The attached worksheet shows a small selection of new and improved results in integration for Maple 2016. Note that integration is a vast topic, so there will always be more improvements that can be made, but be sure that we are working on them.

Maple2016_Integration.mw

Austin Roche
Mathematical Software, Maplesoft

A wealth of knowledge is on display in MaplePrimes as our contributors share their expertise and step up to answer others’ queries. This post picks out one such response and further elucidates the answers to the posted question. I hope these explanations appeal to those of our readers who might not be familiar with the techniques embedded in the original responses.

The Question: Variable Identification

Don Carota wanted to know the best approach for finding variables that are not assigned to a value within an equation. He wrote:

I got a set of equations to solve, like this one:

eq[1]:=W[1,0]*(a*HRa[1,0]+b*ga[1,0]+c)=d*SR[1,1]/grid

a,b,c,d are numbers, like 2.0458 and so on.

When I want to solve the set, I need to tell Maple the command solve:

solve( {seq(eq[i],i=1..N)},{variables});  (N is an integer of course)

To set the variables, one must check each equation to write: {W[1,0],HRa[1,0],ga[1,0]...} and so on.

I know that I can use the command is(variable,assignable) to check if a variable has not a value assigned already and, according to true/false I can construct the set {variables} and solve the set of equations.

That´s an easy solution if I need to check variables under a certain pattern, like: X[1], X[2], X[3] since I can create a loop and check them one after the other. But my problem is that I have different names for variables or that variables change to assignable from assigned according to other conditions, so I can never be sure if W[1,0] is going to be a variable to compute in all steps instead of SR[1,1].

for example:

if a>3 then
SR[1,1]:=1/2;
else
W[1,0]:=300:
end if;

So, when I need to type solve, the {variables} part is different according to each case. Is there any command that allows me to insert an expression and Maple can return me the variables or parameters in the expression that are not numeric already?

(note that the link added to the is command above was added by me, not the original author)

dharr and Carl Love provided solutions that use the indets command.

The code provided by dharr is as follow:

1. indets(eq[1],name);

Result: gives the indeterminates: {a, b, c, d, HRa[1, 0], SR[1, 1], W[1, 0], ga[1, 0]}

The code provided by Carl Love is as follows:

1.       indets(eq[1], assignable(name));

or, doing all equations at once,

2.       indets({entries(eq, nolist)}, assignable(name));

Further Explaining the indets and type commands.

Both dharr and Carl Love provided an answer that used the indets command. In essence the indets command used in this example contains two arguments: indets(expr, typename). Expr is a rational expression that only uses the operations such as addition, subtraction, division, and multiplication. Typename is a parameter used when the desired return is a set containing all subexpressions in expr that are of type typename.

Carl Love used the assignable(name) argument  for the typename parameter in order to return all the variables that can be assigned a value, excluding constants such as Pi that are also considered names. Indeed, assignable is a type and can be used without an additional argument. For example, the command indets(x+f(x)+y=1, assignable) returns {x,y,f(x)} because all three symbols can be assigned values. However, indets(x+f(x)+y=1, assignable(name)) returns just {x,y} because f(x) is of type function, not of type name. Similarly, indets(x+y=Pi, assignable) would return just {x,y} because Pi is not considered to be something that can be assigned to.

Carl’s second command used ({entries(eq, nolist)} as the expr parameter. In this version, eq is the table whose members are the individual equations. Remember, the syntax x[1] creates a table whose name is x, and whose entry is the object assigned to x[1]. The entries(t) function returns a sequence of the table members, each member being placed in list brackets. By including the option nolist, the return is then a sequence of table members without list brackets. 

Finally, note that different programmers will use different approaches to finding “indeterminants” in expressions. Dr. Lopez pointed out that some years ago he asked three different programmers about extracting the “assignable names” in an expression such as q:=x+Pi+cos(a). The naive indets(q) returns {a,x,cos(a)}, whereas indets(q,name) returns {Pi,a,x}. However, select(type,indets(q),name) returns {a,x}, as does indets(q,And(symbol,Not(constant))).

Don Carota’s question is able to showcase some of the different types that are within Maple’s platform. Therefore, it is important to go over what the type-checking function is and what it does. In many contexts, it is not necessary to know the exact value of an expression; instead it is enough to know if the value belongs to a group of expressions that have similarities. Such groups are knows as types.

