Maple Questions and Posts

These are Posts and Questions associated with the product, Maple

If I were to evaluate a single numerical integral, I can use evalf( Int(,method = _d01amc)).

But when the expression say is created by a built in function, Student[VectorCalculus][Hessian], from a complicated expression involve integrals. The resulting expression does not have the option "method = _d01amc". It then takes a long time to evaluate.

See this "HE" variable for example. HE.txt

value(HE); # takes a long time

evalf(HE); # takes a long time

 

Is there a way to evaluate "HE", using ",method = _d01amc" wherever necessary?

 

Thanks!

I'm doing some stuff with Taylor Polynomials on Maple. In particular, I'm trying to find the second order Taylor Polynomial of e^x cos x. I'm using the "taylor" command.

 

This is what I have so far:

 

> f := proc (x) options operator, arrow; exp(x)*cos*x end proc;
x -> exp(x) cos x
> s2 := taylor(f(x), x = (1/3)*Pi, 2);
> p2 := convert(s2, polynom);
1 /1 \
- exp|- Pi| cos Pi
3 \3 /

/ /1 \ 1 /1 \ \ / 1 \
+ |exp|- Pi| cos + - exp|- Pi| cos Pi| |x - - Pi|
\ \3 / 3 \3 / / \ 3 /
> y1 := evalf(subs(x = .5, f(x)));
0.8243606355 cos
> y2 := evalf(subs(x = .5, p2));
-0.208092943 cos

 

Everything looks fine until the last two lines.

 

How do I interpret ".8243606355*cos" ? Cosine should have an argument. Am I messing up something with my code, or is this standard in Maple language?

Hi all,

Assume that we have a vector, namely v:=[1,-2,3,-4] and we want to construct special matrix namely Z, from Vector v as follow:

first row is 1, secnond row is -2,..., the end row is -4 namely
Matrix([[1,1,1,1],[-2,-2,-2,-2],[3,3,3,3],[-4,-4,-4,-4]]);

in the other word every row of matrix is 4 times corresponding componet in vector.(for example v is (1*4))

how can we do this?

best wishes

Mahmood   Dadkhah

Ph.D Candidate

Applied Mathematics Department

n=512, what is number = nops(T) without actual running code?

 

n := 512;
T := [];
for i from 1 to n do
for j from 1 to n do
if i < j then
for k from 1 to n do
if j < k then
T := [op(T), [i,j,k]];
end:
od:
end if:
od:
od:
nops(T);
T;

Good day everyone, please help in writing finite difference algorithm for these coupled nonlinear ODE. See it here Algorithm.mw 

As title.

 

While  I can  decide velocity,charge and Distance.

 

 

 

Hi

I have three system of ODE and i would like to solve it using Homotopy perturbation method. Could you please provide to me the code in Maple or the Maple pachage that used to solve it by Homotopy perturbation method ?

I hope to hear you soon

Sara

 

Hi there. 

I'm kind of new to Maple and i'm trying to solve a Linear Algebra problem for my class of Linear Algebra of the course of Physics. Also, my first language is portuguese so forgive for my not-so-perfect english.

I have a (solved) linear system of 7 equations and 12 variables (A, B, C, D, E, F, G, H, I, J, K, L) that is the following:

  • A = 33 - K - L
  • B = 1 + F - J
  • C = -15 - F + J + K + L
  • D = 15 + H - K
  • E = 16 - F - H + J + K
  • G = 34 - H - J - L
  • I = 18 - J - K

Note: I'm using letters (A, B, ..., L) instead of X1X2, ..., X12 because it's easier to write it like this here and because I don't know if the Xn notation is allowed on Maple (i don't think so).

So, the system is possible but undetermined (with 5 degrees of freedom), being F, H, J, K and L the free variables.

Until here, everything's fine. The problem arises when the professor asks us for every solution of the system that satisfies the condition that all the variables (form A to L) are positive integers (A, B, C, D, E, F, G, H, I, J, K, L ϵ IN → natural numbers).

