<content> plot </content>

fd430e11-2330-4a28-a448-42cf87b6b062 15 4 5 false false 0 0 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 : 13 mat for data r3 r4 r5 augment 21 line :

Part 1 ... scalar systems

t0 1 time :

The function x f 4 x 2^- exp * 1 - 2 x 2^* + (* π sqrt / :

x y axes x y / - :

The derivative x f' x f x diff : 8 x * x 2^- exp * 32 x 2^* - (* π sqrt /

zero's u 1 el 1 - : i 17 range u i 1 + el u i el u i el f u i el f' / - : for 31 line

u 1 - 0.5 - 0.7 - 0.707072135785007 - 0.707106780337926 - 0.707106781186548 - 0.707106781186549 - 0.707106781186549 - 81 mat

◄ range \[REGION[404d5705b541499aa432effc277ab87e]]\ last 2 results =
initialize wisely inspect graph

'i'

intersecttion x Φ x f x f' - : x Φ' x Φ x diff : 21 sys

intersection's ω 1 el 0.25 - : i 17 range ω i 1 + el ω i el ω i el Φ ω i el Φ' / - : for 31 line

ω 0.25 - 0.155759162303665 - 0.160811071426278 - 0.160818587114107 - 0.160818587131648 - 0.160818587131646 - 0.160818587131648 - 0.160818587131648 - 81 mat

Z u u rows row : I ω ω rows row : 21 sys 0.707106781186549 - 0.160818587131648 - 21 sys

plot x f x f' axes Z 1 Z 1 - 22 mat I 1 - I 3 - 22 mat 51 sys :

plot

x f x 1 - roots x f x f' - x 0.25 - roots 21 sys 0.707106781186547 - 0.160818587131648 - 21 sys

<content> multiple roots ... intersections </content>

The ROBUST  multiple roots Solve scanner f x0 x1 TOL Solve ρ x f x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : xn b a + (2 / : n 0 : xn f abs TOL > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 51 line X i el xn : 21 line for X 31 line :

The ROBUST multiple Intersections Solve scanner f q x0 x1 TOL Solve ρ x f x q - x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : 12 mat xn b a + (2 / : n 0 : xn f xn q - abs TOL > xn f xn q - (a f a q - (* 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 41 line X i el xn : 21 line for X 31 line :

multiple roots/Intersections

TOL 1015 -^:

multiple roots f 1 - 1 TOL Solve 0.707106781186547 - 0.707106781186548 21 mat

x q x f x diff :

multiple intersections f q 1.5 - 1.5 TOL Solve 1.42788625331799 - 0.160818587131648 - 1.08870484044964 31 mat

Newton intersection ... analytical style

Adapted from Smath Forum
Possibly unique application.

a 2.5 :

generally enough N 20 :

n 2 N range :

prepare solving demand x f x sinh a x * - :

x f' x f x diff :

U 1 el a :

x y axes x y / - :

U n el U n 1 - el U n 1 - el f U n 1 - el f' / - eval :

xo U N el : 2.55265472372312

xo f 1.6 1014 -^* -

sol x f x 3 roots : 2.55265472372313

plot x sinh a x * xo 50 xo 5 - 22 mat axes 41 sys :

j 210 range :

plot

sol f 2.2 1014 -^*

solving \[REGION[aa49f27a55064bc5b8a8252593882046]]\ for a prescribed intersection
iterate ... manually. Code NOT offered .

'a'

All roots ... accurate

x f x exp x 2^+ 5 x * - :

TOL 1014 -^:

f 02 TOL Solve 0.280490945324612 1.73399571327305 21 mat

x f axes 2 1 sys

x f x 0 roots 0.280490945324612

x f x 1.5 roots 1.73399571327305

x f 6 x 3^* 7 x 2^* + 94 x * - 50 + :

TOL 1012 -^:

x y axes x y / - :

Initialise wisely by observing the plot myroots f 6 - 4 TOL Solve : 4.79576592387119 - 0.567574701316385 3.06152455588812 31 mat

j myroots 0 * :

r myroots j augment : 4.7958 - 0 0.5676 0 3.0615 032 mat

myroots f vectorize 1.7 1013 -^* - 4.8 1013 -^* 9.9 1013 -^* - 31 mat

r r . 4 black plot :

x f x 4 - roots x f x 0 roots x f x 2 roots 31 sys 4.7957659238712 - 0.567574701316391 3.06152455588813 31 sys

x f axes r 2.707 - 236.734 + 40 red 1 5 mat 1.9292 62.2114 - + 40 red 1 5 mat 5 1 sys

solveroots x f x 5 - 4 solve : 4.7957659 - 0.5675745 3.0615245 31 mat

solveroots f vectorize 1.7097 106 -^* 1.8495 105 -^* 6.1469 106 -^* - 31 mat

<content> Golden ratio min/max </content>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x f eval :