Maple’s engine uses the type function in every single procedure to direct and maintain the flow of control in algorithms and to decide if the user’s input is valid. There are about 240 different types that Maple recognizes including: prime, string, symbol, list, etc.  Let’s see some examples of how the function works using elements from this question. 

Type has two parameters: an expression e, and a valid type expression t. To check that the output of the entries(eq,nolist) is indeed not a list, the type command can be used as follows:

As expected, the last command returns false! If you want to learn more about the type and indets commands you can visit their corresponding help pages: ?type, ?indets.

This blog was written by Maplesoft’s intern Pia under the supervision of Dr. Robert Lopez. We both hope that you find this useful. If there is a particular question on MaplePrimes that you would like further explained, please let us know. 

polygon_2_color.mw

Imitation coloring both sides of the polygon in 3d.  We  build a new polygon in parallel with our polygon on a very short distance t. (We need any three points on the polygon plane, do not lie on a straight line.) This place in the program is highlighted in blue.

Paint the polygons are in different colors.

Valery Ochkov and Volodymyr Voloshchuk have developed a series of thermal engineering applications in Maple 2016. The applications explore steam turbine power generation and refrigeration cycles, and use the ThermophysicalData package for fluid properties.

Their work can be found at the following locations on the Application Center.

• Valery Ochkov
• Volodymyr Voloshchuk

I especially like

• this application, which optimizes the extraction pressures of a steam turbine to maximize its efficiency,
• and this application, which plots the state of a two-stage refrigeration cycle on a pressure-enthalpy chart.

This procedure calculate the equations of motions for Euclidean space and Minkowski space  with help of the Jacobian matrix.

Download bsp_jacobi.mw

Download bsp_jacobi_minkowski.mw

Magnet lattices for particle accelerators (the sequence of focusing and bending magnets and drift sections making up a beam line) are often designed numerically using computer codes like MAD that model each beam-line element using either a matrix description or numeric integration, or some other algorithm. The Lattice package for Maple—recently published in Maplesoft's Application Center—allows modelling such beam lines using the full algebraic power of Maple. In this way, analytic solutions to beam-optics problems can be found in order to establish feasibility of certain solutions or study parameter dependencies. Beam lines are constructed in an intuitive way from standard beam-optical elements (drifts, bends, quadrupoles etc.) using Maple's object-oriented features, in particular Records which represent individual elements or whole beam lines. These Records hold properties like the first-order transport matrices, element length and some other properties plus, for beam lines, the elements the line is composed of. Operations like finding matched Twiss (beam-envelope) functions and dispersion are implemented and the results can be plotted. The Lattice package knows about beam matrices and can track such matrices as well as particles in a beam. The tracking function (map) is implemented separately from the first-order matrices thus allowing nonlinear or even scattering simulations; at present sextupole elements, compensating-wire elements and scattering-foil elements take advantage of this feature. Standard textbook problems are programmed and solved easily, and more complicated ones are readily solved as well €”within the limits set by Maple's capabilities, memory and computing time. Many investigations are possible by using standard Maple operations on the relevant properties of the beam lines defined. An interesting possibility is, e.g., using Maple's mtaylor command to truncate the transfer map of a beam line to a desired order, making it a more manageable function.

The package can output a beam-line description in MAD8 format for further refinement of the solution, cross-checking the results and possibly more detailed tracking.

Developed initially for my own use I have found the package useful for a number of accelerator design problems, teaching at the US Particle Accelerator School as well as in modelling beam lines for experiments I have been involved in. Presently at Version 1.0 the package still has limits; esp. the higher-order descriptions are not yet as complete as desirable. Yet already many practical design problems can be tackled in great detail. The package is backwards compatible to Maple 15. It can be found in the Application center together with help database files (old and new style) and a Users Guide.

An example showing the flavor of working with the package is attached (FODO_example.mw). It analyzes a FODO cell, the basic cell used in many ring accelerators.

Uli Wienands, aka Mac Dude

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations. 
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw 
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations. 
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw 
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations. 
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw 
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

In this course you will learn automatically using Maple course Statics applied to civil engineering especially noting the use of components properly. Let us see the use of Maple to Engineering.

