plotG

11d58b6c-f868-4864-9e50-e72e6fa5734a 29 4 5 decimal

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

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 :

The 3 types of "cubic spline"

4 appVersion 0.98.6179.21440 ≡

Essentially, the coefficients of the cubic spline consists in solving the basis tridiagonal matrix wrt the 'br' vector of derivatives . Independently of the type of cubic spline [type: 0, 1, 2 = l, p, c] two steps of preparation are needed [collapsed area]: 0. Install the constant vector 'br' 1. Install the blank tridiagonal matrix.

XY 3.52 - 2.358 - 3 - 2.213 - 2 - 1.256 - 1 - 1.739 - 0 1.482 - 1 1.277 - 62 mat :

1. Assume a physical project evaluates @ points. 2. Assume this project has become a STANDARD. 3. Assume it has been populated 'cubic linear'. 4. Assume your Smath project is OK but can't be made scalar like in [1.] 5. @ this point if you can't populate 'cubic lin' 6. Then, your project does NOT meet the STANDARD.

X XY 1 col :

Y XY 2 col :

N Y rows 1 - : 5

r 2 N range :

Respective Tridiagonal Matrices ... Collect the 's' coefficients

Prep.0 Install the constant vector 'br'

i r br i el 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 - / (- eval : br N 1 + el 0 : 21 line for

br 0 0.6782 1.44 - 0.74 0.052 - 061 mat

Prep.1 Install the tridiagonal matrix

M 11 el 1 :

M N 1 + N 1 + el 1 :

M 10000000000000000000000000000000000166 mat

Observe the population of the rows upper left row ---------------------- ----------------------lower right row respectively wrt linear/parabolic/cubic. The basis tridiagonal is the "linear", further modified to suit user l_p_c .

0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

slin M 1 -^br * :

M 100000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00000166 mat

slin 0 2.419 3.285 - 2.0808 0.5982 - 061 mat

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

sp M 1 -^br * :

M 11 - 0000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00001 - 166 mat

sp 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 mat

2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

sc M 1 -^br * eval :

M 1 1.52 - 0.52 000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00012 - 166 mat

sc 4.0766 1.6415 3.0412 - 1.8833 0.052 - 1.9873 - 61 mat

linear ends slin 0 2.419 3.285 - 2.0808 0.5982 - 061 mat

parabolic ends sp 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 mat

cubic ends sc 4.0766 1.6415 3.0412 - 1.8833 0.052 - 1.9873 - 61 mat

◄ Sanity Mathcad 11

In there, plug the 's' of the selected type of spline.
'slin & sp' join cubic 2 points entrance/departutre.
'sc' joins cubic 3 points entrance/departure. s slin :

abcd

Collect the cubic 's' components for C(j,u) abcd i 1 X rows 1 - range hs i el s i 1 + el s i el - : hy i el Y i 1 + el Y i el - : hx i el X i 1 + el X i el - : 31 line for i 1 X rows 1 - range a i el hs i el 6 hx i el * / : b i el s i el 2 / : c i el hy i el hx i el / hx i el 6 / s i 1 + el 2 s i el * + (* - (: d i el Y i el : 41 line for a b c d augment 31 line :

abcd 0.7753 0 0.0692 2.358 - 0.9507 - 1.2095 0.6981 2.213 - 0.8943 1.6425 - 0.2652 1.256 - 0.4465 - 1.0404 0.3369 - 1.739 - 0.0997 0.2991 - 0.4044 1.482 - 54 mat

Recover segments coeff's [a,b,c,d]

a abcd 1 col :

c abcd 3 col :

b abcd 2 col :

d abcd 4 col :

j u C i j : a i el u X i el - (3^* b i el u X i el - (2^* + c i el u X i el - (* + d i el + 21 line :

ptsSpec

ptsSpec k 1 : i k X rows range data i el Sym size clr 13 mat : k k 1 + : 21 line for v data 1 el : j 2 data rows range v v data j el stack : for v 51 line :

size 14 :

clr black :

Sym . :

support X Y ptsSpec augment :

plot 1 x C 2 x C 3 x C 4 x C 5 x C support 61 sys :

J 3 :

Cubic spline segments through data points

plot

cubic segment(s) observer 'J'

J x C 4 x C support 3 1 sys

Demo: collect the 'j' segments u Spline j local assignment n X rows : u X 1 el ≤ n 1 : u X n el ≥ n n 1 - : i 1 n range u X i el ≤ break return 'j' < (i-1) if for if if i 1 - (41 line : j u C 21 line :

x segments x X 1 el ≥ (x X X rows el ≤ (& x Spline Not interpolated if :

Entrance/departure observer [slin, sp, sc]

