5 General FAQ

5.1 How to convert Mathematica expression to Maple?

restart; 
with(MmaTranslator); #load the package 
FromMma(`Integrate[Cos[x],x]`);
 

Or

restart; 
with(MmaTranslator); #load the package 
convert(`Integrate[Cos[x],x]`, FromMma);
 

5.2 How to debug and stopat internal procedures, such as dsolve?

f:=proc() 
eq:=x*diff(y(x),x)+y(x)=exp(2*x); 
dsolve(eq,y(x)); 
end proc;
 

Then used the command stopat(f); then called the procedure f(); and now the debugger comes up. Did step command and now it steps inside dsolve

Some examples

stopat(`ODEtools/symtest`); 
stopat(`ODEtools/test`); 
stopat(`ODEtools/normal/expanded`); 
stopat(`ODEtools/odepde`); 
stopat(`ODEtools/odeadv`);  #for DEtools:-odeadvisor 
stopat(`odsolve/dAlembert`); 
stopat(`odsolve/dAlembert/integrate`); 
stopat(`odsolve/answer`); 
stopat(`odsolve/homogeneous`); #for all A,C,D,G types 
stopat(`odsolve/homogeneous_C/integrate`); 
stopat(`odsolve/exact`); #for solving exact ODE 
stopat(`odsolve/exact/integrate`); 
stopat(`odsolve/exact/integrate`(f,y,x,M,N)); #where f here is RHS of y'=RHS; 
 
 
DEtools:-symtest([-3,y],ode,y(x)); 
 
`ODEtools/normal/expanded`
 

For exact ode, can also do

ode:=....# write your ode here 
Student:-ODEs:-Solve:-Exact(ode,y(x),output=steps);
 

5.3 How to display source code of a function?

For integration use

infolevel[`evalf/int`]:=5;infolevel[int]:=5;
                                                                                    
                                                                                    
 

Another option

restart; 
interface(verboseproc=3) #(try 2 also)
 

then print(procedure); or eval(procedure_name); for example

restart: 
interface(verboseproc=3): 
print(LinearAlgebra:-GramSchmidt); 
print(lcm);
 

Also can use showstat, in this case interface(verboseproc=3) is not needed. Also showstat gives line numbers and I think it is easier to read. Some examples

showstat(`odsolve/2nd_order`) 
showstat(`evalf/hypergeom`); 
showstat(`evalf/exp/general`); 
showstat(`evalf/Psi`); 
showstat(`evalf/int`); 
showstat(`dsolve/SERIES`); 
 
#these 3 shows the main 3 functions by each solver 
showstat(`odeadv/dAlembert`);  #used by advisor 
showstat(`odsolve/dAlembert`);  # main API. 
showstat(`odsolve/dAlembert/integrate`); #used to integrate the ode 
 
showstat(`ODEtools/odeadv`); 
showstat(DEtools:-odeadvisor); 
showstat(`dsolve/series/froben/inhom`) 
showstat(`dsolve/series/froben`)
 

To stop at anyone of these functions in debugger do

stopat(`dsolve/series/froben/inhom`) 
#code here, say dsolve command.
 

The above will stop in the debugger in the above function.

There is also a function by Joe Riel here here is the post by Joe Riel:

"A disadvantage of showstat, particularly if you want to cut and paste the output, is that it includes line numbers. Here is a simple procedure I threw together to remove the line numbers."

PrintProc := proc(p::name,lines::{posint,posint..posint}) 
local width; 
option `Copyright (C) 2004 by Joseph S. Riel. All rights reserved.`; 
description "Print like showstat, but without line numbers"; 
width := interface('screenwidth'=200); 
try 
printf("%s", 
StringTools:-RegSubs( 
"\n ...." = "\n" 
,debugopts('procdump'= 
`if`(nargs=1,p,[args])))) 
catch "procedure name expected": 
error "%1 is not a procedure name",p 
finally interface('screenwidth'=width) 
end try; 
NULL 
end:
 

To print source code to file using the above, do the following

currentdir("C:\\data"); 
interface('prettyprint'=1): 
interface('verboseproc'=3): 
writeto("listing.txt") 
PrintProc('singular'); 
writeto('terminal'):
 

Now the output will show up in the file "listing.txt" and also no line wrapping. The above I found is the best solution so far to do this.

5.4 How to display trace of a function as it runs in maple?

trace(foo); 
untrace(foo);
 

also see debug(foo);

Also

infolevel[all]:=5: 
printlevel:=10:
 

See http://www.mapleprimes.com/questions/35951-How-To-Debugtrace-Things-In-Maple

Also look at kernelopts(opaquemodules=true)

Here is a useful post by Carl Love from Maple prime forum that summarizes all of these

Here are four things that you can do to get more information. I have listed them in order by how structured the information is, with the most structured first.

1. Set

   infolevel[all]:= 5;
    
   

   That will cause programs to print out additional information of the programmers’ choosing. You can use higher or lower numbers for more or less information. Most programs don’t use levels higher than 5.

2. Print the code of procedures with showstat:

   showstat(int); 
   showstat(sin); 
   showstat(cos);
    
   

3. Trace the execution of particular procedures with trace:

   trace(int); 
   trace(sin);
    
   

4. Trace the execution of everything with printlevel:

   printlevel:= 10000:
    
   

   You can use higher or lower numbers for more or less information.

5.5 How to display a build in function code?

Some examples

interface(verboseproc=3); 
print(DEtools) 
print(`ODEtools/symgen`); 
print(`symgen/methods`); 
print(`symgen/do`);
 

To stop the debugger at symgen do

stopat(`ODEtools/symgen`);
 

To get infolevel on symgen do

infolevel[`symgen`]:=5;
 

Or to see line numbers

interface(verboseproc=3); 
showstat(dsolve)
 

Or can use the Browse(); command

with(LibraryTools); 
Browse();
 

Another option I found is

s:=debugopts(procdump=`showstat`);
 

Then the above produces listing that can be copied as string with line wrapping ok.

5.6 How to build a LIST or a SET on the fly?

One way

L:=[]: 
for i from 1 to 3 do : 
    L:=[op(L),i]; 
end do;
 

But a better way is to use seq if one knows the length

L:=[seq(i,i=1..3)]; 
 
       L := [1, 2, 3]
 

Since list is unmutable, a more efficient method, for long lists, is to use Array, and then convert the result back to list at the end since Array can grow dynamically without preallocation each time something is inserted as follows

L:=Array(): 
for i from 1 to 3 do : 
    L(i):=i; 
end do; 
 
for i from 1 to numelems(L) do : 
    print(L[i]); 
end do; 
 
L := convert(L,list)
 

Which wil print

                            L := [1] 
 
                          L := [1, 2] 
 
                         L := [1, 2, 3] 
 
                               1 
 
                               2 
 
                               3 
 
                         L := [1, 2, 3]
 

Notice that to add to an Array, () is used. But to access an entry in an array [] is used.

And finally, using Array also, it can be done without using any indexing as follows

L:=Array(1..0): 
for i from 1 to 3 do : 
    L ,= i; 
end do; 
 
L := convert(L,list)
                                                                                    
                                                                                    
 

For the above to work, the array must be declared using Array(1..0). The new syntax  A ,= i  will append to the array, and there is no need to write  A(i) := i

5.7 make function display more information

By Carol Devore on the net:

Use infolevel. 
For example, to show what logic dsolve uses, do this: 
 
First try 
> infolevel[all]:= 5; 
 
That will probably give more information than you want, but if not, 
then try 
> printlevel:= 1000; 
 
If you want information about a specific procedure, you can use debug. 
 For example, 
restart; 
debug(`int/int`); 
int(p, x= 0..1); 
 
To find out what procedures are being called without getting too much 
extra information, use excallgraph.

Trying on dsolve

infolevel[dsolve]:= 3; 
dsolve({eq1},y(x)); 
 
Methods for second order ODEs: 
Trying to isolate the derivative d^2y/dx^2... 
Successful isolation of d^2y/dx^2 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
<- double symmetry of the form [xi=0, eta=F(x)] successful
 

5.8 How to plot a function?

Here, I am looking at fouries series expansion of \(f(x)=0\) between \(–\pi \) and 0, and \(f(x)=1\) between 0 and \(\pi \).

The Fouries series expansion is worked out to be as below. This shows that the series approximate the above \(f(x)\) as more terms are added

restart; 
f:=(x)-> 1/2 + (1/Pi)*(sin(x)+sin(3*x)/3+sin(5*x)/5+sin(7*x)/7); 
plot(f(x),x=-10..10);
 

5.9 How to run maple from command line?

From DOS, point to where your cmaple is

>"C:\Program Files\Maple 7\BIN.WNT\"cmaple

To make it execute maple commands use the < foo.txt to pipe maple commands in the file to it.

5.10 How to use matrices in maple?

A:= Matrix( [ [1, 2, 3] , 
              [3, 6, 7] , 
              [5, 6, 9] , 
              [7, 7, 7] 
            ]); 
whattype(A); 
       Matrix 
size:=LinearAlgebra:-Dimension(A); 
     size := 4, 3 
row:=size[1]; 
      row := 4 
col:=size[2]; 
      col := 3
 

You can extract any part of the matrix like this:

B:=A[1..3,2..2];
 

\[ \left [ \begin {array}{c} 2\\ 6\\ 6\end {array} \right ] \]

By Carl Devore http://mathforum.org/kb/message.jspa?messageID=1570678

Maple list and sequence structures are more flexible than Matrices, which are 
highly structured.  A Maple list of lists (called a listlist in Maplese) is akin 
to a matrix in some other languages.  Many matrix 
operations can be performed directly on the listlist form, but to do 
serious linear algebra, you should convert to a Matrix.  Of course, it is 
trivial to convert a listlist to Matrix: 
 
LL:= [[1,2], [3,4]]; 
M:= Matrix(LL); 
 
So here is another solution in line with your original wishes.  This is 
"index free", but the table-based solution I gave earlier should be 
faster.  (It is usually considered bad form to repeatedly append to a list or sequence.) 
 
L:= [][]; # Create a NULL sequence 
do 
   line:= readline(file); 
   if line::string then 
      if line contains valid data then 
         Z:= a list of that data; 
         L:= L, Z 
      fi 
   else 
      break 
   fi 
od 
 
A:= Matrix([L]); # Note []: seq -> list.
 

To move move a column into a matrix: Here, I want to copy 2nd column to the 3rd column:

A;

\[ \left [ \begin {array}{ccc} 1&2&3\\ 3&6&7\\ 5&6&9\\ 7&7&7 \end {array} \right ] \]

B:=A[1..row,2];

\[ \left [ \begin {array}{c} 2\\ 6\\ 6\\ 7 \end {array} \right ] \]

 A[1..row,3]:=B: A;

\[ \left [ \begin {array}{ccc} 1&2&2\\ 3&6&6\\ 5&6&6\\ 7&7&7 \end {array} \right ] \]

5.11 return more than value from a procedure

Maple can return multiple values. Make sure to use the comma "," in the body of the procedure to separate each return value. Example:

size_matrix:=proc(x) 3*x, 4*x; end proc; 
row,col :=size_matrix(5);
 

5.12 How does maple handle procedure arguments?

When passing a variable to maple procesure, the variable VALUE is passed to the procedure (This is different from say Fortran where the default is pass by reference). But this is the same as with Mathematica.

For example, if a variable X had value 10, then you call a procedure FOO passing it X, then inside FOO, X will be the number 10, not the argument variable X. So, this means one can not have X on the left hand side inside FOO. Like this x:=1

The only way to assign new value to the input and return new value, is to use a local variable, like this:

one:= proc(x) 
        local y; 
        print(x); 
        y:=x+ 1; 
        print(x); 
        y; 
        end proc; 
 
z:='z'; 
z:=5; 
f:=one(z); 
 
      f := 6
 

5.13 How to define your own data types?

Use `type/name` to define new type name.

