export the solution wrt DE parameters

bd4659a5-f085-4f0a-b485-a44b33d5a0bb 65 4 5 decimal

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

Maple Laplace ODE . . .

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t ω d q 1 :

s ω d L t ω d q t s laplace maple :

4 appVersion 0.98.6179.21440 ≡

s ω d L 1 s /

s F Y0 s - m - (* Y1 - s ω d L - s 2^s m * + n + (/ (- :

Laplace solver ODE2 normalised c/w forcing function, parameters and initial conditions.

solution is wild wrt parameters and assigned to a function wrt parameters. Y0 Y1 m n ω t fnct s F t 0 > assume s t invlaplace maple 11 line :

export the solution wrt DE parameters

Export the solution wrt the DE parameters  ... a long expansion that will solve for any [Y0,Y1, m,n] Y0 Y1 m n ω t fnct n 41 m t * 2 / - exp 4 n * m 2^- sqrt t * 2 / cos * 1 - n Y0 * + (* + (* m t * 2 / - exp 4 n * m 2^- sqrt * Y0 * m * 4 n * m 2^- sqrt t * 2 / sin * + m t * 2 / - exp Y0 * 4 n * m 2^- sqrt t * 2 / cos * m 2^* - (* m m 1 m t * 2 / - exp 4 n * m 2^- sqrt t * 2 / cos * - (* m t * 2 / - exp 4 n * m 2^- sqrt * 4 n * m 2^- sqrt t * 2 / sin * + (* - 2 m t * 2 / - exp * 4 n * m 2^- sqrt * Y1 * 4 n * m 2^- sqrt t * 2 / sin * n * + 4 n * - m 2^+ (n * / -

Y0 0 :

Y1 0 :

m 0.5 :

n 1.5 :

d 0.5 :

ω 1 :

The RHS algorithm is the solution
wrt user parameters, assigned f(t) t f Y0 Y1 m n ω t fnct :

1 n / 23 /

x L 1 x 0 ≥ (x 20 ≤ (& cases :

Observe: in the conditional the text "" blanks the trace ... same as Mathcad .

x f x L * 2 3 / 2 1 sys

Solving NASA(h) . . . Smath rescue maple

altitude[m] from measured kPa

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h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

0 NASA 101.4009

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 60000 NASA : H 101.4009 : N 101 : 13 mat

u L H N xd :

Recursive seeded Smath solve ... reasonably fast/accurate.
Once 'sol' in hand  ► spline interpolate. sol i 1 u rows range m_altitude i el h NASA u i el - 0 ≡ h 0100000 solve : for u 12 round vectorize m_altitude 1 round vectorize augment 21 line :

x c 10 x ≤ (x 60000 ≤ (& cases :

XY sol eval :

XY rows 101

t0 1 time :

NASA(h) solved [m_altitude] from measured kPa

sol 101.325 10000 101.325 10000 - 2 2 mat 2 1 sys

NASA kPa vs altitude ... useless

x NASA x c *

1 time t0 - min 0.4

Thiele . . . user Engineering application

Thiele user of variable length j:=rows(XY) data set

1. Calculate 'K' from the given XY data set

2. 'X' exist from XY data set

3. Plug the RHS of Cfr(K,X,x) . . . f(x):=

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j 6 :

Thiele on length of support points 'j' K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 51 line :

symbolic expansion of Thiele continued fraction K X x Cfr K 1 el x X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el x X 5 el - K 6 el / + / + / + / + / +