Static_for_Engineering.mw

(in spanish)

Atte.

Lenin Araujo Castillo

Ambassador of Maple

#Most  dediction of  depth of field of optical lens  involves various simplification,  hence cannot be used  for  close up photography.  With Maple, it is easy to obtain  precise   Depth of field  formuar for optical lens  without  any simplification

> restart; h := H = F^2/(N*coc)+F; E0 := 1/d+1/D0 = 1/F; E1 := 1/(d+e)+1/D2 = 1/F; E2 := a/(d+e) = coc/delta; E3 := a = F/N; eq := {E0, E1, E2, E3, h}; var := {D2, N, coc, d, delta}; e := -delta; sol1 := solve(eq, var); t1 := op(sol1)[1]; Dfar := op(t1)[2]; e := delta; sol2 := solve(eq, var); t2 := op(sol2)[1]; Dnear := op(t2)[2];

>

>
>

>
>
>

IntegerPoints2  procedure generalizes  IntegerPoints1  procedure and finds all the integer points inside a bounded curved region of arbitrary dimension.  We also use a brute force method, but to find the ranges for each variable  Optimization[Minimize]  and   Optimization[Maximize]  is used instead of  simplex[minimize]  or  simplex[minimize] .

Required parameters of the procedure: SN is a set or a list of  inequalities and/or equations with any number of variables, the Var is the list of variables. Bound   is an optional parameter - list of ranges for each variable in the event, if  Optimization[Minimize/Maximize]  fails. By default  Bound  is NULL.

If all constraints are linear, then in this case it is recommended to use  IntegerPoints1  procedure, as it is better to monitor specific cases (no solutions or an infinite number of solutions for an unbounded region).

Code of the procedure:

IntegerPoints2 := proc (SN::{list, set}, Var::(list(symbol)), Bound::(list(range)) := NULL)

local SN1, sn, n, i, p, q, xl, xr, Xl, Xr, X, T, k, t, S;

uses Optimization, combinat;

n := nops(Var);

if Bound = NULL then

SN1 := SN;

for sn in SN1 do

if type(sn, `<`) then

SN1 := subs(sn = (`<=`(op(sn))), SN1) fi od;

for i to n do

p := Minimize(Var[i], SN1); q := Maximize(Var[i], SN1);

xl[i] := eval(Var[i], p[2]); xr[i] := eval(Var[i], q[2]) od else

assign(seq(xl[i] = lhs(Bound[i]), i = 1 .. n));

assign(seq(xr[i] = rhs(Bound[i]), i = 1 .. n)) fi;

Xl := map(floor, convert(xl, list)); Xr := map(ceil, convert(xr, list));

X := [seq([$ Xl[i] .. Xr[i]], i = 1 .. n)];

T := cartprod(X); S := table();

for k while not T[finished] do

t := T[nextvalue]();

if convert(eval(SN, zip(`=`, Var, t)), `and`) then

S[k] := t fi od;

convert(S, set);

end proc:

In the first example, we find all the integer points in the four-dimensional ball of radius 10:

Ball := IntegerPoints2({x1^2+x2^2+x3^2+x4^2 < 10^2}, [x1, x2, x3, x4]):  # All the integer points

nops(Ball);  # The total number of the integer points

seq(Ball[1000*n], n = 1 .. 10);  # Some points

                                                                    48945

                  [-8, 2, 0, -1], [-7, 0, 1, -3], [-6, -4, -6, 2], [-6, 1, 1, 1], [-5, -6, -2, 4], [-5, -1, 2, 0],

                                [-5, 4, -6, -2], [-4, -5, 1, 5], [-4, -1, 6, 1], [-4, 3, 5, 6]

In the second example, with the visualization we find all the integer points in the inside intersection of  a cone and a cylinder:

A := <1, 0, 0; 0, (1/2)*sqrt(3), -1/2; 0, 1/2, (1/2)*sqrt(3)>:  # Matrix of rotation around x-axis at Pi/6 radians

f := unapply(A^(-1) . <x, y, z-4>, x, y, z):  

S0 := {4*x^2+4*y^2 < z^2}:  # The inner of the cone

S1 := {x^2+z^2 < 4}:  # The inner of the cylinder

S2 := evalf(eval(S1, {x = f(x, y, z)[1], y = f(x, y, z)[2], z = f(x, y, z)[3]})):