`type/char`:= x-> x::string and length(x)=1; 
 
P:= proc(c::char) print(c) end proc: 
P("x"); 
                           "x" 
P("xy"); 
Error, invalid input: P expects its 1st argument, c, to be of type char, but received xy 
 
> `type/byte`:= x-> x::integer and (x>= 0 and x<256); 
 
#will define a byte (unsigned integer)
 

5.14 find max element and position in matrix

Code from net by Carl Devore:

MMax:= proc(M::{Matrix,matrix}) 
   local C,r,c,mx,L,p; 
   C:= op(`if`(M::Matrix, [1,2], [2,2,2]), eval(M)); 
   L:= map(op, convert(M, listlist)); 
   mx:= max(L[]); 
   member(mx,L,'p'); 
   r:= iquo(p, C, 'c'); 
   mx, `if`(c=0, [r,C], [r+1,c]) 
end;
 

Code below from C W

A:=matrix(12,12,rand(100)); 
Ao:=array((proc(E) 
            local i; [seq(i=(rhs=lhs)(E[i]),i=1..nops(E))]end) 
            (sort(op(3,eval(A)),proc(E1,E2) if rhs(E1)>rhs(E2) 
             then 
                  true 
             else 
                  false 
             fi 
           end))); 
           Ao[1];
 

5.15 How to create a package?

First create the module:

restart; 
 
nma:= module() 
      option package; 
      export getMaxMatrix; 
      getMaxMatrix := proc (M::{matrix, Matrix}) 
          local C, r, c, mx, L, p; 
          C := op(`if`(M::Matrix,[1, 2],[2,2,2]),eval(M)); 
          L := map(op,convert(M,listlist)); 
          mx := max(L[]); member(mx,L,'p'); 
          r := iquo(p,C,'c'); 
          mx, `if`(c = 0,[r, C],[r+1, c]) 
       end proc; 
end module; 
 
A:= Matrix( [ [1, 2, 3] , 
              [3, 6, 7] , 
              [5, 6, 9] , 
              [7, 7, 7] 
            ]); 
 
nma[getMaxMatrix](A);|
 

Gives 9, [3, 3]. Now save the module.

savelibname := "C:/MAPLE_PACKAES"; 
march('create', savelibname, 20);
 

now save the library to disk.  savelib(nma);

Now we can test everything by reinitialize everything and reload the library.

>restart 
#Add my library to LIBNAME 
>libname:="C:/MAPLE_PACKAGES",libname; 
> A:=matrix( [ [1,2,3],[4,6,9] ]); 
>with(nma); 
>nma[getMaxMatrix](A);
 

Now to print a proc() in the package, do

>interface(verboseproc=3); 
> print(nma[getMaxMatrix]);
 

Now you can list what packages exist in the archive:

march('list',savelibname); 
march('extract',savelibname,":-1.m","C:MAPLE_PACKAGES/t.m")
 

Some notes. need to clean later

> module1lib:=`module1\\lib`; 
> system("md "||module1lib); 
> march('create',module1lib,100); 
> makehelp(module1,`module1/module1.mws`,module1lib): 
> makehelp(`module1/export1`,`module1/export1.mws`,module1lib): 
> savelibname:=module1lib: ### doesn't affect current libname 
> savelib(module1); ### no error message 
> restart; 
> module1lib:="module1\\lib": 
> libname:=module1lib,libname; ### now Maple will find module1 
> with(module1); 
> ?module1
 

Also there is a long thread here on Maple prime on making personal packages in Maple How-To-Create-A-Personal-Package

5.16 How to convert from floating point to Hex?

From: Robert Israel ([e-mail]) 
Subject: Re: Getting non-integral results in hex 
Newsgroups: comp.soft-sys.math.maple 
Date: 2003-06-13 00:07:37 PST 
 
I assume you mean floating-point numbers.  Note that 
Maple floats (as opposed to "hardware floats") are 
in fact stored in base 10.  To convert a float to hex 
with n digits after the ".", you can use this: 
 
> `convert/hexfloat`:= proc(x::numeric, n::nonnegint) 
  local A,B,ax,R; 
  if nargs = 1 then return procname(x,round(Digits*log[16](10))) fi; 
  if x = 0 then return cat(`0.`,`0`$n) fi; 
  ax:= abs(x); 
  A:= floor(ax); 
  B:= round(frac(ax)*16^n); 
  if B = 16^n then A:= A+1; B:= 0 fi; 
  R:= cat(convert(A,hex),`.`); 
  if x < 0 then R:= cat(`-`,R) fi; 
  cat(R,substring(convert(16^n+B,hex),2..-1)); 
  end; 
 
And then, e.g.: 
 
> convert(1234.5678, hexfloat, 4); 
 
                               4D2.915B
 

5.17 How to find taylor series expansion of functions?

mtaylor(sin(x),[x],10);

\[ x-1/6\,{x}^{3}+{\frac {{x}^{5}}{120}}-{\frac {{x}^{7}}{5040}}+{\frac { {x}^{9}}{362880}} \]

5.18 How to print elements of a matrix?

restart; 
a:=Matrix([  [2,3,4],[4,5,6]   ]); 
nRow,nCol :=LinearAlgebra[Dimension](a); 
for i from 1 to nRow do 
    for j from 1 to nCol do 
        printf("a(%d,%d)=%d\n",i,j,a[i,j]); 
    end do; 
end do; 
 
a(1,1)=2 
a(1,2)=3 
a(1,3)=4 
a(2,1)=4 
a(2,2)=5 
a(2,3)=6
 

5.19 How to find determinant of matrix?

restart; 
a:=Matrix([ [2,4],[5,7] ]); 
LinearAlgebra:-Determinant(a); 
         -6
 

5.20 How to generate Hilber matrix?

H := LinearAlgebra:-HilbertMatrix(5);

\[ \left [ \begin {array}{ccccc} 1&1/2&1/3&1/4&1/5\\ \noalign {\medskip }1/ 2&1/3&1/4&1/5&1/6\\ \noalign {\medskip }1/3&1/4&1/5&1/6&1/7 \\ \noalign {\medskip }1/4&1/5&1/6&1/7&1/8\\ \noalign {\medskip }1/5&1/6&1 /7&1/8&1/9\end {array} \right ] \]

5.21 How to plot matrix data?

Matlab is much easier here. In maple, need to covert the matrix to a list of list of points first.

restart; 
H := LinearAlgebra:-HilbertMatrix(5): 
nRow,nCol :=LinearAlgebra[Dimension](H): 
L:=[seq([seq( [i,j,H[i,j]], i=1..nRow) ], j=1..nCol)]: 
plots:-surfdata(L);
 

5.22 How to catch an error from a proc()?

An error in maple raises an exception. So, use try catch to trap it as follows:

try 
   v,pos:=MMax(4); 
catch: 
   printf("an error is cought\n"); 
end try;
                                                                                    
                                                                                    
 

5.23 How to convert 3456 to 3,456 ?

From the net, by Carl Devor:

`print/commas`:= proc(N::integer) 
    local n,s,i,b; 
    n:= ListTools:-Reverse(convert(abs(N), base, 1000)); 
    if N<0 then n:= subsop(1= -n[1], n) fi; 
    nprintf("%s", sprintf(cat("%d", ",%03d" $ nops(n)-1), n[])) 
 end proc: 
 
commas(456554); 
 
         456,554
 

To convert a string to array of chars use array(StringTools:-Explode(S))

s:="Nasser M. Abbasi": 
r:=array(StringTools:-Explode(s)); 
              r:=["N" "a" "s" .......]
 

Now can use the string as normal array

r[4]; 
      "s"
 

5.24 How to use units ?

Units[GetDimensions](base); 
 amount_of_information, amount_of_substance, currency, electric_current, length, 
 logarithmic_gain, luminous_intensity, mass, thermodynamic_temperature, time
 

5.25 How to evaluate catlan number and other sums?

Use the Sum command.

restart; 
expr:= (-1)^i/(2*i+1)^2; 
Sum(expr,i=0..infinity); 
evalf(%,50); 
          0.91596559417721901505460351493238411077414937428167
 

Notice, if I used the sum command instead of the Sum command I get this result:

sum(expr,i=0..infinity); 
          Catalan
 

5.26 write a text file that contains a package, and load it

This shows how to do a simple package and use it without building a library. Just using a plain text file.

Create this nma_pkg1.txt file:

 
nma_pkg1 := module() 
  export f1; 
  option package; 
 
  f1:= proc() 
       print("in pakcage nma_pkg1"); 
  end proc; 
 
end module;
 

now save it, and from maple do

>read("c:\\nma_pkg1.txt");
 

now execute f1() as this:

>nma_pkg1[f1](); 
          "in pakcage nma_pkg1"
 

now put it in a library (so that we can use with, instead of read)

> savelibname:=("c:/maple"); 
> march('create', savelibname, 20); 
> savelib(nma_pkg1); 
>restart; 
> libname := "c:/maple",libname; 
> with(nma_pkg1); 
> f1(); 
    "in pakcage nma_pkg1"
 

now make changes to the nma_pkg1.txt file and updated again as above.

5.27 How to find what packages are included in maple

?index,package

5.28 How to plot the gradiant vector field?

restart; 
f:=3*x^2 + y* cos(x*y); 
the_grad :=linalg[grad](f,[x,y]); 
plots[fieldplot](the_grad,x=-2..2,y=-2..2);
 

or

or can do it in just one command: plots[gradplot](f,x=-2..2,y=-2..2);

5.29 How to put the digits of Pi into a list?

Suppose you want the 100 digits of Pi put in a list. This is one way to do it:

restart; 
L:=evalf(Pi,100); 
S:=convert(L,string); 
the_list:=[seq(parse(S[i]),i=3..length(S))]; 
 
    the_list := [1, 4, 1, 5, 9, 2, 6, 5, 3, ..
 

This below now tells how many times each digits occurs.

>stats[transform,tally](the_list); 
 
[Weight(0, 8), Weight(1, 8), Weight(2, 12), Weight(3, 11), 
Weight(4, 10), Weight(5, 8), Weight(6, 9), Weight(7, 7), 
Weight(8, 13), Weight(9, 13)]
 

5.30 Digits of PI in maple and mma

Written sometime in 2005? I should really record the time when I write something.

I just run these now, Auust 2014, and now Maple 18 as very fast. So this all below is no longer valid. I will leave it here for now for reference until I update it all later

I have written a few lines of code, which counts how many times each digit occurs after the decimal points of \(\pi \)

Written this in maple first. Then did similar thin in mma 5.0. Both are run on the same PC. No other applications are running at the time when I run the code.

The basic idea of the algorithm is to use evalf(Pi,digits) in maple to find \(\pi \) for any number of decimal digits, and to use N[Pi,digits] in mma for doing the same. (Where the variable digits above is the number of digits)

Then in maple convert the above \(\pi \) to a string, and generate a sequence of the characters to right of decimal point, then use stats[transform,tally] to do the actual counting.

In mma, I use RealDigits[] to get a list of the digits, and then use Count[] to do the counting.

This is result of some of the runs to find Pi to some digits, and the total time (to find Pi and do the counting)

All times are in cpu seconds, machine is P4, 2.8 Ghz, 500 MB of RAM, single CPU, hyperthreading enabled, running XP home edition. Maple 9.03 student version, and mma 5.0 student version.

Below is the result, and below that I show the maple code and the mma code.

Because of this, before each run in mma, I exited the application and started it fresh. In maple, it does not matter for the above reason.

100,000 digits: 
             Find_Pi     Total 
Maple 9.0      55          84 
Mma 5.0        0.9         1.54
 

Mma is 60 times faster in finding pi and about 56 times faster overall

300,000 digits: 
             Find_Pi     Total 
Maple 9.0     309         781 
Mma 5.0       3.7         6
 

Mma is 300 times faster in finding Pi, and 130 times faster overall.

3,000,000 digits 
             Find_Pi     Total 
Maple 9.0 
Mma 5.0        85        118 
 
Maple time in hours ! Still running.
 

