Geometrical entertainment in the form of rolling without slipping, now inside a torus.
You can also print out the corresponding equations - these graphs are # in the text, but it is better to do this separately from the geometric animation, because textplot3d takes up a lot of resources.

Please consider it not as a Maple program, but simply as an idea for a corresponding algorithm.
for_TORUS_IN_TORUS_for.mw
 

Grupo6_Tc1.mw

Should maple have given solution to x as zero in the following?

solve(x^n=0,x)

What if n is zero? what if n is negative?  

Another program I tried this on gives

Maple 2025

Hello,

I am working with the following overdetermined system of nonlinear equations in Maple and attempting to solve it for the unknowns xi[3] and xi[8].

Maple returns the following solution:

However, when I substitute the solution back into the original equations, the left-hand side should equal the right-hand side. Yet, using simplify(subs(solution, equation)) does not reduce the expression sufficiently to verify the equality.

Am I missing something in the simplification process?

Many thanks,

Ed

We can say, a test example. Probably a continuation of this topic.
An equation with one variable X, and let 0<X. To try to find such a largest value of X, in the neighborhood of which not a single solution will be missed in the amount of at least 50 solutions in a row. And in this case, the residual value of the equation discrepancy will be no more than 10^-7 .

f :=0.995-cos(0.25*x)*sin(x^3);

Purely for fun, of course.

I am working on a script (attached sheet) where I use a FOR loop to iterate over different values of varepsilon. Within each iteration, I perform the following steps:

1. I optimize the function R_out and obtain a result, denoted as Pc

2. I then substitute Pc into another function, L_out and optimize it to find the values of p1,p2, and the corresponding function value.

I follow a similar procedure for a second case as well.

However, I'm encountering an issue: I'm unable to successfully substitute the result from R_out into L_out within the loop. This is causing an error in execution.

Finally, I intend to generate a plot with varepsilon  on the x-axis and certain result variables on the y-axis in a single plot (as indicated at the end of the sheet).

Sheet: Question_New.mw

Can someone help me:

1. Fix the substitution error in the loop, and

2. Provide the correct syntax for generating the desired plot?

Hello,

i try to create a quiz. Is it possible to set a comlex math expression like the first expression in my document for a choice? In the Helppage I did not see the oppertunity to use an Integral expression for example. Or is it posssible to use 2D Math Input for the choices ? 

Download 1.mw

I'm optimizing a function with constraints, meaning the decision variable has both a lower and an upper bound. When I use the  (MAX) syntax in Maple, it returns the lower bound as the optimal value (Pc) along with the corresponding function value. However, when I plot the graph, it shows that the function actually reaches its maximum at the upper bound. What could be causing this discrepancy

Sheet:Q_result_1.mw

Should solution to a first order ode with IC not have any constant of integration in it? This is what the teacher said at school.

But Maple in this example returns a solution to first order Riccati ode with c1 still in the solution even though it is given IC.

How is this possible? This is problem from Differential equations and their applications, 3rd ed., M. Braun, Section 1.10. Page 80, problem #5

If dsolve was not able to resolve c1 from IC for some reason, should it not have returned any solution in this case? 

btw, I could not verify the solution on the ode itself using odetest, but may be assumptions are needed. Will try and see...

Download why_c_in_solution_may_23_2025.mw

Update

Looked up the textbook, it says solution exist and unique over 0<=x<=1/3, using these now Maple verifies the ode itself, but does not verify the IC (because c__1 is there). Here is updated worksheet. The bottom line, I think the solution is wrong as it should not have any constant of integration in it. Textbook also does say what the solution should be.

Download why_c_in_solution_may_23_2025_v2.mw

The workbook includes the excel data file but I also included the worksheet and data file separate for versions that can't load workbooks.