From my understanding, that gives rise to a system of linear inequalities with 12 variables and the following inequalities:

  • A = 33 - K - L > 0
  • B = 1 + F - J > 0
  • C = -15 - F + J + K + L > 0
  • D = 15 + H - K > 0
  • E = 16 - F - H + J + K > 0
  • G = 34 - H - J - L > 0
  • I = 18 - J - K > 0
  • > 0
  • > 0
  • > 0
  • > 0
  • > 0                            (and A,B,C,D,E,F,G,H,I,J,K,L ϵ IN)



After some research, i found that a possible way to solve this type of system of linear inequalities is trough a method of elimination (analog to Gauss-Jordan's elimination method for systems of linear equations) named Fourier-Motzkin. But it's hardwork and i wanted to do it on the computer. After some research, i came across with the following Maple command:

SolveTools[Inequality][LinearMultivariateSystem]

http://www.maplesoft.com/support/help/Maple/view.aspx?path=SolveTools%2fInequality%2fLinearMultivariateSystem

So, I tried to use that command to solve my system, with the following result (or non-result):

with(SolveTools[Inequality]);
LinearMultivariateSystem({F > 0, H > 0, J > 0, K > 0, L > 0, 1+F-J > 0, 15+H-K > 0, 18-J-K > 0, 33-K-L > 0, 34-H-J-L > 0, -15-F+J+K+L > 0, 16-F-H+J+K > 0}, [F, H, J, K, L]);

Error, (in SolveTools:-Inequality:-Piecewise) piecewise takes at least 2 parameters


So, i really need help solving this as the professor told us that the first one to solve would win a book, eheh. I don't know what I'm doing wrong. Maybe this Maple command is not made for 12 variables? Or maybe i'm just writing something on a wrong form. I've never used Maple before so i can be doing something really stupid without knowing it.

I would really apreciate an answer, as my only goal as a future physicist is to unveil the secrets of the Cosmos to us all.

Thank you again.

Miguel Jesus





This application calculates the number of photons reaching a camera sensor for a given exposure. A blackbody model of the sun is generated. The "Sunny 16" rule for exposure is demonstrated. Calculations are done using units.Photon_Exposure_Array.mw

Photon ExposureNULLNULL

Blackbody Model of the Sun

    h := Units:-Standard:-`*`(Units:-Standard:-`*`(0.6626069e-33, Units:-Standard:-`^`(Unit('m'), 2)), Units:-Standard:-`*`(Unit('kg'), Units:-Standard:-`/`(Unit('s')))): 

Plank Constant       

  kb := Units:-Standard:-`*`(Units:-Standard:-`*`(0.1380650e-22, Units:-Standard:-`*`(Units:-Standard:-`^`(Unit('m'), 2), Units:-Standard:-`/`(Units:-Standard:-`^`(Unit('s'), 2)))), Units:-Standard:-`*`(Unit('kg'), Units:-Standard:-`/`(Unit('K')))): 

Boltzman Constant  

c := Units:-Standard:-`*`(0.2997925e9, Units:-Standard:-`*`(Unit('m'), Units:-Standard:-`/`(Unit('s')))):  ``

Light Speed

Rsun := Units:-Standard:-`*`(Units:-Standard:-`*`(6.955, Units:-Standard:-`^`(10, 8)), Unit('m')): ``

Sun Radius  

Re_orb := Units:-Standard:-`*`(Units:-Standard:-`*`(1.496, Units:-Standard:-`^`(10, 11)), Unit('m')): ``

Earth Orbit

Tsun := Units:-Standard:-`*`(5800, Unit('K')): ``

Sun Color Temperature     

 tf_atm := .718: 

Transmission Factor  

 

Sun: Spectral Radiant Exitance to Earth: Spectral Irradiance                   

  "M(lambda):=(2*Pi*h*c^(2))/((lambda)^(5))*1/((e)^((h*c)/(lambda*kb*Tsun))-1)*(Rsun/(Re_orb))^(2)*tf_atm:" NULL

evalf(M(Units:-Standard:-`*`(555, Unit('nm')))) = 1277414308.*Units:-Unit(('kg')/(('m')*('s')^3))"(->)"1.277414308*Units:-Unit(('W')/(('nm')*('m')^2))NULL

