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MaplePrimes Announcement

Maple 2023 is here!

by: Karishma 685 Product: Maple 2023

March 09 2023

15 0

The moment we've all been waiting for has arrived: Maple 2023 is here!

With this release we continue to pursue our mission to provide powerful technology to explore, derive, capture, solve and disseminate mathematical problems and their applications, and to make math easier to learn, understand, and use. Bearing this in mind, our team of mathematicians and developers have dedicated the last year to adding new features and enhancements that not only improve the math engine but make that math engine more easily accessible within a user-friendly interface.

And if you ever wonder where our team gets inspiration, you don't need to look further than Maple Primes. Many of the improvements that went into Maple 2023 came as a direct result of feedback from users. I’ll highlight a few of those user-requested features below, and you can learn more about these, and many, many other improvements, in What’s New in Maple 2023.

The Plot Builder in Maple 2023 now allows you to build interactive plot explorations where parameters are controlled by sliders or dials, and customize them as easily as you can other plots

Plot Builder Explore

In Maple 2023, 2-D contour and density plots now feature a color bar to show the values of the gradations.

For those who write a lot of code: You can now open your .mpl Maple code files directly in Maple’s code editor, where you can view and edit the file from inside Maple using the editor’s syntax highlighting, command completion, and automatic indenting.

Programming Improvements

Integration has been improved in many ways. Here’s one of them: The definite integration method that works via MeijerG convolutions now does a better job of checking conditions on parameters so that they are only applied under proper assumptions. It also tells you the conditions under which the method could have produced an answer, so if your problem does meet those conditions, you can add the appropriate assumptions to get your result.
Many people have asked that we make it easier for them to create more complex interactive Math Apps and applications that require programming, such as interactive clickable plots, quizzes that provide feedback, examples that provide solution steps. And I’m pleased to announce that we’ve done that in Maple 2023 with the introduction of the Quiz Builder and the Canvas Scripting Gallery.
- The new Quiz Builder comes loaded with sample quizzes and makes it easy to create your own custom quiz questions. Launch the quiz builder next time you want to author interactive quizzes with randomized questions, different response types, hints, feedback, and show the solution. It’s probably one of my favorite features in Maple 2023.

The Scripting Gallery in Maple 2023 provides 44 templates and modifiable examples that make it easier to create more complex Math Apps and interactive applications that require programming. The Maple code used to build each application in the scripting gallery can be easily viewed, copied and modified, so you can customize specific applications or use the code as a starting point for your own work

Finally, here’s one that is bound to make a lot of people happy: You can finally have more than one help page open at the same time!

For more information about all the new features and enhancements in Maple 2023, check out the What’s New in Maple 2023.

P.S. In case you weren’t aware - in addition to Maple, the Maplesoft Mathematics Suite includes a variety of other complementary software products, including online and mobile solutions, that help you teach and learn math and math-related courses. Even avid Maple users may find something of interest!

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jthomson

Probability Trees

The areas of statistics and probability are my favorite in mathematics. This is because I like to be able to draw conclusions from data and predict the future with past trends. Probability is also fascinating to me since it allows us to make more educated decisions about real-life events. Since we are supposed to get a big snow storm in Waterloo, I thought I would write a blog post discussing conditional probability using the Probability Tree Generator, created by Miles Simmons.

If the probability of snowfall on any given day during a Waterloo winter is 0.75, the probability that the schools are closed given that it has snowed is 0.6, and the probability that the schools are closed given that it has hasn’t snowed is 0.1, then we get the following probability tree, created by Miles’s learn document:

From this information we can come to some interesting conclusions:

What is the probability that the schools are closed on a given day?

From the Law of total probability, we get:

Thus, during a very snowy Waterloo winter, we could expect a 0.475 chance of schools being closed on any given day.

One of the features of this document is that the node probabilities are calculated. You can see this by comparing the second last step to the number at the end of probability trees' nodes.

What is the probability that it has snowed given that the schools are closed?

From Bayes’ Theorem, we get:

Thus, during a very snowy Waterloo winter, we expect there to be a probability of 0.947 that it has snowed if the schools are closed.

We can also add more events to the tree. For example, if the students are happy or sad given that the schools are open:

Even though we would all love schools to be closed 47.5% of the winter days in Waterloo, these numbers were just for fun. So, the next time you are hoping for a snow day, make sure to wear your pajamas inside out and sleep with a spoon under your pillow that night!

To explore more probability tree fun, be sure to check out Miles’s Probability Tree Generator, where you can create your own probability trees with automatically calculated node probabilities and export your tree to a blank Maple Learn document. Finally, if you are interested in seeing more of our probability collection, you can find it here!

March 10

5 0

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mmcdara

4520

Centered Divided Difference...

This code enables building Centered Divided Difference (CDD) approximations of derivatives of a univariate function.
Depending on the stencil we choose we can get arbitrary high order approximations.

