without submatrix(...) n k gencm n 1 ≡ k gencm1 M n 1 + 10 genm : i 0 k range N n 1 - k 1 + (i - gencm : w N idx 1 - : O 1 w 1 + i genm : N O N stack : M M N augment : 51 line for w M idx 0 - : M 1 n 1 + (range 2 w range el 41 line if 11 line :

15 gencm 012344321025 mat

Only true for n=2 and any 'k'

24 gencm 11 mat 000001111222334012340123012010432103210210100315 mat 11 mat

n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - (i^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

sanity 15 D 32 weights 0.050793650793651 0.096428571428571 - 0.044444444444444 31 mat

n s pts d 2 s * 1 + : i 0 s range M n s 1 + (i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for n=1

n s 1 + (i - gencm

13 pts 0.8333 0.5 0.1667 13 mat 0.75 0.25 12 mat 0.5 11 mat 31 mat

n s pts d 2 s * 1 + : i 0 s range M n s i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for n=2

n s i - gencm

show 27 pts : simplex show 1 el transpose : vx simplex 1 col : vy simplex 2 col : 41 line

23 pts 0.143 0.429 0.714 0.143 0.429 0.143 0.714 0.429 0.143 0.429 0.143 0.143 26 mat 0.2 0.6 0.2 0.6 0.2 0.2 23 mat 0.333 0.333 21 mat 31 mat

S vx vy o 5 red plotG :

simplex 0 0 0 1 1 1 1 0 0 0 5 2 mat S 0.2 1 0.2 0 2 2 mat 5 1 sys

The doctored triple integration

23 gencm 000011122301230120103210210100310 mat

n s pts d 2 s * 1 + : v 11113 mat transpose : d 2 s * 1 + : i 0 s range M n s i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + v augment : Pt s 1 + el v : scale i 1 + el 1 d n + 2 i * - / : 51 line for Pt scale * vectorize 51 line :

Specific for n=3

's' pilots accuracy of ∫(f)

n 3 : s 4 : 12 mat

pts n s pts :

t0 1 time :

Simplex  triple integration algorithm f n s ∫∫∫ v 0 : Pt pts : w n s weights : i 1 w rows range P Pt i el : v v w i el P j col f j 1 P idx sum * + : 21 line for v 51 line :

Simplex integration is on the 0 .. 1

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x f e x 1 el x 2 el +^0.5 x 3 el (3^- (* :

sanity ... 0.129622968748757 f 34 ∫∫∫ 0.129622968748766

1 time t0 - 0.2 s *

Revisit Simpson: poor SQRT(x) ... exact b-spline

Romberg-Simpson

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 :

x f x sqrt :

acc 1000 poor Integration
doctored Romberg-Simpson u f u 010 int 21.0792837853217

2010 sqrt * 3 / 21.0818510677892

f 0101000 Int 21.0818510677907

u b 0 u 2 - ≤ 16 / 2 u + (3^* 2 - u ≤ (u 1 - ≤ (& 16 / 2 u + (3^* 23 / 1 u + (3^* - 1 - u ≤ (u 0 ≤ (& 23 / 1 u - (3^* - 16 / 2 u - (3^* + 0 u ≤ (u 1 ≤ (& 16 / 2 u - (3^* 1 u ≤ (u 2 ≤ (& 0 cases :

The Gaussian b-spline

exact 1 by design ... acc 100 u b u 2 - 2 int 0.999999715555557

U 2 - 2 1.99 - range :

Σ Cumulative Integration i 1 U rows range Cumul i el x b x U 1 el U i el int eval : for U Cumul augment 31 line :

Gaussian basis 'b-spline'

U U b vectorize augment Σ 2 1 sys Romberg-Simpson

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 :

DOUBLE: from maple pre-algorithm ================

x y F x 2^y 2^+ sqrt :

x c 0 x * :

x d x :

x y F y x c x d int x int

# # int # int

optimiz: Symbolic x y F y x c x d int x int maple x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + ln * + x x ln * - (* 2 / x int

x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + ln * + x x ln * - (* 2 / x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + x / ln * + (* 2 / ≡

inner integrand x f x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + x / ln * + (* 2 / :

ε 1012 -^:

x c 1 ε x ≤ (x 10 ≤ (& cases :

integrand is sanity checked

x f

x int u f u ε x int :

Cumulative canvas ... acc 50 native Simpson adequate.