ε 1015 -^:

X f 3 - 2 - y1 y2 ≥ ε Extrema f 13 y1 y2 ≤ ε Extrema 21 mat :

X X f vectorize 12 mat 2.707 - 1.9292 21 mat 236.734 62.2114 - 21 mat 12 mat

Accurate intersections ...

TOL 1014 -^:

x f x 2^:

x q 1 0.25 x * + :

Xsect f q 1.25 - 1.25 TOL Solve :

Xsect 0.882782218537316 - 1.13278221853732 21 mat

Ysect Xsect f vectorize : 0.779304445365665 1.283195554634340 21 mat

x f x q Xsect 1 el Ysect 1 el + 20 black 1 5 mat Xsect 2 el Ysect 2 el + 20 black 1 5 mat 4 1 sys

x f 6 x 3^* 7 x 2^* + 94 x * - 50 + :

TOL 1011 -^:

x q 14010 x * + :

Bracket wisely, observing the plot ZACroots f q 5 - 5 TOL Solve :

ZACroots 4.35556637414308 - 0.852208956357678 - 4.04110866383414 31 mat

ZACroots f vectorize 96.4443362585761 131.477910436423 180.411086638342 31 mat

r Clear 1

4 ½ decimals r x f x q - x 5 - 5 solve : 4.3555666 - 0.852209 - 4.0411087 31 mat

X ZACroots : Y ZACroots f vectorize : 12 mat

x f x q X 1 el Y 1 el + 20 black 1 5 mat X 2 el Y 2 el + 20 black 1 5 mat X 3 el Y 3 el + 20 black 1 5 mat axes 6 1 sys

x f x q X 1 el Y 1 el + 20 black 15 mat X 2 el Y 2 el + 20 black 15 mat X 3 el Y 3 el + 20 black 15 mat axes 61 sys

X Y augment 4.356 - 96.444 0.852 - 131.478 4.041 180.411 32 mat

poor accuracy ... r f vectorize 96.444295 131.477911 180.411086 31 mat

0.75 x BesselJ eval axes 2 1 sys

plot eval(,,,)

Approximation of the Bessel function J0(x) by trigonometric functions x P.0 19128 x 2^* / - 367532768 x 4^* / + 24012454194304 x 6^* / - 130439058752147483648 x 8^* / + : x Q.0 18 x * / - 751024 x 3^* / + 59535262144 x 5^* / - 5797291533554432 x 7^* / + : x J.0maxX 2 π x * / sqrt x π 4 / - cos x P.0 * x π 4 / - sin x Q.0 * - (* (: x J.0minX 1 x 2 / (2^- 122^/ x 2 / (4^(* + 162^/ x 2 / (6^(* - : Approximation of the Bessel function J0(x) x J.0 x J.0minX x 2 ≤ (x 2 - ≥ (& (* x J.0maxX x 2 > ((* + x - J.0maxX x 2 - ≤ (* + : 61 line

x J.0 0 0 + 200 red 1 5 mat 2 1 sys

TOL 1013 -^:

x f x J.0 :

The ROBUST 'zizag' root(s) Solve scanner f x0 x1 TOL Solve ρ x f x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : xn b a + (2 / : n 0 : xn f abs TOL > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 51 line X i el xn : 21 line for X 31 line :

range wisely ... avoid freak collection f 042 TOL Solve 2.40230052662502 5.52007492175481 8.65372784221755 11.7915344342011 14.9309177078841 18.0710639678 21.211636629852 24.3524715307414 27.4934791320382 30.6346064684311 36.9170983536645 40.0584257646281 43.199791713177 131 mat

<content> Γ(x) [15 D] Alvaro Same otherwise </content>

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

Valid for Re(ζ)>0

n x J 1 - (k^k! (k n + 1 + Γ * / x 2 / (n 2 k * +^* (k 045 sum :

x λ 1 x 32 - ≥ (x 32 ≤ (& cases :

x y axes x y / - :

mid accuracy roots

0 x J x λ * 1 x J x λ * axes 3 1 sys

TOL 108 -^:

x f 0 x J eval :

f 030 TOL Solve 2.40482557 5.52007811 8.65372792 11.79153444 14.93091771 18.07106397 21.21163672 24.35246553 27.49376202 91 mat

Complementary visit ...

