Carl Love

## 26508 Reputation

11 years, 186 days
Himself
Wayland, Massachusetts, United States
My name was formerly Carl Devore.

## Missing multiplication...

You need to change s(e+i) to s*(e+i).

And although it doesn't cause a problem here, please don't use with(linalg).

## Just one loop...

It can be done like this:

```restart:
model:= [x^2*y*alpha[1, 11], x*z^2*alpha[2, 15], y^2*z*alpha[3, 17] + y*z*alpha[3, 8]]:
M:= [
alpha[i, 0], alpha[i, 1]*x, alpha[i, 2]*y, alpha[i, 3]*z, alpha[i, 4]*x^2,
alpha[i, 5]*y*x, alpha[i, 6]*z*x, alpha[i, 7]*y^2, alpha[i, 8]*z*y, alpha[i, 9]*z^2,
alpha[i, 10]*x^3, alpha[i, 11]*y*x^2, alpha[i, 12]*z*x^2, alpha[i, 13]*y^2*x,
alpha[i, 14]*z*y*x, alpha[i, 15]*z^2*x, alpha[i, 16]*y^3, alpha[i, 17]*z*y^2,
alpha[i, 18]*z^2*y, alpha[i, 19]*z^3
]:
#Construct new models as a list of lists:
models:= CodeTools:-Usage([
for i,m in model do
map(subs, m=~ m+~ subs({`if`}(m::`+`, op(m), m)=~ (), M), model)[]
od
]);
memory used=41.66KiB, alloc change=0 bytes,
cpu time=0ns, real time=0ns, gc time=0ns
#Lenghty output omitted
nops(%);
56
```

## Small change...

The lines

p2 := with(plots);
inequal(58 < x, x = 0 .. 100, y = 0 .. 2);

should be changed to

with(plots):
p2:= nequal(58 < x, x = 0 .. 100, y = 0 .. 2);

## Save as a table in a module...

I suppose that vmcofs is very large (millions of numeric entries or tens of thousands of symbolic entries), and you don't need to save every slice? Is that right?

Create a module that contains a single export, a table:

VM:= module() export vmcofs::table:= table(); end module;

Then every time that you have a slice k that you want to save, instead of doing savelib, do instead

VM:-vmcofs[k]:= vmcofs[k];

When you're done choosing slices, save the whole module with

savelib(VM, lib);

If you change any entries in the array vmcofs[k] after putting that slice into the table but before using savelib, then the table will automatically contain the newly updated slice k, not the original slice k, and only the updated one will be saved to the library. If you don't want this to happen, let me know.

The module is superfluous in this simplistic version; all that's needed is the table. I just included the module because you might find it useful later (for storing other info related to your project). The added overhead for using a module is infinitesimal.

A module with no locals, only exports, is also called a Record. You can create it with the Record command, if you wish. It hardly makes any difference.

## The backquotes are superfluous...

Since test is not a reserved word and doesn't contain any special characters, the backquotes don't do anything: test and `test` mean exactly the same thing. So yes, you're right, the help page's usage of backquotes is unnecessarily confusing.

If I do it without Physics Updates, then there's no problem.

## Type identical(...)...

The type given as 2nd argument to subsindets can use a specific symbol by referring to it as identical(xi). So, this works:

S:= ex-> subsindets(ex, identical(xi)^integer, e-> H(op(2,e))*e);

## maplemint...

Yes, the command is called maplemint.

## A simple flag would solve this problem...

Great point @sursumCorda ! That is something that I noticed and explained to my students on day 1 of the very first Maple class that I ever taught, some 25 years ago (2nd-semester calculus, freshman students with no Maple experience). Now that you've described the problem, I've thought of an easy solution that Maplesoft could implement: Put a flag on the status bar that indicates which of the following two things has occurred most recently:

1. the worksheet has been edited;
2. the worksheet has been sequentially executed with the Execute-Entire-Worksheet command.

This flag could be as unobtrusive as the single "*" that indicates whether the worksheet is unsaved.

## eval(..., int= 1)...

Like this:

eval(expr, int= 1)

## interface(screenwidth)...

The output width of both lprint and showstat is controlled by interface(screenwidth). Its default value is 79, which I recall was a typical actual screenwidth in the 1980s. Modern screens are much wider.

