<p> Integrand ... NL </p>

48861828-7044-443d-b216-27650e686dc0 53 4 5 decimal

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

Example 1: the non-linear integration ... Integrand is function

4 appVersion 0.98.6179.21440 ≡ Timing 5 ... 8 min ≡ 21 sys

k 0.975 :

elliptic integral 1rst kind u F 11 k 2^u sin (2^* - sqrt / :

x f 101 x 2^+ / :

escape division by 0 ε 0.001 :

f x f :

'x' is the running  canvas ...
as well the running discrete x Int u F u ε f int :

0 Int 18.0932

user 1 :

lo 4 - : hi 4 : N 199 : 13 mat

symbol below ʃ

interpolate «» user Int : 9.8262

recast 'f' plot x f f :

INT u lo hi lo hi lo - (N / + range : i 1 u rows range ʃ i el u i el Int eval : for u ʃ augment 31 line :

Scalar/Interpolate NON-linear

x F x Int x f user «» + 20 black 1 5 mat 4 1 sys

Export discrete NON-linear

INT 2 1 sys

data INT :

t0 1 time :

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

Finite differences Cumulative Integration

data Σ data 2 1 sys

1 time t0 - 0.59 s *

Example 2: Resume the non-linear integration [splined path]

μ 0 :

σ 1 :

x λ 14 - x ≤ (x 4 ≤ (& cases :

u F u μ σ pdf.Normal :

lo 4 - : hi 4 : N 199 : 13 mat

shape range NON-linear path s 5 :

x q s 1 x 2^+ / :

ω u lo hi lo hi lo - (N / + range : i 1 u rows range vy i el u i el q eval : for u vy augment 31 line :

x g ω 1 col ω 2 col x ainterp :

ω x g 2 1 sys

non-linear path of integration f x g :

'x' is the running  canvas ...
as well the running discrete x Int u F u 0 f int :

user 0.75 :

0 Int 0.5

lo 4 - : hi 4 : N 199 : 13 mat

symbol below ʃ

interpolate «» user Int : 0.4993

INT x lo hi lo hi lo - (N / + range : i 1 x rows range ʃ i el x i el Int eval : for x ʃ augment 31 line :

Scalar/Interpolate NON-linear

x F x Int x λ * 0.125 x g * x λ * user «» + 20 black 1 5 mat 4 1 sys

Export discrete NON-linear

INT user «» + 20 black 1 5 mat 3 1 sys

Σ u lo hi lo hi lo - (N / + range : i 1 u rows range ʃ i el x q x u 1 el u i el int eval : for u ʃ augment 31 line :

PDF style NOT normalized

x q Σ 2 1 sys

Observe . . .

1. 's' has NO influence on the plateau magnitude

however sets the range shape.

2. Do not extrapolate Int(x).

]]>

Example 3: NOT compatible spline companions ... ......... Reconciliate integrand from model function.