Photopic Relative Response VP vs λ

 

csvFile := FileTools[Filename]("/VPhotopic.csv")NULL = "VPhotopic.csv"NULL

VPdata := ImportMatrix(csvFile) = Vector(4, {(1) = ` 471 x 2 `*Matrix, (2) = `Data Type: `*float[8], (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})NULLNULL

 

`&lambda;P` := [seq(1 .. 4000)]:

VP := ArrayInterpolation(VPdata, `&lambda;P`):             (ArrayInterpolation for x,y data VPdata returns y' for new x data lambdaP)

NULLVParray := [`$`([`&lambda;P`[n], VP[n]], n = 1 .. 4000)]:                     

Mearth := [`$`([n, Units:-Standard:-`*`(Units:-Standard:-`*`(M(Units:-Standard:-`*`(n, Unit('nm'))), Unit('nm')), Units:-Standard:-`*`(Units:-Standard:-`^`(Unit('s'), 3), Units:-Standard:-`/`(Unit('kg'))))], n = 1 .. 4000)]:````

``

dualaxisplot(plot([Mearth], lambda = 300 .. 900, style = line, color = [blue], labels = ["&lambda; (nm)", "M (W/nm m^2)"], title = "Spectral Radiant Exitance of the Sun", titlefont = ["ARIAL", 15], legend = [Exitance], size = [800, 300]), plot([VParray], style = line, color = [green], labels = ["&lambda; (nm)", "Relative Response"], legend = [Units:-Standard:-`*`(Units:-Standard:-`*`(Photopic, Relative), Response)]))

 

``

 

 

 

Illuminance in Radiometric and Photometric Units:

E__r := sum(Units:-Standard:-`*`(M(Units:-Standard:-`*`(lambda, Unit('nm'))), Unit('nm')), lambda = 200 .. 4000) = 984.7275549*Units:-Unit(('kg')/('s')^3)"(->)"984.7275549*Units:-Unit(('W')/('m')^2)NULL

NULL

E__po := Units:-Standard:-`*`(Units:-Standard:-`*`(683.002, Units:-Standard:-`*`(Unit('lm'), Units:-Standard:-`/`(Unit('W')))), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(VP[lambda], M(Units:-Standard:-`*`(lambda, Unit('nm')))), Unit('nm')), lambda = 200 .. 4000)) = HFloat(91873.47376063903)*Units:-Unit('lx')NULL

Translation from Illuminance to Luminance for Reflected Light;

 

Object Reflectance          R__o:      

Object Luminance           L__po := proc (R__o) options operator, arrow; R__o*E__po/(Pi*Unit('sr')) end proc:                evalf(L__po(1)) = HFloat(29244.234968360346)*Units:-Unit(('cd')/('m')^2) 

 

Illuminance of a Camera Sensor  Eps applied for time texp determines Luminous Exposure Hp;

Ideal Illuminance is determined by the exposure time texp, effective f-number N and to a less extent the angle to the optical axis θ;

 

• 

H       Luminous Exposure

• 

Eps     Illuminance to the Camera

• 

N                                               Effective F-Number

• 

texp             Exposure Time

• 

θ        Angle to the Optical Axis    

 

E__ps_ideal = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), L__po), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))):

H__p_ideal = Units:-Standard:-`*`(E__ps_ideal, t__exp):

 

The camera meter determines the exposure time texp to balance the object luminance, reflectance and effective f-number. It does this based on an internal constant k and the camera ISO s.

