Numerical Inverse Laplace Transform ... [a,ns,nd]

6bdc1429-1c42-4913-90c5-cbe319750d9c 14 3 12 decimal

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

Parametric invLaplace ... adapted from Oscar

in program ... user a 6 : ns 20 : nd 19 : 13 mat

s F t invLaplace t 1 el 105 -^: t.ini 0 ≡ t cases [a,ns,nd] user comman line n 1 ns nd + 1 + (range α n el a n 1 - (π * i * + : β n el e a^1 - (n^* - : 21 line for bdif 1 el nd 1 + Gamma nd 2 + 1 - Gamma / 1 Gamma / (: β2 β ns 2 + (ns 1 + nd + (range 11 range el : n 2 nd range bdif n el nd 1 + Gamma nd 2 + n - Gamma / n Gamma / (bdif n 1 - el + : for bdif bdif reverse 2 nd^/ : n 1 nd range β2 n el β2 n el bdif n el * : for β β 1 ns 1 + (range 11 range el β2 stack : β 1 el β 1 el 2 / : 41 line kt 1 pts range tt t kt el : s α tt / : βt β tt / : n 1 α length range βtF n el βt n el s n el F * : for RβtF βtF Re : ft kt el RβtF eval sum : 61 line for OUT t ft augment : 91 line :

s F 1 L / s s 2^s R L / * + 1 L C * / + / (* :

Laplace algebra

maple assume(t>0) often not needed, meet all F(s)

t RLC s F t 0 > assume s t invlaplace maple :

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

user range/mesh t.ini 0 : t.end 3 : pts 100 : 13 mat

ε 1012 -^≡

sometimes =>

t.ini

in-situ incremental t t.ini t.end t.end t.ini - (pts 1 - / range :

command-line user parameters R 1 : L 1 : C 0.01 : 13 mat

x c 10 x ≤ cases :

Export s F t invLaplace :

User XFR parameters [H, γ, T]

Export x RLC x c * 2 1 sys

Export 01 0.03 0.926 0.061 0.77 0.091 0.55 0.121 0.288 0.152 0.007 0.182 0.266 - 72 mat

Export maple parametric

R t * 2 L * / - exp 4 L * - R 2^C * + L 2^C * / - sqrt t * 2 / cos 4 L * - R 2^C * + (* 4 L * - R 2^C * + L 2^C * / - sqrt R * 4 L * - R 2^C * + L 2^C * / - sqrt t * 2 / sin * L * C * + (* 4 L * - R 2^C * + (L * /

Reluctant companions ! ... partwise

ttt q2 ttt - exp - : σ L2 ttt q2 ttt σ laplace maple : 21 sys

ttt q1 1 : σ L1 ttt q1 ttt σ laplace maple : 21 sys

s λaplace s L1 s L2 + : 11 s + (s * /

1 s / 11 s + / - 11 s + (s * / ≡

More about Γ(x) . . .

Laplace numerical inverser is built around Smath Gamma(x), 0 < x, 4 decimals.

Quite convenient for inverser applications ... for more demanding projects,

here are the more generalized Γ(x)

Γ(x) Reconciliation . . . typical ~15 D

Though slightly different, the two approximations are considered ~

Alvaro & H.T. Davis support Smath native Simpson integrator.

H.T. Davis is tracable to Abramowitz & Stegun

]]> H. T. Davis Γa(x)

Abramowitz & Stegun, p. 256 c 1 0.5772156649015329 0.6558780715202538 - 0.0420026350340952 - 0.1665386113822915 0.0421977345555443 - 0.0096219715278770 - 0.0072189432466630 0.0011651675918591 - 0.0002152416741149 - 0.0001280502823882 0.0000201348547807 - 0.0000012504934821 - 0.0000011330272320 0.0000002056338417 - 0.0000000061160950 0.0000000050020075 0.0000000011812746 - 0.0000000001043427 0.0000000000077823 0.0000000000036968 - 0.0000000000005100 0.0000000000000206 - 0.0000000000000054 - 0.0000000000000014 0.0000000000000001 261 mat :

1 z Γ / c k el z k^* k 1 ∞ sum ≡

z Γpoly c k el z k^* k 126 sum eval :

z Γa 1 z Γpoly / :

x c 10 x ≤ (x 1 ≤ (& cases :

x Γa x c * 1 1 + 15 red 1 5 mat 2 1 sys if/otherwise ... (x ≤ 0)&(x ≤ 12) suite IterativeProduct

Gamma(x)

constant 15 D accuracy @ integers ... can be made iterated product(s) x Γ x Γa x 0 ≤ x Γa 0 x ≤ (x 1 ≤ (& x 1 - (x 1 - Γa * 1 x ≤ (x 2 ≤ (& x 2 - (x 1 - (* x 2 - Γa * 2 x ≤ (x 3 ≤ (& x 3 - (x 2 - (* x 1 - (* x 3 - Γa * 3 x ≤ (x 4 ≤ (& x 4 - (x 3 - (* x 2 - (* x 1 - (* x 4 - Γa * 4 x ≤ (x 5 ≤ (& x 5 - (x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 5 - Γa * 5 x ≤ (x 6 ≤ (& x 6 - (x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 6 - Γa * 6 x ≤ (x 7 ≤ (& x 7 - (x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 7 - Γa * 7 x ≤ (x 8 ≤ (& x 8 - (x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 8 - Γa * 8 x ≤ (x 9 ≤ (& x 9 - (x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 9 - Γa * 9 x ≤ (x 10 ≤ (& x 10 - (x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 10 - Γa * 10 x ≤ (x 11 ≤ (& x 11 - (x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 11 - Γa * 11 x ≤ (x 12 ≤ (& x 12 - (x 11 - (* x 10 - (* x 9 - (* x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 12 - Γa * cases :

acc 100 x Γa x 1 e / 1 int 0.910301006

x Γ

12 Γ 11! 2 π * Γ 31 mat 3991680039916800 195.9361 31 mat

acc 100 x Γ x 1 e / e int 2.7196687

Γ(x) [15 D] Alvaro

Γ.κ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh 1 - 2 ζ * + 2 / : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ.κ 1 el Γ.κ κ 1 + el z κ + / κ 114 sum + (* zp 2^* e zg -^* eval 21 line :

Valid for Re(ζ)>0

acc 100 x Γ x 1 e / e int 2.7196687

considered sanity
Gamma(x) does NOT
exist exactly ! 0.5 - Γ 0.0001 Γ 2 π * Γ 12 Γ 41 mat 3.54490770181108 - 9999.42288323158 195.936115724443 39916799.9999992 41 mat

Fails integrator

x = 1.461632144968362, [y = 0.885603194410889]
Extended decimal '± x' ... & factorial [!]
Rejects Smath native Simpson Integrator x Γ i 1 x (Floor range r i el i : for x ε + (r rows - Γa x ε + (r n el - (n 1 r rows product * eval 21 line x 1 - (! try :

ε 1012 -^:

Mutual sanity MCD negative range 9 Decimals
..........Who is right/wrong ? .......... 0.5 - Γ 10 Γ 12.123456789 Γ 31 mat 3.54490770178686 - 362880.000001026 54001961.0748096 31 mat

The extended Γ(x).. recast from
               Gilberto E. Urroz x Γ x Γ x Gamma : x 0 ≥ x Γ π x abs 1 + (* x abs 2 + Γ π x abs 1 + (* sin * / if 21 line :

x Γ