plot(XY,a,b,c) ..... Deriv(data,type,x)

94053bef-e8ac-4b56-b142-18f333f748c5 57 3 5 decimal

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

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 :

data type x Deriv X XY 1 col : Y XY 2 col : x f X Y x linterp type 0 ≡ X Y x ainterp type 1 ≡ X Y x cinterp cases : h 0.001 : x deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x f x deriv 1 X 1 el x ≤ (x X X rows el ≤ (& cases * 21 mat 61 line :

a b N x algo Export u xstep b a - N / : i N 1 range xx i 1 + el xstep i * : for xx a + 31 line : i 1 u rows range d i el u i el algo eval : for u d augment 31 line :

XY 3.52 - 0.358 - 3 - 0.213 - 2 - 0.744 1 - 0.261 0 0.518 1 0.323 62 mat :

=== user inspect ===

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

X Y x linterp type 0 ≡ X Y x ainterp type 1 ≡ X Y x cinterp cases

splined data XY type x Deriv 1 el

derivative XY type x Deriv 2 el

The three types splined derivatives

XY 0 x Deriv 2 el XY 1 x Deriv 2 el XY 2 x Deriv 2 el 3 1 sys

The three types splined data set

XY 0 x Deriv 1 el XY 1 x Deriv 1 el XY 2 x Deriv 1 el X Y . 14 black plotG 4 1 sys

4

=== user export ===

XY 3.52 - 0.358 - 3 - 0.213 - 2 - 0.744 1 - 0.261 0 0.518 1 0.323 62 mat :

X XY 1 col :

Y XY 2 col :

x f X Y x cinterp :

x c 14 - x ≤ (x 1.5 ≤ (& cases :

typical increment h 0.001 :

x deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* :

L 3.52 - : H 1 : N 100 : 13 mat Φ L H N x f Export : Δ L H N x deriv Export : 31 sys

x f x c * x deriv x c * XY . 14 black plot 3 1 sys Φ Δ XY . 14 black plot 3 1 sys

About the quadratic spline

Nothing governs the choice between cubic/qudaratic.

That's not so true.The conventional cubic is analytical up to

the 2nd order derivative. The quadratic => up to the 1rst order.

Conventional cubic is the worst to interpolate the foot of

Gaussian. The Akima [Smath ainterp] is superb for those

Gaussians. The quadratic is as good as Akima ainterp.

The quadratic scalar interpolator is immediate as given in this

toolbox document ready to use.

Cubic junctions are smooth, quadratic junctions look smooth

but aren't analyticaly [discontinuous]... they just meet.

]]> plotG ...Hatch

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 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 :

XY 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 :

X XY 1 col : Y XY 2 col : data X Y . 14 black plotG : 31 line

D 1 el 0 : i 26 range D i el 2 Y i el Y i 1 - el - X i el X i 1 - el - / (* D i 1 - el - : for 21 line

Smath 6179 takes index as argument ... eval(,) j x Q D j el 2 / x X j 1 - el - (2^X j el X j 1 - el - (/ * D j 1 - el 2 / x X j el - (2^X j el X j 1 - el - (/ * - Y j el + D j el 2 / X j el X j 1 - el - (* - eval :

2 x Q 3 x Q 4 x Q 5 x Q 6 x Q data 6 1 sys Scalar interpolation

The companions scalar interpolation

Local interpolation wrt segments.
The "if" does not recognise. x Quad j local assignment n 6 : x X 2 el ≤ n 1 : x X n el ≥ n n : i 2 n range x X i el ≤ break return 'j' < (i-1) if for if if i (41 line : j x Q 21 line :

Scalar algo style interpolation 
over all 'J' segment(s) x Xdata J search index n 6 : x Xdata 2 el ≤ n 1 : x Xdata n el ≥ n n : resume next if if i 2 n range x Xdata i el ≤ break close if for J i (: 51 line :

u Interp u X J u Q :

Identical interpolation 0.375 Interp 0.375 Quad 21 mat 0.12246 0.12246 21 mat

Quad t 0 0.5 0.005 range : i 1 t rows range vy i el t i el Interp eval : for t vy augment t 0.5 + vy reverse augment 21 mat 31 line :

Estimated model function β x φ β 1 el β 1 el x β 2 el - (2^+ / :

β 0.002 0.5 21 mat :

Quad 1 el rows 101

An excellent Quadratic spline scalar interpolation.

Quad 1 el data 0.4585 0.5 + 14 red 1 5 mat Quad 2 el 4 1 sys data β x φ 2 1 sys X Y x cinterp β x φ Quad 1 el data 4 1 sys

Assuming no model could be found, the quadratic fit would be good enough for purpose ... => Set "ainterp" to help the casual Gaussian foot.

The 5 linear derivatives of the Quadratic h 0.001 : x f x Interp : x deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x c 10 x ≤ (x 0.5 ≤ (& cases : 41 line

x deriv x c * 0.1 1 0.1 1 - 22 mat 0.2 1 0.2 1 - 22 mat 0.3 1 0.3 1 - 22 mat 0.4 2.5 0.4 1 - 22 mat 51 sys

x deriv x c * 0.1 1 0.1 1 - 2 2 mat 0.2 1 0.2 1 - 2 2 mat 0.3 1 0.3 1 - 2 2 mat 0.4 2.5 0.4 1 - 2 2 mat 5 1 sys

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Note Numerical integrator supports neither ordinary Quadratic nor Holistic. Both will be integrated via the FD [Finite Differences] discrete algorithm.

β x φ x 0 0.5 int 0.0663

CumulΣ Quad 1 el Σ Quad 1 el Σ rows row : 0.5 0.0631 12 mat

Quad 1 el Σ