• 

s        ISO Gain (Based on saturation at 3 stops above the average scene luminance)

• 

k       Reflected Light Meter Calibration Constant      k__m := Units:-Standard:-`*`(Units:-Standard:-`*`(12.5, Unit('lx')), Unit('s')):  

                                                                                                  for Nikon, Canon and Sekonic

• 

c        Incident Light Meter Calibration Constant       c__m := Units:-Standard:-`*`(Units:-Standard:-`*`(250, Unit('lx')), Unit('s')):        

                                                                                                  for Sekonic with flat domeNULL

N^2/t__exp = `#mrow(mi("\`E__po\`"),mo("&sdot;"),mi("s"))`/c__m                        (Incident Light Meter)  NULL 

Units:-Standard:-`*`(Units:-Standard:-`^`(N, 2), Units:-Standard:-`/`(t__exp)) = Units:-Standard:-`*`(`#mrow(mi("\`L__po\`"),mo("&sdot;"),mi("s"))`, Units:-Standard:-`/`(k__m)):                        (Reflected Light Meter)

NULL

Solve for H in terms of the Camera Meter Constant k and s

 

Es = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))): NULL

t = Units:-Standard:-`*`(Units:-Standard:-`*`(km, Units:-Standard:-`^`(N, 2)), Units:-Standard:-`/`(Units:-Standard:-`*`(Lo, s))):NULL

NULL

NULL

H = Es*t

H = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))), Units:-Standard:-`*`(Units:-Standard:-`*`(km, Units:-Standard:-`^`(N, 2)), Units:-Standard:-`/`(Units:-Standard:-`*`(Lo, s))))"(=)"H = (1/4)*Pi*cos(theta)^4*km/sNULLNULL

 t = H/Es

t = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), km), Units:-Standard:-`/`(s))), Units:-Standard:-`/`(Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2))))))"(=)"t = km*N^2/(Lo*s)NULLNULL

H__p := proc (s, theta) options operator, arrow; (1/4)*Pi*k__m*cos(theta)^4/s end proc:                                              

  evalf(H__p(100, 0)) = 0.9817477044e-1*Units:-Unit(('cd')*('s')/('m')('radius')^2)"(->)"0.9817477044e-1*Units:-Unit(('lx')*('s'))NULL

 

Note:  Meters are typically set for a scene reflectance 3 stops below 100% or 12.5%.

           

  E__ps := proc (N, R__o, theta) options operator, arrow; (1/4)*Pi*Unit('sr')*R__o*E__po*cos(theta)^4/(Pi*Unit('sr')*N^2) end proc:               

 evalf(E__ps(16, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)), 0)) = HFloat(11.215023652421756)*Units:-Unit('lx')                                                                                                   

t__exp_ideal := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/E__ps(N, R__o, theta) end proc:                                     

  evalf(t__exp_ideal(16, 100, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)))) = HFloat(0.008753862094289947)*Units:-Unit('s') NULL NULL

 

 

Actual exposure time includes typical lens losses;

 m := Units:-Standard:-`/`(80):``

Magnification  

  T := .9:``

Lens Transmittance

 F := 1.03:``

Lens Flare

V := 1: ``

Vignetting

 

                                                  ``

Total Lens Efficiency

q := Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(T, F), V), Units:-Standard:-`^`(Units:-Standard:-`+`(1, Units:-Standard:-`-`(m)), 2)):                                      evalf(q) = .9039698438NULL

 

Replacing Eps with q*Eps we get the "Sunny 16" relation between exposure time and ISO;  NULL

t__exp := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/(q*E__ps(N, R__o, theta)) end proc:NULL               evalf(t__exp(16, 100, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)))) = HFloat(0.009683798806264942)*Units:-Unit('s')NULL

t__exp_alt := proc (N, s, R__o) options operator, arrow; k__m*N^2*Pi/(s*q*R__o*E__po) end proc:                  evalf(t__exp_alt(16, 100, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)))) = HFloat(0.00968379880412244)*Units:-Unit('s') 

• 

The Number of Photons NP Reaching the Sensor Area A;

• 

Circle of confusion for 24x36mm "Full Frame" for 1 arcminute view at twice the diagonal:

                          A__cc := Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`^`(Units:-Standard:-`*`(12.6, Unit('`&mu;m`')), 2)), Units:-Standard:-`/`(4)):    

     

• 

  Sensor Bandwidth                                          Photopic Response VP

• 

  Exposure Time for Zone 5: Rscene=12.5% , Saturation in Zone 8 Rscene=100%

• 

  Camera ISO differs from Saturation ISO. Typical Saturation ISO is 2300 when the camera is set to 3200. See DxoMark.