0.01 x int * x c * 10 6 10 2 - 2 2 mat 2 1 sys

Romberg-Simpson
ordinary Simpson f ε 10100 Int 10 int 21 mat 382.597859 382.597858 21 mat

x y Clear 1

a b L u Clear 1

'I' maple is 'i' imaginary x L δ F L x - (* E u * / :

optimiz NONE x L δ L 0 x int x 0 L int maple F L 3^* 6 E * u * / -

F 1 : E 1 : I 1 : 13 mat

x F E I flexion F x 3^* 6 E * I * / - :

x flex 10 x ≤ cases :

x F E I flexion x flex *

x y f 6 x 2^* 2 y * + :

x c x 2^:

x d 4 :

x y f y x c x d int x a b int maple 7 b 5^* 5 / - 7 a 5^* 5 / + 8 b 3^* + 8 a 3^* - 16 b * + 16 a * -

x y Double 7 y 5^* 5 / - 7 x 5^* 5 / + 8 y 3^* + 8 x 3^* - 16 y * + 16 x * - :

Double

x c x :

x d 4 :

x y f y x c x d int x a b int maple 3 b 4^* 2 / - 3 a 4^* 2 / + 23 b 3^* 3 / + 23 a 3^* 3 / - 16 b * + 16 a * -

x y double 3 y 4^* 2 / - 3 x 4^* 2 / + 23 y 3^* 3 / + 23 x 3^* 3 / - 16 y * + 16 x * - :

double

Iterative integration ... as piece wise companion, unexplored.
May rescue the project if no other method is profitable. f a b TOL ∫singular Y1 b a - (a b + 2 / f * : Y2 b a - 6 / 2 a * b + 3 / f a 2 b * + 3 / f + (* : N 1 TOL / : Y1 Y2 - abs TOL ≥ Y1 Y2 : N 2 N * : Y2 0 : h b a - N 1 - / : j 1 N range x0 a j h * + : x1 x0 h + : 12 mat Y2 Y2 h x0 f x1 f + 2 / (* + : 21 line for 51 line while 2 Y2 * Y1 - 51 line eval :

x f x ln :

t0 1 time :

TOL 0.001 :

a 1015 -^: b 0.1 : c 0.25 : d 0.5 : 14 mat

7 min  ...  TOL 0.0001 ∫1 f a b TOL ∫singular : 0.3302 -

x ln b 1 b 6 - 2 2 mat c 1 c 4 - 2 2 mat d 1 d 2 - 2 2 mat 1 1 1 2 - 2 2 mat 5 1 sys

∫2 f b c TOL ∫singular : 0.2663 -

∫3 f c d TOL ∫singular : 0.25 -

∫4 f d 1 TOL ∫singular : 0.1534 -

hoover to see inside

Σ ∫1 ∫2 + ∫3 + ∫4 + : 0.999966 -

1 time t0 - min 0.8

Romberg Simplex 1D kernel

k gencm1 N 2 k matrix : i 1 k range N 1 i el i 1 - : N 2 i el k i - : 21 line for N 31 line :

nr nc v genm M nr nc matrix : i 1 nr range j 1 nc range M i j el v : for for M 31 line :

11 gencm1 012345678910109876543210211 mat

330 genm 00000000033 mat

Carlos added,
enables recursive N 2 k matrix ≡ M nr nc matrix ≡ 21 sys

M idx M transpose 1 col rows :

5 gencm1 idx 5

Carlos  'submat' M 1 n 1 + (range 2 w range el M 1 n 1 + (2 w submatrix ≡

330 genm idx 3

without submatrix(...) n k gencm n 1 ≡ k gencm1 M n 1 + 10 genm : i 0 k range N n 1 - k 1 + (i - gencm : w N idx 1 - : O 1 w 1 + i genm : N O N stack : M M N augment : 51 line for w M idx 0 - : M 1 n 1 + (range 2 w range el 41 line if 11 line :

15 gencm 012344321025 mat

Only true for n=2 and any 'k'

24 gencm 11 mat 000001111222334012340123012010432103210210100315 mat 11 mat

weights(n,s)is universal for any dimension n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - (i^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

Simplex integration algorithm f n s ∫ v 0 : Pt n s pts : w n s 1 - weights : i 1 w rows range P Pt i el : v v w i el P j col f j 1 P idx sum * + : 21 line for v 51 line :

n s pts d 2 s * 1 + : i 0 s range M n s 1 + (i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for 1D