Maple code

> restart; 
startingTime :=time(); 
L:=evalf(Pi,100000): 
timeToFindPiInSecs:=time()-startingTime; 
S:=convert(L,string): 
the_list:=[seq(parse(S[i]),i=3..length(S))]: 
stats[transform,tally](the_list); 
endingTime :=time(): 
cpuTimeInSecs := endingTime - startingTime;
 

mma code

Clear[] 
startingTime=TimeUsed[] 
t1=N[Pi,100000]; 
timeToFindPiInSecs=TimeUsed[]-startingTime 
{c,d}=RealDigits[t1]; 
theList=c[[Range[2,Length[c]]]]; 
f[digit_]:=Count[theList,digit]; 
r=Range[0,9]; 
Map[f,r] 
cpuTimeInSecs=TimeUsed[]-startingTime
 

update 12/25/03 Changed maple code on how to do the counting : To use

StringTools[CharacterFrequencies](S)
 

Now the counting in maple is much faster. It is always hard to know which is the best function to use.

restart; 
 startingTime :=time(); 
 L:=evalf(Pi,300000): 
 timeToFindPiInSecs:=time()-startingTime; 
 S:=convert(L,string): 
StringTools[CharacterFrequencies](S); 
endingTime :=time(): 
cpuTimeInSecs := endingTime - startingTime;
 

5.31 How to find where functions are?

From: Ken Lin ([e-mail]) 
Subject: Re: how to find which package a function belongs to? 
Newsgroups: comp.soft-sys.math.maple 
Date: 2003-12-04 03:49:26 PST 
 
When Maple first loaded, There are only two kinds of "internal" 
commands which can be called directly. One is the "kernal" commands 
coded in C, and the other includes many "internal" prodecures 
programmed by the kernal commands which lies in the "Main Library", 
There are also many other "external" procedures which were categorized 
into so called "packages", plots[display](...) for example, plots[] is 
a package(Library), and display() is the prodecure inside plots[]. All 
the packages can be loaded by with() command, like 
> with(plots); 
 
 
Because Different Packages include user library might have the same 
procedure name, Maple doesn't realize the "procedure_name" you type 
in, it took it for a "symbol". If you really want to know which 
packages provided by Maple the external procedure lies in, just mark 
the procedure_name and press F1 key, the Maple Help Browser will show 
you the packages you might be interested. 
 
By the way, plot3d() is a "internal" procedure lies in the Main 
Library. You can confirm that by: 
> op(0, eval(plot3d)); 
                         procedure 
or in Maple 9 
> type( plot3d, 'std' );      #Is it internal?                      true 
> type( plot3d, 'stdlib' );   #Does is lie in "Standard(Main) Library"? 
                         true 
If you are interested the codes inside plot3d()... 
> interface(verboseproc=2):   #Turn on verboseproc 
> print(plot3d);              #eval() also works 
> interface(verboseproc=1):   #Turn off verboseproc 
 
I hope this will give you some help. Have fun with Maple. 
 
Ken Lin
 

5.32 how to use laplace transform?

restart; 
f:= t->sin(omega*t) ; 
L:=convert(inttrans[laplace](f(t),t,s),int);
 

\[ {\frac {\omega }{{\omega }^{2}+{s}^{2}}} \]

To find the inverse, do:

 inttrans[invlaplace](L,s,t);
 

\[ \sin \left ( \omega \,t \right ) \]

For unit step, use

_EnvUseHeavisideAsUnitStep:=true; 
f:=Heaviside(t-a); 
INV:=inttrans:-laplace(f,t,s) assuming a>0;  #make sure to use a>0
 

\[ \frac {{\mathrm e}^{-s a}}{s} \]

Another example

_EnvUseHeavisideAsUnitStep:=true; 
f:=Heaviside(t)-Heaviside(t-a); 
INV:=inttrans:-laplace(f,t,s) assuming a>0;  #make sure to use a>0
 

\[ \frac {1-{\mathrm e}^{-s a}}{s} \]

5.33 questions I have

Any difference between using 
 
`diffalg/Rosenfeld_Groebner`(args) 
or 
diffalg[Rosenfeld_Groebner](args)
 

5.34 3D plotting

restart; 
f:= (x,y)->x^3-3*x*y^2; 
plot3d(f,-1..1,-1..1,numpoints=2500,style=patchcontour);
 

5.35 How to raise each element in a list to a power?

Use map

map(`^`,{1,2,3},3); 
      {1, 8, 27}
 

5.36 How to generate a sequence with any increment?

incr:=.25; start:=0; last:=3; 
seq(start+i*incr,i=1..(last/incr));
 

5.37 What shortcuts are there for matrix manipulation?

read ?MVshortcut, ?MVassignment, and ?Mvextract and Transpose(R) can be shortened to R^%T

5.38 How to solve a set of equations for the derivative?

Written feb 20, 2004

This is problem 7.4 chapter 4, in the Mary Boas book. Given

\begin{align*} x s^2+y t^2 &= 1\\ x^2 s+y^2 t &= xy-4 \end{align*}

Find \(\frac {dx}{dt}, \frac {dx}{ds}, \frac {dy}{dt}, \frac {dy}{ds}\) at \(x=1,y=-3,s=2,t=-1\)

This is how I did it in maple:

restart; 
alias(x=x(s,t)); 
alias(y=y(s,t)); 
alias(Xt= diff(x(s,t), t)); 
alias(Xs= diff(x(s,t), s)); 
alias(Yt= diff(y(s,t), t)); 
alias(Ys= diff(y(s,t), s)); 
 
eq1:= x*s^2+y*t^2=1; 
eq2:= x^2*s+y^2*t=x*y-4; 
 
r1:=diff(eq1,t); 
r2:=diff(eq1,s); 
r3:=diff(eq2,t); 
r4:=diff(eq2,s); 
 
sol:=solve({r1,r2,r3,r4},{Xt,Xs,Yt,Ys});
 

\begin{align*} {\frac {\partial }{\partial s}}x \left ( s,t \right ) &= -{\frac {x \left ( s,t \right ) \left ( x \left ( s,t \right ) {t}^{2}-4\,y \left ( s,t \right ) st+2\,x \left ( s,t \right ) s \right ) }{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t{s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial t}}x \left ( s,t \right ) &=-{\frac {y \left ( s,t \right ) t \left ( -3\,y \left ( s,t \right ) t+2\,x \left ( s,t \right ) \right ) }{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t{ s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial s}}y \left ( s,t \right ) &=-{\frac {x \left ( s,t \right ) \left ( 3\,x \left ( s,t \right ) s-2\,y \left ( s,t \right ) \right ) s}{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t {s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial t}}y \left ( s,t \right ) &=-{\frac {y \left ( s,t \right ) \left ( 4\,x \left ( s,t \right ) st-y \left ( s,t \right ) {s }^{2}-2\,y \left ( s,t \right ) t \right ) }{2\,x \left ( s,t \right ) s{t} ^{2}-2\,y \left ( s,t \right ) t{s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}} \end{align*}

points:= {x=1,y=-3,s=2,t=-1}; 
subs(points,sol);
 

5.39 How to solve a set of equations for differentials?

This is problem 7.15 chapter 4 in Boas:

Given \(x^2 u-y^2 v=1\) and \(x+y=uv\) Find \(\frac {dx}{du},v\) and \(\frac {dx}{du},y\)

This is the maple code to solve this:

restart; 
eq1:=x^2*u-y^2*v=1; 
eq2:=x+y=u*v; 
r1:=D(eq1); 
r2:=D(eq2); 
r1_:=subs(D(v)=0,r1); 
r2_:=subs(D(v)=0,r2); 
sol:=solve({r1_,r2_},{D(x),D(u)}); 
print("dx/du,v="); 
rhs(sol[1])/rhs(sol[2]); 
 
r1_:=subs(D(y)=0,r1); 
r2_:=subs(D(y)=0,r2); 
sol:=solve({r1_,r2_},{D(x),D(u)}); 
print("dx/du,y="); 
 
rhs(sol[1])/rhs(sol[2]);
 

5.40 How to plot binary tree

by http://www.math.fsu.edu/~bellenot

restart; 
t2 := proc(i, x, y) 
    if i < 2 then [[x, y], [x, y - 1]], [[x, y], [x + 2^i, y - 1]] 
    else [[x, y], [x, y - 1]], [[x, y], [x + 2^i, y - 1]], 
        t2(i - 1, x, y - 1), t2(i - 1, x + 2^i, y - 1) 
    end if 
end proc; 
 
PLOT(CURVES(t2(6,0,0)));
 

5.41 Problem 12.4 chapter 4, Boas book

restart; 
z:=   Int( sin(t)/t, t=sin(x)..cos(x)); 
diff(z,x);
 

\[ -{\frac {\sin \left ( x \right ) \sin \left ( \cos \left ( x \right ) \right ) }{\cos \left ( x \right ) }}-{\frac {\cos \left ( x \right ) \sin \left ( \sin \left ( x \right ) \right ) }{\sin \left ( x \right ) }} \]

5.42 example of doing convergence test in maple

restart; 
c:='c': C:='C': n:='n': P:='P': 
C := n -> ((n+2)/(3*n+1))^n: 
### WARNING: calls to `C` for generating C code should be replaced by codegen[C] 
`The general term is `, c[n]= C(n); ` `; 
`The n-th root is:`; 
### WARNING: calls to `C` for generating C code should be replaced by codegen[C] 
P := C(n)^(1/n): 
abs(c[n])^(1/n) = P; 
P := simplify(P, assume=positive): 
abs(c[n])^(1/n) = P;
 

5.43 Problem ch 14, 3.18, Boas book. contour integration

restart; 
f:= 1/(  (1-2*z)*(5*z-4) ); 
residue(f,z=4/5);
 

\[ \frac {-1}{3} \]

5.44 How to find multiple roots to an equation such as \(sin(x) = 0\)

_EnvAllSolutions:=true; 
solve(sin(x)=0);
 

Pi _Z1~

5.45 Dr Basti Associated Legendre

Subject:      Associated Legendre 
Author:       Mehran Basti <[e-mail]> 
Organization: AT&T Worldnet 
Date:         Mon, 25 Nov 2002 02:48:15 GMT 
 
 
Dear newsgroup: 
 
I had mentioned that my methods will solve classical equations without 
the use of  infinite series. 
 
The following is a Maple code of my old files. Those days I had Maple2 but the 
general idea is the same in the process  and you see that we can also 
solve the integrals involved. 
 
It does not make sense how are the theory behind it but eventually it will come into light. 
 
Just read the procedures and you can see the solution of associated legendre AL at the end. 
 
> s1:=-diff(p(t),t)+p(t)^2; 
> 
> s2:=exp(2*int(p(t),t))*T(t); 
> s3:=s1+s2; 
> s4:=diff(T(t),t)/T(t); 
> s5:=-(1/2)*(diff(s4,t))+(1/4)*s4^2; 
> s6:=s5+s2; 
> p(t):=-1/t+(1)/(2-t); 
> s1:=simplify(s1); 
> s1:=collect(%,t); 
> s2:=simplify(s2); 
> s1+s2=(2*t^2-4*t+m^2-1)/(t*(-2+t))^2; 
> solve(%,T(t)); 
> T(t):=simplify(%); 
> s2:=simplify(s2); 
> s2+s1; 
> s3:=simplify(%); 
> 
> s6:=simplify(s6); 
> t*(-2+t); 
> simplify(%); 
> z:=(r3*t^3+r2*t^2+r1*t+r0)/(%); 
> 
> simplify(diff(z,t)+z^2-s6); 
> s7:=collect(numer(%),t); 
> 
> coeff(%,t,0); 
> solve(%,r0); 
> r0:=op(1,{%}); 
> coeff(s7,t,1); 
> solve(%,r1); 
> r1:=simplify(%); 
> coeff(s7,t,2); 
> solve(%,r2); 
> r2:=simplify(%); 
> coeff(s7,t,3); 
> solve(%,r3); 
> r3:=simplify(%); 
> simplify(s7); 
> s3:=simplify(s3); 
> s4:=simplify(s4); 
> s6:=simplify(s6); 
> T(t):=simplify(T(t)); 
> z:=simplify(z); 
> 1/2*s4+2*p(t)+z; 
> s8:=simplify(%); 
> exp(int(%,t)); 
> expand(%); 
> g:=(%); 
> simplify(g,power); 
> g:=%; 
> Int(%,t); 
> Integralg:=(%); 
> int(g1(t),t); 
> x1:=-p(t)+g1(t)/(%); 
> diff(x1,t)+x1^2-s3; 
> simplify(%); 
> s10:=numer(%); 
> solve(%,int(g1(t),t)); 
> Ing:=(%); 
> simplify(subs(g1(t)=g,%)); 
> 
>  Ing:=(%); 
> expand(%); 
> Ing:=simplify(%); 
> simplify(diff(%,t)-g); 
> expand(%); 
> simplify(%); 
> x:=-p(t)+g/Ing; 
> simplify(diff(x,t)+x^2-s3); 
>  int(x,t); 
> exp(%); 
> expand(%); 
> s11:=simplify(%); 
> ALT:=t*(2-t)*diff(u(t),t$2)+2*(1-t)*diff(u(t),t)+(2-m^2/(1-(1-t)^2))*u(t); 
> -2*(1-t)/(2*t*(2-t)); 
> int(%,t); 
> exp(%); 
> s12:=simplify(%,power); 
> 
> u1:=s12*s11; 
> u1:=simplify(%,power); 
>  simplify(subs(u(t)=u1,ALT)); 
> AL:=(1-nu^2)*diff(u(nu),nu$2)-2*nu*diff(u(nu),nu)+(2-m^2/(1-nu^2))*u(nu); 
> 
> u2:=subs(t=1-nu,u1); 
> simplify(subs(u(nu)=u2,AL)); 
> 
 
The advantage of these methods are that there are ample rooms for advances. 
 