The function x f 4 x 2^- exp * 1 - 2 x 2^* + (* π sqrt / :

r Clear 1

The derivative x f' x f x diff : 8 x * x 2^- exp * 32 x 2^* - (* π sqrt /

zero's u 1 el 1 - : i 17 range u i 1 + el u i el u i el f u i el f' / - : for 31 line

u 1 - 0.5 - 0.7 - 0.707072135785007 - 0.707106780337926 - 0.707106781186548 - 0.707106781186549 - 0.707106781186549 - 81 mat

f x init myroots zero's u 1 el init : x f' x f x diff : i 111 range u i 1 + el u i el u i el f u i el f' / - : for 41 line indes 1 to explode u u rows row 1 el 31 line :

All roots at once ...same accuracy 'roots' f x 1 - 121 mat myroots vectorize 0.707106781186547 - 0.707106781186547 21 mat

sanity check f x 1 - 121 mat myroots f vectorize 21015 -^* - 21015 -^* - 21 mat

0.707106781186547 - f 21015 -^* -

NOT compatible r x f x 1 - 121 mat roots vectorize :

Part 2 ... discrete systems

<content> all spline stuff </content>

Prepare the pseudo-integrand s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i el h i el 2 / Y i el Y i 1 + el + (* h i el (3^24 / s i el s i 1 + el + (* - : 21 line for f 21 line :

s X Y t dspline t X 2 el ≤ k 1 : t X X rows 1 - el ≥ k X rows 1 - : k 1 : t X k 1 + el > k k 1 + : while 21 line if if c Y k 1 + el Y k el - X k 1 + el X k el - / s k 1 + el 2 s k el * + (X k 1 + el X k el - 6 / (* - : s k 1 + el s k el - X k 1 + el X k el - / (t X k el - (2^2 / * s k el t X k el - (* + c + 31 line eval :

The while loop form of the spline interpolation function s X Y t interpspline Find the knots that bracket 't' i0 1 : j X rows : j i0 - 1 > i i0 j + 2 / Floor : X i el t > j i : i0 i : if 21 line while Evaluate the spline based on those knots h X j el X i0 el - : a X j el t - h / : b t X i0 el - h / : a Y i0 el * b Y j el * + h 2^6 / a a 2^1 - (* s i0 el * b b 2^1 - (* s j el * + (* + 91 line :

The long cubic spline [lpc], scalar interpolation. s X Y t InterpSpline t X 1 el ≤ idx 1221 mat : t X N el ≥ idx N N 1 + 21 mat : i 1 : X i el t < i i 1 + : while idx i 1 - i 21 mat : 31 line if if hx X idx 2 el el X idx 1 el el - (: hy Y idx 2 el el Y idx 1 el el - (: hs s idx 2 el el s idx 1 el el - (: 31 line a hs 6 hx * / : b s idx 1 el el 2 / : c hy hx / hx 6 / s idx 2 el el 2 s idx 1 el el * + (* - (: d Y idx 1 el el : 41 line a t X idx 1 el el - (3^* b t X idx 1 el el - (2^* + c t X idx 1 el el - (* + d + 41 line :

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 :

L H N xd range discretise U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 51 line :

data 03 - 0.2 2.951056 - 0.4 2.589484 - 0.6 1.581057 - 0.8 0.05423 1 1.5 1.2 2.112192 1.4 2.110927 1.6 1.8785 1.8 1.607987 2 1.363636 2.2 1.15897 2.4 0.991538 2.6 0.855127 2.8 0.743554 3 0.651639 3.2 0.575282 3.4 0.511303 3.6 0.457245 3.8 0.411207 4 0.371707 212 mat :

data

Part 1. get the 's' spline coefficients

<content> l_p_cspline sanity Mathcad </content>

X data 1 col : Y data 2 col : n data rows : 31 line

Initialise the blank
diagonal matrix

M 11 el 1 :

M n n el 1 :

N n 1 - : r 2 N range : 21 line

Prepare the cubic splines

Install the constant vector 'br' i r br i el Y i 1 + el Y i el - X i 1 + el X i el - / (Y i el Y i 1 - el - X i el X i 1 - el - / (- eval : br N 1 + el 0 : 21 line for

Fill-in the respective l_p_c splines tridiagonal matrices

0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

lspline M :