111 pts 0.955 0.864 0.773 0.682 0.591 0.5 0.409 0.318 0.227 0.136 0.045 111 mat 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05 110 mat 0.944 0.833 0.722 0.611 0.5 0.389 0.278 0.167 0.056 19 mat 0.938 0.812 0.688 0.562 0.438 0.312 0.188 0.062 18 mat 0.929 0.786 0.643 0.5 0.357 0.214 0.071 17 mat 0.917 0.75 0.583 0.417 0.25 0.083 16 mat 0.9 0.7 0.5 0.3 0.1 15 mat 0.875 0.625 0.375 0.125 14 mat 0.833 0.5 0.167 13 mat 0.75 0.25 12 mat 0.5 11 mat 111 mat

Recast the simplex domain

Romberg-Simplex 1D integrator  ... domain [a,b] ... f a b s ∫ u b a - (34 / 2 x 1 el * 1 - (* 14 / 2 x 1 el * 1 - (3^* - 12 / + (* a + : x f u f 12 x 1 el * 1 - (2^- (* : simplex 32 / b a - (* f 1 s ∫ * : simplex 41 line :

x f x ln :

t0 1 time :

sanity 5 D f 0133 ∫ 1.000096 -

1 time t0 - 1.7 s *

Explore f 0130 ∫ f 0132 ∫ f 0133 ∫ f 0134 ∫ 41 mat : 1.000015 - 1.000014 - 1.000096 - 0.999728 - 41 mat

Simpson iterative

Simpson iterative TOL ... [10e-6 n=10] => ~ 30 min ... -1.00000056802215  
independent acc ... n = 5 works well ... may be useful occsionally. f a b TOL n ΣS h b a - n / : ε x f x a b int : S1 ε : ε abs TOL > n 2 n * : h b a - n / : S2 13 / h 2 / a f b f + (* h a k h * + f k 1 n 1 - sum * + (* 23 / h * a h 2 / - k h * + f k 1 n sum * + eval : ε S2 S1 - 2 / : S1 S2 : 51 line while S2 S2 ε + : 51 line :

x f x ln :

0.0001 ... 30 sec TOL 0.0001 :

n 10 :

f 1015 -^1 TOL n ΣS 1.000081 -

"Romberg" ... integral accuracy: 10000 ... -1.00000000 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 :

x f x ln :

f 1015 -^1 Int 0.99991 -

Vectorize these two
... maybe others ? x erf vectorize x # # int x Gamma vectorize x # # int 21 mat

ODE integration ... f(x)

x f x ln :

end 6 :

ε 1015 -^:

x Y D x ln :

x c 10 x ≤ (x end ≤ (& cases :

sol 0 ε end 1002 x Y D dn_GearsBDF :

x f x c * sol e 2 e 1 - 2 2 mat 3.858 1.35 + 100 black 1 5 mat 4 1 sys

X sol 1 col : Y sol 2 col : 12 mat x q X Y x linterp : 21 sys

jnct x ln x q - 0 ≡ x 3.5 4 solve : 3.8579

jnct ln 1.3501

sol sol 2 col Min 2 findrows 1 1.00038 - 12 mat

Cumulative integation- Trapeze Method f a b N Trapeze h b a - N / : h 2 / a f b f + 2 a k h * + f k 1 N 1 - sum * + (* eval 21 line :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 0 : Y 1 el 0 : 12 mat 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 :

a 109 -^: b 0.1 : c 0.25 : 13 mat

x f x 0.5 1 -^x - exp * :

Legendre wr 63 ... 64

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.

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

x c 1 a x ≤ (x b ≤ (& cases :

x f x c * b 50 b 50 - 2 2 mat 2 1 sys

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

weight, wi w wr 1 col :

abscissea, xi x wr 2 col :

m wr rows : 64

Legendre-Gauss quadrature a b Quad i 1 m range Leg i el b a - 2 / (x i el * b a + 2 / (+ eval : for Leg b a - (2 / w i el Leg i el f * (i 1 m sum eval * 31 line :

a b Quad 0.6077222

xx 0.1 :

π sqrt xx sqrt erf * 0.61199136611177

1 0.6077222 0.61199136611177 / - 0.007

x f x a b int 11.1131150335738