dilog(x0 ... Dilog(x)

8603dfac-fa78-4f8c-966d-4171d43c6bfa 111 4 5 decimal

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

Numerical Integrators

]]>

3 appVersion 0.98.6179 ≡ Timing 16 min ≡ 21 mat

dilog(x0 ... Dilog(x)

Set integral accuracy 100 ε 1014 -^: x dilog t ln 1 t - / t 1 ε + x int : x Dilog 1 t - ln t / t ε x int - : 31 line

x dilog x Dilog 1 - π 2 ^ 6 / 2 π 2 ^ 6 / 2 2 mat 3 1 sys Int(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 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 :

∫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 :

Logarithmic integrals; dilog & Dilog

t f t ln 1 t - / :

t q 1 t - ln t / :

ε 1014 -^:

x c 1 ε x ≤ (x e ≤ (& cases :

t0 1 time :

x d f 1 ε + x 100 Int :

[200000, 8.4 min]... 1.64493406682487
[100, 0.3 min] ..... 1.64491089757522 ε d 1.64491089757522

x d x c * x f 1 1 1 2 - 2 2 mat 3 1 sys

1 time t0 - min 2.3

ε dilog 1.64493406684808 ≡

Zeta(2) 0 dlog 1.644934066848226 ≡

ε 1014 -^:

Infinite  limit x f x ln 1 x - / :

x f 1 1 1 5 - 2 2 mat 2 1 sys

'N' iterations
precision N 2000 : p 1016 -^: 21 sys

User inputs

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

t0 1 time :

Sanity -π^2/6 [0 .. 1] .....   -1.644934066848226
Wolfram stalls @ 5 D ... -1.64493 f a b N p ∫Adaptive 1.64493406684793 -

1 time t0 - min 1.3

1.6 min f a b 20000 Int 1.64493406627038 - ≡

Long range Romberg-Simpson Emeritus

]]>

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

x q 4 x 10 ≤ x x 10 ≥ cases :

x 016 0.1 range : XY x x q vectorize augment : data XY o 3 black plot : 31 sys

Numerical engine does not recognize > 15 D input

..... OK 
..... OK
from otherwise 9.999999999999999 q 10.000000000000001 q 5 q 31 mat 410431 mat

data

NOT exact [100] x q x 1016 int 77.94

x f x q :

f 1016100 Int 78.000000000000100

x c 10 x ≤ (x 16 ≤ (& cases :

Cumulative integration discontinuous

t q t 0 x int x c * 10 100 10 100 - 2 2 mat 2 1 sys plotG

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 :

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

Gaussian basis 'b-spline'

2 - x ≤ x 2 ≤ & x b Undefined if

0 b 0.6667

u v Bsp v m el u m - b * (m 1 v rows sum :

exact 1 by design u b u 2 - 2 int 1

X 1.875 1.875 1.95 2 2.05 2.125 2.125 2.125 18 mat : Y 11101 - 1 - 1 - 1 - 18 mat : 21 line

The XY plot allows to plot in the reverse X direction. The choice of the control points is by "user". You may have to trick the ranging of 't'. "data" is an export. There is no limit in the length of X, Y. With this kind of spline, you always have to at least duplicate end points [here RHS is triplicate]

XY X transpose Y transpose augment : u oy u Y transpose Bsp : u ox u X transpose Bsp : 31 sys

Points X transpose Y transpose . 14 black plotG :

data t 1.75 6.75 1.8 range : i 1 t rows range vx i el t i el ox : vy i el t i el oy : 21 line for vx vy augment 31 line :

data rows 101

XY data 2 1 sys XY data Points 3 1 sys

Finite differences derivative 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 - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : D X deriv augment : Arc integrand .................... xx D 1 col : δx D 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 - : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 181 line

Derivative of the function

Arc integrand

Arc

Cumulative arc length

Because

of the ranging of 't' that is difficult

to adjust, spooling to file is a

desirable conservative option.

]]>

CWD C:\SmathFile CurrentDirectory :

Σ CWD ArcRobotics wfile 1

Arc CWD ArcRobotics rfile :

in Auto-calc [once work sheet done] graph and arc(α) will update α 2 :

arc length data
arc length from interpolation
.....
plot range X Σ 1 col : Y Σ 2 col : 12 mat x arc X Y x cinterp : x c 1 1.875 x ≤ (x α ≤ (& cases : 31 sys

Cumulative arc length

x arc x c * 1.875 0.5 1.875 0.5 - 2 2 mat 2.125 2.5 2.125 1.5 2 2 mat 3 1 sys

α arc 1.005

1.875 α ≤ (α 2.125 ≤ (&

... as explained before, required Σ are resident thus in hand for next demand cinterp(X,Y,x), graph, arc(α)

http://www.geometrictools.com/Documentation/FiniteDifferences.pdf

https://www.scilab.org/sites/default/files/numericalderivatives-v0.4.pdf

x c 12 - x ≤ (x 2 ≤ (& cases :

The infinitesimal companion [b-spline] . . .

b_spline represents a discontinuous function that you may have

otherwise from project design.