## Horner's is effective in Maple...

There are many stumbling blocks that can occur when trying to time Maple commands, and I think that you've been fooled by one. For example, it's invalid to compare two measurements at least one of which has been rounded down to 0. The same is true for any measurements, even outside of computers.

The following example shows that using Horner's rule is much faster:

```restart:
#degree, number of terms, and number of test-data points:
deg:= 2^6:  nterms:= deg+1:  nData:= 2^12:
p:= x^deg + randpoly(x, degree= deg-1, terms= nterms-1):
(P,Ph):= unapply~([p, convert(p, horner)], x)[]:
#Test data is rational numbers in fraction form:
Data:= `~`[`/`]('rtable(1..nData, random(), subtype= Vector[row])' \$ 2):
gc(); Rh:= CodeTools:-Usage(map(Ph, Data)):
memory used=454.41MiB, alloc change=64.00MiB,
cpu time=359.00ms, real time=702.00ms, gc time=93.75ms

gc(); R:= CodeTools:-Usage(map(P, Data)):
memory used=1.06GiB, alloc change=0 bytes,
cpu time=3.28s, real time=7.22s, gc time=140.62ms

max(abs~(R-Rh)); #Check that results are identical.
0

```

## Updated, simpler Factrix...

There have been many innovations in Maple since Robert Israel wrote Factrix (circa 24 years ago). The following works for any rtable (i.e., vector, matrix, array), of any number of dimensions, whose entries are all exact explicit rational numbers (integers or fractions). It factors out the largest possible constant such that the remaining rtable is all integer.

Factrix:= (M::rtable)->
(C-> `if`(C=0, M, C %* (M/~C)))(((igcd@op@numer~)/(ilcm@op@denom~))({seq}(M)))
:
Factrix(b);
#b is the matrix from the original question.

This is not meant to be a complete replacement for the original Factrix because this only handles rational numbers.

## try ... catch, max[index], densityplot...

The first step is to adjust the functions that you want to plot so that they return an extended_numeric value for all numeric inputs from the plot intervals (alpha= 0..1, delta= 0..1). There's no need for you to understand the subtle distinction between numeric and extended_numeric. You just need to know that undefined is an extended_numeric value that all plot commands can handle gracefully. Currently, diffrP1 and diffrP2 return errors over a large portion of the plot intervals. Most plot commands can handle those errors and can convert them to undefined, but this doesn't work very well when you're trying to indicate the max of the 3 functions.

Here is how to trap the errors and convert them to undefined:

for k to 3 do
DiffrP||k:= subs(
_D= diffrP||k,
proc(a,d) try _D(a,d) catch: undefined end try end proc
)
od:

It is simply defining 3 new functions that call the original 3, "catch" any errors, and return undefined in those cases.

This next command plots those 3 functions in side-by-side plots. This is not strictly necessary for your final plot, but it'll help you understand it.

plots:-display(
plots:-densityplot~(
`<|>`(DiffrP||(1..3)), 0..1, 0..1, title=~ ["diffrP"||(1..3)],
labels= [alpha, delta], labeldirections= ["horizontal", "vertical"],
)
);

I've omitted a direct display of those plots here.

Finally, we create a new function that returns either 12, or 3 to indicate which of the functions returns the maximum value and excludes undefined from consideration. A simple way to write that new function is

max[index, defined]@[DiffrP||(1..3)]

And we plot it like this:

plots:-densityplot(
max[index, defined]@[DiffrP||(1..3)], 0..1, 0..1,
labeldirections= ["horizontal", "vertical"], labels= [alpha, delta],
);

So, the third function is usually the maximum, and there are small regions where the first or second are the maximum.

The colorbar on the right of the plot was a feature introduced in Maple 2023. Without having the bar, you just need to know that red = 3, white = 2, blue = 1.

## applyrule is much older than seq...

[The title refers to the seq parameter modifier, not the seq command.]

Addressing your question about why this inconsistency exists: I guess that a large part of the reason is that applyrule is many years older than parameter modifiers. I'd guess about 10 years.

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