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))])

Dear Power Users,

This is probably a simple question for you but I am stuck. In the attached worksheet I would like to plot (the plot is called P5) a vector with units (mm) against a vector without units. This didn't work out, so I removed the units from the vector with units, and this plotted but after the conversion the numeric values are no longer in mm but inches. How can I correct this?

Thank you in advance for your time and willingness to help.

testSheetHorseShoe.mw

Hello,

I have a few questions.

1. When using ordered list, it keeps getting out of order. It will do 1, 2, then 5, 6 sometimes. If I tried to delete incorrectly numbered list, instead of restarting from the correct order, it will keep incrementing.
2. How do I create a indented bullet list?
3. Documentation for Maple 2025 doesn't seem to be up to date. I am looking to use "Tabkey" to indent text. It seems really difficult to get it done. Where do I find "Format>Tab Navigation"?
   1. https://www.maplesoft.com/support/help/Maple/view.aspx?path=worksheet/documenting/tabkey
      1. 

Thanks for the help.

in some of my function i have a big problem which i can't plot thus function even i know what is the shape of plot, i have two type of ploting directly giving parameter and using explor but for this kind of plot we can't use explor so i have to give the function directly parameter but is wasting my time a lot and i can't get my plot even spending a days by changing parameter one by one, my questions is this how i can plot this kind of function without bieng a singular i need this function to be non singular is not importan about the parameter can be any number 
thanks for any help 

plots-long_term_.mw

the shape of plot must be like this but must have two of them 

Dear all

I get error when solving Riccati equation. I want to get six different solution. 

ricatti_equation.mw

thank you for your help