The extension to bivariate functions is based upon what is often named tensorization in numerical analysis: for instance diff(f(x, y), [x, y] is obtained this way (the description here is purely notional)

Let CDD_x the CDD approximation of diff( f(x), x) ) .
CDD_x is a linear combination of shifted replicates of f(x)
Let s one of this shifted replicates
Let CDD_y(s) the CDD approximation of diff( s(y), y) ) .
Replace in CDD_x all shifted replicates by their corresponding expression CDD_y(s)

REMARKS:

When I write for instance "approximation of diff(f(x), x)", this must be intended as a short for "approximation of diff(f(x), x) at point x=a"
I agree that a notation like, for instance, diff(f(a), a) is not rigourous and that something like a Liebnitz notation would be better. Unfortunately I don't know how to get it when I use mtaylor.

restart:

CDDF stands for Cendered Divided Difference Formula

CDDF := proc(f, A, H, n, stencil)
  description "f = target function,\nA = point where the derivatives are approximated,\nH = step,\nn = order of the derivative,\nstencil = list of points for the divided differenceCDDF\n";
  local tay, p, T, sol, unknown, Unknown, expr:

  tay := (s, m) -> convert(
                     eval(
                       convert(
                         taylor(op(0, f)(op(1, f)), op(1, f)=A, m),
                         Diff
                       ),
                       op(1, f)=A+s*H),
                     polynom
                   ):

  p   := numelems(stencil):
  T   := add(alpha[i]*tay(i, p+1), i in stencil):
  T   := convert(%, diff):

  if p > n+1 then
    sol := solve([seq(coeff(T, h, i)=0, i in subsop(n+1=NULL, [$0..p]))], [seq(alpha[i], i in stencil)])[];
  else
    sol := solve([seq(coeff(T, H, i)=0, i in subsop(n+1=NULL, [$0..n]))], [seq(alpha[i], i in stencil)])[];
  end if:

  if `and`(is~(rhs~(sol)=~0)[]) then
    WARNING("no solution found"):
    return
  else
    unknown := `union`(indets~(rhs~(sol))[])[];
    Unknown := simplify(solve(eval(T, sol) = Diff(op(0, f)(A), A$n), unknown));
    sol     := lhs~(sol) =~ eval(rhs~(sol), unknown=Unknown);
    expr    := normal(eval(add(alpha[i]*op(0, f)(A+i*H), i in stencil), sol));
  end if:

  return expr
end proc:

>	Describe(CDDF)

# f = target function,
# A = point where the derivatives are approximated,
# H =
# step,
# n = order of the derivative,
# stencil = list of points for the divided
# differenceCDDF
#
CDDF( f, A, H, n, stencil )

>	# 2-point approximation of diff(f(x), x) \| x=a CDDF(f(x), a, h, 1, [-1, 1]); convert(simplify(mtaylor(%, h=0, 4)), Diff);

Diff(f(a), a)+(1/6)*(Diff(Diff(Diff(f(a), a), a), a))*h^2

(1)

>	# 3-point approximation of diff(f(x), x$2) \| x=a CDDF(f(x), a, h, 2, [-1, 0, 1]); convert(simplify(mtaylor(%, h=0)), Diff);

Diff(Diff(f(a), a), a)+(1/12)*(Diff(Diff(Diff(Diff(f(a), a), a), a), a))*h^2

(2)

>	# 5-point pproximation of diff(f(x), x$2) \| x=a CDDF(f(x), a, h, 2, [$-2..2]); convert(simplify(mtaylor(%, h=0, 8)), Diff);

Diff(Diff(f(a), a), a)-(1/90)*(Diff(Diff(Diff(Diff(Diff(Diff(f(a), a), a), a), a), a), a))*h^4

(3)

>	# 7-point approximation of diff(f(x), x$2) \| x=a CDDF(f(x), a, h, 2, [$-3..3]); # simplify(taylor(%, h=0, 10)); convert(simplify(mtaylor(%, h=0, 10)), Diff);

Diff(Diff(f(a), a), a)+(1/560)*(Diff(Diff(Diff(Diff(Diff(Diff(Diff(Diff(f(a), a), a), a), a), a), a), a), a))*h^6

(4)

>	# 4-point staggered approximation of diff(f(x), x$3) \| x=a CDDF(f(x), a, h, 3, [seq(-3/2..3/2, 1)]); convert(simplify(mtaylor(%, h=0, 6)), Diff);

-(f(a-(3/2)*h)-3*f(a-(1/2)*h)+3*f(a+(1/2)*h)-f(a+(3/2)*h))/h^3

Diff(Diff(Diff(f(a), a), a), a)+(1/8)*(Diff(Diff(Diff(Diff(Diff(f(a), a), a), a), a), a))*h^2

(5)

>	# 6-point staggered approximation of diff(f(x), x$3) \| x=a CDDF(f(x), a, h, 3, [seq(-5/2..5/2, 1)]); # simplify(taylor(%, h=0, 8)); convert(simplify(mtaylor(%, h=0, 8)), Diff);

