<p> plotG, Hatch </p>

8b230bd0-598f-43f1-8967-74eceafb33aa 111 4 5 decimal

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

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 :

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) 
between functions f(x) and g(x) in x-range from x1 to x2 f g u v N hatch dx v u - N / (: h u u f u u g 22 mat : x u v dx - u dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

The Quadratic Spline [Holistic]

t0 1 time :

This Smath work sheet is adapted from a Mathcad work sheet published in the web, years ago. It works fine for the given data set. The construction of the matrix 'A' is complex in the last two "for loops". This demo is given as a classroom session that will be useful in real works.

data 1012 -^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 :

<= Holistic matrix does not take '0'

X data 1 col : Y data 2 col : 21 line

xy X Y . 14 black plotG :

Matrix whose inverse is needed to solve for the quadratic 
coefficients A n X rows : i 13 n 1 - (* range j 13 n 1 - (* range M i j el 0 : for for i 1 n 1 - range j 12 range k 13 range M 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 M 2 n 1 - (* i + 3 i * 8 - k + j 3 * + (el 1 - (j^- (2 X i el * (k 1 -^* : for for for M 13 el 1 : M 61 line :

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

A 1 -^S * sometimes needed ≡ 11 line

C A 1 -^S * :

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

CWD C:\SmathFile CurrentDirectory :

C CWD C_QuadraticHolistic wfile 1

C CWD C_QuadraticHolistic rfile :

j 02 range d d β 3 i * j + el u j^* + : for

In this expansion, 'd' is dummy so that the "stack is not empty" to get the symbolic expansion. Further down, "d = 0"

d d β 3 i * el + u β 23 i * + el u * β 13 i * + el + (* +

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

11243741051352 mat

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

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

xy data 1 col data 2 col . 14 black plotG :

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

Collect and solve individually 13 λ 0.6 + f 0.167 ≡ λ 1 - 0 solve 1 el 10 λ 0.7 + f 0.048 ≡ λ 1 - 0 solve 1 el 7 λ 0.8 + f 0.022 ≡ λ 1 - 0 solve 1 el 4 λ 0.9 + f 0.012 ≡ λ 1 - 0 solve 1 el 1 λ 1 + f 0.00794 ≡ λ 1 - 0 solve 1 el 51 mat

10 λ 0.7 + f 0.048 ≡ λ 1 - 0 solve 1 el 0.443476 -

x collect xy 2 1 sys

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

0.375 Q 0.121699

left½ t 0 0.5 0.005 range : i 1 t rows range H i el t i el Q eval : for t H augment 31 line :

left½ xy 2 1 sys

d 0 :

t Q n X rows : i 1 n 1 - (range t X i 1 + el ≤ (t X i el ≥ (& d 0 : j 02 range d d C 3 i * 2 - j + el t j^* + : for 21 line N/A if for t Q d : 31 line :

To view the complete project, let's discretise Q(t) so that we can add the data points and appreciate.

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 0 :

H 0.5 :

U L H N xd :

N 99 :

Quad i 1 U rows range vy i el U i el Q eval : for U vy augment U 0.5 + vy reverse augment 21 mat 21 line :

vx data 1 col : vy data 2 col : 21 line

global vx vy o 8 black plotG :

The final Quadratic spline

Quad 1 el Quad 2 el global 3 1 sys

The quadratic spline is by far not obsolete. Not knowing more at the point X = 0.5, the sole first fit session is the "Quadratic". It goes in one direction only. The solve block is there just to view all segments.

t f t Q :

h 0.0001 :

Recast the FD deriv in scalar form to support
the  internal "if"  ... success. f x h deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* eval :

f 0.5 h deriv 14.634289

Quadratic derivative

0 x ≤ ( x 0.5 ≤ ( & f x h deriv zombie if 0 2 1 sys

The cumulative Integral transit procedure

u F x Q :

z Int u F u X 1 el x int :

Does NOT support compactum

x Int z Int :

1 time t0 - 20.9 s *