Altitude(m) vs kPa

XY 0.0279 60000 1.0416 30742 2.0554 26212 3.0691 23750 4.0828 21932 5.0966 20520 6.1103 19364 7.124 18387 8.1378 17539 9.1515 16791 10.1652 16122 11.1789 15517 12.1927 14964 13.2064 14455 14.2201 13984 15.2339 13545 16.2476 13135 17.2613 12750 18.275 12386 19.2888 12042 20.3025 11716 21.3162 11406 22.33 11110 23.3437 10824 24.3574 10551 25.3712 10288 26.3849 10032 27.3986 9785 28.4123 9545 29.4261 9312 30.4398 9085 31.4535 8864 32.4673 8649 33.481 8439 34.4947 8235 35.5085 8035 36.5222 7840 37.5359 7649 38.5496 7462 39.5634 7279 40.5771 7100 41.5908 6924 42.6046 6752 43.6183 6583 44.632 6418 45.6458 6255 46.6595 6095 47.6732 5939 48.6869 5784 49.7007 5633 50.7144 5483 51.7281 5337 52.7419 5192 53.7556 5050 54.7693 4910 55.7831 4772 56.7968 4636 57.8105 4502 58.8242 4369 59.838 4239 60.8517 4110 61.8654 3984 62.8792 3858 63.8929 3735 64.9066 3613 65.9204 3493 66.9341 3374 67.9478 3256 68.9615 3140 69.9753 3025 70.989 2912 72.0027 2800 73.0165 2689 74.0302 2580 75.0439 2471 76.0577 2364 77.0714 2258 78.0851 2153 79.0988 2049 80.1126 1947 81.1263 1845 82.14 1744 83.1538 1645 84.1675 1546 85.1812 1448 86.195 1351 87.2087 1256 88.2224 1161 89.2361 1066 90.2499 973 91.2636 881 92.2773 789 93.2911 699 94.3048 609 95.3185 519 96.3323 431 97.346 343 98.3597 256 99.3734 170 100.3872 85 101.4009 01012 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 :

data list ListCollect ct 1 : i list a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 31 line :

list 1236101525406080101111 mat :

XY XY list ListCollect : data XY o 5 black plot : X XY 1 col : Y XY 2 col : 12 mat 31 sys

XY 0.027912 60000 1.041641 30742.4 2.055371 26212 5.096561 20519.7 9.15148 16791.4 14.22013 13984.1 24.357429 10551.3 39.563377 7279 59.837975 4239.1 80.112572 1946.6 101.4009 0112 mat

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

data XY 2 1 sys

Thiele K's continued fraction K X Y 0 Thiele 2 col :

K K 12 round vectorize :

J XY rows :

x c 1 X 1 el x ≤ (x 101.4 ≤ (& cases :

Thiele on length of support points 'j' K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 51 line :

x f K X x Cfr :

data x f x c * 2 1 sys

NASA Atmospheric Pressure vs altitude kPa(m)... as published h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

0 NASA 101.4009

NASA Altitude vs Pressure m(kPa)... from Smath/Maple companions. kPa alt 20001440714404100 kPa * 10129 / ln 5.256 / exp * - (* 649 / 101.4009 kPa ≥ (kPa 22.7074 ≥ (& 1000017310020 kPa * 453 / ln * - (* 157 / 22.7074 kPa ≥ (kPa 2.5223 ≥ (& 100014189 - 21660125 kPa * 311 / ln 11.388 / - exp * + (* 299 / cases :

101.4009 alt 0

A380 cokpit elevation ~[m] 101.325 alt 6.3

collect/plot NASA regions x alt x c11 * x alt x c25 * x alt x c60 * 22.7 11000 + 20 black 15 mat 2.52 25000 + 20 black 15 mat 51 sys

x c11 1 101.4009 x ≥ (x 22.7074 ≥ (& cases : x c25 1 22.7074 x ≥ (x 2.5223 ≥ (& cases : x c60 1 2.5223 x ≥ (x 0.0279 ≥ (& cases : 31 sys

x alt x c11 * x alt x c25 * x alt x c60 * 22.7 11000 + 20 black 1 5 mat 2.52 25000 + 20 black 1 5 mat 5 1 sys

NASA atmosphere . . . [3 regions . . . m(kPa)]

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list 152540608010116 mat :

Thiele . . . Φ(x)

Export an extrapolated commercial segment.

The global fit f(x) fails to fit the all range, this

is known. NASA overlaps 3 regions of much

different DE. In f(x), the low kPa region has

too much effect on the high kPa.

NASA first segment is hardly convenient for

commercial flight. Thiele Φ(x) is preferred.

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XY sol list ListCollect : data XY o 5 black plot : X XY 1 col : Y XY 2 col : 12 mat 31 sys

Thiele K's continued fraction K X Y 0 Thiele 2 col :

K K 12 round vectorize :

x c 110 x ≤ (x 101.4 ≤ (& cases :

x Φ K 1 el x X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el x X 5 el - K 6 el / + / + / + / + / + :

x alt x c11 * x Φ x c * 22.7 16000 22.7 2000 - 2 2 mat data 4 1 sys

header kPa m alt 12 mat : XY header XY stack : 21 sys

XY kPa m alt 14.2201 13984.1 24.3574 10551.3 39.5634 7279 59.838 4239.1 80.1126 1946.6 101.4009 072 mat