(1/8)*(f(a-(5/2)*h)-13*f(a-(3/2)*h)+34*f(a-(1/2)*h)-34*f(a+(1/2)*h)+13*f(a+(3/2)*h)-f(a+(5/2)*h))/h^3

Diff(Diff(Diff(f(a), a), a), a)-(37/1920)*(Diff(Diff(Diff(Diff(Diff(Diff(Diff(f(a), a), a), a), a), a), a), a))*h^4

(6)

>	# 5-point approximation of diff(f(x), x$4) \| x=a CDDF(f(x), a, h, 4, [$-2..2]); convert(simplify(mtaylor(%, h=0, 8)), Diff);

Diff(Diff(Diff(Diff(f(a), a), a), a), a)+(1/6)*(Diff(Diff(Diff(Diff(Diff(Diff(f(a), a), a), a), a), a), a))*h^2

(7)

>	# 7-point approximation of diff(f(x), x$4) \| x=a CDDF(f(x), a, h, 4, [$-3..3]); convert(simplify(mtaylor(%, h=0, 10)), Diff);

Diff(Diff(Diff(Diff(f(a), a), a), a), a)-(7/240)*(Diff(Diff(Diff(Diff(Diff(Diff(Diff(Diff(f(a), a), a), a), a), a), a), a), a))*h^4

(8)

A FEW 2D EXTENSIONS

# diff(f(x, y), [x, y]) approximation over a (2 by 2)-point stencil

stencil := [-1, 1]:

# step 1: approximate diff(f(x, y), x) over stencil < stencil >

fx := CDDF(f(x), a, h, 1, stencil):
fx := eval(% , f=(u -> f[u](y))):
ix := [indets(fx, function)[]]:

# step 2: approximate diff(g(y), y) over stencil < stencil > where
# g represents any function in fx.

fxy := add(map(u -> CDDF(u, b, k, 1, stencil)*coeff(fx, u), ix)):

# step 3: rewrite fxy in a more convenient form

[seq(u=op([0, 0], u)(op([0, 1], u), op(1, u)), u in indets(fxy, function))]:
fxy := simplify( eval(fxy, %) );

convert(mtaylor(fxy, [h=0, k=0]), Diff)

(9)

# Approximation of diff(f(x, y), [x, x, y, y] a (3 by 3)-point stencil

stencil := [-1, 0, 1]:

# step 1: approximate diff(f(x, y), x) over stencil < stencil >

fx := CDDF(f(x), a, h, 2, stencil):
fx := eval(% , f=(u -> f[u](y))):
ix := [indets(fx, function)[]]:

# step 2: approximate diff(g(y), y) over stencil < stencil > where
# g represents any function in fx.

fxy := add(map(u -> CDDF(u, b, k, 2, stencil)*coeff(fx, u), ix)):

# step 3: rewrite fxy in a more convenient form

[seq(u=op([0, 0], u)(op([0, 1], u), op(1, u)), u in indets(fxy, function))]:
fxy := simplify( eval(fxy, %) );

convert(mtaylor(fxy, [h=0, k=0], 8), Diff)

-(2*f(a, b-k)-4*f(a, b)+2*f(a, b+k)-f(a-h, b-k)+2*f(a-h, b)-f(a-h, b+k)-f(a+h, b-k)+2*f(a+h, b)-f(a+h, b+k))/(h^2*k^2)

(10)

# Approximation of diff(f(x, y), [x, x, y] a (3 by 2)-point stencil

stencil_x := [-1, 0, 1]:
stencil_y := [-1, 1]:

# step 1: approximate diff(f(x, y), x) over stencil < stencil >

fx := CDDF(f(x), a, h, 2, stencil_x):
fx := eval(% , f=(u -> f[u](y))):
ix := [indets(fx, function)[]]:

# step 2: approximate diff(g(y), y) over stencil < stencil > where
# g represents any function in fx.

fxy := add(map(u -> CDDF(u, b, k, 1, stencil_y)*coeff(fx, u), ix)):

# step 3: rewrite fxy in a more convenient form

[seq(u=op([0, 0], u)(op([0, 1], u), op(1, u)), u in indets(fxy, function))]:
fxy := simplify( eval(fxy, %) );

convert(mtaylor(fxy, [h=0, k=0], 6), Diff)

(11)

# Approximation of the laplacian of f(x, y)

stencil := [-1, 0, 1]:

# step 1: approximate diff(f(x, y), x) over stencil < stencil >

fx := CDDF(f(x), a, h, 2, stencil):
fy := CDDF(f(y), b, k, 2, stencil):

fxy := simplify( eval(fx, f=(u -> f(u, b))) + eval(fy, f=(u -> f(a, u))) );

convert(mtaylor(fxy, [h=0, k=0], 6), Diff)