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

pspline M :

2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

cspline M :

Discrete as needed L X 1 el : H X n el : Num 100 : 13 mat U L H Num xd : 21 mat

s pspline 1 -^br * :

select spline type

assign/isolate s s :

Part 2. The complimentary cumulative integrate

Σ s X Y fintegrate :

Σ rows X rows 21 sys 202121 sys

doctored

Export the cumulative integral Σcumul n X rows 1 - : Sum 1 el 0 : Sum n 1 + el 0 : i 1 n range Sum i 1 + el Sum i el Σ i el + eval : for X Sum augment 51 line :

Σcumul Σcumul :

Σ Σcumul o 4 black plot :

plot Σ Σcumul data 041 sys :

plot

The discrete Golden ratio

vx Σcumul 1 col : vy Σcumul 2 col : 12 mat x Σspline vx vy x cinterp : x f x Σspline : 31 sys

start end mesh # f Xend Algo

L H N # f range Algo u y Clear U 0 : dx H L - N 1 - / : 12 mat i 1 N 1 + (range U i el L dx - (dx i * + : for j 1 range range u j el U j el : y j el U j el f eval : 21 line for u y augment 51 line :

NO need to over populate Σgold 021000 x f 1000 Algo :

<content> lookup(value,table,col) </content>

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

value table col lookup

ROW Σgold 2 col Min Σgold 2 lookup : 39811 mat

Σgold ROW row 0.795 1.748 - 12 mat

Part 3. The complimentary scalar dspline

x D s X Y x dspline :

3 D 0.417676 -

D2 X s augment :

x plotD L x ≤ (x H ≤ (& (x D out spline if :

x D2 D2 1 col D2 2 col x linterp :

myD i 1 U rows range deriv i el U i el D eval : for U deriv augment 21 line :

x λ 1 L x ≤ (x H ≤ (& cases :

Part 4. The [lpc] spline interpolation

1 time t0 - 23.1 s *

x Spline s X Y x interpspline :

Data data o 5 black plot :

x Spline x λ * x D x λ * 0 2 x + 4 1 sys x D2 x λ * 0 2 1 sys

vx myD 1 col : vy myD 2 col : x s vx vy x cinterp : 31 sys

x f x s eval :

x q 2 x + eval :

The ROBUST multiple Intersections Solve scanner f q x0 x1 TOL Solve ρ x f x q - x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : 12 mat xn b a + (2 / : n 0 : xn f xn q - abs TOL > xn f xn q - (a f a q - (* 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 41 line Ω i el xn : 21 line for Ω 31 line :

TOL 1012 -^:

Xs f q 0 1.5 TOL Solve :

accepted Xs 0.3485 1.0942 21 mat

NOT recognized ... x Spline x 04 int

Integrate . . . what-so-ever . . .

<content> Int(f,a,b,n) ... Simpson/Romberg Kernel </content>

Basic Romberg-Simpson ... integration acc from options f a b Int 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 :

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 Integration accuracy from user command line 'n' 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 / : 14 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 :

x Spline x λ * x D x λ * 0 3 1 sys

x f x D eval :

f 04100 ∫kernel 3.3718

x f x Spline :

f 04100 ∫kernel 1.709

view the cumulative integrations

Σ Clear 1

x f x Spline :

x λ 10 x ≤ (x 5 ≤ (& cases :

x ∫ f 0 x 40 ∫kernel x λ * :

x ∫ eval 0 x f x λ * 3 1 sys

x f x D eval :

x ∫ f 0 x 40 ∫kernel x λ * :

u π :

user u π ∫ augment :

x ∫ eval user x f x λ * 3 1 sys

Canvas \[REGION[edcf234143d94283aed0789127cefb31]]\ need not be meshed fine
unless you want to explore accurately.

x ∫

user 3.1416 3.5962 12 mat

data x 04 0.01 range : i 1 x rows range vy i el x i el D eval : for x vy augment 31 line :

data Σ Y 0 : 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 :

Σ data Σ eval :

exΣlorer J 201 :

XY Σ J row (: 2 4.3636 12 mat

data Σ 0 XY 4 1 sys

Additionally, you can via splines
1. Golden ratio Min/Max
2. Intersect " data, cumul "

XY 2 4.36 12 mat

x Clear 1

vx data 1 col : vy data 2 col : 12 mat x sd vx vy x cinterp : ux Σ 1 col : uy Σ 2 col : 12 mat x sΣ ux uy x cinterp : 41 sys