S := IntegerPoints2(`union`(S0, S2), [x, y, z]);  # The integer points inside of the intersection of the cone and the rotated cylinder

Points := plots[pointplot3d](S, color = red, symbol = solidsphere, symbolsize = 8):

Sp := plot3d([r*cos(phi), r*sin(phi), 2*r], phi = 0 .. 2*Pi, r = 0 .. 5, style = surface, color = "LightBlue", transparency = 0.7):

F := plottools[transform]((x, y, z)->convert(A . <x, y, z>+<0, 0, 4>, list)):

S11 := plot3d([2*cos(t), y, 2*sin(t)], t = 0 .. 2*Pi, y = -4 .. 7, style = surface, color = "LightBlue", transparency = 0.7):

plots[display]([F(S11), Sp, Points], scaling = constrained, orientation = [25, 75], axes = normal);

In the third example, we are looking for the integer points in a non-convex area between two parabolas. Here we have to specify ourselves the ranges to enumeration (Optimization[Minimize] command fails for this example):

P := IntegerPoints2([y > (-x^2)*(1/2)+2, y < -x^2+8], [x, y], [-4 .. 4, -4 .. 8]);

A := plots[pointplot](P, color = red, symbol = solidcircle, symbolsize = 10):

B := plot([(-x^2)*(1/2)+2, -x^2+8], x = -4 .. 4, -5 .. 9, color = blue):

plots[display](A, B, scaling = constrained);

 IntegerPoints2.mw

This post is my attempt to answer the question from   here : how to find all integer points (all points with integer coordinates) in the intersection of two cubes. The following procedure  IntegerPoints  solves a more general problem: it finds all the integer points of a bounded polyhedral region of arbitrary dimension, defined by a system of linear inequalities and / or equations.

Required parameters of the procedure: SN is a set or a list of linear inequalities and/or equations with any number of variables, the Var is the list of variables. The procedure returns the set of all integer points, satisfying the conditions  SN .

Code of the procedure:

restart;

IntegerPoints := proc (SN::{list, set}, Var::list)

local SN1, sn, n, Sol, k, i, s, S, R;

uses PolyhedralSets, SolveTools[Inequality];

SN1 := convert(evalf(SN), fraction);

for sn in SN1 do

if type(sn, `<`) then SN1 := subs(sn = (`<=`(op(sn))), SN1)

end if; end do;

if IsBounded(PolyhedralSet(SN1)) = false then error "The region should be bounded" end if;

n := nops(Var);

Sol := LinearMultivariateSystem(SN, Var);

if Sol = {} then return {} else

k := 0;

for s in Sol do if nops(indets(s[1])) = 1 then

S[0] := [[]];

for i to n do

S[i] := [seq(seq([op(j1), op(j2)], j2 = [isolve(eval(s[i], j1))]), j1 = S[i-1])] end do;

k := k+1; R[k] := op(S[n]);

end if; end do;

convert(R, set);

map(t->rhs~(t), %);

end if;

end proc:

Examples of use:

IntegerPoints({x > 0, y > 0, z > 0, 2*x+3*y+z < 12}, [x, y, z]);

  {[1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 1, 4], [1, 1, 5], [1, 1, 6], [1, 2, 1], [1, 2, 2], [1, 2, 3], [2, 1, 1], [2, 1, 2],

                                   [2, 1, 3], [2, 1, 4], [2, 2, 1], [3, 1, 1], [3, 1, 2]}

IntegerPoints({x > 0, y > 0, z > 0, 2*x+3*y+z = 12}, [x, y, z]);

                                    {[1, 1, 7], [1, 2, 4], [1, 3, 1], [2, 1, 5], [2, 2, 2], [3, 1, 3], [4, 1, 1]}

IntegerPoints([x > 0, y > 0, z > 0, 2*x+3*y+z = 12, x+y+z <= 6], [x, y, z]);

                                                           {[1, 3, 1], [2, 2, 2], [4, 1, 1]}

isolve({x > 0, y > 0, z > 0, 2*x+3*y+z < 12});  #  isolve fails with these examples

              Warning, solutions may have been lost

isolve({x > 0, y > 0, z > 0, 2*x+3*y+z = 12});