]]>

x f x b :

Algo style Memo formula of incremental analysis of this demo example h 0.001 : (increment ≡ x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / : (1rst Derivative ≡ x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : (2nd Derivative ≡ 41 line

x Φ x h f D1 :

x Φ x h f D2 2 1 sys

a function in  ...
infinitesimal mode x q x h + f x h - f + 2 / :

x ∫ u b u x - x int :

x ∫ u q u x - x int ≡

xo 2 - :

normalized CDF style x cumul 0.5 x ∫ x c * xo ∫ - (* :

x f x cumul : x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : 21 sys

∫(x) integrator low acc quick plot

x q x cumul 2 1 sys

u b u 2 - 2 int 1

x q x cumul ≡ x 0 0.4 solve 0.1998

0 h f deriv 0.6667

0. plot cumulative arc length on canvas.
1. assign arc length integrator as function.
2. plot/calculate arc length b_spline. x arc_spline 1 u Φ (2^+ sqrt u xo x int eval :

About real projects . . .

Not all projects are based on function(s). More commonly

they will be data sets. At this point, make sure you are

given raw data. Never trust data processed by others as

under the hood you lose tracability.

As much as possible, attempt to fit a model function or

at worst smooth wisely. Remember: FD derivative is a

noise amplifier. Differently, FD integrator is almost

insensible to noisy data set. The minimalist FD integrator

has been tested vs F.B. Hildebrand more advanced FD,

not to avail tested on the most noisy "Corn Thermies".

]]>

2 arc_spline 4.31

x h D3 x 3 h * - f 8 x 2 h * - f * - 13 x h - f * + 13 x h + f * - 8 x 2 h * + f * + x 3 h * + f - 8 h 3^* / :

x h D4 x 3 h * - f - 12 x 2 h * - f * + 39 x h - f * - 56 x f * + 39 x h + f * - 12 x 2 h * + f * + x 3 h * + f - 6 h 4^* / :

An otherwise discontinuous function

min(,) undocumented x f 2110 x - (* 260 x - 2 / 5128 x - (* 4 / 31 mat min :

x f x bound 412004164 - 22 mat 802008064 - 22 mat 41 sys

x bound 1080 x - (* 7 / :

x f x bound 41 200 41 64 - 2 2 mat 80 200 80 64 - 2 2 mat 4 1 sys

x f x bound - (x 16.923 - 185 int 6911.497

x f x bound - 0 ≡ x 32 - 0 solve x f x bound - 0 ≡ x 160200 solve 21 mat 16.923 - 18521 mat

The efficient FD cumulative

raw/smoothed Thermies xw 02116 2113.3 12121 2111.7 22099 2109.2 32120 2105.2 42113 2099.4 52098 2092.1 62067 2084.7 72071 2078.6 82070 2075.1 92056 2073.9 102083 2074.1 112090 2074.4 122072 2074.0 132064 2072.5 142081 2070.1 152074 2067.2 162054 2064.4 172056 2062.0 182058 2060.5 192061 2059.6 203 mat :

Cumulative corn Thermies vx xw 1 col : vy xw 2 col : 12 mat ∫ 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : ∫ i el ∫ i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx ∫ augment : vΣ ∫ sum : 61 line

vΣ 4105^*

1396983397145 / - 0.000408

raw xw 1 col xw 2 col augment : smooth xw 1 col xw 3 col augment : 21 line

raw 397145 ≡ smoothed 396983 ≡ 21 mat

raw smooth 2 1 sys

. . . . . More Adaptive Emeritus . . . . .

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

<= Wolfram Alpha

MCD 11 default adaptive ... 0.611991 x 0.5 1 -^x - exp * x 1015 -^0.1 int Don't even attempt ≡

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

'N' iterations
precision N 2000 : p 1016 -^: 21 sys

lower limit
upper limit a 1014 -^: b 0.1 : 21 sys

t0 1 time :

f a b N p ∫Adaptive 0.611991166111776

π sqrt b sqrt erf * 0.611991366111770

1 time t0 - min 2.6

5 minutes f a b 40000 Int 0.6119868 ≡

Discrete Integral Curve [AntiDerivative]

There will be instances of:

1. No reply from Maple 2. Otherwise not exploitable

Isolate this section . . .

Copy/paste in blank sheet,

run integrator acc 1000 typical.