@ 10kPa, you are flying  16371 m 10 Φ 16347

100 Φ 100 alt 21 mat 117.5 117.4 21 mat

101.4009 Φ 101.4009 alt 21 mat 0 0.003 21 mat

10 Φ 10 alt 21 mat 163471622721 mat

Lambert W . . . solving Gumbel PDF

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LambertW two branches ... x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : 21 sys

Gumbel PDF μ σ x Gp 1 σ / x μ - x μ - (- σ / exp σ * + (- σ / exp * :

yy μ σ x Gp - x solve maple μ yy σ * - LambertW - ln σ * -

μ 1 : σ 0.625 : 12 mat

solve Gumbel for two X's Ω μ σ x Gp 0.375 ≡ x 02 solve : 0.4873 1.7042 21 mat

Y μ σ 2 Gp : 0.264

X μ Y σ * - W0 - ln σ * - : 2

μ 0.375 σ * - W1 - ln σ * - μ 0.375 σ * - W0 - ln σ * - 21 mat 0.4873 1.7042 21 mat

μ σ Ω Gp vectorize 0.375 0.375 21 mat

μ x σ * - W0 - ln σ * - μ x σ * - W1 - ln σ * - 0.25 - 1 0.75 1 2 2 mat 3 1 sys μ σ x Gp X Y X 0 2 2 mat 2 1 sys

x W w w exp * x - 0 ≡ w 1 e / - roots :

range roots wisely
NormalPDF [μ=0, σ=1] x G 12 π * sqrt / x 2^- 2 / exp * :

relevant Lambert calculus

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x Inverse xx G x - 0 ≡ xx 1 - roots : x inverse xx G x - 0 ≡ xx 1 roots : 21 sys

x W x diff x W 1 x W + (x * / ≡

x W x int x - x x W / + x W x * + ≡

Lambert identity x W y W + x y * 1 x W / 1 y W / + (* W ≡

In the main document COMPENDIUM, we have inversed Fréchet. Here APPENDIX Gumbel and Normal PDF. Different techniques for 3 different PDF's ... How many more techniques for so many PDF's ?

sys(Scalar(x),-e,2,1)

x Inverse x inverse π 10 x * G * 3 1 sys x W 1 x W + ( x * / x - x x W / + x W x * + 1 e / - 10 1 e / - 2 - 2 2 mat 3 1 sys

Maple LambertW two branches ... x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : x W w w exp * x - 0 ≡ w 1 e / - roots : 31 sys

# LambertW 1 - # LambertW x W # ProductLog ≡ 31 sys

X 0.25 - :

Lambert complementary branches

x W0 x W1 1 e / - 1 1 e / - 5 - 2 2 mat X 1 X 3 - 2 2 mat 4 1 sys

e W 1.000000000000000

X W0 X W1 21 mat 0.357403 - 2.153292 - 21 mat

ωω expr ωω 3^e ωω^- :

ωω expr ωω solve maple 313 / - LambertW * - 31 - 13 / - LambertW * - 31 i 3 sqrt * - 6 / LambertW * - 31 i 3 sqrt * + 6 / LambertW * - 41 mat

... wrt Maple 31 - branch -1 ≡ 13 / - W * -

313 / - W * - (3^313 / - W * - exp - 0

Solving the expression "completely"

313 / - W * - 313 / - W * - (3^313 / - W * - exp - 313 / - W1 * - 313 / - W1 * - (3^313 / - W1 * - exp - 61 mat 1.85718386020784 6.40567207898112 6.40567207898109 - 4.53640365497352 93.3544608350032 93.3544608350029 - 61 mat

31 i 3 sqrt * - 6 / abs W * - 31 i 3 sqrt * - 6 / abs W * - (3^31 i 3 sqrt * - 6 / abs W * - exp - 31 mat 0.772882959 - 0.461680143 - 0.461680143 - 31 mat

more algebraic Lambert

collapsed in there

]]> Lambert W solving algebraic expressions

more algebraic u 2^e u^- 0 ≡ a b v * + c 10 v -^* ≡ a b v * + c e v -^* ≡ 3 x^x 2^≡ α α exp * 1 e / - ≡ a x^b x * + c + 0 ≡ 2 t^5 t * - 0 ≡ x t^z - 0 ≡ z z b log * a - 0 ≡ 91 sys

y 1 υ e υ^* + ≡ υ solve maple y υ υ e υ^* + υ υ e υ^* - / ≡ υ solve maple y 1 υ e υ^* + 1 υ e υ^* - / ≡ υ solve maple 31 mat 1 - y + LambertW 1 - y + 1 y + / ln 1 - y + 1 y + / LambertW 31 mat