Today my skills for solving classical equations such as Riccati is much advanced. 
 
Highly complicated and more general Riccati equations in its billions now possible. 
 
Sincerely 
 
Dr.M.Basti
 

5.46 Understanding conformal mapping in maple

To plot mapping of complex function in maple, use [plots]conformal The trick is to how to specify the quadrant in the x-y plane. This example shows how.

Suppose we want to map the first quadrent. Then we specify the DIAGONAL points in the range, from the lower left corner to the upper right corner, which then should be 0..1+I Because 0 is the lower left corner, and \((1,i)\) is the upper right corner. Example:

restart; 
assume(y,real); 
assume(x,real); 
#f:= z->I+z*exp(I*Pi/4); 
f:= z->z^2; 
w:=f(x+I*y); 
u:=Re(w); 
v:=Im(w); 
plots:-conformal(f(z),z=0..1+I,grid=[16,16],numxy=[16,16],scaling=constrained);
 

This below uses the first TWO quadents, i.e. the upper half of the x-y plane

restart; 
assume(y,real); 
assume(x,real); 
#f:= z->I+z*exp(I*Pi/4); 
f:= z->z^2; 
w:=f(x+I*y); 
u:=Re(w); 
v:=Im(w); 
plots:-conformal(f(z),z=-1-I..1+I,grid=[16,16],numxy=[16,16],scaling=constrained);
 

This below puts the plots next to each others so to see them

restart; 
assume(y,real); 
assume(x,real); 
f:= z->I+z*exp(I*Pi/4); 
#f:= z->z^2; 
w:=f(x+I*y); 
u:=Re(w); 
v:=Im(w); 
A := array(1..2): 
A[1]:=plots:-conformal(z,z=0..1+I/2,grid=[16,16],numxy=[16,16],scaling=constrained): 
A[2]:=plots:-conformal(f(z),z=0..1+I/2,grid=[16,16],numxy=[16,16],scaling=constrained): 
plots:-display(A);
 

5.47 Hide tilda character when using assumption

interface(showassumed=0) removes all tildas and interface(showassumed=1) adds the tildas.

5.48 Fourier series in maple

I wrote this to generate FS in Maple for some HW I was doing. I think this was for Math 121A at UC Berkeley in 2003

restart; 
f:=x->piecewise(-Pi<x and x<Pi/2,-1, 
                Pi/2<x and x<1,0,1); 
 
assume(n,integer); 
 
nmaFourier2:=proc(f,freq,from_,to_,maxN) 
       local n::integer,denomC,denomS,a,b; 
       denomC:=( to_ - from_ ) / 2; 
       denomS:=( to_ - from_ ) / 2; 
 
       a:=proc(n) 
         int(f(x)*cos(n*freq*x),x=from_..to_) /denomC; 
       end proc; 
 
       b:=proc(n) 
         int(f(x)*sin(n*freq*x),x=from_..to_) / denomS; 
       end proc; 
 
       evalf(denomC); 
 
       1/2*a(0) + sum( a(n) * cos(n*freq*x) ,n=1..maxN) 
                + sum( b(n) * sin(n*freq*x) ,n=1..maxN) 
end proc; 
 
r:=[seq(nmaFourier2(f,1,-Pi,Pi,nIter),nIter=1..10)]; 
plot(r,x=-Pi..Pi);
 

To animate do

g:=n->plot(nmaFourier2(f,1,-Pi,Pi,n),x=-2*Pi..2*Pi); 
plots:-animate(g,[n],n=1..40);
 

Here is the animation from the Maple notebook:

Another version

restart; 
f:=x->piecewise(-Pi<x and x<Pi/2,-1, 
                Pi/2<x and x<1,0,1); 
 
assume(n,integer); 
nmaFourier2:=proc(f,freq,from_,to_,maxN::integer) 
       local n::integer,denomC,denomS,a,b; 
 
       denomC:=( to_ - from_ ) / 2; 
       denomS:=( to_ - from_ ) / 2; 
 
       a:=proc(n) 
         int(f(x)*cos(n*freq*x),x=from_..to_) /denomC; 
       end proc; 
 
       b:=proc(n) 
         int(f(x)*sin(n*freq*x),x=from_..to_) / denomS; 
       end proc; 
 
       1/2*a(0) + sum( a(n) * cos(n*freq*x) ,n=1..maxN) 
                + sum( b(n) * sin(n*freq*x) ,n=1..maxN) 
end proc; 
 
plots[setoptions](title=` `, axesfont=[SYMBOL,8] ,font=[COURIER,1], 
   xtickmarks=[seq(evalf(k*Pi/2)=sprintf("%a %s", k/2 ,"pi" ),k= -3..3)], 
   ytickmarks=[-1.0="-1",-0.5="",0.0="0",0.5="",1.0="1"]); 
 
B:=array(1..3,1..3); 
k:=0; 
for i from 1 to 3 do 
    for j from 1 to 3 do 
       k:=k+1; 
       B[i,j]:=plot({f(x),nmaFourier2(f,1,-Pi,Pi,k)},x=-Pi..Pi,size=[200,100]); 
    end do; 
end do; 
 
plots:-display( B);
 

5.49 How to plot graphs next to each others in a grid like fashion

restart; 
v:=1; 
B:=Matrix(3,3); 
for i from 1 to 3 do 
   for j from 1 to 3 do 
       v:=v+1; 
       B[i,j]:= plot(x^v,x=-2..2,thickness=3,size=[200,100] ); 
   end do; 
end do; 
plots:-display(B);
 

5.50 How to generate Pi on X-axis

From book Maple animation by John Putz

plot( sin(x), x=0..2*Pi, xtickmarks=evalf([Pi/2="p/2", Pi="p", 
3*Pi/2="3p/2", 2*Pi="2p"]), ytickmarks=[-1,1], axesfont=[SYMBOL,16], labels=["",""] );
 

5.51 How to make output from FunctionAdvisor look better?

From Preben Alsholm

res:=FunctionAdvisor(sin): 
res2:=op(2,eval(res)): 
map(print,res2);
 

or answer by Thomas Richard

> FunctionAdvisor( display, sin );
 

5.52 How to do partial fractions?

Use convert(expr,parfrac) or convert(f,fullparfrac)

5.53 How to generate sequence sum symbolically

n := 7; 
f:=sum('a[k]*b[k]','k'=1..n);
 

\[ a_{{1}}b_{{1}}+a_{{2}}b_{{2}}+a_{{3}}b_{{3}}+a_{{4}}b_{{4}}+a_{{5}}b_{ {5}}+a_{{6}}b_{{6}}+a_{{7}}b_{{7}} \]

5.54 Nice plot from Maple

from Serge from the net:

restart; 
with(geom3d): 
plane(OYZ,x=0,[x,y,z]): 
plane(OXZ,y=0,[x,y,z]): 
plane(OXY,z=0,[x,y,z]): 
c:=1/2:r:=1/4: 
L:=combinat[permute]([-c$3,c$3],3): 
S:=seq(sphere(s||i,[point(A||i,op(op(i,L))),r]),i=1..8): 
draw([OYZ,OXZ,OXY,S]);
 

5.55 How to check if 2 expressions are the same?

Use evalb(). For example evalb(I*sinh(x)=sin(I*x)); gives true

The above does not always work. Only sure way is to do this

> m1 := exp(I*n*x); 
m2 := (cos(n*x)+I*sin(n*x)); 
simplify(m1-m2); 
simplify(m1-convert(m2,exp));
 

5.56 converting series to factorials

Function by Robert Israel from the net:

restart; 
 
thefacts:= [seq(i!,i=2..20)]: 
  getfacts:= proc(x::{algebraic,series}) 
    local i; 
    if type(x, {`+`,`*`,series}) then 
      map(getfacts,x) 
    elif type(x, fraction) then 
      getfacts(numer(x))/getfacts(denom(x)) 
    elif type(x,`^`) then 
      getfacts(op(1,x))^op(2,x) 
    elif type(x,negint) then 
      -getfacts(-x) 
    elif type(x,posint) then 
      for i from 1 to 19 while irem(x, thefacts[i]) = 0 do od: 
      if i = 1 then x 
      elif thefacts[i-1] = x then ``(i)! 
      else ``(i-1)!*getfacts(x/thefacts[i]) 
      fi 
    else x 
    fi 
  end; 
 
getfacts(series(sin(x),x));
 

\[ \text {series} \left ( x-{\frac {{x}^{3}}{ \left ( \left ( 3 \right ) \right ) !}}+{\frac {{x}^{5}}{ \left ( \left ( 5 \right ) \right ) !}}+O \left ( {x}^{7} \right ) ,x,7 \right ) \]

5.57 How to find what new additions made to Maple?

 ?updates,maple10

5.58 Maple can’t solve laplace equation and numerically

Maple 2020.

restart; 
PDE := diff(u(x,y), y$2 ) + diff(u(x,y), x$2) = 0; 
BC:= u(x,0)=0, u(x,100)=100, u(0,y)=0, u(10,y)=0; 
sol:=pdsolve(PDE,[BC] ,numeric); 
 
Error, (in pdsolve/numeric) unable to handle elliptic PDEs
 

Compare to

restart; 
PDE := diff(u(x,y), y$2 ) + diff(u(x,y), x$2) = 0; 
BC:= u(x,0)=0, u(x,100)=100, u(0,y)=0, u(10,y)=0; 
sol:=pdsolve([PDE,BC]);
 

\[ u \left ( x,y \right ) =\sum _{n=1}^{\infty }-200\,{\frac { \left ( \left ( -1 \right ) ^{n}-1 \right ) {{\rm e}^{10\,\pi \,n}}\sin \left ( 1/10\,n\pi \,x \right ) \left ( {{\rm e}^{1/10\,n\pi \,y}}- {{\rm e}^{-1/10\,n\pi \,y}} \right ) }{\pi \,n \left ( {{\rm e}^{20\,\pi \,n}}-1 \right ) } } \]

5.59 Some Maple Matrix operations

Create a new matrix, by appending some rows of one matrix to rows from another matrix:

restart; with(LinearAlgebra): 
A:=< <1|2|3> , <4|5|6> >;
 

\[ \left [ \begin {array}{ccc} 1&2&3\\ \noalign {\medskip }4&5&6 \end {array} \right ] \]

B:=< <7|8|10> , <11|12|13> , <14|15|16>  >;
 

\[ \left [ \begin {array}{ccc} 7&8&10\\ \noalign {\medskip }11&12&13 \\ \noalign {\medskip }14&15&16\end {array} \right ] \]

Now append first row of A to last 2 rows of B

C:=<  A[1,1..-1] ,  B[2..-1,1..-1] >;
 

\[ \left [ \begin {array}{ccc} 1&2&3\\ \noalign {\medskip }11&12&13 \\ \noalign {\medskip }14&15&16\end {array} \right ] \]