              Warning, solutions may have been lost

In the following example (with a visualization) we find all integer point in the intersection of a square and a triangle:

S1 := {x > 0, y > 0, x < 13/2, y < 13/2}:

S2 := {y > (1/4)*x+1, y < 2*x, y+x < 12}:

S := IntegerPoints(`union`(S1, S2), [x, y]):

Region := plots[inequal](`union`(S1, S2), x = 0 .. 7, y = 0 .. 7, color = "LightGreen", nolines):

Points := plot([op(S)], style = point, color = red, symbol = solidcircle):

Square := plottools[curve]([[0, 0], [13/2, 0], [13/2, 13/2], [0, 13/2], [0, 0]], color = blue, thickness = 3):

Triangle := plottools[curve]([[4/7, 8/7], [4, 8], [44/5, 16/5], [4/7, 8/7]], color = blue, thickness = 3):

plots[display](Square, Triangle, Points, Region, scaling = constrained);

In the following example (with a visualization) we find all integer point in the intersection of two cubes. The second cube is obtained from the first cube by rotation with orthogonal matrix  A  and by a translation:

A := <1/3, 2/3, 2/3; -2/3, 2/3, -1/3; -2/3, -1/3, 2/3>:

f := unapply(A^(-1).<x+5, y-4, z-7>, x, y, z):

S1 := {x > 0, y > 0, z > 0, x < 6, y < 6, z < 6}:

S2 := eval(S1, {x = f(x, y, z)[1], y = f(x, y, z)[2], z = f(x, y, z)[3]}):

S := IntegerPoints(`union`(S1, S2), [x, y, z]);

Points := plots[pointplot3d](S, color = red, symbol = box):

Cube := plottools[cuboid]([0, 0, 0], [6, 6, 6], color = blue, linestyle = solid):

F := plottools[transform]((x, y, z)->convert(A.<x, y, z>+<-5, 4, 7>, list)):

plots[display](Cube,  F(Cube), Points, scaling = constrained, linestyle = solid, transparency = 0.7, orientation = [25, 75], axes = normal);

In the example below, all the ways to exchange $ 1 coins of 1, 5, 10, 25 and 50 cents, if the number of coins no more than 8, there is no pennies and there is at least one 50-cent coin:

IntegerPoints({x1 = 0, x2 >= 0, x3 >= 0, x4 >= 0, x5 >= 1,  x1+5*x2+10*x3+25*x4+50*x5 = 100, x1+x2+x3+x4+x5 <= 8}, [x1, x2, x3, x4, x5]);

nops(%);

                              {[0, 0, 0, 0, 2], [0, 0, 0, 2, 1], [0, 0, 5, 0, 1], [0, 1, 2, 1, 1], [0, 2, 4, 0, 1],

                                                 [0, 3, 1, 1, 1], [0, 4, 3, 0, 1], [0, 5, 0, 1, 1]}

                                                                                    8

Integer_points.mw

Addition: Below in my comments another procedure  IntegerPoints1  is presented that solves the same problem.

Some Maple 18 short (and I believe elegant) code for doing gravitational simulations with N bodies in space:

N_body_problem.mw

Initial velocities have been tweaked to keep the system stable for the duration of the animation.

Please feel free to fiddle with its parameters, velocities and positions and/or N itself, to produce more interesting animations or re-use the code therein (You can safely ignore the (c), it's there just for archiving purposes).

The following are animations from three runs with N=4, N=3 and N=2, no other parameters changed.

There has been a spate of Questions posted in the past week about computing eigenvalues. Invariably, the Questioners have computed some eigenvalues by applying fsolve to a characteristic polynomial obtained from a floating-point matrix via LinearAlgebra:-Determinant. They are then surprised when various tests show that these eigenvalues are not correct. In the following worksheet, I show that the eigenvalues computed by the fsolve@Determinant method (when applied to a floating-point matrix) are 100% garbage for dense matrices larger than about Digits x Digits. The reason for this is that computing the determinant introduces too much round-off error into the coefficients of the characteristic polynomial. The best way to compute the eigenvalues is to use LinearAlgebra:-Eigenvalues or LinearAlgebra:-Eigenvectors. Furthermore, very accurate results can be obtained without increasing Digits.

Download Eigenvalues.mw