<p> Integrand ... NL </p>

Integrand 4 - 1.4816 3.9598 - 1.4235 3.9196 - 1.3714 3.8794 - 1.3247 3.8392 - 1.2828 3.799 - 1.2451 3.7588 - 1.2113 3.7186 - 1.1809 3.6784 - 1.1536 3.6382 - 1.1292 3.598 - 1.1075 3.5578 - 1.0881 3.5176 - 1.071 3.4774 - 1.056 3.4372 - 1.043 3.397 - 1.0318 3.3568 - 1.0224 3.3166 - 1.0147 3.2764 - 1.0087 3.2362 - 1.0043 3.196 - 1.0014 3.1558 - 1.0001 3.1156 - 1.0003 3.0754 - 1.0021 3.0352 - 1.0054 2.995 - 1.0103 2.9548 - 1.0168 2.9146 - 1.025 2.8744 - 1.0349 2.8342 - 1.0466 2.794 - 1.0602 2.7538 - 1.0758 2.7136 - 1.0936 2.6734 - 1.1136 2.6332 - 1.1361 2.593 - 1.1613 2.5528 - 1.1895 2.5126 - 1.2209 2.4724 - 1.2558 2.4322 - 1.2946 2.392 - 1.3379 2.3518 - 1.3861 2.3116 - 1.4399 2.2714 - 1.5 2.2312 - 1.5674 2.191 - 1.643 2.1508 - 1.7282 2.1106 - 1.8244 2.0704 - 1.9334 2.0302 - 2.0574 1.9899 - 2.1988 1.9497 - 2.3604 1.9095 - 2.5453 1.8693 - 2.7564 1.8291 - 2.9958 1.7889 - 3.2634 1.7487 - 3.5542 1.7085 - 3.8548 1.6683 - 4.1383 1.6281 - 4.3645 1.5879 - 4.4877 1.5477 - 4.4775 1.5075 - 4.3366 1.4673 - 4.0991 1.4271 - 3.8108 1.3869 - 3.5104 1.3467 - 3.2223 1.3065 - 2.9587 1.2663 - 2.7236 1.2261 - 2.5166 1.1859 - 2.3353 1.1457 - 2.1768 1.1055 - 2.0381 1.0653 - 1.9165 1.0251 - 1.8094 0.9849 - 1.715 0.9447 - 1.6313 0.9045 - 1.557 0.8643 - 1.4908 0.8241 - 1.4316 0.7839 - 1.3787 0.7437 - 1.3312 0.7035 - 1.2887 0.6633 - 1.2504 0.6231 - 1.216 0.5829 - 1.1852 0.5427 - 1.1575 0.5025 - 1.1326 0.4623 - 1.1105 0.4221 - 1.0908 0.3819 - 1.0734 0.3417 - 1.0581 0.3015 - 1.0447 0.2613 - 1.0333 0.2211 - 1.0237 0.1809 - 1.0158 0.1407 - 1.0095 0.1005 - 1.0048 0.0603 - 1.0017 0.0201 - 1.0002 0.0201 1.0002 0.0603 1.0017 0.1005 1.0048 0.1407 1.0095 0.1809 1.0158 0.2211 1.0237 0.2613 1.0333 0.3015 1.0447 0.3417 1.0581 0.3819 1.0734 0.4221 1.0908 0.4623 1.1105 0.5025 1.1326 0.5427 1.1575 0.5829 1.1852 0.6231 1.216 0.6633 1.2504 0.7035 1.2887 0.7437 1.3312 0.7839 1.3787 0.8241 1.4316 0.8643 1.4908 0.9045 1.557 0.9447 1.6313 0.9849 1.715 1.0251 1.8094 1.0653 1.9165 1.1055 2.0381 1.1457 2.1768 1.1859 2.3353 1.2261 2.5166 1.2663 2.7236 1.3065 2.9587 1.3467 3.2223 1.3869 3.5104 1.4271 3.8108 1.4673 4.0991 1.5075 4.3366 1.5477 4.4775 1.5879 4.4877 1.6281 4.3645 1.6683 4.1383 1.7085 3.8548 1.7487 3.5542 1.7889 3.2634 1.8291 2.9958 1.8693 2.7564 1.9095 2.5453 1.9497 2.3604 1.9899 2.1988 2.0302 2.0574 2.0704 1.9334 2.1106 1.8244 2.1508 1.7282 2.191 1.643 2.2312 1.5674 2.2714 1.5 2.3116 1.4399 2.3518 1.3861 2.392 1.3379 2.4322 1.2946 2.4724 1.2558 2.5126 1.2209 2.5528 1.1895 2.593 1.1613 2.6332 1.1361 2.6734 1.1136 2.7136 1.0936 2.7538 1.0758 2.794 1.0602 2.8342 1.0466 2.8744 1.0349 2.9146 1.025 2.9548 1.0168 2.995 1.0103 3.0352 1.0054 3.0754 1.0021 3.1156 1.0003 3.1558 1.0001 3.196 1.0014 3.2362 1.0043 3.2764 1.0087 3.3166 1.0147 3.3568 1.0224 3.397 1.0318 3.4372 1.043 3.4774 1.056 3.5176 1.071 3.5578 1.0881 3.598 1.1075 3.6382 1.1292 3.6784 1.1536 3.7186 1.1809 3.7588 1.2113 3.799 1.2451 3.8392 1.2828 3.8794 1.3247 3.9196 1.3714 3.9598 1.4235 1992 mat :