# Now append first column of A to first 2 rows  of B 
A[1..-1,1]; 
B[1..2,1..-1]; 
C:=< A[1..-1,1] | B[1..2,1..-1]  >;
 

\[ \left [ \begin {array}{cccc} 1&7&8&10\\ \noalign {\medskip }4&11&12&13 \end {array} \right ] \]

#Now remove the middle row of B 
B; 
B:=<B[1,1..-1] , B[-1,1..-1] >;
 

\[ \left [ \begin {array}{ccc} 7&8&10\\ \noalign {\medskip }14&15&16 \end {array} \right ] \]

#now set the diagonal elements of B to be 0 
B:=RandomMatrix(3); 
for i from 1 to 3 do 
    B[i,i]:=0; 
end do: 
B;
 

\[ B:=\left [ \begin {array}{ccc} 0&99&92\\ \noalign {\medskip }8&0&-31 \\ \noalign {\medskip }69&44&0\end {array} \right ] \]

\[ \left [ \begin {array}{ccc} 0&99&92\\ \noalign {\medskip }8&0&-31 \\ \noalign {\medskip }69&44&0\end {array} \right ] \]

To find inverse.

restart; 
with(LinearAlgebra): 
A:=Matrix( [ [2,0],[4,2] ]); 
MatrixInverse(A);
 

\[ \left [ \begin {array}{cc} 1/2&0\\ \noalign {\medskip }-1&1/2 \end {array} \right ] \]

To check that for any matrix A, then A*transpose(A) is always a matrix which is symmetrical

A:=RandomMatrix(2,3); 
A.Transpose(A);
 

\[ A:=\left [ \begin {array}{ccc} 99&44&-31\\ \noalign {\medskip }29&92&67 \end {array} \right ] \]

\[ \left [ \begin {array}{ccc} 99&44&-31\\ \noalign {\medskip }29&92&67 \end {array} \right ] \]

how to create a random lower triangular matrix?

restart; 
with(LinearAlgebra); 
A:=RandomMatrix(4,4,outputoptions=[shape=triangular[lower]]);
 

\[ \left [ \begin {array}{cccc} 67&0&0&0\\ \noalign {\medskip }-31&92&0&0 \\ \noalign {\medskip }44&29&99&0\\ \noalign {\medskip }69&8&27&-4 \end {array} \right ] \]

5.60 How set diagonal elements to some value, say 1?

restart; 
with(LinearAlgebra); 
A:=RandomMatrix(5); 
LinearAlgebra:-Map[(i,j)->evalb(i=j)](x->1,A);
 

\[ A:= \left [ \begin {array}{ccccc} 1&-98&-76&-4&29\\ \noalign {\medskip }-38& 1&-72&27&44\\ \noalign {\medskip }-18&57&1&8&92\\ \noalign {\medskip }87& 27&-32&1&-31\\ \noalign {\medskip }33&-93&-74&99&1\end {array} \right ] \]

\[ \left [ \begin {array}{ccccc} 1&-98&-76&-4&29\\ \noalign {\medskip }-38& 1&-72&27&44\\ \noalign {\medskip }-18&57&1&8&92\\ \noalign {\medskip }87& 27&-32&1&-31\\ \noalign {\medskip }33&-93&-74&99&1\end {array} \right ] \]

5.61 How to multiply roots of a polynomial?

eq:=3*x^3+2*x^2+x+5=0; 
s:=[evalf(solve(eq,x))]; 
mul(s[i],i=1..nops(s));
 

Gives

5.62 How to plot a surface in 3D?

restart; 
eq:=3*x+4*y+2*z=10; 
plot3d(solve(eq,z),x=-5..5,y=-5..5,axes=normal);
 

One can also use impliticplot3d

restart; 
with(plots): 
implicitplot3d(3*x+4*y+2*z=10, x=-5..5,y=-5..5, z=-20..20,axes=normal);
 

5.63 How to convert trigs to sinc function in an expression

From http://www.mapleprimes.com/questions/40470-Trigonometric-Function-To-Sinc-Function

Maple doesn’t have a sinc function. If you mean the function sinc(x) = sin(x)/x, you could say something like

> eval(expr, {sin = (x -> x*sinc(x)), 
              cos = (x -> (x+Pi/2)*sinc(x+Pi/2)), 
              tan = (x -> x*sinc(x)/(x+Pi/2)/sinc(x+Pi/2))});
 

5.64 How to find NullSpace and ColumnSpace of a matrix?

restart; 
with(LinearAlgebra): 
A:=Matrix([[1,0,1,0,1],[0,1,0,1,0]]); 
NullSpace(A); 
ColumnSpace(A);
 

5.65 How to fix the interface to using Maple notation for input?

Go to tools->optiopn, and Display, and select Maple notation for input display.

5.66 How to find all solutions using allvalues ?

solve(x^2-sin(x),x); 
RootOf(-sin(_Z)+_Z^2) 
 
allvalues(%); 
RootOf(-sin(_Z)+_Z^2, 0.), RootOf(-sin(_Z)+_Z^2, .8767262154) 
 
evalf(%); 
0., .8767262154
 

5.67 Adding only to diagonal of a matrix

Use Map with filter

A:=< 1,2,3;4,5,6;7,8,9>; 
LinearAlgebra:-Map[(i,j)->evalb(i=j)](x->x+1,A);
 

5.68 How to search help for updates on some package

Go to http://www.maplesoft.com/support/help/search.aspx

and type say updates,Maple17,DE in the small box there.

5.69 How to work with groups in worksheet

From http://www.mapleprimes.com/questions/201092-How-To-Insert-New-Paragraph-On-Its-Own by Carl Love:

I use these special keystrokes constantly in my Maple worksheet typing: 
 
    Ctrl-J: Insert execution group below cursor. 
    Ctrl-K: Insert execution group above cursor. 
    Ctrl-T: Switch from executable code mode to text mode (for entering extended formatted comments). 
    Ctrl-M: Switch from text mode to executable code mode. 
    Shift-Enter (or Shift-Return): Begin a new line in the same execution group. 
    Func-3: Split execution group into two (at cursor). 
    Func-4: Join cursor execution group with execution group below.

5.70 How to read code into worksheet?

Use the read command, as in read "mycode.mpl" where mycode.mpl is plain text file that contains maple code

5.71 Code editors for Maple

1. http://www.mapleprimes.com/forum/codeeditormaple
2. http://www.mapleprimes.com/blog/joe-riel/emacs-mode-for-maple
3. http://www.mapleprimes.com/blog/jacquesc/vim-mode-for-maple
4. http://www.maplesoft.com/products/toolboxes/IDE/index.aspx

5.72 How to find if package is module or table?

New packages are module, which allows using packageName:-function() since it is easier. Old packages use tables which needs packageName[function]() which is not common.

To find if package is based on module or not, use the command

           type(combstruct,'`module`');
 

This will return true or false. To know if name is package use the command

          type(combstruct,'package');
 

5.73 How to replace a string?

file_name :=StringTools:-SubstituteAll(file_name,":","-");
 

5.74 How to use geometry and plottools ?

restart; 
c:= i->([i/(1+i),0],1/(1+i)): 
d:= i->([1,1/i],1/i): 
geometry:-circle(c1,[geometry:-point(o,2/3,0),1/3],[x,y]): 
geometry:-circle(c2,[geometry:-point(o,1,1),1],[x,y]): 
geometry:-intersection(o,c1,c2,[u,v]): 
plots:-display(plottools:-circle(c(2)),plottools:-circle(d(1)),geometry:-draw(o));
 

To know more about the intersection, use this:

geometry:-detail(o);
 

5.75 How to simplify log expressions ?

Use symbolic option

restart; 
simplify(ln(3^x/2^y) =ln(n),symbolic);
 

5.76 How to simplify hyperbolic expression ?

How to convert

\[ \frac {3+2\sinh (x)^2}{\sinh (x)^2\tanh (x)} \]

to

\[ 3 \coth ^3(x)-\coth (x) \]

restart; 
e := (3+2*sinh(x)^2)/(sinh(x)^2*tanh(x)); 
expand(student[changevar](sinh(x)^2=tanh(x)^2/(1-tanh(x)^2),e));
 

5.77 How to create text file and append string to it?

restart; 
try 
   fd :=-1; 
   fd := fopen("C:\\output3.txt",APPEND,TEXT); 
catch: 
   print(`Unable to open file, error is`); 
   print(StringTools:-FormatMessage(lastexception[2])); 
end try: 
 
if not(evalb(fd=-1)) then #file open ok 
   str:="hello world"; 
   try 
      fprintf(fd,"%s\n",str); 
   catch: 
      print(`failed to append to file, error is`); 
      print(StringTools:-FormatMessage(lastexception[2])); 
   finally: 
     close(fd); 
   end try; 
fi:
 

5.78 How to search packages and libraries?

To find in which library a command is do

with(LibraryTools); 
FindLibrary('int',all); #find which library command int is in 
 
"C:\Program Files\Maple 18\lib\update.mla", 
"C:\Program Files\Maple 18\lib\DEsAndMathematicalFunctions18.mla", 
"C:\Program Files\Maple 18\lib\maple.mla"
 

To get content of library do

restart; 
with(LibraryTools): 
LibLocation:=cat(kernelopts(mapledir),"/lib/maple.mla"); 
c:=ShowContents(LibLocation);
 

Then can use this to print the name of each symbol/command, and then use whattype command to find its type

seq(c[i,1],i=1..20);
 

To get list of Maple kernel builtin commands and symbols, use this. Written by Acer from Maple prime site:

restart: 
interface(warnlevel=0): 
started := false: 
T := 'T': 
for i from 1 to 1000 do 
  f := eval(parse(cat("proc() option builtin=",i,"; end proc"))); 
  p := (s->StringTools:-Take(s,StringTools:-Search(";",s)-1))(convert(eval(f),string)[26..]); 
  if not type(parse(p),posint) then 
    T[i] := p; 
    started := true; 
  else 
    if started then i:=1000; next; end if; 
  end if; 
end do: 
i; 
[ entries(T,nolist) ]; 
nops(%);
 