υ 2^e υ^- 0 ≡ υ solve maple 212 / - LambertW * - 212 / LambertW * - 21 mat

a b v * + c e v -^* ≡ v solve maple c a b / exp * b / LambertW b * - a + b / -

a b v * + c 10 v -^* ≡ v solve maple 10 ln a * b 10 ln c * 10 ln a * b / exp * b / LambertW * - 10 ln b * / -

a xx^b xx * + c + 0 ≡ xx solve maple a ln c a ln * b / - exp * b / LambertW b * c a ln * + a ln b * / -

3 xx^xx 2^≡ xx solve maple 23 ln 2 / - LambertW * 3 ln / - 23 ln 2 / LambertW * 3 ln / - 21 mat

a a exp * 1 e / - ≡ a solve maple 1 -

2 tt^5 tt * - 0 ≡ tt solve maple 2 ln 5 / - LambertW 2 ln / - 1 - 2 ln 5 / - LambertW 2 ln / - 21 mat

xx tt^z - 0 ≡ tt solve maple z ln xx ln /

xx tt^z - 0 ≡ xx solve maple z ln tt / exp

xx tt^z - 0 ≡ z solve maple xx tt^

z z log * a - 0 ≡ z solve maple a a LambertW /

MCD 11 a 10 ln a 10 ln * W / *

a 3 :

a a W / a 10 ln a 10 ln * W / * 21 mat 2.857390784 4.555535705 21 mat

Maple RootOf . . .

applied for solving PDF in statistics..

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Fréchet distribution
α > 0 ... β > 0 α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

w α β x Fr ≡ x solve maple w β * α β_Z / (1 α +^* β_Z / (α^- exp * - RootOf

recast RootOf

Fr 51 x Fr 11 x Fr 21 x Fr 0.5 1 x Fr 41 sys :

plot Fr 1 - vy 1.5 vy 22 mat vx 1 el vy . 14 black 15 mat vx 2 el vy . 14 black 15 mat 41 sys :

α 5 : β 1 : vy 1 : 13 mat w t ipdf w β * α β t / (1 α +^* β t / (α^- exp * - : w t Sol w t ipdf 0 ≡ t 1 - 10 solve : vx vy t Sol (: 41 sys

'X' blue peak
'Y' blue peak 0.96419 1.87428 21 sys

x c 1 0.001 x ≤ (x 1.874278 ≤ (& cases :

t0 1 time :

Fréchet distribution ... α >0 ... β > 0

plot x t Sol 1 el x c * x t Sol 2 el x c * 1 1.5 1 0.5 2 2 mat 3 1 sys

1 time t0 - min 2

vx α β vx Fr vectorize 21 mat 0.809 1.234 21 mat 1121 mat 21 mat

By inspection ... Sol(■ ,t)produces two sols Forcibly each canvas branch each indexed on the plot.

0.25 t Sol 0.5 t Sol 1 t Sol 31 mat 0.728 1.623 21 mat 0.76 1.427 21 mat 0.809 1.234 21 mat 31 mat

NOTE ... Canvas Sol(x,t) is wrt [α, β]. For oter Frechet [α, β], consider ranging the solve bloc to suit the PDF as well as finding optimum c(x). This can be done using Golden ratio.

∫kernel(f,a,b,n) ... Simpson/Romberg Kernel [Jean]

Basic Romberg-Simpson ... integration acc from options ... NUMERIC f a b Romberg w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

f a b n ∫kernel

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline, Y^X 
Integration accuracy from user command line 'n' ... NUMERIC.
The integration variate is in Romberg [0,1] within 'n' ... 'n' independent of work sheet option. 
Thus, on long range of the integration variate ∫kernel(f,a,b,n) improves the convergence. f a b n ∫kernel w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : ... α 0 : β 1 : h β α - n / : 15 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

Some Fréchet PDF will not integrate Simpson 
Under the hood, Fr is the Power fnct X^Y.
Recast to suit ∫kernel(f,a,b,n) α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

