[VA] SRC #012d - Then and Now: Area
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01-12-2023, 07:31 AM
Post: #10
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RE: [VA] SRC #012d - Then and Now: Area
All calculators used (42S, DM42 and Free42) have
1. define the rectangle Let's first determine the y-values of the rectangle. It's clear that EXP(-((x-d)^3-y)^2) <= 1, and it is equal to 1 for all y=(x-d)^3 so y^2/M + EXP(-SIN(y)) = 1 is a border: anything < 1 may belong to the area, > 1 does not. that turns out to be the case for y=0 and y=2.8274.., found with the 42S solver program. So our area is between these two y values. 00 { 27-Byte Prgm } 01▸LBL "VA4Y" 02 MVAR "Y" 03 RCL "Y" 04 ENTER 05 X^2 06 RCL÷ "M" 07 X<>Y 08 SIN 09 +/- 10 E^X 11 + 12 1 13 - 14 END Incidentally (and accidentally) I also found a tiny region -4.0851.. <= y <= -4.0492.. where the original inequality will hold (eg. y=-4.05 and x=0.004). There are no other regions since EXP(-SIN(y)) >= 1/e, abs(y) < SQRT((1 - 1/e)*M) = 4.3598. If we rewrite the orginal formula as an equality, we can isolate x for a given y: (1) x = d + CBRT(y +/- SQRT(-LN(y^2/M + EXP(-SIN(y))))) When the result of LN is positive, we're outside of the area and there are no real solutions for x. Within the area, there are two results for x (that coincide at the edges), describing the shape. A bit of trial and error gives the following rectangle boundaries, just for the purpose of graphing the area (I used Y=-0.1 to have the shape come clear off the bottom edge). Main Tiny X0: 0.97 -0.001 X1: 3.05 0.0045 Y0: -0.1 -4.049 Y1: 2.85 -4.086 2. graph the shape I graphed the main shape on my DM42 with the following drawing routine, with GrMod=2 (I enlarged the X range to X0=0.58 and X1=3.44 to have the X and Y scale be the same) Set the values with VARMENU "VA4D", EXIT the menu and do XEQ "VA4D". (the program can be made a lot more efficient, but that was not the scope here). VA4.bmp (Size: 12.31 KB / Downloads: 20) 00 { 167-Byte Prgm } 01▸LBL "VA4D" 02 MVAR "X0" 03 MVAR "X1" 04 MVAR "Y0" 05 MVAR "Y1" 06 MVAR "GrMod" 07 CLLCD 08 RCL "X1" 09 RCL- "X0" 10 RCL "ResX" 11 LSTO "X" 12 DSE ST X 13 ÷ 14 LSTO "Sx" 15 RCL "Y0" 16 RCL- "Y1" 17 RCL "ResY" 18 DSE ST X 19 ÷ 20 LSTO "Sy" 21▸LBL 10 22 RCL "Sx" 23 RCL× "X" 24 LASTX 25 - 26 RCL+ "X0" 27 LSTO "Xc" 28 RCL "ResY" 29 LSTO "Y" 30▸LBL 11 31 RCL "Sy" 32 RCL× "Y" 33 LASTX 34 - 35 RCL+ "Y1" 36 ENTER 37 ENTER 38 X^2 39 RCL÷ "M" 40 X<>Y 41 SIN 42 +/- 43 E^X 44 + 45 RCL "Xc" 46 RCL- "d" 47 3 48 Y^X 49 R^ 50 - 51 X^2 52 +/- 53 E^X 54 X≤Y? 55 GTO 00 56 RCL "Y" 57 RCL "X" 58 PIXEL 59▸LBL 00 60 DSE "Y" 61 GTO 11 62 DSE "X" 63 GTO 10 64 END 3. calculate area Well then. All that is needed is to integrate the shape over the Y-range. That can be done with a standard 42S integration routine, integrating dx = x2-x1, with x2 the positive root of (1): c := SQRT(-LN(y^2/M + EXP(-SIN(y)))); dx := CBRT(y + c) - CBRT(y - c); Area := integral(y=0 to 2.8274,dx); Here I use LLIM=0 and ULIM=2.82740261413 for Main area LLIM=-4.08514674764 and ULIM=-4.092122644 for Tiny area the accurate limits are calculated with the solver program VA4Y above The integration is done in Free42. 00 { 55-Byte Prgm } 01▸LBL "VA4" 02 MVAR "Y" 03 RCL "Y" 04 ENTER 05 X^2 06 RCL÷ "M" 07 X<>Y 08 SIN 09 +/- 10 E^X 11 + 12 LN 13 X>0? 14 CLX 15 X=0? 16 RTN 17 +/- 18 SQRT 19 ENTER 20 RCL+ "Y" 21 XEQ 13 22 X<>Y 23 RCL- "Y" 24 XEQ 13 25 + 26 RTN 27▸LBL 13 @ Cube Root 28 SIGN 29 LASTX 30 ABS 31 3 32 1/X 33 Y^X 34 × 35 END with ACC=1E-5 we get 2.07663 for the Main area 7.19762E-5 for the Tiny area This accuracy needs 16383 function evaluations (for the Main area). That would take quite some time on a real 42S. The DM42 on USB takes about 94s. Since the constant M is only given to 4 sig. digits, probably the Tiny area can be considered negligible. If not, I took the calculation of the integral a bit further, only on Free42 this time: ACC #evals Main Tiny 1E-5 16383 2.07662797558 7.19761929874-5 1E-7 131071 2.07662603505 7.19761930422E-5 1E-9 524287 2.07662630959 = 1E-11 = = 1E-15 = = So the sum of both areas is 2.07669828578 And now I'm going to try and understand Albert's contribution ;-) Cheers, Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE |
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