The above gives on Maple 18.02 the following

["crinterp", "equation", "`{}`", "even", "debugopts", 
  "embedded_imaginary", "define_external", "embedded_real", 
  "coeff", "cx_zero", "coeffs", "embedded_axis", "conjugate", 
  "constant", "convert", "cx_infinity", "dlclose", "identical", 
  "divide", "hfloat", "`done`", "function", "`$`", "fraction", 
  "denom", "float", "degree", "finite", "disassemble", 
  "extended_rational", "diff", "extended_numeric", "frem", 
  "`union`", "frontend", "upperbound", "exports", "writeto", 
  "factorial", "`xor`", "evalgf1", "type", "expand", "typematch", 
  "entries", "unames", "evalb", "unbind", 
  "`evalf/hypergeom/kernel`", "atomic", "hfarray", "anything", 
  "hastype", "complex", "has", "boolean", "goto", "`:-`", 
  "gmp_isprime", "`!`", "genpoly", "anyfunc", "gc", "algebraic", 
  "SFloatMantissa", "ssystem", "Scale10", "`stop`", "Scale2", 
  "sort", "SearchText", "`[]`", "`~`", "`subset`", "~Array", 
  "subsindets", "~Matrix", "streamcall", "~Vector", "subs", 
  "Unordered", "table", "ToInert", "system", 
  "_hackwareToPointer", "substring", "UpdateSource", "subsop", 
  "_maplet", "trunc", "_jvm", "`kernel/transpose`", "_treeMatch", 
  "tcoeff", "_savelib", "taylor", "abs", "rtable_num_dims", 
  "addressof", "rtable_num_elems", "_unify", "rtable_options", 
  "_xml", "rtable_redim", "`and`", "rtable_scale", "andmap", 
  "rtable_scanblock", "alias", "rtable_size", "anames", 
  "rtable_sort_indices", "assign", "savelib", "assemble", 
  "rtable_zip", "array", "select", "appendto", "searchtext", 
  "cat", "series", "callback", "selectremove", "bind", "sign", 
  "attributes", "setattribute", "ormap", "ArrayOptions", "order", 
  "Array", "parse", "`**`", "overload", "`*`", "`::`", "numer", 
  "CopySign", "numelems", "`^`", "`or`", "`||`", "op", "nops", 
  "seq", "normal", "time", "`not`", "piecewise", "numboccur", 
  "`?[]`", "userinfo", "modp2", "inner", "mods", "timelimit", 
  "mvMultiply", "traperror", "negate", "rtable_normalize_index", 
  "call_external", "rtable_is_zero", "assigned", "rtable_indfns", 
  "evalf", "rtable_histogram", "eval", "evaln", "rtable_eval", 
  "truefalse", "evalhf", "rtable_convolution", "tabular", "mul", 
  "rtableInfo", "zppoly", "`if`", "rtable", "uneval", "remove", 
  "sfloat", "rhs", "specfunc", "readlib", "string", "reduce_opr", 
  "symbol", "ASSERT", "`?()`", "realcons", "TRACE", "`quit`", 
  "relation", "_local", "pointto", "sequential", "add", "print", 
  "set", "SFloatExponent", "iolib", "radical", "SDMPolynom", 
  "`int/series`", "protected", "Record", "irem", "procedure", 
  "Re", "iquo", "poszero", "isqrt", "real_infinity", "RETURN", 
  "is_gmp", "ratpoly", "`+`", "lcoeff", "rational", "OrderedNE", 
  "kernelopts", "range", "Object", "NumericEventHandler", 
  "icontent", "numeric", "NumericStatus", "igcd", "odd", 
  "NumericClass", "ilog10", "nonpositive", "NumericEvent", 
  "ilog2", "nonreal", "`implies`", "posint", "NameSpace", 
  "indets", "positive", "NextAfter", "indices", "polynom", 
  "MPFloat", "`intersect`", "pos_infinity", "MorrBrilCull", 
  "`<`", "member", "neg_infinity", "Im", "maxnorm", "name", 
  "`<>`", "max", "negint", "`<=`", "map2", "negative", "modp1", 
  "nonnegative", "FromInert", "modp", "negzero", 
  "EqualStructure", "`minus`", "nonposint", "`>=`", "min", 
  "nonnegint", "`>`", "DefaultUnderflow", "lexorder", 
  "imaginary", "`=`", "lhs", "indexable", "ERROR", "ldegree", 
  "indexed", "EqualEntries", "length", "integer", "macro", 
  "list", "DEBUG", "map", "literal", "`..`", "lowerbound", 
  "`module`", "Default0", "lprint", "moduledefinition", 
  "DefaultOverflow"] 
                              296 

5.79 How to numerically solve a BVP ode and plot the solution?

This one has one solution

eq:=diff(u(z),z$2)+(k-1)*diff(u(z),z)/z+lambda*exp(u(z))=0; 
sol:=dsolve({subs({k=1,lambda=2},eq),u(0)=1,u(1)=0},numeric,u(z), 
            method=bvp[midrich],'abserr'=0.001); 
plots[odeplot](sol);
 

This solved coupled ODE’s, so there are 2 solutions. Say \(x_1(t)\) and \(x_2(r)\), It is a little tricky to plot all solutions generated, but here is an example

restart; 
R := 0.4; px := 32000; Mm := 0.1; Ds := 9; DO2 := 7.2; YXS := 0.3; KS := 10; 
Sp := 30; Cb := 8; KO2 := 0.2; R0 := 0.000001; YXO := 0.42857; 
Vs := px*1/YXS*(Mm*x2(r))/(KS + x2(r))*x1(r)/(KO2 + x1(r)); 
Vo := px*1/YXO*(Mm*x2(r))/(KS + x2(r))*x1(r)/(KO2 + x1(r)); 
 
eqs := diff(x1(r),r$2) + 2/r*diff(x1(r),r)= Vo/DO2, 
diff(x2(r),r$2) + 2/r* diff(x2(r),r)= Vs/Ds; 
ic:=D(x1)(R0)=0,x1(R) = Cb,D(x2)(R0)= 0, x2(R) = Sp; 
sol:=dsolve({eqs,ic},numeric,{x1(r),x2(r)},'abserr'=.52,'maxmesh'=1000,output=listprocedure);
 

And now to plot do

x1Sol:=rhs(sol[2]); 
plot(x1Sol(r),r=0..0.4); 
 
x2Sol:=rhs(sol[4]); 
plot(x2Sol(r),r=0..0.4);
 

5.80 How to display on screen for specific width?

This below by Axel Vogt posted on sci.math.symbolic which does a nice job of formatting output to specific width.

split_for_print:=proc(expr, len) 
  # expr = some Maple expression 
  # len  = length to split with line breaks 
  local L,s,tmp,j; 
  s:=convert(expr, string); 
  L:=[StringTools:-LengthSplit(s, len)]; 
  for j from 1 to nops(L) do 
  #  if j = nops(L) then printf("%s ;", L[-1]) 
    if j = nops(L) then printf("%s", L[-1]) 
    else printf("%s\\\n", L[j]); 
    end if; 
  end do: 
end proc; 
 
evalf[100](Pi); 
split_for_print(%, 40); 
 
3.14159265358979323846264338327950288419\ 
7169399375105820974944592307816406286208\ 
998628034825342117068
 

5.81 Maple IDE links

for VIM

1. https://code.google.com/p/maplevim/source/browse/trunk/syntax/ma ple.vim

in vim, type set syntax=maple after putting the file maple.vim in ~/.vim/syntax/maple.vim. I found maple.vim in above link.

For Maple IDE

MapleIDE18

5.82 loading, remove and finding what packages loaded

use packages(); to find what packages loaded. use unwith to remove package

packages(); 
                               [] 
 
with(DynamicSystems): 
packages(); 
                        [DynamicSystems] 
 
unwith(DynamicSystems); 
packages(); 
                               []
 

5.83 some rules of thumbs when using Maple

1. put restart in separate execution group
2. do not use with inside proc(). Use uses instead.

5.84 How to make multiple assumptions on a symbol?

assume( A::AndProp(NonZero,constant) );
 

Now can use is(A,constant);

5.85 How to check if expression is an equation?

check for ‘=‘ as follows

eq:= x=1; 
whattype(eq);   #  `=` 
 
if whattype(eq) = `=` then 
   print("yes"); 
else 
   print("no"); 
fi; 
 
      "yes"
 

5.86 How to check if expression is a set?

check for ‘set‘ as follows

eq:= {diff(y(x),x)=1,x(0)=1}; 
 
if whattype(eq) = `set` then 
   print("yes"); 
else 
   print("no"); 
fi; 
                             "yes"
 

5.87 How to export a plot to PDF?

I could only find a way to export to eps

plotsetup(default): 
plotsetup(postscript, plotoutput=`t.eps`, plotoptions=`color,portrait,height=300`); 
plot(sin(x),x=-Pi..Pi,'gridlines'); 
plotsetup(default):
 

Make sure not to put : at the end of the plot command! else it will not be exported. It has to end with ;

This will same it to t.eps in the currentdir() location. Then used ps2pdf t.eps t.pdf to convert it to PDF. Or just ps2pdf t.eps it will automatically create t.pdf

Or ps2pdf -dCompatibilityLevel=1.4  t.eps but may it is best to do

ps2pdf -dCompatibilityLevel=1.4 -dEmbedAllFonts=true  t.eps

Also try adding

-dPDFSETTINGS=/printer

to the above. This tells it to optimize it for printing.

Another example of a direction field for an ODE

plotsetup(postscript, plotoutput=`t0.eps`, plotoptions=`color,portrait, height=300` ); 
ode:= diff(y(x),x) = 3*x^2 - 1; 
DEtools:-DEplot( ode, y(x), x=-2..2, [y(0) = 0], y=-2..2, 
              linecolour=red, color = blue, stepsize=.05,arrows=MEDIUM ); 
plotsetup(default);
 

5.88 How to find all roots of complex number

To find roots of \( (3+4 i)^{1/3}\), do

fsolve(z^3=(3+4*I),z); 
 
#gives 
 
-1.26495290635775+1.15061369838445*I, 
-.363984239564424-1.67078820068900*I, 
1.62893714592218+.520174502304545*I
 

5.89 How to convert matrix of matrices to a matrix?

A:= Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 2}); 
f:=x->`if`(x<>0,x*LinearAlgebra:-IdentityMatrix(2),0*Matrix(2)); 
B:=map(f,A);
 

Which gives

\[ \left [ \begin {array}{cc} \left [ \begin {array}{cc} 0&0 \\ \noalign {\medskip }0&0\end {array} \right ] & \left [ \begin {array} {cc} 0&0\\ \noalign {\medskip }0&0\end {array} \right ] \\ \noalign {\medskip } \left [ \begin {array}{cc} 0&0 \\ \noalign {\medskip }0&0\end {array} \right ] & \left [ \begin {array} {cc} 2&0\\ \noalign {\medskip }0&2\end {array} \right ] \end {array} \right ] \]

now

r:=Matrix(convert(B,listlist))
 

Gives

\[ \left [ \begin {array}{cccc} 0&0&0&0\\ \noalign {\medskip }0&0&0&0 \\ \noalign {\medskip }0&0&2&0\\ \noalign {\medskip }0&0&0&2\end {array} \right ] \]

5.90 How to do pattern matching in Maple?

Maple has a simple but easy to use pattern matching, which works well. Here are some example. For each case, will show what pattern to detect and how to do it. I am still not very good at pattern matching in Maple and will need to make improvement in this with time.

5.91 Example 1

Detect \(\sqrt {x y}\) in expression.

restart; 
expr:= sin(x)*sqrt(x*y); 
if patmatch(expr,a::anything*(b::anything*x*y)^(c::anything),'la') then 
    assign(la); 
    if c =1/2 or c=-1/2 then 
       print("found sqrt(x*y)"); 
    else 
       print("did not find sqrt(x*y)"); 
    fi; 
 fi;
 

But if the expression was \(\sin (x)\sqrt {x y}+3\) then the above would fail, because there are a term after \(\sqrt {x y}\), so the pattern has to change to

restart; 
expr:= sin(x)*sqrt(x*y)+3; 
if patmatch(expr,a::anything*(b::anything*x*y)^(c::anything)+d::anything,'la') then 
    assign(la); 
    if c =1/2 or c=-1/2 then 
       print("found sqrt(x*y)"); 
    else 
       print("did not find sqrt(x*y)"); 
    fi; 
 fi;
 

5.92 Example 2

There was a case where I wanted to detect form \(f(x) g(\frac {y}{x})\), i.e. \(f(x)\) can be any expression which is function of \(x\) only (it can be constant also) multiplied by a function whose argument must be \(\frac {y}{x}\) or a constant multiplied by \(\frac {y}{x}\).

This means something like \(x g(\frac {y}{x})\) or \(x^2 e^{3 \frac {y}{x}}\) or \(f(x) \sin {\frac {y}{x}}\) or \(\cos {\frac {y}{x}}\) where in this last case \(f(x)=1\) which is allowed.

TO FINISH.

5.93 How to find trig indetities?

use trigsubs, very useful command. For example

trigsubs(cos(theta)^3)
 

Gives

\[ [1/2\,\cos \left ( \theta \right ) +1/2\,\cos \left ( 2\,\theta \right ) \cos \left ( \theta \right ) ,1/4\,\cos \left ( 3\,\theta \right ) +3/4\,\cos \left ( \theta \right ) ] \]

5.94 How to find directional derivative of scalar function?

Given \(f(x,y,z)=x^2 z+y^3 z^2-xyz\) we want to find its directional derivative along the vector \(n\).

One way

n:=<-1,0,3>; 
g:=VectorCalculus[Gradient](x^2*z+y^3*z^2-x*y*z, [x,y,z]); 
Student[VectorCalculus][DotProduct](g,n/LinearAlgebra[Norm](n,2))
 

Gives

\[ -{\frac { \left ( 2\,xz-yz \right ) \sqrt {10}}{10}}+{\frac { \left ( 6\,{y}^{3}z+3\,{x}^{2}-3\,xy \right ) \sqrt {10}}{10}} \]

Another is

Student[MultivariateCalculus][DirectionalDerivative](x^2*z+y^3*z^2-x*y*z, [x,y,z], [-1,0,3]);
 

Gives the same result.