α 5 : β 1 : 12 mat

recast to suit integrator x f α β x Fr :

f 0.01 550 ∫kernel 0.9997

Smath vectorize oddity . . . illustrate circular algo

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The Gram-Schmidt orthogonal polynomials

N 7 :

i 1 N range f i el x i 1 -^: for

0. The two integrals must be vectorized but can't. ... round(■,n) does it otherwise. 1. The true error message is "Syntax is incorrect". 2. The kernel algo becomes active once in call from the suite Φ(x):= Φ ... providing Φ is initialised. This is a closed loop circular algorithmic structure. 3. To make the demo fast, Integration accuracy 50. 4. acc 1000 ► 10 times.

f transpose 1 x x 2^x 3^x 4^x 5^x 6^17 mat

Optimiz NONE Φ 1 el f 1 el :

Circular kernel algo: optimiz NUMERIC Φ i 2 f rows range d 0 : j 1 i 1 - (range a f i el Φ j el * x 1 - 1 int Φ j el Φ j el * x 1 - 1 int / 15 round : d d a Φ j el * - : 21 line for Φ i el f i el d + (: 31 line for :

Optimiz SYMBOLIC x Φ Φ :

x N x Φ 1 el x Φ 2 el x Φ 3 el x Φ 4 el x Φ 5 el x Φ 6 el 61 sys :

x N Expansion N(x) collapsed

x N 1 x 500000000000000 x 2^* 166666666666667 - 500000000000000 / x 150000001 - 250000000 x 2^* + (* 250000000 / 5568434188608080100000000666667 - 500000000000000 x 4^* + (* 487229337000119166666666666667 - 500000000000000 x 2^* + (* - (* 1421085471520200000000000000000 / x 74505805969238542857144857 - 100000000000 x 4^* + (* 331137025750460000150000001 - 250000000 x 2^* + (* - (* 74505805969238300000000000 / 61 sys

Export Lagrange roots style

Integral acc   ▼  1000 0.906179722448874 - 0.538469381723912 - 0 0.538469298199461 0.906179801921587 51 mat

recast for the solve bloc x Φ x Φ 6 el :

r x Φ x 1 - 1 solve : 0.90618 - 0.53847 - 0 0.538469 0.90618 51 mat

r 0.9062 - 0.5385 - 0 0.5385 0.9062 51 mat :

roots i 1 r rows range pts i el . : for i 1 r rows range zero i el 0 : for r zero pts augment 31 line :

convenient roots

local definition mode x Lagrange x r i el - (i 1 r rows product 11 line :

x Lagrange 5000 x * 4531 + (2000 x * 1077 + (* x * 2000 x * 1077 - (* 5000 x * 4531 - (* 100000000000000 /

roots x Lagrange 2 1 sys

The ultimate and gracious export

Define Symbolic ... Optimiz = NONE x collect x Lagrange parfrac x convert evalf maple :

Expand Symbolic ... Optimiz = NONE x collect x 5^111118069 x 3^* 100000000 / - 2381329713 x * 10000000000 / +

x L x 5^1.11118069 x 3^* - 0.2381329713 x * + :

j 5 :

x N j el x L roots 3 1 sys

maple nuance . . . erroneous-different code . . .

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x N 1 x 500000000000000 x 2^* 166666666666667 - 500000000000000 / x 9375001 - 15625000 x 2^* + (* 15625000 / 5568434188608080100000010666667 - 500000000000000 x 4^* + (* 487229824037453166666666666667 - 500000000000000 x 2^* + (* - (* 1421085471520200000000000000000 / x 745058059692385669643357 - 1562500000 x 4^* + (* 827846696120510009375001 - 15625000 x 2^* + (* - (* 1164153218269350000000000 / 61 mat :

x f x N 5 el :

x f parfrac x convert maple 24361545748802794445691809791284217094304040000000000000000 / x 4^+ 487229824037453 x 2^* 568434188608080 / -

x f x N 6 el :

x f parfrac x convert evalf maple 1 x 5^* 1111117027 x 3^* 1000000000 / - 2380985387 x * 10000000000 / +

x f x N 6 el :

x f parfrac x convert maple 745058059692385 x 5^* 745058059692384 / 413923348060255 x 3^* 372529029846192 / - 55436636029066531662911 x * 232830643653870000000000 / +

x f x N 5 el :

x f parfrac x convert evalf maple Le format de la chaîne d'entrée est incorrect.