Euler Math Toolbox 2022-05-18

This version involves some internal changes which make calling a function or finding a global variable slightly more efficient. The main effect is that global variables are now mutable.

>function test ...
global a;
a=1:10;
endfunction
In the function, the variable must still be declared as global.

>a=1:4; test; a,
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
It is also possible to generate global variables.

>function test ...
global b=1:10;
endfunction
>test; b,
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Note that variables passed by reference cannot be changed in type or size even if they are global.

>function test (%x) ...
%x=1:5;
endfunction
>a=5; test(a)
Cannot change type of non-local variable a!
test:
%x=1:5;
Fixes

It is now more difficult to forget "return" in functions that are meant to return a value.

>function f(x) ...
if x<0 then return -1
elseif x~=0 then return 0
else 1
endif;
endfunction
That seems to work, because 1 prints.

>f(6)
1
But in fact the function returns a special version of "none".

>a=f(6)
1
A function returned no value. Cannot assign this to a variable.
Error in:
a=f(6) ...
^
Note that it is possible to vectorize functions with no return value.

>function map test(x) ...
"output : "+x,
endfunction
>test(1:5)
output : 1
output : 2
output : 3
output : 4
output : 5
But they too return the special "no result", which cannot be assinged to a variable.

>v=test(1)
output : 1
A function returned no value. Cannot assign this to a variable.
Error in:
v=test(1) ...

EMT can now include LaTeX formulas in the plot if LaTeX is installed. The function latex(string) will parse the string through latex, generate a PNG file and return the image as an RGB matrix. This can be used instead of a simple string in labels and titles.

>plot2d("exp(-x)*sin(x)",0,2pi,color=blue, ...
xl=latex("\phi"),yl=latex("\Psi(\phi)"), ...
title=latex("\text{Function $\Psi$}")); ...
textbox(latex("\Psi(\phi) = e^{-\phi}\,\sin\phi"),x=0.8,y=0.5); ...
label(latex("\Psi",color=blue),2,0.15):
Changes

Large Matrices

EMT will no longer bore you with the print of large matrices. It will show the matrix size and type and eclipses.

>id(10)
Real 10 x 10 matrix

1 0 0 0 ...
0 1 0 0 ...
0 0 1 0 ...
0 0 0 1 ...
0 0 0 0 ...
0 0 0 0 ...
0 0 0 0 ...
0 0 0 0 ...
0 0 0 0 ...
0 0 0 0 ...
To see the a full matrix, you can use

showlarge A
To toggle seeing all matrices in full size, enter

largematrices on
largematrices off
Tracing

You can put a "trace on" everywhere. This will enable tracing. If you leave tracing with cursor-left, evaluation will continue until "trace on" is met again.

Now, there is "traceif ...". This will conditionally break execution. E.g, the following example will begin tracing, if |x| gets too large.

>function f(x) ...
repeat
xnew=1+x^2/5;
until xnew~=x;
traceif abs(x)>1000;
x=xnew;
end;
return xnew;
endfunction
If you call it wit start values, where the iteration converges, you see no tracing. Otherwise, execution would stop as if "trace on" is called.

>f(2)
1.38196601125

I added some useful functions to handle matrices.

>seed(2); A=random(4,2)
0.765761 0.401188
0.406347 0.267829
0.13673 0.390567
0.495975 0.952814
The function mnonzeros() works like nonzeros() but for matrices, not just for vectors. It returns a list of indices.

>k=mnonzeros(A<0.5)
1 2
2 1
2 2
3 1
3 2
4 1
You can use these indices to get a vector of corresponding values from A.

>v=mget(A,k)
[0.401188, 0.406347, 0.267829, 0.13673, 0.390567, 0.495975]
Or you can set the elements at these indices to a specific value.

>mset(A,k,0)
0.765761 0
0 0
0 0
0 0.952814
Another matrix can also be used.

>mset(A,k,-A)
0.765761 -0.401188
-0.406347 -0.267829
-0.13673 -0.390567
-0.495975 0.952814
This might also be useful to construct thin matrices from full matrices. The function cpxset() also uses a list of indices and values. We can construct such a list with mnonzeros() and mget().

>A=random(100,100); k=mnonzeros(A<0.001); ...
cpxset(cpxzeros(100,100),k|mget(A,k)')
Compressed 100x100 matrix
6 5 0.000279092
20 46 0.000508132
32 60 4.08995e-005
81 11 0.000347999
81 51 0.000129522
82 84 0.000923609
88 93 0.000575104
94 50 0.00045528
3D Plots

The functions plot3d() and plot2d() got some updates. First of all, there is a new plot style in plot3d() a contour plane (cp).

