<p> Truncated 'C' for visible export </p>

3baae596-520b-4b5a-94bd-e29ec6c493ab 17 6 12 decimal

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

1. The Quadratic-Symmetric Holistic

t0 1 time :

<= Holistic matrix does not take '0' ε 1012 -^:

data ε 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 ε - 0.00794 112 mat :

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 :

X data 1 col : Y data 2 col : xy data . 14 black plot : 31 line

Tridiagonal matrix whose inverse is needed to solve for 
the quadratic coefficients, given the vector 'S' ... => A n X rows : i 13 n 1 - (* (range j 13 n 1 - (* (range T i j el 0 : for for i 1 n 1 - (range j 12 range k 13 range T 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 T 2 n 1 - (* i + 3 i * 8 - k + j 3 * + (el 1 - (j^- (2 X i el * (k 1 -^* : for for for T 13 el 1 : T 61 line :

Vector associated with the 
Holistic tridiagonal 'A'. S n X rows : i 13 n 1 - (* (range V i el 0 : for i 1 n 1 - (range j 01 range V 2 i 1 + (* 2 - j + el Y i j + el : for for V 41 line :

A 1 -^S * sometimes needed ≡ 11 line

C A 1 -^S * :

'S' the quadratic solving vector associated with 'A' for solving 'C' vector of the coefficients

CWD C:\SmathFile CurrentDirectory :

We construct individual segments with their respective 'α' set of coefficients. As well, we isolate 'C' by spooling to file.

C CWD C_QuadraticHolistic wfile 1

C CWD C_QuadraticHolistic rfile :

α u f u C 2 α + el u * C 1 α + el + (* C α el + :

C rows 30

1 X 1 el f 4 X 2 el f 7 X 3 el f 10 X 4 el f 13 X 5 el f 13 X 6 el f 61 mat 0.00794 0.012 0.022 0.048 0.167 161 mat

<= Sanity check

x c 10 x ≤ (x 1 ≤ (& cases :

x collect 1 x f x c * 4 x f x c * 7 x f x c * 10 x f x c * 13 x f x c * 13 0.232 - x + f x c * 10 0.443477 - x + f x c * 7 0.758449 - x + f x c * 4 0.86835 - x + f x c * 11 - x + f x c * 101 sys :

x collect xy 2 1 sys

Identify the pertaining quadratic segments for the suite below ...

1 x f 4 x f 7 x f 10 x f 13 x f 5 1 sys xy 2 1 sys <p> Truncated 'C' for visible export </p>

C 0.00794 0.0406 0 0.01388 0.0782 - 0.594 0.03036 0.243 - 1.006 0.68628 4.6158 - 8.294 9.4562 48.4654 - 63.106 63.7468 - 244.3466 229.706 - 127.34552 392.6278 - 301.106 163.18136 - 437.449 291.806 - 211.35432 498.8902 - 293.406 263.48226 - 556.3022 292.812 - 301 mat :

...  'α' the symmetric
segment recognotion . α u f u C α 1 + el C α 2 + el u * + (* C α el + :

1 u f 4 u f 7 u f 10 u f 13 u f 51 mat 3972030 u * + 50000 / 3475 u * 391 - 2970 u * + (* + 25000 / 75925 u * 243 - 1006 u * + (* + 25000 / 171575 u * 23079 - 41470 u * + (* + 25000 / 47281 u 242327 - 315530 u * + (* + 5000 / 51 mat

2. The Quadratic Holistic segments

x s1 1 x f X 1 el x ≤ (x X 2 el ≤ (& cases :

x s4 10 x f X 4 el x ≤ (x X 5 el ≤ (& cases :

x s1 x s2 x s3 x s4 x s5 xy 61 sys

x s2 4 x f X 2 el x ≤ (x X 3 el ≤ (& cases :

x s5 13 x f X 5 el x ≤ (x X 6 el ≤ (& cases :

x s3 7 x f X 3 el x ≤ (x X 4 el ≤ (& cases :

Expose the weak Akima region that can be replaced by the better Holistic segment(s). They will join smooth for a final reconstruction ... left as home work exercise.

x s1 x s2 x s3 x s4 x s5 xy 6 1 sys

3. The Quadratic Holistic Reconciliation

x Holistic 1 x f X 1 el x ≤ (x X 2 el ≤ (& 4 x f X 2 el x ≤ (x X 3 el ≤ (& 7 x f X 3 el x ≤ (x X 4 el ≤ (& 10 x f X 4 el x ≤ (x X 5 el ≤ (& 13 x f X 5 el x ≤ (x X 6 el ≤ (& cases :

x Holistic

x spline X Y x ainterp :

x range 10 x ≤ (x 0.5 ≤ (& cases :

x s 1 X 5 el x ≤ (x X 7 el ≤ (& cases :

x f x Holistic :

h 0.001 :

X 5 el X 6 el X 7 el 31 mat 0.4 0.5 0.6 31 mat

Y 5 el Y 6 el Y 7 el 31 mat 0.167 1 0.167 31 mat

x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / eval :

x h f D1 x range * xy x Holistic x range * x spline x s * 3 1 sys

x r5 1 X 1 el x ≤ (x X 5 el ≤ (& cases :

x r6 1 X 5 el x ≤ (x X 7 el ≤ (& cases :

x spline x r6 * x Holistic x r5 * 1 x - Holistic 1 x - r5 * xy 41 sys

Fitted from the proposed symmetric reconciliation

x spline x r6 * x Holistic x r5 * 1 x - Holistic 1 x - r5 * xy 4 1 sys

At this point: discretize the three components, export. There are advanced techniques to re-fit ... not exposed here.

δ 0.01 :

sanity the symmetry X 5 el δ + spline X 7 el δ - spline 21 mat 0.203021 0.203021 21 mat

1 time t0 - s 9.2