TOL 109 -^:

x f x sd :

x q x sΣ :

fails to solve this splines system f q 1 1.1 TOL Solve

sol x f x q - x 1 1.1 solve : 1.026611

sol f sol q - 91010 -^*

xy sol sol f augment : 1.0266 4.6319 12 mat

<content> Golden ratio min/max </content>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

from original discrete l_p_c X Clear 1

ε 106 -^:

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x f :

X f 0.5 1 y1 y2 ≥ ε Extrema f 1.5 2 y1 y2 ≤ ε Extrema 21 mat :

X X eval :

X 1 el f X 2 el f 21 sys 8.6034 1.3661 - 21 sys

X 0.7563 1.6707 21 mat

Max X 1 el X 1 el f augment : Min X 2 el X 2 el f augment : 21 sys 0.7563 8.6034 12 mat 1.6707 1.3661 - 12 mat 21 sys

plot x f x q xy 0 Max Min 61 sys :

plot

1 time t0 - 36 s *

This Grand Tour ends here.
Rich of solving techniques.
Solving is a long discours
in Engineering maths ...

jmG: 20231003

Part 3 ... more systems

1. Circle/line .... Normal, normal-parallel

C 6 : Circle ≡

L 1.5 : (Line ≡

Φ vector equalities Φ x 2^y 2^+ C - 0 ≡ x y + L - 0 ≡ 21 mat :

init x 1.5 ≡ y 1.5 - ≡ 21 mat :

TOL 1013 -^:

Does not support assignment Φ init TOL FindRoot 2.311249 0.811249 - 21 mat

Assign the solutions[x,y] coordinates ofthe point of normalcy sol x y 21 mat : 2.311249 0.811249 - 21 mat

normal(x) separation parameter θ 0.725 :

L x Normal L x + 2 sol 1 el * - :

x normal L x + θ sol 1 el * - :

I C x 2^- sqrt - L x Normal - 0 ≡ x 03 solve : 0.8112 2.3112 21 mat

r C sqrt : 2.4495

X I 1 el : Y I 2 el : 12 mat 0.8112 2.3112 12 mat

N C x 2^- sqrt x normal - 0 ≡ x 03 solve : 1.8177

Normal is tangent to circle
for L = 0 only ... style 3 below

C x 2 ^ - sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 0 L + 20 red 1 5 mat L x Normal x normal X Y - o 7 black 1 5 mat N N normal o 7 black 1 5 mat L x + 9 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 2.3112 0.8112 - 21 mat

sanity x 2^y 2^+ x y + 21 mat 6.000000000000 1.500000000000 21 mat

C x 2^- sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 25 mat 0 L + 20 red 15 mat L x Normal x normal X Y - o 7 black 15 mat N N normal o 7 black 15 mat L x + 91 sys

N N normal 21 mat 1.8177 1.642 21 mat

'roots' . . . the super companion

]]>

U v x y Clear 1

Style 1 ...

U y 3 ≡ 3 x * y + 5 ≡ 21 mat :

v x y 21 mat :

x y 21 mat U v roots (: 0.666667 321 mat

sanity check y 3 x * y + 21 mat 3521 mat

Style 2 ...

x y Clear 1

x y z f 5 x * 6 y z + (* - 8 - 3 x y - (* 2 z * + 6 + x y + z + 2 - 31 mat :

x y z 31 mat x y z f x y z 31 mat roots : 1.818182 2.363636 2.181818 - 31 mat

sanity x y z f 0.000000000000018 0.00000000000002 031 mat

Circle/line .... intersections

x y Clear 1

C 6 : Circle ≡

L 0 : (Line ≡

Φ vector equalities Φ x 2^y 2^+ C - 0 ≡ x y + L - 0 ≡ 21 mat :

init x 1.5 ≡ y 1.5 - ≡ 21 mat :

TOL 1013 -^:

Does not support assignment Φ init TOL FindRoot 1.732051 1.732051 - 21 mat

Assign the solutions[x,y] coordinates ofthe points of normalcy sol x y 21 mat : 1.732051 1.732051 - 21 mat

r C sqrt : 2.4495

Ciecle/Line ... intrsections

C x 2 ^ - sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 3 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 1.7321 1.7321 - 21 mat

sanity x 2^y 2^+ x y + 21 mat 6.000000000000 0.000000000000 21 mat

C x 2^- sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 25 mat 31 sys

x y Clear 1

XY 00 0.1 0.0244 0.2 0.0834 0.3 0.1618 0.4 0.2555 0.5 0.362 0.6 0.48 0.7 0.6083 0.8 0.7461 0.9 0.8928 1 1.0478 112 mat :

vx XY 1 col : vy XY 2 col : n XY rows : 31 line

Solver 30405 kaput
slope/intercept ...

