rlopez

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14 years, 343 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

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These are replies submitted by rlopez

Simply work in the Student LinearAlgebra package where all quantities are assumed to be real. Also, the VectorCalculus packages do not conjugate for dot products.

RJL Maplesoft

I've looked at the two worksheets supplied in this question, but they import data from Excel files that weren't provided. I've privately requested access to the data. If I get the data, I'll explore the calculations to try finding out why there's a problem. I'd like the conversation to be off-line until I have something to report. Meanwhile, if anyone else can spot what's happening in the failed calculation, please enlighten us all.

 RJL Maplesoft

Unfortunately, Gem 12 contains a typo that leads my discussion into error. The typo appears in the definition of LEFT, the left side of the equation being investigated. This left side is the product of two radicals, and under the second radical I typed the number 1 where I should have typed the imaginary unit "i". I used the resulting graphs to deduce an erroneous conclusion. The correct conclusion is obvious from the (colorful) graph provided by Alec Mihailovs.

I've revised the worksheet upon which this blog is based, and will replace the download with the revised copy. Sorry for that, folks. And thanks to Alec for sparking my second look at that Gem.

RJL Maplesoft

I found Robert Israel's investigation of guessgf interesting and puzzling. It was interesting to see that ultimately, the limit of the sequence of Picard iterates was found by solving the very differential equation we started with. It was puzzling because it raised the question: What math is Maple implementing to recover the ODE from whatever information is in "L"?

Unfortunately, I could not see where either he or I had defined a quantity "L" in that Gem. Robert, what information was in "L" and have you any idea what mathematics it takes to go from L back to the ODE?

RJL Maplesoft

frem(75,evalf(2*Pi)) => -0.39822370, which is 75-12*evalf(2*Pi). On the other hand, 75-11*evalf(2*Pi) = 5.88496161, so is the problem of subtracting multiples of 2 Pi solved yet? I guess it depends on what the user actually wants.

RJL Maplesoft

frem(75,evalf(2*Pi)) => -0.39822370, which is 75-12*evalf(2*Pi). On the other hand, 75-11*evalf(2*Pi) = 5.88496161, so is the problem of subtracting multiples of 2 Pi solved yet? I guess it depends on what the user actually wants.

RJL Maplesoft

 Thanks, Alejandro. I appreciate your continued development of this topic.

RJL Maplesoft

 I am edified by the comment demonstrating the use of rules and the applyrule command. However, the rules for sine and cosine are quadrant-dependent. This approach is useful when the quadrant of the angle is known. Otherwise, a sign error could be introduced.

But the illustration of the use of applyrule is quite valuable and has been added to the Little Red Book.

RJL Maplesoft

 The assignment to "f" is made in the yellow cell. The Table Properties dialog has a checkbox "Show Input" which is selected by default. If this is unselected, then input in the table is hidden and all you see is the output. Of course, this has to be done in the Maple worksheet, not in the html rendering read through MaplePrimes.

I have found that this device in a 1x1 table allows me to make an assignment and have just the echo displayed, much like one sees in a textbook or journal article. The cell is tinted with some color to indicate that there is hidden input. If the !!! button is used to execute the whole worksheet, these hidden inputs will execute.

I've filed a request with Maplesoft to add a facility to make the hidden inputs in all tables in a file visible at once. Presently, each table has to be adjusted to see the hidden inputs. It would be a lot more convenient to be able to make inputs visible or invisible with the click of a single button.

Note too, that each figure is drawn in a table with hidden input. I think this mechanism is a workable solution to being able to write a mathematical exposition without having the Maple syntax cloud the discussion. The details of the calculations can then be seen by exposing the hidden inputs.

RJL Maplesoft

Elementary manipulations designed to solve q(x,y,z)=0 for x lead to an equation quartic in x. Hence, one expects four solutions for x=x(y,z).

In Maple 13, "solve" does not return any explicit solutions for x. In Maple 14 it does, but at least 3 of the 4 branches don't seem to satisfy the original equation. Hence, at least one of the local minima supposedly found by analyzing these branches doesn't satisfy the original equation.

