<p> Genfit Utilities </p>

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Abstract: The HyperSine Function ...

HyperSine Identity a b Φ x f a b x Φ - (*^≡ x β f ≡ β 1 el x * exp β 2 el x * β 3 el + sin * ≡

... The HyperSine concludes more underlying mathematical considerations. The complete plot became possible with Mathcad 8 [typical]. Parameters [a,b,Φ] are demo in this document. HyperSine was not published in the MathSoft Collaboratory. I granted Theodore B. a personal copy [Mathcad 11 De Prony project]. However, it became public in my web site "mathengjmg" [~2012]. One may wonder how "sine" can appear out of a function absent of "sine" itself ? Euler had found his own relationship wrt "exp". Hypersine is another "maths mystics". The proof HyperSine generates "sine/cosine" is given below.

Part: 1 => Experimental HyperSine

a 0.5 : b 0.25 - : Φ π 1 / : 31 line

HyperSine/HyperExponential a b Φ x f a b x Φ - (*^:

XY a b Φ x f a b Φ x f - a - b Φ x f Re a - b Φ x f Im 00505 - 22 mat Φ 2 Φ 2 - 22 mat 71 sys :

HyperSineCosine Explorer, HyperExponential Envelopes

u 2 : user phase ≡

Constants @ any phase Φ,
specific to HyperSine a - b Φ Φ f Re a - b Φ Φ f Im 21 mat 1021 mat

Ordinates HyperSineCosine ...
... @ user 'u' search a - b Φ u f Re a - b Φ u f Im - 21 mat 0.512 0.641 - 21 mat

HyperExponentials @ Origin a b Φ 0 f a b Φ 0 f - 21 mat 0.58 0.58 - 21 mat

Part: 2 => Collect Relevant HyperSine Samples

a 0.5 : b 0.25 - : Φ 0 : 31 line

HyperSine/HyperExponential a b Φ x f a b x Φ + (*^:

XY a b Φ x f a b Φ x f - a - b Φ x f Re a - b Φ x f Im 00505 - 22 mat 61 sys :

=================================================================================

size 10 :

color blue :

k f a - b Φ k f Re :

n 20 :

n pts data data Clear : clear previous ≡ data collects in nested form k 0 : k n ≤ k k 1 + : data k 1 + el k k f . size color 15 mat : for Unest 'data' compatible to plot v data 1 el : j 2 data rows range v v data j el stack : for v 71 line :

mixed plot Plot n pts a - b Φ x f Re 21 sys :

X n pts 1 col : Y n pts 2 col : Samples X Y augment : 31 line

1. collect samples, 2. plug in XY Genfit

Plot

Part: 3 => Fit HyperSine Samples

<p> Genfit Utilities </p>

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 :

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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Alt+0126 data n ~ data n col :

Applicable parts are: Optimiz "numeric" XY f φ β minimize X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx-> raw init vector fit i 1 X length range vx i el X i el β f eval : for M Cholesky GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver 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 :

Samples expected exact 15 D XY 011 0.840896415253715 203 1.18920711500272 - 42 - 5 1.68179283050743 - 607 2.37841423000544 849 3.36358566101486 10011 4.75682846001089 - 128 - 13 6.72717132202972 - 14015 9.51365692002176 161617 13.4543426440594 18019 19.0273138400436 - 2032 - 212 mat :

x β f β 1 el x * exp β 2 el x * β 3 el + sin * :

Model function

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

x β φ x β f 3 φ :

<= Gradient Vector Function

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

β 0.17328679514 π 4 / π 2 / 31 mat :

0.15 π 4 / π 2 / 31 mat

<= ..... initial β ........ .... iterate manually ..... <= <= re-inject the last 'β'

re-inject to iterate manually

β 0.17328679514 0.785398163397448 1.5707963267949 31 mat

Refresh this coefficient β 1 el 0.173286795139986 :

Genfit coeff's of fit β XY f φ β minimize :

0.173286795139986327354 Smathonly ≡ 0.173286795139986 ≡

data X Y . 12 black plotG :

Notes about this fitting session: 1. The phase shift β3 is π/2. 2. We try β2 magnitude π/4 3. If we are correct, only the exponential growth β1 needs be found. This will arise when the Δ's will equate 0

data x β f β 1 el x * exp β 1 el x * exp - 4 1 sys

Δ π 4 / β 2 el - 15 round π 2 / β 3 el - 15 round 21 mat : 21015 -^* 21015 -^* - 21 mat

21015 -^* 21015 -^* - 21 mat 0021 mat ≡

Observe this particular example: β 1 el 24 nthroot ln ≡ β 3 el denom β 2 el denom nthroot ln ≡ 2 ln 4 / ≡

X Y^Y X ln * exp ≡

<= remember this relationship

sanity check sample(s) 19 β f 19.0273138400431 -

a b x Φ - (*^b x Φ - (* a ln * exp ≡

Numerical curiosity 24 nthroot ln 2 ln 4 / 21 mat 0.173286795139985 0.173286795139986 21 mat

a b Φ 31 mat 0.5 0.25 - 031 mat

x ε x β f a - b Φ x f Re - :

b x Φ - ( * a - ln * exp Im b x Φ - ( * a - ln * exp Re π - 1 π 1 2 2 mat 3 1 sys

Absolute error ε(x) [expsine] Strictly numerical noise & ULP accumulation. At some different numerical figures, Mathcad produces identical ε(x) pattern.

x ε

a sense of scale: 5.635882*e-15 m ≡ 5.6 fm ≡ 5.6 fermi classical diameter of an electron.

Note that ε(x) is in absolute value rather than "relative error". Both f(x,β) & Re(f(-a,b,Φ,x)) refer to numerical approximations given in "absolute error" [Hart et al.]

Part: 4 => The creation of HyperSine

From the form ...

b x Φ - ( * a - ln * exp Re b x Φ - ( * a - ln * exp Im a b Φ x f a b Φ x f - 0 0 5 0 5 - 2 2 mat 6 1 sys

a b x Φ - (*^b x Φ - (* a ln * exp ≡

... obtain HyperSineCosine...

b x Φ - (* a - ln * exp Re b x Φ - (* a - ln * exp Im 21 mat a - b Φ x f Re a - b Φ x f Im 21 mat ≡

... and the envelopes

a b Φ x f a b Φ x f - 21 mat

End of HyperSine Q.E.D [Quod Erat Demonstrandum]