x y axes x y / - :

β vx transpose vy * 1 n / vx sum (* vy sum (* - vx transpose vx * 1 n / vx sum (2^* - / : α 1 n / vy sum (β vx sum (* - (* : 21 line

interceptslope α β 21 mat 0.113 - 1.074 21 mat

XY α β x * + axes 3 1 sys

x y α β Clear 1

Human Eye model function ... Maple companion

α 0.842 :

β 2 :

x y f 1 x 2^y α - (y α + (* + (2^2 x * y α - (* (2^+ / :

x y axes x y / - :

y F x y f β ≡ x solve maple :

Iso-curve position U 0.875 :

switch iso blue/red n 2 :

X 0.5 021 mat :

Y U U F n el 21 mat : 0.875 0.805 - 21 mat

Green distance Y 1 el X 1 el - (2^Y 2 el X 2 el - (2^+ sqrt 0.888

u 0.5 - :

plot x F 1 el x F 2 el X 1 el X 2 el Y 1 el Y 2 el 22 mat U U F n el . 14 black 15 mat X 1 el X 2 el . 14 black 15 mat u 0.5 u 0.5 - 22 mat 1.18 - 0.25 1.18 - 0.25 - 22 mat 0.044 - 0.25 0.044 - 0.25 - 22 mat 81 sys :

Human eye ... β level Iso-contour

plot

radus U 2^Y 1 el 2^+ sqrt : 1.2374

u u F 1 el 12 mat 0.5 - 0.2138 12 mat

check strip complex x 1.18 - 0.044 - 1.179 - range : i 1 x rows range uh i el x i el F 1 el eval : for x uh augment 41 line :

check 1.18 - 0.25 1.18 - 0.125 - 2 2 mat 0.044 - 0.25 0.044 - 0.125 - 2 2 mat 0 0 + 200 black 1 5 mat 4 1 sys

1 time t0 - 38 s *

Part 4 ... curvature/normal/tangent

<content> hatch </content>

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 x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 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 :

Graphing the Analytics of a function

x a xo Clear 1

The function a x f a x 3^* :

The curvature a x K a x f x 2 diff 1 a x f x diff (2^+ (32 /^/ :

The normal a x N a xo K xo diff (1 -^x xo - (* - a xo K + :

The tangent a x T a xo K xo diff (x xo - (* a xo K + :

a 0.5 : (user parameter ≡ xo 0.45 - : (point of normalcy ≡ 21 line

x f a x K :

x g a x N :

X a u N u 2 - 0 solve : T a u T u 01 solve : 21 mat

X T 21 mat 1.9828 - 0.4625 21 mat

a 0 N a 0 T 21 mat 1.5298 - 0.5994 - 21 mat

a x K a x N a x T xo a xo K . 15 black 1 5 mat a x f x f x g xo X 20 hatch 7 1 sys

Intersection of "normal/tangent" with curvature a u N a u K - u 2 - 0 solve a u T a u K - u 0.5 1.5 solve 21 mat 1.943 - 0.7913 - 0.45 - 31 mat 0.9391 21 mat

a 1.943 - 0.7913 - 0.45 - 31 mat K vectorize 0.0307 - 0.9193 - 1.1826 - 31 mat

<content> QuickPlot image </content>

x y axes x y / - :

a x K a x N a x T xo a xo K . 4 black 1 5 mat a x f axes 6 1 sys

a x K a x N a x T xo a xo K . 15 black 15 mat a x f x f x g xo X 20 hatch 71 sys

Solving for the "normalcy" of the triplet junctions

specific jonction of "f & K" is unique fK a u f a u K - 0 ≡ u 0.9 1.1 solve : 1.0039

curvature @ 'fK' a fK K 0.50581999575087

determine a solving intersection a fK Φ 0.505819995750424 :