Average_World_Temperature_workbook-.maple

Average_World_Temperature_since_1850_worksheet.mw

HadCRUT5.zip

Suppose we have two equations, and all parameters involved are assumed to be positive. Is there a systematic way or syntax to determine whether C1>C2​ or vice versa? Additionally, can we derive specific conditions under which C1>C2​ holds

Sheet: Q_greater.mw

Last week, DeepMind announced their new AlphaEvolve software along with a number of new results improving optimization problems in a number of areas. One of those results is a new, smaller, tensor decomposition for multiplying two 4 × 4 matrices in 48 scalar multiplications, improving on the previous best decomposition which needed 49 multiplications (Strassen 1969). While the DeepMind paper was mostly careful about stating this, when writing the introduction, the authors mistakenly/confusingly wrote:

For 56 years, designing an algorithm with rank less than 49 over any field with characteristic 0 was an open problem. AlphaEvolve is the first method to find a rank-48 algorithm to multiply two 4 × 4 complex-valued matrices

This gives the impression that this new result is the way to multiply 4 × 4 matrices over a field with the fewest number of field multiplications. However, it's not even the case that Strassen's 1969 paper gave a faster way to multiply 4 × 4 matrices. In fact, Winograd's 1968 paper on computing inner products faster could be applied to 4 × 4 matrix multiplication to get a formula using only 48 multiplications. Winograd uses a trick 

which changes two multiplications of ab's into one multiplication of ab's and two multiplications involving only a's or only b's. The latter multiplications can be pre-computed and saved when calculating all the innerproducts in a matrix multiplication

    # i=1..4, 4*2 = 8 multiplications
    p[i] := -A[i, 1]*A[i, 2] - A[i, 3]*A[i, 4];
    # 4*2 = 8 multiplications
    q[i] := -B[1, i]*B[2, i] - B[3, i]*B[4, i];

    # i=1..4, j=1..4, 4*4*2 = 32 multiplications
    C[i,j] = p[i] + q[j] 
            + (A[i,1]+B[2,j])*(A[i,2]+B[1,j])
            + (A[i,3]+B[4,j])*(A[i,4]+B[3,j])

It is simple to verify that C[i,j] = A[i,..] . B[..,j] and that the above formula calculates all of the entries of C=A.B with 8+8+32=48 multiplications.

So, if Winograd achieved a formula of 48 multiplications in 1968, why is the DeepMind result still interesting? Well, the relative shortcoming of Winograd's method is that it only works if the entries of A and B commute, while tensor decomposition formulas for matrix multiplication work even over noncommutative rings. That seems very esoteric, but everyone's favorite noncommutative ring is the ring of n × n matrices. So if a formula applies to a matrix of matrices, that means it gives a recursive formula for matrix multiplication - an tensor decomposition formulas do just that.

The original 1969 Strassen result gave a tensor decomposition of 2 × 2 matrix multiplication using only 7 scalar multiplications (rather than the 8 required by the inner product method) and leads to a recursive matrix multiplication algorithm using O(N^log[2](7)) scalar multiplications. Since then, it has been proved that 7 is the smallest rank tensor decompostion for the 2 × 2 case and so researchers have been interested in the 3x3 and larger cases. Alexandre Sedoglavic at the University of Lille catalogs the best tensor decompositions at https://fmm.univ-lille.fr/ with links to a Maple file with an evaluation formula for each.