1 x C 5 x C support 3 1 sys Mathcad 8 Pro. code conversion

s X Y t InterpSpline Mathcad 8 Pro. code conversion t X 1 el ≤ idx 1221 mat : t X N el ≥ idx N N 1 + 21 mat : i 1 : X i el t < i i 1 + : while idx i 1 - i 21 mat : 31 line if if hx X idx 2 el el X idx 1 el el - (: hy Y idx 2 el el Y idx 1 el el - (: hs s idx 2 el el s idx 1 el el - (: 31 line a hs 6 hx * / : b s idx 1 el el 2 / : c hy hx / hx 6 / s idx 2 el el 2 s idx 1 el el * + (* - (: d Y idx 1 el el : 41 line a t X idx 1 el el - (3^* b t X idx 1 el el - (2^* + c t X idx 1 el el - (* + d + 51 line :

recast the code x cubic s X Y x InterpSpline :

Discretise for export L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

L X 1 el : H X X rows el : Npts 100 : 13 mat U L H Npts xd : 21 mat

... I don't have Akima code x Akima X Y x ainterp :

Ω i 1 U rows range vy i el U i el cubic eval : for U vy augment 21 line :

Discrete ranged data user

Ω x Akima support 3 1 sys

x blue 1 3.75 - x ≤ (x 2.375 - ≤ (& cases :

x fushia 1 1.75 - x ≤ (x 1.25 ≤ (& cases :

Cubic spline segments through data points ... [slin .. sp] join 2 points segments

1 x C x blue * 2 x C 3 x C 4 x C x fushia * 5 x C support 8 1 sys

data 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X data 1 col : Y data 2 col : 12 mat data X Y + 15 black plotG : 21 line

x Akima X Y x ainterp : x Cinterp X Y x cinterp : x λ 10 x ≤ (x 1 ≤ (& cases : 31 sys

Akima has about unique application for some Gaussian foot

x Akima x λ * x Cinterp x λ * data 3 1 sys

Example 3 . . . Short form Quadratic Holistic . . .

]]> Tridiagonal System: [A, S]

Matrix whose inverse will solve for the quadratic coeff's X A n X rows : i 13 n 1 - (* (range j 13 n 1 - (* (range Hol i j el 0 : for for i 1 n 1 - (range j 12 range k 13 range Hol 2 i * 1 - j + (3 i * 3 - k + (el X i 1 - j + el (k 1 -^: for for for i 2 n 1 - (range j 12 range k 12 range Hol 2 n 1 - (* i + 3 i * 8 - k + j 3 * + (el 1 - (j^- (2 X i el * (k 1 -^* : for for for Hol 13 el 1 : Hol 61 line :

X Y S n X rows : i 13 n 1 - (* (range Hol i el 0 : for i 1 n 1 - (range j 01 range Hol 2 i 1 + (* 2 - j + el Y i j + el : for for Hol 41 line :

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 :

Quadratic/Holistic adapted Mathcad 11 jmG: 2021-05-16

data 1015 -^0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 162 mat :

X data 1 col : Y data 2 col : 12 mat support data . 14 black plot : 21 mat

4 appVersion 0.98.6179.21440 ≡

A X A :

S X Y S :

x f C 1 el x C 2 el C 3 el x * + (* + C 4 el x C 5 el C 6 el x * + (* + C 7 el x C 8 el C 9 el x * + (* + C 10 el x C 11 el C 12 el x * + (* + C 13 el x C 14 el C 15 el x * + (* + 51 mat :

C A 1 -^S * :

A 1 -^S * sometimes needed ≡ 11 line

isolate C C eval :

C rows 15

X data 1 col : Y data 2 col : 12 mat x h 10 x ≤ (x 0.5 ≤ (& cases : 21 sys

x y axes x y / - :

Collect piecewise plot for solving ...

x f 1 el x f 2 el x f 3 el x f 4 el x f 5 el data 0.4584 0.5 1 2 mat 7 1 sys

Scalar companion interpolator ω 0 : t Q n X rows : i 1 n 1 - (range t X i 1 + el ≤ (t X i el ≥ (& ω 0 : j 02 range ω ω C 3 i * 2 - j + el t j^* + : for 21 line out range if for t Q ω : 31 line : 21 line

x f 5 el 0.5 - 0 ≡ x 0.375 0.5 solve 0.4584

preset longer Holistic C 1 el x C 2 el C 3 el x * + (* + C 4 el x C 5 el C 6 el x * + (* + C 7 el x C 8 el C 9 el x * + (* + C 10 el x C 11 el C 12 el x * + (* + C 13 el x C 14 el C 15 el x * + (* + C 16 el x C 17 el C 18 el x * + (* + C 19 el x C 20 el C 21 el x * + (* + # 81 mat

x Q x h * 0 0.0079 . 14 black 0.1 0.012 . 14 black 0.2 0.022 . 14 black 0.3 0.048 . 14 black 0.4 0.167 . 14 black 0.5 1 . 14 black 6 5 mat 3 1 sys