linfit modules

8ad9916f-f23a-4c97-b8cb-3164ceabb66b 12 15 12 decimal

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

s 25 :

Height of source and receiver

t 25 :

δdegrees 45 :

δ δdegrees π 180 / * :

a s δ tan * :

c 50 :

Horizontal separation of source and receiver

a 25

b c a - (: 25

t L t 2^b 2^+ (sqrt s 2^a 2^+ sqrt + (:

Plot L as a function of δdegrees.

δdegrees = 0 to 60 in increments of .1

x L

X 0 b 21 mat :

Y s 021 mat :

X Y Intercept X Y Slope 21 mat 251 - 21 mat

X b c 21 mat :

Y 0 t 21 mat :

x F x t - b x ≤ (x c ≤ (& x - s + 0 x ≤ (x b ≤ (& cases :

X Y Intercept X Y Slope 21 mat 25 - 121 mat

x t - x - s + b 0 + 40 black 1 5 mat 3 1 sys x F a 10 a 10 - 2 2 mat 0 0 . 20 black 1 5 mat c 100 c 10 - 2 2 mat 0 s + 20 red 1 5 mat 5 1 sys

a b s t 41 mat 2525252541 mat

Line of reflection ... demo user x f 1.23456789 x * - 2.56789 + :

Xo x f 0 ≡ x 05 solve : Yo 0 f : 21 mat

Xo Yo 21 mat 2.079991 2.56789 21 mat

°incidence Yo Xo / : ° 70.736

°reflection 90 °incidence ° / - : 19.264

Yo sin Yo * 1 - 0.393711

U 0.393711 Xo * u - 0 ≡ u 0.5 1 solve : 0.819

x f Yo sin Yo * 1 - ( x * U - 2 1 sys

0.393711 Xo * U - 61016 -^* -

========== Else consideration [alyles]

Height of source and receiver s 25 : t 25 : 12 mat

δdegrees 45 :

δ δdegrees π 180 / * :

a s δ tan * :

Horizontal separation of source and receiver c 50 :

b c a - (:

L t 2^b 2^+ (sqrt s 2^a 2^+ sqrt + (:

The PROJECT: Plot L as a function of δdegrees. δdegrees = 0 to 60 in increments of .1

Replace b with (c-a)

δ L t 2^c a - (2^+ (sqrt s 2^a 2^+ sqrt + ≡

recast the project
Replace a with s*tan(δ*pi/180) δ L t 2^c s δ π 180 / * tan * - (2^+ sqrt s 2^s δ π 180 / * tan * (2^+ sqrt + :

X 080 0.1 range :

k 1 X length range Y k el X k el L : for

XY X Y augment :

XY 1 row XY XY rows row 21 mat 0 80.902 12 mat 80 239.095 12 mat 21 mat

recast the project ... Replace a with s*tan(δ*π/180) δ L t 2^c s δ π 180 / * tan * - (2^+ sqrt s 2^s δ π 180 / * tan * (2^+ sqrt + :

x c 10 x ≤ (x 90 ≤ (& cases :

x L x c * 60 96 60 64 2 2 mat 90 1000 90 64 2 2 mat 3 1 sys

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

XY XY 40 Decimate :

CWD C:\SmathFile CurrentDirectory :

XY CWD reflection wfile 1

XY CWD reflection rfile :

linfit modules

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Applicable parts are: Optimiz "numeric" XY f φ β βlinfit 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 Gradient Matrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for Cholesky solver [LS] Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

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 :

Original XY

XY 10 1.16 0.136 1.32 0.269 1.48 0.39 1.64 0.494 1.8 0.572 1.96 0.623 2.12 0.641 2.28 0.625 2.44 0.576 2.6 0.493 2.76 0.378 2.92 0.238 3.08 0.069 3.24 0.115 - 3.4 0.313 - 3.56 0.516 - 3.72 0.718 - 3.88 0.913 - 4.04 1.092 - 4.2 1.251 - 4.36 1.382 - 4.52 1.481 - 4.68 1.542 - 4.84 1.564 - 5 1.543 - 262 mat :

The linfitCheby Procedure

0. reserve for backscale in original domain xy XY : minXd xy 11 el : maxXd xy xy rows 1 el : Origin xy . 10 red plot : 41 line

minXd maxXd 21 mat 07821 mat

order 7 :

1. Convert Original XY => Cheby domain [±1] v i 1 XY rows range v i el i : for v 2 v * v Min - v Max - v Max v Min - / : 21 line : XY v XY 2 col augment : vx XY 1 col : vy XY 2 col : data XY . 10 red plot : 51 line

2. Create ChebyShev Optimiz: Symbolic x β p n 0 order range F n 1 + el β n 1 + el n x acos * cos * : for F 21 line :

x β p β 1 el β 2 el x * β 3 el 2 x acos * cos * β 4 el 3 x acos * cos * β 5 el 4 x acos * cos * β 6 el 5 x acos * cos * β 7 el 6 x acos * cos * β 8 el 7 x acos * cos * 81 mat

0. Create the original reserve 1. The original XY is converted in ChebyShev domain. 2. The ChebyShev vector function is created. 3. Collect the model f(x,β). 4. Cheby coefficients are spooled to file. 5. Collocate the "linfitCheby" Colln(x) 6. Reverse back the fit & the data set

Prepare the 'Unary β' i 1 order 1 + range idx i el 1 : for

3. Collect  the model function x β f x β p n el n 1 order 1 + sum 11 line :

idx transpose 1111111118 mat

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

set the 'Unary β' & length 'n' of the vector gradient

β idx : n idx rows : 21 line

x β φ x β f n φ :

<= vector gradient

The "linfit" coefficients Cheby XY f φ β βlinfit :

CWD C:\SmathFile CurrentDirectory :

Cheby CWD Cheby wfile 1

c CWD Cheby rfile :

c 93.29848 31.509821 30.846456 18.616241 10.428553 4.950607 1.959425 0.61993 81 mat

minXd maxXd 21 mat 07821 mat

x p x β p :

5. Collocate the "linfitCheby" x Colln c k el x p k el * (k 1 order sum 11 line :

6. reverse scale 'x domain' ω X 2 ω * minXd - maxXd - maxXd minXd - / :

test 72 :

5. Data set & the fit in ChebyShev domain [-1..1]

x Colln data 2 1 sys

6. linfitCheby in the original domain

x X Colln Origin test test X Colln + 40 black 1 5 mat 3 1 sys

test X Colln 118.015

0 80.902 12 mat 80 239.095 12 mat 21 mat