plot(data,a,b,c)

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plot(data,a,b,c)

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 :

ChebyShev Polynomials

◄  reserved for "linfitCheby" n x T n x acos * cos :

j 05 range :

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 :

j x T vectorize 1 x 2 x 2^* 1 - x 3 - 4 x 2^* + (* 8 x 2^* 1 - x 2^+ (* 1 + x 54 x 2^* 5 - 4 x 2^* + (* + (* 61 mat

λ Cheby Ω λ x T x coeffs maple eval : n order the collection n 1 Ω rows 2 range stack : ct 2 if λ odd ct 1 if λ even ct λ 2 mod 1 ≡ 21 if : output coeffs ordered wrt 'n' i n New ct el Ω i el : ct ct 2 + : 21 line for New 81 line 21 line :

maxorder 30 ≡ n 9 : (order ≡ C n Cheby : r C polyroots : r r eval : 51 line

r 0 0.342 - 0.342 0.6428 - 0.6428 0.866 - 0.866 0.9848 - 0.9848 91 mat

C 090120 - 04320576 - 0256101 mat

Plot roots of ChebyShev roots r r 0 * augment : roots roots . 14 black plot : 21 line

0 x T 1 x T 2 x T 3 x T roots n x T 6 1 sys

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

Abramowitz x f x 1 - 1 int 2 n / x i el f i 1 n sum * ≡

11½ Decimals Abramowitz ... Mathcad 11 Wr 0 0.174532925199 0.342020143326 - 0.523598775598 0.342020143326 0.872664625997 0.642787609687 - 1.221730476396 0.642787609687 1.570796326795 0.866025403784 - 1.919862177194 0.866025403784 2.268928027593 0.984807753012 - 2.617993877991 0.984807753012 2.96705972839 92 mat

F.B. Hildebrand
.... 8.8.10 x f 1 x 2^- sqrt / x 1 - 1 int W i el x i el f * i 1 n sum ≡

Collect vector otherwise ... spool to file

start 1 : end 9 : 12 mat j start end range : k j vT k 1 + start - el k 1 - x T : for 31 line

vT 1 x 2 x 2^* 1 - x 3 - 4 x 2^* + (* 8 x 2^* 1 - x 2^+ (* 1 + x 54 x 2^* 5 - 4 x 2^* + (* + (* 1 - 2 x 2^* 98 x 2^* 3 - 2 x 2^* + (* + (* + x 7 - 8 x 2^* 72 x 2^* 7 - 4 x 2^* + (* + (* + (* 32 x 2^* 1 - x 2^54 x 2^* 2 - x 2^+ (* + (* + (* 1 + 91 mat

CWD C:\SmathFile CurrentDirectory :

insert the formula to store vT vT :

vT CWD Cheby wfile 1

x Cheby CWD Cheby rfile :

Example from linfitCheby Treasury . . .

]]> linfit modules

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 :

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

order 7 :

0. reserve the 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 1521 mat

1. Convert  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 o 5 black 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 isolated. 5. Collocate the "linfitCheby" Colln(x) 6. Reverse back the fit & the data set

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

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

idx transpose 1111111118 mat

unary initial 'β'
'n' vector gradient β idx : n idx rows : 21 sys

<= vector gradient x β φ x β f n φ :

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

c Cheby 6 round vectorize : c c eval : 21 sys

5. Collocate the "linfitCheby" x Colln x p x β p : c k el x p k el * (k 1 order sum 11 line 21 line :

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

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

x Colln data 2 1 sys

6. linfitCheby in the original XY domain

x X Colln Origin 2 1 sys

Extrapolate recovers the orginal range of 
the DE model test not given in XY data set Extrapolate ω 0.75 6 0.8 range : n 1 ω rows range Ex n el ω n el X Colln eval : for ω Ex Re augment 31 line :

Smath 6179 creates extra imaginary residual of low value, clean export.

CWD C:\SmathFile CurrentDirectory :

x V x X Colln op maple :

x V CWD Chebyfit wfile 1

x Φ CWD Chebyfit rfile :

x Q x Φ n el n 17 sum 11 line :

Extrapolate Origin 2 1 sys x Q 2 1 sys

Recall from a new or existing document

plot(data,a,b,c)

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 :

CWD C:\SmathFile CurrentDirectory :

x Cheby CWD Cheby rfile :

j 19 range :

x Cheby j el 1 x 2 x 2^* 1 - x 3 - 4 x 2^* + (* 8 x 2^* 1 - x 2^+ (* 1 + x 54 x 2^* 5 - 4 x 2^* + (* + (* 1 - 2 x 2^* 98 x 2^* 3 - 2 x 2^* + (* + (* + x 7 - 8 x 2^* 72 x 2^* 7 - 4 x 2^* + (* + (* + (* 32 x 2^* 1 - x 2^54 x 2^* 2 - x 2^+ (* + (* + (* 1 + 91 mat

ChebyShev polyroots otherwise . . . for export . . . User Applications

]]> plot(data,a,b,c)

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 :

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 :

clear from prevuous r 0 :

1. very fast increasing 'n' 2. @ high 'n' uncertified D

user 'n' n 11 :

x N n x T x expand maple : x N x N op maple : x 1 : v x N reverse : 41 line

i 1 v rows range n 2 mod 0 ≡ N 2 i * 1 - el v i el : N 2 i * el v i el : if for

r N polyroots : r r eval : 21 sys

'n' even

Plot roots of ChebyShev roots r r 0 * augment : roots roots . 14 black plot : 21 line

'n' odd

roots n x T 2 1 sys

N 011 - 022001232 - 0281602816 - 01024121 mat

r 0 0.281732556841 - 0.281732556841 0.540640817456 - 0.540640817456 0.755749574354 - 0.755749574354 0.909631995355 - 0.909631995355 0.989821441881 - 0.989821441881 111 mat