NL 4 - 0.5882 3.9598 - 0.5995 3.9196 - 0.6111 3.8794 - 0.6231 3.8392 - 0.6353 3.799 - 0.648 3.7588 - 0.661 3.7186 - 0.6744 3.6784 - 0.6882 3.6382 - 0.7024 3.598 - 0.7171 3.5578 - 0.7322 3.5176 - 0.7478 3.4774 - 0.7638 3.4372 - 0.7804 3.397 - 0.7975 3.3568 - 0.8151 3.3166 - 0.8334 3.2764 - 0.8522 3.2362 - 0.8716 3.196 - 0.8917 3.1558 - 0.9125 3.1156 - 0.934 3.0754 - 0.9562 3.0352 - 0.9792 2.995 - 1.003 2.9548 - 1.0277 2.9146 - 1.0532 2.8744 - 1.0797 2.8342 - 1.1071 2.794 - 1.1356 2.7538 - 1.1651 2.7136 - 1.1957 2.6734 - 1.2275 2.6332 - 1.2605 2.593 - 1.2948 2.5528 - 1.3304 2.5126 - 1.3674 2.4724 - 1.406 2.4322 - 1.446 2.392 - 1.4878 2.3518 - 1.5312 2.3116 - 1.5765 2.2714 - 1.6236 2.2312 - 1.6728 2.191 - 1.7241 2.1508 - 1.7775 2.1106 - 1.8334 2.0704 - 1.8917 2.0302 - 1.9525 1.9899 - 2.0162 1.9497 - 2.0827 1.9095 - 2.1522 1.8693 - 2.225 1.8291 - 2.3011 1.7889 - 2.3808 1.7487 - 2.4642 1.7085 - 2.5516 1.6683 - 2.6432 1.6281 - 2.7391 1.5879 - 2.8397 1.5477 - 2.9451 1.5075 - 3.0556 1.4673 - 3.1715 1.4271 - 3.293 1.3869 - 3.4205 1.3467 - 3.554 1.3065 - 3.6941 1.2663 - 3.8408 1.2261 - 3.9946 1.1859 - 4.1555 1.1457 - 4.324 1.1055 - 4.5001 1.0653 - 4.684 1.0251 - 4.8759 0.9849 - 5.0759 0.9447 - 5.284 0.9045 - 5.5001 0.8643 - 5.7239 0.8241 - 5.9553 0.7839 - 6.1938 0.7437 - 6.4387 0.7035 - 6.6892 0.6633 - 6.9445 0.6231 - 7.2032 0.5829 - 7.4639 0.5427 - 7.7248 0.5025 - 7.9839 0.4623 - 8.2391 0.4221 - 8.4877 0.3819 - 8.7271 0.3417 - 8.9544 0.3015 - 9.1667 0.2613 - 9.3608 0.2211 - 9.5339 0.1809 - 9.6831 0.1407 - 9.8059 0.1005 - 9.9 0.0603 - 9.9638 0.0201 - 9.996 0.0201 9.996 0.0603 9.9638 0.1005 9.9 0.1407 9.8059 0.1809 9.6831 0.2211 9.5339 0.2613 9.3608 0.3015 9.1667 0.3417 8.9544 0.3819 8.7271 0.4221 8.4877 0.4623 8.2391 0.5025 7.9839 0.5427 7.7248 0.5829 7.4639 0.6231 7.2032 0.6633 6.9445 0.7035 6.6892 0.7437 6.4387 0.7839 6.1938 0.8241 5.9553 0.8643 5.7239 0.9045 5.5001 0.9447 5.284 0.9849 5.0759 1.0251 4.8759 1.0653 4.684 1.1055 4.5001 1.1457 4.324 1.1859 4.1555 1.2261 3.9946 1.2663 3.8408 1.3065 3.6941 1.3467 3.554 1.3869 3.4205 1.4271 3.293 1.4673 3.1715 1.5075 3.0556 1.5477 2.9451 1.5879 2.8397 1.6281 2.7391 1.6683 2.6432 1.7085 2.5516 1.7487 2.4642 1.7889 2.3808 1.8291 2.3011 1.8693 2.225 1.9095 2.1522 1.9497 2.0827 1.9899 2.0162 2.0302 1.9525 2.0704 1.8917 2.1106 1.8334 2.1508 1.7775 2.191 1.7241 2.2312 1.6728 2.2714 1.6236 2.3116 1.5765 2.3518 1.5312 2.392 1.4878 2.4322 1.446 2.4724 1.406 2.5126 1.3674 2.5528 1.3304 2.593 1.2948 2.6332 1.2605 2.6734 1.2275 2.7136 1.1957 2.7538 1.1651 2.794 1.1356 2.8342 1.1071 2.8744 1.0797 2.9146 1.0532 2.9548 1.0277 2.995 1.003 3.0352 0.9792 3.0754 0.9562 3.1156 0.934 3.1558 0.9125 3.196 0.8917 3.2362 0.8716 3.2764 0.8522 3.3166 0.8334 3.3568 0.8151 3.397 0.7975 3.4372 0.7804 3.4774 0.7638 3.5176 0.7478 3.5578 0.7322 3.598 0.7171 3.6382 0.7024 3.6784 0.6882 3.7186 0.6744 3.7588 0.661 3.799 0.648 3.8392 0.6353 3.8794 0.6231 3.9196 0.6111 3.9598 0.5995 1992 mat :