5.95 How to check if name is assigned a value?

For simple variable, use assigned

restart; 
x:=10: 
assigned(x) 
                              true 
assigned(y) 
                             false
 

For a field in table do

restart; 
A:=table(["x"=10,"y"=20]): 
 
assigned(A["x"]) 
                              true 
assigned(A["z"]) 
                             false
 

For field in Record, I do not know how yet, other than using try catch, as assigned does not seem to work for Record fields.

restart; 
A:=Record('x'=10,'y'=20); 
try 
   assigned(A:-x) 
catch: 
   print("no such field in record") 
end try; 
 
                              true 
 
try 
   assigned(A:-z) 
catch: 
   print("no such field in record") 
end try; 
 
                   "no such field in record"
 

5.96 How to simplify \(e^{\ln (x)+\ln (y)}\)

given

\[ {\mathrm e}^{\frac {2 \ln \left (\sqrt {p^{2}+1}+p \right )+2 \ln \left (a \right )+\ln \left (p^{2}+1\right ) a}{2 a}}+{\mathrm e}^{3 x} \]

simplify(expr) does not work. So tried subsindets

restart; 
expr := exp((2*ln(sqrt(p^2 + 1) + p) + 2*ln(a) + ln(p^2 + 1)*a)/(2*a))+ exp(3*x); 
subsindets(expr,'specfunc( anything, exp )',f->(`if`(has(op(1,f),'ln'),expand(f),f)))
 

\[ \left (\sqrt {p^{2}+1}+p \right )^{\frac {1}{a}} a^{\frac {1}{a}} \sqrt {p^{2}+1}+{\mathrm e}^{3 x} \]

It is possible to also try simplify(expr,exp) in some cases, but for the above example, this did not work, i.e. it did not simplify it.

Update december 2023. Trying Maple 2023.2.1, it simplifies the above using simplify(expr,exp)

restart; 
expr := exp((2*ln(sqrt(p^2 + 1) + p) + 2*ln(a) + ln(p^2 + 1)*a)/(2*a))+ exp(3*x); 
simplify(expr,exp)
 

\[ sqrt{p^{2}+1}\, \left (\sqrt {p^{2}+1}+p \right )^{\frac {1}{a}} a^{\frac {1}{a}}+{\mathrm e}^{3 x} \]

And

restart; 
expr:=exp(ln(x)+ln(y)); 
simplify(expr)
 

\[ x y \]

5.97 Basis for Null space, Row space and column space of matrix

Given

\[ \left [\begin {array}{cccc}1 & -1 & 0 & 2 \\1 & 2 & 2 & -2 \\0 & 2 & 3 & -1 \end {array}\right ] \]

Find its Null, Row and Column space basis vectors.

restart; 
A:=Matrix([[1,-1,0,2],[1,2,2,-2],[0,2,3,-1]]); 
LinearAlgebra:-NullSpace(A)
 

\[ \left \{ \left [\begin {array}{c}0 \\2 \\-1 \\1 \end {array}\right ] \right \} \]

restart; 
A:=Matrix([[1,-1,0,2],[1,2,2,-2],[0,2,3,-1]]); 
LinearAlgebra:-RowSpace(A)
 

\[ \left [\left [\begin {array}{cccc}1 & 0 & 0 & 0 \end {array}\right ], \left [\begin {array}{cccc}0 & 1 & 0 & -2 \end {array}\right ], \left [\begin {array}{cccc}0 & 0 & 1 & 1 \end {array}\right ]\right ] \]

restart; 
A:=Matrix([[1,-1,0,2],[1,2,2,-2],[0,2,3,-1]]); 
LinearAlgebra:-ColumnSpace(A)
 

\[ \left [\left [\begin {array}{c}1 \\0 \\0 \end {array}\right ], \left [\begin {array}{c}0 \\1 \\0 \end {array}\right ], \left [\begin {array}{c}0 \\0 \\1 \end {array}\right ]\right ] \]

5.98 How to do Gaussian elimination on a Matrix?

Given

\[ \left [\begin {array}{cccc}1 & -4 & -3 & -7 \\2 & -1 & 1 & 7 \\1 & 2 & 3 & 11 \end {array}\right ] \]

Find the new form after Gaussian elimination

restart; 
A:=Matrix([[1,-4,-3,-7],[2,-1,1,7],[1,2,3,11]]); 
LinearAlgebra:-GaussianElimination(A);
 

\[ \left [\begin {array}{cccc}1 & -4 & -3 & -7 \\2 & -1 & 1 & 7 \\1 & 2 & 3 & 11 \end {array}\right ] \]

5.99 How to find Reduced Echelon form of a Matrix?

Given matrix

\[ \left [\begin {array}{ccc}5 & 2 & 18 \\0 & 1 & 4 \\4 & 1 & 12 \end {array}\right ] \]

Find its Reduced Echelon form.

restart; 
A:=Matrix([[5,2,18],[0,1,4],[4,1,12]]); 
Student:-LinearAlgebra:-ReducedRowEchelonForm(A)
 

\[ \left [\begin {array}{ccc}1 & 0 & 2 \\0 & 1 & 4 \\0 & 0 & 0 \end {array}\right ] \]

Another option is

restart; 
A:=Matrix([[5,2,18],[0,1,4],[4,1,12]]); 
MTM:-rref(A)
 

\[ \left [\begin {array}{ccc}1 & 0 & 2 \\0 & 1 & 4 \\0 & 0 & 0 \end {array}\right ] \]

5.100 How add a new row to bottom of matrix?

Given matrix

\[ \left [\begin {array}{cc}1 & 1 \\2 & 3 \\4 & 5 \end {array}\right ] \]

How to add row

\[ [a, b] \]

to end of the matrix?

restart; 
A:=Matrix([[1,1],[2,3],[4,5]]); 
the_row:=convert([a,b],Vector['row']); 
ArrayTools:-Concatenate(1,A,the_row);
                                                                                    
                                                                                    
 

\[ \left [\begin {array}{cc}1 & 1 \\2 & 3 \\4 & 5 \\a & b \end {array}\right ] \]

5.101 How to find the cofactor matrix of a matrix?

Use LinearAlgebra:-Adjoint and then transpose the result. Since the Adjoint is the transpose of the cofactor.

Given

\[ \left [\begin {array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end {array}\right ] \]

then

restart; 
A:=Matrix([[1,2,3],[4,5,6],[7,8,10]]); 
LinearAlgebra:-Transpose(LinearAlgebra:-Adjoint(A))
 

\[ \left [\begin {array}{ccc} 2 & 2 & -3 \\ 4 & -11 & 6 \\ -3 & 6 & -3 \end {array}\right ] \]

5.102 How to normalize eigenvectors?

When finding eigenvectors of matrix, using LinearAlgebra, the vectors are not normalized. How to normalized them so the length is one?

One way is

restart; 
LA:=LinearAlgebra; 
Sx:=Matrix([[0,1,0],[1,0,1],[0,1,0]]); 
 
#this finds eigenvectors in v 
lam,v:=LA:-Eigenvectors(Sx); 
 
#this normalize it 
B:=map(n -> v[.., n]/norm(v[.., n], 2), [$1..LA:-RowDimension(v)]): 
B:=`<|>`(op(B)); #this converts the list back to matrix.
 

\[ v=\left [\begin {array}{ccc} -1 & 1 & 1 \\ 0 & \sqrt {2} & -\sqrt {2} \\ 1 & 1 & 1 \end {array}\right ] \]

\[ B=\left [\begin {array}{ccc} -\frac {\sqrt {2}}{2} & \frac {1}{2} & \frac {1}{2} \\ 0 & \frac {\sqrt {2}}{2} & -\frac {\sqrt {2}}{2} \\ \frac {\sqrt {2}}{2} & \frac {1}{2} & \frac {1}{2} \end {array}\right ] \]

5.103 How to typeset \(\hslash \)?

expr:=`ℏ`*x
 

gives

\[ \hslash x \]

Notice, the ; is needed. This `&hbar`*x will not work. It must be `ℏ`*x

5.104 How to find the Curl of a vector?

First example

restart; 
VectorCalculus:-SetCoordinates( 'cartesian'[x,y,z] ); 
F:=VectorCalculus:-VectorField(<y,-x,0>);
 

\begin{align*} F &= y \mathbf {\bar {e}_{x}}-x \mathbf {\bar {e}_{y}} \end{align*}

And now

VectorCalculus:-Curl(F);
 

\[ -2 \mathbf {\bar {e}_{z}} \]

Second example

restart; 
VectorCalculus:-SetCoordinates( 'cartesian'[x,y,z] ); 
F:=VectorCalculus:-VectorField(<y*z^2,x*z^2+2,2*x*y*z-1>);
 

\begin{align*} F &= y \,z^{2} \mathbf {\bar {e}_{x}}+(x \,z^{2}+2) \mathbf {\bar {e}_{y}}+(2 x y z -1) \mathbf {\bar {e}_{z}} \end{align*}

And now

VectorCalculus:-Curl(F);
 

\[ 0 \]

Since Curl is zero, field is conservative.

Third example, in cylinderical coodinates

restart; 
VectorCalculus:-SetCoordinates( 'cylindrical'[rho,phi,z] ); 
F:=VectorCalculus:-VectorField(<0,-rho,2>);
 

\begin{align*} F &= -\rho \mathbf {\bar {e}_{\phi }}+2 \mathbf {\bar {e}_{z}} \end{align*}

And now

VectorCalculus:-Curl(F);
 

\[ 2 \mathbf {\bar {e}_{z}} \]

5.105 See all steps in RREF of an augmented matrix

Use  Student:-LinearAlgebra:-GaussJordanEliminationTutor( A, output=steps )  Where \(A\) is your augmented matrix.

5.106 How to find column space of matrix?

Do not use the Maple command LinearAlgebra:-ColumnSpace for this. it gives the columns in the RREF. The correct way is to obtain the corresponding columns of the pivot columns in the original matrix \(A\). Hence use the command Basis like this

A:=Matrix([[1,0,0],[1,1,1]]); 
LinearAlgebra:-Basis([seq(A[..,i],i=1..LinearAlgebra:-ColumnDimension(A) )]);
 

Which gives

\[ \left [\left [\begin {array}{c} 1 \\ 1 \end {array}\right ], \left [\begin {array}{c} 0 \\ 1 \end {array}\right ]\right ] \]

If you use ColumnSpace command you’ll get this

A:=Matrix([[1,0,0],[1,1,1]]); 
LinearAlgebra:-ColumnSpace(A);
 

\[ \left [\left [\begin {array}{c} 1 \\ 0 \end {array}\right ], \left [\begin {array}{c} 0 \\ 1 \end {array}\right ]\right ] \]

These are different. Basis is the correct command to use, which matches the standard definition in textbooks.

5.107 Show step by step. Calculus problem and differential equations.

For integration do

Student:-Calculus1:-ShowSolution(Int(x*sin(x),x));
 

The steps are displayed. This does not work all the time. For example

integrand:=x*y(x)*diff(y(x),x$2)+x*(diff(y(x),x))^2-y(x)*diff(y(x),x); 
Student:-Calculus1:-ShowSolution(Int(integrand,x));
                                                                                    
                                                                                    
 

gives

Error, (in Student:-Calculus1:-ShowSolution) unable to determine which calculus operation is being applied in this problem; you can provide this information as the 2nd argument on your call to Rule or Hint

For differential equations, support is limited but these are the steps

restart; 
ode:=diff(y(x),x)=sin(x); 
Student:-ODEs:-ODESteps(ode)
 

Prints the steps. If IC is there, then

restart; 
ode:=diff(y(x),x)=sin(x); 
ic:=y(0)=1; 
Student:-ODEs:-ODESteps([ode,ic])
 

5.108 How to obtain list of files with some extension in folder?

Use FileTools:-ListDirectory

dir_name:="C:/tmp"; 
currentdir(dir_name); #cd to directory 
files_to_process := FileTools:-ListDirectory(dir_name,'all','returnonly'="*.tex"): 
numelems(files_to_process) 
    100
 

In the above, files_to_process is a list of the files in the current folder with extension .tex

5.109 How to delete lines from text file that contains some string?

There was a case when I needed to delete lines from text file that contains a say "foo" as an example.