You may consider more profitable integrators.

]]>

x f x ln :

∫

z 1015 -^5 0.1 range :

Discrete  ∫ over linear path ∫Curve i 1 z rows range C i el x f x z 1 el z i el int : for z C augment 21 line :

∫Curve => x*(-1+ln(x))

∫Curve x f 2 1 sys Anti_Derivative of ln(x)

Anti_Deriv 11015 -^* 0 0.1 0.331 - 0.2 0.523 - 0.3 0.662 - 0.4 0.768 - 0.5 0.849 - 0.6 0.909 - 0.7 0.952 - 0.8 0.982 - 0.9 0.998 - 1 1.004 - 1.1 1 - 1.2 0.986 - 1.3 0.964 - 1.4 0.935 - 1.5 0.898 - 1.6 0.855 - 1.7 0.805 - 1.8 0.749 - 1.9 0.688 - 2 0.622 - 2.1 0.551 - 2.2 0.475 - 2.3 0.394 - 2.4 0.309 - 2.5 0.22 - 2.6 0.127 - 2.7 0.03 - 2.8 0.071 2.9 0.175 3 0.283 3.1 0.394 3.2 0.508 3.3 0.626 3.4 0.746 3.5 0.87 3.6 0.996 3.7 1.125 3.8 1.257 3.9 1.391 4 1.528 4.1 1.667 4.2 1.809 4.3 1.954 4.4 2.1 4.5 2.249 4.6 2.4 4.7 2.553 4.8 2.709 4.9 2.866 5 3.026 512 mat :

Finite differences derivative X 0 : Y 0 : deriv 0 : D 0 : 14 mat X Anti_Deriv 1 col : Y Anti_Deriv 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 - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : D X deriv augment : 101 line

D Anti_Deriv 2 1 sys ∫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 :

'n' intervals in range adapted for singular
admissible low limit in ∫Adaptive(f,a,b,N,ε) n range interval j 1 n 1 + (range h j el range - range j * + n / : for h 1 el 0.001 : h 31 line :

z 4005 interval :

z 0.001 0.0125 0.025 0.0375 0.05 0.0625 0.075 0.0875 0.1 91 mat

Infinite  limit
'N' iterates
TOL vs timing x f x ln : N 2000 : ε 0.0001 : 31 sys

t0 1 time :

Discrete ∫Adaptive(f,a,b,N,ε) over linear path ∫Curve a z 1 el : count 1 : i 1 z rows range b z i el : C count el f a b N ε ∫Adaptive : count count 1 + : 31 line for z C augment 41 line :

∫Curve x f 2 1 sys

1 time t0 - min 1.6

indefinite integral x ln x int maple x 1 - x ln + (* ≡

x 1 - x ln + (* x ln - 0 ≡ x 01 solve x 1 - x ln + (* x ln - 0 ≡ x 3.75 4 solve 21 mat 0.446432953 3.857334826 21 mat

0.446432978 3.857334826 21 mat ln vectorize 0.806465995 - 1.349976485 21 mat

XY 0.446432978 0.806465995 - 3.857334826 1.349976485 22 mat :

vx XY 1 col : vy XY 2 col : 21 line

a vx vy Intercept :

b vx vy Slope :

x 1 - x ln + ( * x ln a b x * + e 0 + 20 red 1 5 mat 4 1 sys

a b 21 mat 1.089 - 0.632 21 mat

x f x 1 - x ln + (* : N 100 : ε 1012 -^: a 1015 -^: b e : 51 sys

f a b N ε ∫Adaptive 1.847264025 -

∫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 :

x f x ln : N 100 : ε 103 -^: a 1015 -^: b e : 51 sys

∫Adaptive(f,a,x,N,ε) . . . canvas

This integrator manages either/both ends [singular/infinite limit]

Check first if the the 'n' Romberg-Simpson is adequate.

]]>

sanity the canvas plot accuracy f a b N ε ∫Adaptive 0.001 -

x Canvas f a x N ε ∫Adaptive :

x c 1 a x ≤ (x 4 ≤ (& cases :

x Canvas x c * x ln 2 1 sys

ZedGraph complains

∫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 :

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

eval(,) gains 1 min
N 100 plenty
'ε' canvas accuracy
.....
'a' infinite limit
.....
user end canvas plot x f x 0.5 1 -^x - exp * eval : N 100 : ε 0.001 : a 1015 -^: b e : 51 sys

t0 1 time :

x Canvas f a x N ε ∫Adaptive :

x c 1 a x ≤ (x e ≤ (& cases :

Canvas(x) is scalar wrt pixels runs 126 *∫Adaptive(f,a,x,N,ε)

x f x Canvas x c * 2 1 sys

1 time t0 - min 3.9