The tangent @ xo a xo x T a xo K xo diff (x xo - (* a xo K + :

normalcy of the triplet a xo fK T a fK Φ - xo 0.47 - 0.45 - solve 0.4607 -

a t x a t * : a t y a 2 / t 2^* : 12 mat

a 1 :

t 2 - 2 1.99 - range :

a t X a t x a t y t diff a t x t diff 2^a t y t diff 2^+ a t x t 2 diff a t y t diff * a t y t 2 diff a t x t diff * - / * + :

a t Y a t y a t x t diff a t x t diff (2^a t y t diff (2^+ a t y t 2 diff a t x t diff * a t x t 2 diff a t y t diff * - / * + :

a t parabola a t x vectorize a t y vectorize augment : a t evolute a t X vectorize a t Y vectorize augment : 21 sys

evolute of the parabola ...

a t parabola a t evolute 2 1 sys

parabola...normal...tangent a x f a x 2^* : a x N a u f u diff (1 -^x u - (* - a u f + : a x T a u f u diff (x u - (* a u f + : 31 sys

a 0.5 :

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

U 1.25 - 1.25 100 u : Q i 1 U rows range vx i el U i el : vy i el a U i el f : 21 line for vx vy augment 21 line : 21 line

range/populate tangents/normals lines x 1.5 - 1.5 50 u :

x x : u 0.75 - 0.75 0.5 - range : 12 mat i 1 u rows range tan i el a u i el f u i el diff (x u i el - (* a u i el f + eval : for 21 line

Not used ortransparent x y axes x y / - :

Avoid divide  ▼ '0' 1 - 1 0.801 - range

x x : u 1 - 1 0.801 - range : 12 mat i 1 u rows range norm i el a u i el f u i el diff (1 -^x u i el - (* - a u i el f + eval : for 21 line

j tan x tan j el augment : j norm x norm j el augment : 21 sys

Tan Q 1 tan 2 tan 3 tan 4 tan 5 tan 6 tan 7 tan 81 sys :

Norm Q 1 norm 2 norm 3 norm 4 norm 5 norm 6 norm 7 norm 8 norm 9 norm 10 norm 11 norm 121 sys :

7 nests [tan]11 nests [norm]50 values per nest tan rows norm rows tan 1 el rows 31 mat 7115031 mat

Mathcad creates a ONE col of nests.
Each nest of the ranged Tangents/Normals.
The plot explodes each corresponding nest.
Done otherwise Smath X_Y plot by collection.

1 time t0 - 40 s *

J 4 :

vx Q 1 col : vy Q 2 col : 12 mat x splineQ vx vy x cinterp : ux Norm J el 1 col : uy Norm J el 2 col : 12 mat x splNorm ux uy x cinterp : 41 sys

sol x splineQ x splNorm - x 1 - 1 solve :

Norm_j sol sol splineQ augment :

Tan Norm Norm J el Norm_j 3 1 sys

Norm_j 0.602 - 0.1812 12 mat

J 8 :

vx Q 1 col : vy Q 2 col : 12 mat x splineQ vx vy x cinterp : ux Tan J el 1 col : uy Tan J el 2 col : 12 mat x splineTan ux uy x cinterp : 41 sys

sol x splineQ x splineTan - x 1 - 1 solve :

sol sol splineQ 12 mat 0.75 0.2812 12 mat

Q Tan J el sol sol splineQ augment 0 0 + 100 black 1 5 mat 4 1 sys

Q Tan J el sol sol splineQ augment 00 + 100 black 15 mat 41 sys

Part 5 ... conditional roots canvas

The all thing is solving for 'M'
that appears on both sides.
Perfect Smath "roots"
Resize QuickPlot.

x k U Clear 1

γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Plot limit x sup 1 k x ≤ (x 4 ≤ (& cases :

Solver 30405 does not save some quick plot
Slide the graph to presrve subsonic supersonic
Delete the graph ... call for a new one, resize

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases : x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - U 1 roots k x ≤ Subsonic cases : 21 line

<content> image Pitot </content> x Subsonic x Supersonic x sup * k 1 + 20 black 1 5 mat 3 1 sys

subsonic

supersonic

Solving otherwise wrt circular datum

α 0.842 :

β 1.75 :

NOT effective ε 109 -^:

r 0.25 :

x y f1 1 x 2^y α - (y α + (* + (2^2 x * y α - (* (2^+ / β - :

make ellipse (2*y)^2 x y circle x 2^y 2^+ r - :

f1 circle 2 1 sys

r 0.25 :