 

NULL

The average number of photons for exposure time based on Reflectance of the scene  relative to the metered value:    

Zone 5;   R__meter := R__scene: 

NP := proc (s, R__o, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc: 

                                                                               evalf(NP(2300, 1, Units:-Standard:-`*`(0, Unit('deg')))) = HFloat(2191.5645712603696)  NULL

Zone 8;       R__meter := Units:-Standard:-`*`(R__scene, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3))):   NULL

NP__sat := proc (s, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc:  NULL

                                                                              evalf(NP__sat(2300, Units:-Standard:-`*`(0, Unit('deg')))) = HFloat(17532.516570082957)NULL

NULL

 

Approximate Formula

 

H__sat := proc (s__sat) options operator, arrow; H__p(s__sat, 0)*E__ps(N, 1, 0)/E__ps(N, 1/8, 0) end proc:      

                                                                                       evalf(H__sat(s__sat)) = HFloat(78.53981635)*Units:-Unit(('cd')*('s')/('m')('radius')^2)/s__satNULLNULL

Average Visible Photon Energy

P__e_ave := Units:-Standard:-`*`(Units:-Standard:-`/`(Units:-Standard:-`+`(850, -350)), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(h, c), Units:-Standard:-`/`(Units:-Standard:-`*`(lambda, Unit('nm')))), lambda = 350 .. 850)):                    evalf(P__e_ave) = 0.3533174192e-18*Units:-Unit('J') 

NPtyp := proc (s__sat) options operator, arrow; H__sat(s__sat)*A__cc/(683.002*(Unit('lm')/Unit('W'))*P__e_ave) end proc: 

                               evalf(NPtyp(2300)) = HFloat(17644.363333654386)"(->)"HFloat(17644.363333654386)NULL

NULL

 

Download Photon_Exposure_Array.mw

teliko.mwMapl_doc.docxHello,

After I have substitute all the variables by hand ,maple still continious  processing(27 hours and still evaluating)for a simple

summarisation by the time that live math2 needed only 2 mins.Have i done something wrong?I want to solve the following equation which is shown in the attached file with the experimental data

OK. Perfect. It runs.

Now, I would like to remove all the set of solutions where for two variables (for example a and b) are negative.

For example, if I have this :

{a=1,b=-1,c=1}, {a=1,b=1,c=1}, {a=-2,b=-1,c=1}

After filtering, I would like to have only :

{a=1,b=1,c=1}

Thanks a lot for your help.

 

Hello,

After a solve on a system composed with trigonometric functions, i obtain several group of solutions.

Here a picture of my solutions :



How can I do to extract from thses solutions only the group composed with real components.

Thanks a lot for your help

I read this, but there was nothing on how to write a help associated with it.

Does anyone know where I can find it?

 

Many thanks,

 

casperyc

i got kernel connection lost error when memory used reach 2.87GB

how to make it fully used 3.8GB ram, total 4GB ram

indexlistresult := [];
gaga := [seq(i, i=1..512];
nops(indexlistresult);
indexlistresult := [op(indexlistresult), op(gaga)];
nops(indexlistresult);
indexlistresult := [op(indexlistresult), op(gaga)];
nops(indexlistresult);
indexlistresult := [op(indexlistresult), op(gaga)];
nops(indexlistresult);
with(combinat):
gg := permute(indexlistresult, 3);

I actually had the same problem on Maple 16. When i go to plot an equation (usually on implicit plots) it just repeats the equation in blue text rather than showing any sort of plot. Im sure i am just missing a simple option somewhere, but for the life of me I cant find it. Any and all help would be appreciated. 

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