The previous best 4 × 4 tensor decomposition was based on using the 2 × 2 Strassen decomposition recursively (2 × 2 matrix of 2 × 2 matrices) which led to 7 2 × 2 matrix multiplications each requiring 7 scalar multiplications, for a total of 49. The new DeepMind result reduces that to 48 scalar multiplications, which leads to a recursive algorithm using O(N^log[4](48)) scalar multiplications: O(N^2.7925) vs. O(N^2.8074) for Strassen. This is a theoretical improvment over Strassen but in 2024 the best known multiplication algorithm has much lower complexity: O(N^2.3713) (see https://arxiv.org/abs/2404.16349). Now, there might be some chance that the DeepMind result could be used in practical implementations, but its reduction in the number of multiplications comes at the cost of many more additions. Before doing any optimizations I counted 1264 additions in the DeepMind tensor, compared to 318 in the unoptimized 4×4 Strassen tensor (which can be optimized to 12*7+4*12=132). Finally, the DeepMind tensor decomposition involves 1/2 and sqrt(-1), so it cannot be used for fields of characteristic 2, or fields without sqrt(-1). Of course, the restriction on characteristic 2 is not a big deal, since the DeepMind team found a 4 × 4 tensor decomposition of rank 47 over GF2 in 2023 (see https://github.com/google-deepmind/alphatensor).

Now if one is only interested in 4 × 4 matrices over a commutative ring, the Winograd result isn't even the best possible. In 1970, Waksman https://ieeexplore.ieee.org/document/1671519 demonstrated an improvement of Winograd's method that used only 46 multiplications as long as the ring allowed division by 2. Waksman's method has since been improved by Rosowski to remove the divisions see https://arxiv.org/abs/1904.07683. Here's that nice compact straight-line program that computes C = A · B in 46 multiplications.

And, here attached are five Maple worksheets that create the explicit formulas for the 4 × 4 matrix methods mentioned in this post, and verify their operation count, and that they are correct.

Strassen_444.mw  Winograd.mw  DM_444.mw Waksman.mw Rosowski.mw

If you are interested in some of the 2022 results, I posted some code to my GitHub account for creating formulas from tensor decompositions, and verifying them on symbolic matrix multiplications: https://github.com/johnpmay/MapleSnippets/tree/main/FastMatrixMultiplication

The clear followup question to all of this is:  does Maple use any of these fast multiplication schemes?  And the answer to that is mostly hidden in low level BLAS code, but generally the answer is: No, the straightforward inner-product scheme for multiplying matrices optimized memory access thus usually ends up being the fastest choice in most cases.

I have just started to use Maple workbook (*.maple). In the workbook, there are 4 worksheets: "main.mw","calc1.mw","calc2.mw","calc3.mw". In "main.mw", I plot variables saved from "calc?.mw". This works. However, when I make changes in one of the worksheet "calc?.mw" and get the new expressions for the variables, they are not updated in "main.mw". I deleted the variables being changed but "main.mw" still run and show the old expressions of these variables.

How do I refresh so that "main.mw" will show the updated variables from "calc?.mw"?

Thanks.

Hi,

I’ve [K] & [M] matrices in sparse format for FE code. How do I convert them to be able to do eigensolution?

restart;

with(LinearAlgebra);

stiffness_data := ImportMatrix("d:/Simulation_1_STIF1.csv");

stiffness_matrix := Matrix(max(stiffness_data[() .. (), 1]), max(stiffness_data[() .. (), 2]), sparse = [seq([stiffness_data[i, 1], stiffness_data[i, 2], stiffness_data[i, 3]], i = 1 .. nops(stiffness_data))]);

Error, (in Matrix) argument `sparse = [[1, 1, 328.0000000], [1, 5, -20434.40000], [5, 1, -20434.40000]]` is incorrect or out of order

Even if I try with a simple matrix example it gave an error.

with(LinearAlgebra);

A := Matrix(5, 5, sparse = [[1, 1, 328.0], [1, 5, -20434.4], [5, 1, -20434.4]]);

Error, (in Matrix) argument `sparse = [[1, 1, 328.0], [1, 5, -20434.4], [5, 1, -20434.4]]` is incorrect or out of order

Please help.

Thanks,

Shashi

Trying to produce two different font sizes within a plot title.  I can achieve it with Typesetting but I'm getting brackets and fluff I don't want.  Any other ideas?

For example

with(Typesetting):
plot(x^2, title = [cat(mtext("sort of", font = "TimesNewRoman", size = 30), mtext("works", size = 10))])