Hatch

810cf922-5ebb-46ef-b8a7-0ac9e48dc25e 37 6 9 decimal

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Hatch

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 :

plotG & discretise Spline, FD cumulative

vx vy chr sz clr plotG n vx length : plot vx 1 el vy 1 el chr sz clr augment : i 2 n range plot plot vx i el vy i el chr sz clr augment stack : for plot 41 line :

Low step High x fnct inc U Low High step range : i 1 U rows range f i el U i el fnct eval : for U f augment 31 line :

data Σ FD 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 :

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - range deriv i el 0.5 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 - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X deriv augment 101 line :

The Smath "normalised" integrator companion
... enables integration of the 'ainterp, cinterp' f a b Int x f x a b int 32 / b a - (* w G w 01 int * ≡ memo ≡ 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 * 41 line :

Example 1

Test function x f x sqrt :

a 0 : b 2 : 21 line

Smath/Maple/Mathematica x f x int maple 2 x 32 /^* 3 /

x f x a b int 1.885615515881

x m 23 / x 32 /^* :

sanity check in  absolute accuracy b m a m abs + 1.88561808316413

x f a b Int 1.88561808316427

x f x m :

x g 0 :

<= activate the hatching "algo"

ω x f x m : x g 0 : blank the green hatching H x f x g a b 30 hatch : 41 sys :

Example 2

Test function x f x sqrt exp :

Smath/Maple agree with Mathematica x f x int maple 2 x sqrt exp * 1 - x sqrt + (*

x m 2 x sqrt exp * 1 - x sqrt + (* :

x f x a b int 5.407526

sanity check in  absolute accuracy b m a m abs + 5.40752818465627

x f a b Int 5.40752818465657

x f x m :

x g 0 :

<= activate the hatching "algo"

ω x f x m : x g 0 : blank the green hatching H x f x g a b 30 hatch : 41 sys :

sanity check the quadruplet Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Smath Romberg 5.40752818465657 ≡ Smath Integrator 5.407526 ≡ 41 line

Clear from previous

x f x f Clear :

Integrate the splines, derive incremental

XY 1 1.3 1.5 2.5 2 5.2 2.5 0.8 42 mat :

X XY 1 col : Y XY 2 col : x f X Y x cinterp : 31 line

Spline 1 1.01 2.51 x f inc : data X Y . 14 red plotG : 21 line

Spline Σ 151 row 2.5 4.72486 12 mat

Scalar derivative of the spline. 
Must be called below f(x):= cinterp(X,Y,x) f x h deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* :

h 0.001 :

f 1.5 h deriv 5.61

Spline Spline Σ data 3 1 sys 1 x ≤ x 2.5 ≤ & x f x h deriv out range if 1 1.3 . 14 red 1.5 2.5 . 14 red 2 5.2 . 14 red 2.5 0.8 . 14 red 4 5 mat 2 1 sys

f 1 2.5 Int 4.725

x f x f Clear :

Conclude this integration session:

1. If you can obtain the indefinite integral function, this approach is exact. 2. From the 3 demos, the normalised Int(f(x),a,b)is ~ 2 decimals from exact 3. The Smath "unknown" integrator ranges from exact [rarely] to ~ 5/6 decimals.

Example 3

u Φ 11 x 2^+ / :

x f 11 x 2^+ / :

a 5 - : b 5 : 21 line

x P x f x int maple : x atan

8 - P 1.44644133224814 -

x Φ x P x f x diff 3 1 sys

b P a P abs + 2.74680153389004

x f x a b int 2.74680153388998

x f a b Int 2.74680153388992

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

an associated integral 
of unspecified utility. z int u Φ u x - x int :

x q u Φ u x - x int maple : 2 x * 1 x 2^+ /

x int z int :

x int x int 2 x * / 2 1 sys

x q 2 x * x f (* ≡

u Φ u 0 x int maple x 1 x 2^+ /

0 f 1

x f x f Clear :

Example 4

x f 13 x x 2^- (* 3 x * - 2 / exp * :

a 0 : b 4 : 21 line

x m x f x int maple : 263 x * 2 / - exp * 23 x * 13 x * + (* + (* 27 /

x f x m 2 1 sys

x f x a b int 1.54878837254612 -

x f a b Int 1.54878837253552 -

sanity check in  absolute accuracy b m a m - 1.54878837252795 -

http://mathfaculty.fullerton.edu/mathews//n2003/adaptivequad/AdaptiveQuadProof.pdf

Matlab_Adaptive 1.54878823413 - ≡ depends upon TOL ≡ Smath_Native_Integrator 1.54878837254612 - ≡ Smath_Companion_Integrator 1.54878837253552 - ≡ True_Value 1.54878837252794813327 - ≡ 41 mat

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

Example 5

a 1016 -^: b 1 : 21 line

x m x ln x int maple : x 1 - x ln + (*

x ln x a b int 1.004452 -

Ω x f x 1 - x ln + (* : x g 0 : N 10 : blank green blank magenta G x f x g 0 b N hatch : e 0.15 black 15 mat 71 sys :

15 decimals a m b m + 1 -

Disambiguation

x m x a b int 0.749999980745912 -

x m a b Int 0.749999999999454 -

x f x ln :

6 true decimals x f a b Int 1 -

x m a 1 Int 0.999999768890007 -

x m 1 e Int 1

x m a 1 Int abs x m 1 e Int + 1.99999976888993

Specific suggestive procedure

certify/adapt procedure

f a b TechnicalArea x m x f x int maple : Area b m a m abs - : 21 line :

Example 1

Test function x f x sqrt :

a 0 : b 2 : 21 line

x f x a b int 1.885616

f a b TechnicalArea 1.88561808316413

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

Example 2

x f x sqrt exp :

x f x a b int 5.407526

f b a TechnicalArea abs 5.40752818465627

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

x f 11 x 2^+ / :

a 5 - : b 5 : 21 line

Example 3

x f x a b int 2.746802

f b a TechnicalArea abs 2.74680153389004

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

x f 13 x x 2^- (* 3 x * - 2 / exp * :

a 0 : b 4 : 21 line

Example 4

x f x a b int 1.548788 -

f a b TechnicalArea 1.54878837252795 -

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

Example 5

Is a case of "on error" not supported by Smath 5346 OFFicial release.

a 1016 -^: b 1 : 21 line

x ln x a b int 1.004452 -

x f x ln :

f a b TechnicalArea 1 -

x f x a b int ranges from 2 decimals to excat, depending upon f(x) ≡ f a b Int normalised companion ranges 12 decimals to exact ≡ f a b Int normalised companion integrates splines ≡ f a b MapleTechnicalArea is full 15 decimals ≡ 41 mat