NOT effective ε 109 -^:

x y axes x y / - : 11 sys

"FindRoot" needs toexplicit the two specific algorithms eq_1 1 x 2^y α - (y α + (* + (2^2 x * y α - (* (2^+ / β - : eq_2 x 2^y 2^+ r - : 21 sys

guess wisely ... init 0.25 0.5 - 21 mat 0.25 - 0.5 - 21 mat 0.5 - 0.25 21 mat 0.5 0.25 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

roots roots eval : 0.2339 0.4419 - 0.2339 - 0.4419 - 0.4618 - 0.1917 0.4618 0.1917 42 mat

roots roots eval :

plot f1 circle roots axes 41 sys :

set the solving circle
from above 'r' ...

plot

Solving wrt [X,Y] datum ...

x y Xdatum 0 x * y - + (0.5 + : x y Ydatum x - 0 y * + (2.5 + : 21 sys

x y axes x y / - :

Xdatum Ydatum axes 3 1 sys

x y eyedatum x - 0 y * + (0.625 + :

f1 eyedatum axes 3 1 sys

eq_1 1 x 2^y α - (y α + (* + (2^2 x * y α - (* (2^+ / β - : eq_2 x y eyedatum : 21 sys

guess wisely ... init 0.5 0.25 21 mat 0.5 1.25 21 mat 12 mat :

header X solvedY 12 mat :

eye j 1212 mat : ct 1 : j j EYE j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el FindRoot : ct ct 1 + : 21 line for all EYE 1 el transpose EYE 2 el transpose stack : 41 line :

eye header eye stack :

solved pairwisewrt search Ydatum eye X solvedY 0.625 0.27 0.625 1.0202 32 mat

The 'eyedatum' is fully automated.
Watch/manage wrt 'init'.

x y eyedatum 0 x * y - (0.625 + :

f1 eyedatum axes 3 1 sys

eq_1 1 x 2^y α - (y α + (* + (2^2 x * y α - (* (2^+ / β - : eq_2 x y eyedatum : 21 sys

guess wisely ... init 1 - 0.5 21 mat 1 0.5 21 mat 12 mat :

header X solvedX 12 mat :

eye j 1212 mat : ct 1 : j j EYE j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el FindRoot : ct ct 1 + : 21 line for all EYE 1 el transpose EYE 2 el transpose stack : 41 line :

eye header eye stack :

solved pairwisewrt search Xdatum eye X solvedX 0.9724 - 0.625 0.9724 0.625 32 mat

The 'eyedatum' is fully automated.
Watch/respect wrt 'init'.

Multi solve scalar intersections ...

a 1 :

b 1 :

as 5 :

x y Clear 1

x y Hyperbola x 2^a 2^/ y 2^b 2^/ - (1 - :

ortho-axes x y axes x y / - :

x y Astroide x 2^y 2^+ as 2^- (3^27 as 2^* x 2^* y 2^* + :

"FindRoot" needs toexplicit the two specific algorithms eq_1 x y Hyperbola : eq_2 x y Astroide : 21 sys

guess wisely ... init 2 1.5 21 mat 2 - 1.5 21 mat 2 - 1.5 - 21 mat 2 1.5 - 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

roots 1.9109 1.6284 1.9109 - 1.6284 1.9109 - 1.6284 - 1.9109 1.6284 - 42 mat

isolate/store roots roots eval :

plot Hyperbola Astroide roots axes 41 sys :

plot

sanity roots 11 el roots 12 el Hyperbola :

sanity 61014 -^*

sanity roots 11 el roots 12 el Astroide roots 12 el roots 11 el Astroide 21 sys :

sanity 21010 -^* - 21010 -^* - 21 sys

Implicit 'roots' solve datum ...

m r x y Clear 1

m 1 : r 1 : g 9.8066 : 13 mat

x y P PE.D m g * x r - (* : PE.B l 2 r * (2^x r - (2^- sqrt : L 2 l π r * + 2 r * + (* : ΔL L 2 r * 3 π + (* - : y ΔL 2^2 / * 41 line : PE.D PE.B + 31 line :

recast/accelerate the system x y P x y P :

command line 'roots' Y0 Y1 21 mat 3020 - 21 mat :

r x Y0 P x Y1 P 21 mat x 0221 mat roots vectorize : 0.3554 - 2.2575 21 mat

r r eval :

xy r 2 el Y1 augment :

xy xy eval : 2.2575 20 - 12 mat

You can manage more 'roots' ...
from the command line vectorized.
Suggestive demo to be adapted
wrt other P(x,y) & 'init'.

all r 1 el Y0 augment r 2 el Y1 augment stack (:

all 0.3554 - 30 2.2575 20 - 22 mat

P all axes 3 1 sys

cumulative time 1 time t0 - 74 s *