It appears that the minimum value of -4 is correct, but the analysis based on an explicit solution for x=x(y,z) is highly suspect.

The function q was given as

q:=(x,y,z)-> -5 + ((x*y*z/z+1-x)+ sqrt( (x*(x+2*z^2-2*y)/x+3*z-1))/ (4*x*z^2-y));

Taking this literally means that "z" in the product x*y*z cancels out, and that under the radical, the factor of "x" also cancels out. (I have to wonder if the author meant to enclose some of these divisors with parentheses.) Hence, the radicand is actually x+2*z^2-2*y+3*z-1. For real values, this expression has to be nonnegative. The feasible region lies above a surface x(y,z) easily found explicitly. Over the square 0<=y,z<=1, the lowest point on this surface is (x,y,z)=(-4,0,1).

RJL Maplesoft

Elementary manipulations designed to solve q(x,y,z)=0 for x lead to an equation quartic in x. Hence, one expects four solutions for x=x(y,z).

In Maple 13, "solve" does not return any explicit solutions for x. In Maple 14 it does, but at least 3 of the 4 branches don't seem to satisfy the original equation. Hence, at least one of the local minima supposedly found by analyzing these branches doesn't satisfy the original equation.

It appears that the minimum value of -4 is correct, but the analysis based on an explicit solution for x=x(y,z) is highly suspect.

The function q was given as

q:=(x,y,z)-> -5 + ((x*y*z/z+1-x)+ sqrt( (x*(x+2*z^2-2*y)/x+3*z-1))/ (4*x*z^2-y));

Taking this literally means that "z" in the product x*y*z cancels out, and that under the radical, the factor of "x" also cancels out. (I have to wonder if the author meant to enclose some of these divisors with parentheses.) Hence, the radicand is actually x+2*z^2-2*y+3*z-1. For real values, this expression has to be nonnegative. The feasible region lies above a surface x(y,z) easily found explicitly. Over the square 0<=y,z<=1, the lowest point on this surface is (x,y,z)=(-4,0,1).

RJL Maplesoft

 Actually, the third and fourth branches of the explicit solution x=x(y,z) are real at exactly one point, namely, (y,z) = (0,1). At this point, x = -4 on one branch, and -3.9960860944187867195 on the other. Indeed, evaluating the given function q at (y,z) = (0,1) gives sqrt(x+4)/4/4-x-4=0, the solutions of which are -4 and 0.123119558.

Note that the solution found by the GlobalSearch command in the DirectSearch package is

 (x,y,z) = (-3.61380491750659472, .193097504987588819, .99999998964025316).

However, substitution of this point into the given function q gives the complex number

-1.084011795-0.2158799325e-5*I and not the number zero. I find this puzzling.

RJL Maplesoft

 Actually, the third and fourth branches of the explicit solution x=x(y,z) are real at exactly one point, namely, (y,z) = (0,1). At this point, x = -4 on one branch, and -3.9960860944187867195 on the other. Indeed, evaluating the given function q at (y,z) = (0,1) gives sqrt(x+4)/4/4-x-4=0, the solutions of which are -4 and 0.123119558.

Note that the solution found by the GlobalSearch command in the DirectSearch package is

 (x,y,z) = (-3.61380491750659472, .193097504987588819, .99999998964025316).

However, substitution of this point into the given function q gives the complex number

-1.084011795-0.2158799325e-5*I and not the number zero. I find this puzzling.

RJL Maplesoft

 Thanks for the feedback.

I, too, would like to see inline math baselined correctly. I've made sure that Will got a copy of your remarks.

The button issue happens in Windows Explorer, too.

Finally, note that in the Matrix Action: 2D task template, the Math Container to the right of A= can be supplied with a matrix, or with the name of a matrix defined elsewhere. That's what I did in this blog. Earlier in the worksheet, the matrix was given the name A, and hence, in the task template, I needed only to refer to the input matrix by its name, A.

RJL Maplesoft

 No one has yet pointed to the CenterOfMass command in the Student Precalculus package. It is certainly simpler to use than the one in the geometry package, and it will compute the centroid of discrete points. It also handles weighted points.

RJL Maplesoft

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