X Integrand 1 col : Y Integrand 2 col : 12 mat vx NL 1 col : vy NL 2 col : 12 mat 21 mat

x λ 14 - x ≤ (x 4 ≤ (& cases : x f vx vy x ainterp : ζ F X Y ζ linterp : 31 mat

DECEPTIVE EXAMPLE ... No matter how faithful 'splines' may look they don't replace the function or model. spline [7 min] ... F(u) [0.6 min] ... N199

elliptic  integral 1rst kind k 0.975 : u F 11 k 2^u sin (2^* - sqrt / : 21 line

recast NON-linear path argument f x f :

x f x λ * x F x λ * 2 1 sys

escape division by 0 ε 1015 -^:

shape/magnitude m 1 :

'x' is the running  canvas ...
as well the running discrete x Int u F u ε m f * int :

0 Int 18.094

same from either
elliptic/spline ζ F ζ 4 - 4 int u F u 4 - 4 int 21 sys 13.6162 13.6162 21 sys

lo 4 - : hi 4 : N 199 : 13 mat

t0 1 time :

INT u lo hi lo hi lo - (N / + range : i 1 u rows range ʃ i el u i el Int eval : for u ʃ augment 31 line :

INT INT :

1 time t0 - min 0.6

Export discrete NON-linear

INT 2 1 sys

Spline canvas ... 8 min plot

x Int x λ * 2 1 sys

Example 4: Resume the non-linear integration

x λ 14 - x ≤ (x 4 ≤ (& cases :

μ 0 :

σ 1 :

u F u μ σ pdf.Normal :

x F x erf 2 1 sys

x f 4 x sin * :

non-linear path of integration f x f :

'x' is the running  canvas ...
as well the running discrete x Int u F u 0 f int :

lo 4 - : hi 4 : N 199 : 13 mat

symbol below ʃ

user 1.5 :

0 Int 0

1 Int 0.4996

interpolate «» user Int : 0.5

INT u lo hi lo hi lo - (N / + range : i 1 u rows range ʃ i el u i el Int eval : for u ʃ augment 31 line :

Scalar/Interpolate NON-linear

x F x Int x λ * user «» + 20 black 1 5 mat 3 1 sys

Export discrete NON-linear

INT user «» + 20 black 1 5 mat 3 1 sys

Example 5: Resume the non-linear erf(x) integration

μ 0 :

σ 1 :

x λ 14 - x ≤ (x 4 ≤ (& cases :

u F u μ σ pdf.Normal :

NON-linear path x f 4 x sin * :

x F x erf 2 1 sys

NON-linear path rejected x f x erf :

a 2 : k 1.25 : updn 1 - : xo 0 : 14 mat

4 parameters 
erf(x) close
+ manipulation. x f a k x xo - (* a * exp * 1 k x xo - (* a * exp + / updn + :

0 f 0.000000000000000

x erf x f 2 1 sys

non-linear path of integration f x f :

'x' is the running  canvas ...
as well the running discrete x Int u F u 0 f int :

1 Int 0.3019

0 Int 0

3 Int 0.3411

user 1 :

lo 4 - : hi 4 : N 199 : 13 mat

symbol below ʃ

interpolate «» user Int : 0.3019

INT u lo hi lo hi lo - (N / + range : i 1 u rows range ʃ i el u i el Int eval : for u ʃ augment 31 line :

Scalar/Interpolate NON-linear

x F x Int x λ * user «» + 20 black 1 5 mat 3 1 sys

Export discrete NON-linear

INT user «» + 20 black 1 5 mat 3 1 sys x erf vectorize

x 1.5 - 1.5 1.49 - range :

x erf