This is what I did. use readline to read the lines, check, and if the line contains "foo" skip, else write the line to a temporary file. At the line, use Rename to rename the temporary file to the file being read.

dir_name:="C:/tmp"; 
currentdir(dir_name); 
 
tmp_file_name      := "TMP.txt"; 
source_file_name   := "source.txt"; 
file_id            := fopen(tmp_file_name,WRITE): 
line               := readline(source_file_name): 
 
while line<>0 do 
 
   if not StringTools:-Has(line,"foo") then 
      fprintf(file_id,"%s\n",line); 
   fi; 
 
   line := readline(source_file_name): 
od: 
 
fclose(file_id); 
FileTools:-Rename(tmp_file_name,source_file_name,force=true);
 

5.110 How to truncate a polynomial?

Given \(9 x^{5}+4 x^{4}+3 x^{3}+x^{2}+x +1\) how to truncate it, so that all terms of \(x^3\) and higher are removed?

This can be done as follows

restart; 
p:=1+x+x^2+3*x^3+4*x^4+9*x^5; 
simplify(p,{x^3=0})
 

\[ x^{2}+x +1 \]

5.111 How to make a local declare like block inside a proc?

Sometimes it is useful to make a small local piece of code inside a proc, with its own local variables that do not interfer with the variables of the proc. In Ada, this is done using declare clause for example. In Maple on can do the same as follows

restart; 
 
foo:=proc() 
   local n; 
   n:=10; 
   proc() 
      local n; 
      n:=99; 
      print("inside inner proc, n=",n); 
   end proc(); 
   print("n=",n); 
end proc; 
 
foo();
 

Which prints

                  "inside inner proc, n=", 99 
                            "n=", 10
 

Notice the end of the inner anonymous proc above. It has end proc(); and not end proc; as normal proc. This defines the inner proc and calls it at same time. All the local variables inside the anonymous proc only exist inside that proc and go away after the call, and they do not interfer with the outer proc variables. This way we can declare temporary variables and use them where they are needed.

5.112 Using short name for a proc inside nested modules intead of long name

There was a case where I was making lots of calls from many places to pne specific proc inside a module. I did not want to keep using the long name each time.

the command alias did not work. After some trial and error, found that using use works. Here is the solution. First this is the original layout

restart; 
 
A:=module() 
   export B:=module() 
        export foo:=proc() 
            print("in A:-B:-foo()"); 
        end proc; 
   end module; 
 
   export C:=module() 
      export boo:=proc() 
              print("in A:-C:-boo()"); 
              A:-B:-foo(); 
      end proc; 
   end module; 
end module;
 

In the above, the goal is replace A:-B:-foo(); with just foo() and have it bind to A:-B:-foo(); automatically.

This is done by modifying the above to

restart; 
A:=module() 
   export B:=module() 
        export foo:=proc() 
            print("in A:-B:-foo()"); 
        end proc; 
   end module; 
 
   use foo=A:-B:-foo in #add this line here 
   export C:=module() 
      export boo:=proc() 
              print("in A:-C:-boo()"); 
              foo(); #now can just use the short name 
      end proc; 
   end module; 
   end use; #add this line here. 
end module;
 

Wrapping the whole module where the short name is used worked.

Any module that needs to use the short name, can do the same. This solved the problem.

5.113 Remove duplicates objects in a list based on condition on a field

I had case where there is list of Objects, and wanted to removed duplicate entries in the list based on if some field is the same among the objects.

This can be done using the command ListTools:-MakeUnique and using a proc which checks for the condition. In this example, we want to remove objects where the field age in each object is the same.

restart; 
 
#create data type 
module person_type() 
    option object; 
    export name::string:="me"; 
    export age:=25; 
end module: 
 
#make few objects 
o1:=Object(person_type); 
o2:=Object(person_type); 
o3:=Object(person_type); 
o3:-age:=46; 
o4:=Object(person_type); 
 
#make list of them 
list_of_people:=[o1,o2,o3,o4]; 
 
nops(list_of_people);  #this will print 4 
 
#now delete object if age is same 
list_of_people:=ListTools:-MakeUnique( 
            list_of_people, 
            1, 
            proc(a,b) 
                evalb(a:-age=b:-age); 
            end proc 
        ); 
 
nops(list_of_people); #this will print 2
 

5.114 How to remove duplicates Vectors from a list?

Converting a list of Vectors to set will not remove duplicates, as each Vector occupies different memory address, even if the structure is the same. To remove duplicate vector, use ListTools:-MakeUnique as follows

restart; 
 
my_list:=[Vector([1,0]),Vector([1,0]),Vector([2,0])]; 
convert(my_list,set); #this will still show the 3 vectors. 
 
ListTools:-MakeUnique(my_list,1,proc(a,b) LinearAlgebra:-Equal(a,b) end proc) 
 
#now only 2 vectors will remain. Duplicate one was removed
 

5.115 How to find all poles and their order of a rational function?

Gives a rational function in \(x\), such as

\[ \frac {1}{10 \left (x -4\right ) \left (x -5\right )^{3}} \]

How to find all its poles which are \(x=4\) and \(x=5\) and the order of each pole which will be \(1\) and \(3\) in this example?

Using  sqrfree as follows

restart; 
get_poles_and_order:=proc(r_in,x::symbol)::list; 
  local r:=r_in,N::posint; 
  local the_poles::list; 
  local item; 
 
  r:=normal(r); 
  if not type(r,'ratpoly'(anything,x)) then 
     error("Not be a polynomial or a rational function in ",x) 
  fi; 
 
  the_poles := sqrfree(denom(r),x); 
  the_poles := the_poles[2,..]; #we do not need the overall factor 
  for N,item in the_poles do 
      the_poles[N]:=[solve(item[1]=0,x),item[2]]; 
  od; 
  return the_poles; 
 
end proc:
 

The above proc get_poles_and_order returns back a list of lists. Each sublist has its first entry the pole and the second entry the order.

Here are some examples

r:=1/(10*(x-4)*(x-5)^3); 
get_poles_and_order(r,x) 
 
     #[[4, 1], [5, 3]]
 

The above says there is a pole at \(x=4\) of order 1 and pole at \(x=5\) of order 3.

5.116 find series of function with specific number of terms

Doing series in Maple with specific order value, the number of terms generated ofcourse depends on the function. I had need to have the series generated always with same number of terms. I could not find an option in Maple to do that. This function does this. It keeps finding the series for the function with increasing order until the number terms that comes out is what requested. There is an upper limit that can be changed if needed to protect against pathological cases.

restart; 
 
get_series_by_terms:=proc(expr,x::symbol,at::numeric,number_terms_needed::posint) 
local keep_running::boolean:=true; 
local current_order::integer:=0; 
local MAX_ORDER_TO_TRY::posint:=100; #change as needed 
local result; 
 
    do 
        current_order := current_order+1; 
        result        := convert(series(expr,x=at,current_order),polynom); 
        if nops(result) >=  number_terms_needed or current_order>MAX_ORDER_TO_TRY then 
           keep_running:=false; 
        fi; 
    until keep_running=false; 
 
    return result; 
end proc:
 

And now

get_series_by_terms(sin(x),x,0,10)
 

returns

\begin{equation} \begin {aligned} & x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}-\frac {1}{5040} x^{7}+ \frac {1}{362880} x^{9} -\\ & \frac {1}{39916800} x^{11}+\frac {1}{6227020800} x^{13}-\frac {1}{1307674368000} x^{15}+\\ & \frac {1}{355687428096000} x^{17}-\frac {1}{121645100408832000} x^{19} \end {aligned} \end{equation}

5.117 How to call sibling’s proc without making the sibling module exported?

Given a parent module \(A\) and inside it there are two child modules (local modules) with names say \(B\) and \(C\). To call a proc foo inside \(B\) from another proc inside \(C\), the proc foo has to be exported. But the module \(B\) does not have to be exported, if we make sure to use B:-foo() call instead of full name A:-B:-foo() call.

So make sure to use child:-proc() from other sibilings to avoid having to make each child exported. Making children exported means they can be seen and called directly from outside the parent which his not what we want.

Here is an example

restart; 
 
A:=module()  #parent 
 
  export main:=proc() 
     C:-foo(); 
  end proc; 
 
  local B:=module()  #child 
     export foo:=proc() 
        print("in A:-B:-foo() proc"); 
     end proc; 
  end module; 
 
  local C:=module() #child 
     export foo:=proc() 
        print("in A:-C:-foo(). About to call A:-B:-foo()"); 
        B:-foo();  #do this and NOT A:-B:-foo() 
     end proc; 
  end module; 
end module;
 

and now

A:-main() 
     "in A:-C:-foo(). About to call A:-B:-foo()" 
     "in A:-B:-foo() proc"
 

If instead we have written  A:-B:-foo() in the above call, then Maple will complain with the error Error, (in foo) module does not export `B`

5.118 Convert time to use seconds instead of milliseconds

Maple’s command Value(Time()) returns 13 digits number, which is number of milliseconds from epoch. I wanted this value to be in seconds, to match the file changed time from  FileTools[Status]("A.txt" ) which uses seconds and not milliseconds. I could not find an option to tell Date or Time to do this. Here is one way to do this.

r:=Value(Time());  #r := 1652677498870 
length(r);   #13 
r:=convert(r, base, 10); 
r:=ListTools:-Reverse(r); 
r:=r[1..-4]; #remove last 3 digits 
nops(r); 
r:=parse(cat(op(r)))  #r := 1652677498 
length(r); #10
 

This can be made into a function

get_time_in_seconds:=proc()::integer; 
local r; 
r:=Value(Time()); 
r:=convert(r, base, 10); 
r:=ListTools:-Reverse(r); 
r:=r[1..-4]; 
r:=parse(cat(op(r))); 
return r; 
end proc; 
 
get_time_in_seconds()  #1652679222
 

5.119 Change the summation index letter

I noticed that Maple returns the summation index variable using leading underscore as in _n or _m which makes the latex looks not as good. Here is an example

restart; 
dsolve(diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x),'formal_series'); 
 
y(x) = _C1*Sum((-1/2 - sqrt(3)*I/2)^_n*x^_n/_n!, _n = 0 .. infinity) + 
       _C2*Sum((-1/2 + sqrt(3)*I/2)^_n*x^_n/_n!, _n = 0 .. infinity)
 

The latex of the above is

\[ y \left (x \right ) = \textit {\_C1} \left (\sum _{\textit {\_n} =0}^{\infty }\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{\textit {\_n}} x^{\textit {\_n}}}{\textit {\_n} !}\right )+\textit {\_C2} \left (\sum _{\textit {\_n} =0}^{\infty }\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{\textit {\_n}} x^{\textit {\_n}}}{\textit {\_n} !}\right ) \]

Not seeing an option to change _n to n, I wrote the following function which takes in the solution, use subsindets and remove the leading underscore.

This is the above example showing how to use the function

restart; 
 
fix_summation_index:=proc(expr) 
local fix_it:=proc(the_sum) 
  local the_letter::symbol,the_new_letter::symbol,the_letter_as_string::string; 
  the_letter:= op([2,1],the_sum); 
  the_letter_as_string:=String(the_letter); 
  if the_letter_as_string[1]="_" then 
     the_new_letter:=parse(the_letter_as_string[2..]); 
     RETURN(subs(the_letter=the_new_letter,the_sum)); 
  else 
     RETURN(the_sum); 
  fi; 
end proc; 
 
if not(has(expr,Sum)) then 
   RETURN(expr); 
else 
   RETURN(subsindets( expr, 'specfunc( anything, Sum )', f->fix_it(f))); 
fi; 
 
end proc; 
 
sol:=dsolve(diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x),'formal_series'): 
sol:=fix_summation_index(sol); 
 
y(x) = _C1*Sum((-1/2 - sqrt(3)*I/2)^n*x^n/n!, n = 0 .. infinity) + 
       _C2*Sum((-1/2 + sqrt(3)*I/2)^n*x^n/n!, n = 0 .. infinity)
 

The latex now is

\[ y \left (x \right ) = \textit {\_C1} \left (\sum _{n =0}^{\infty }\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{n} x^{n}}{n !}\right )+\textit {\_C2} \left (\sum _{n =0}^{\infty }\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{n} x^{n}}{n !}\right ) \]