>plot3d("x^2+y^2",>spectral,>contour,>cp,n=100):
Changes

As you see the default style is for level lines is now a vector of level ranges.

To get the previous look, you can use the flag >levels or level="thin". This thin style for level lines will also be applied if level is a vector of values. If level is a 2xn matrix of values, or the parameter dl is not 0, level ranges will be shown.

The examples does also show the style cp=2, which shows only level lines.

>plot3d("x^2+x*y",>levels,cp=2,cpdelta=0.3,frame=3,color=green):

Changes

You can use a contour plane for all 3D plots, where it makes sense. There are some parameters for this style of the plot.

>plot3d("x^2+x*y",>cp,cpdelta=0.2,cplevel=-1:0.2:2,cpcolor=green
):
Changes

Likewise the default level style of implicit 2D plots is now a vector of ranges. The following plot shows also the new grid style 9.

>plot2d("x^2+2*y^2-x*y",>contour,contourcolor=blue,>hue,grid=
9,n=100):
Changes

EMT does not plot true 3D scenes. It can currently only show one triangle mesh with hidden surfaces. However, you can hold the plot to add another plot. To make this easier, I enabled the >add parameter for plot3d().

Here is an example. We wish to plot a random path.

>seed(2); x=random(20); y=random(20); z=random(20);
Then we need a plot function, which does all the plotting.

>function plotxyz (x,y,z) ...
plot3d(x,y,z,>wire,<frame,wirecolor=red);
plot3d(x,y,z,>points,style="o#",color=red,<frame,>add);
endfunction
This would plot without a frame.

>plotxyz(x,y,z):
Changes

Now, we can use framedplot() to get a frame.

>function myplot (x,y,z) ...
framedplot("plotxyz",[-1,1,-1,1,-1,1];x,y,z);
endfunction
>myplot(x,y,z):
Changes

Then we can use plot3d() with >user or >anaglyph.

>plot3d("myplot";x,y,z,>own,>anaglyph):
Changes

Default Parameters

Often, a parameter is handled with the same name to a function where it has a default value by an assignment. There is now a special shorter syntax for this.

>function test(f$,color=black) := plot2d(f$,=color);
Here "=color" behaves like a shortcut for "color=color". (Internally, it is handled more efficiently.)

>test("x^3-x",red):

There are new grid style 7 and 8.

>load over plot; reset; ...
figure(3,3); ...
loop 1 to 8; figure(#); plot2d("x^3-x",r=1.2,grid=#); end; ...
figure(0):
Changes

>load over plot; reset;
The ticks depend on the current aspect.

>aspect(3/2); plot2d("sin(x)",-2pi,2pi,grid=8,<frame,>insimg); aspect();
Changes

Draw Package

Maxima has a package "draw". For this, you have to set the output format correctly.

>&load(draw); mxmsetdraw(600,400); ...
&draw2d(explicit(x^4-x,x,-1,1)):

The date and time functions got some extensions. The function day() can now recognize day and time in ISO 6801 format.

>day("2014-01-01 12:00")
735664.5
The time string can be omitted. Moreover, the function understands english names of months. (You can provide your own names by changing the global variable englishmonths$).

>day("2014-Jul-02")-day("2014-jan-02")
181
Moreover, the function understands an alternative format for the date. The printdate() function got the parameter >time and <german.

>printdate(date(day("1.1.2015")),<time,>german)
01.01.2015
The readtable() function can now detect dates in files. It uses the day() for this. The writetable() function can print dates.

We demonstrate this with a file in the user directory.

>T=[day(2014,1,1),435;day(2014,2,1),440]
735664 435
735695 440
We print our small table to test.dat in the user home directory.

>filename=userhome()+"test.csv"; ...
writetable(T,date=[1],separator=",",file=filename); ...
printfile(filename)
2014-01-01,435
2014-02-01,440

Now we can read the file back.

>readtable(filename,<clabs,separator=",",date=[1])
735664 435
735695 440
You could open the file with Excel unless you have a German system, which expects a decimal comma and semicolons.

>writetable(T,date=[1],separator=";",file=filename,>comma); ...
printfile(filename)
2014-01-01;435
2014-02-01;440

Let us remove this file.

>fileremove(filename);
The function writematrix() can also have another separator and a decimal comma.

>writematrix(random(5,3),>comma,separator=";")
0,6554163483485531;0,2009951854518572;0,8936223876466523
0,2818865431288053;0,5250003829714993;0,3141267749950177
0,4446156782993733;0,2994744556282315;0,2826898577756425
0,8832274852666707;0,2709059757306278;0,7044190158488305
0,217693385437102;0,4453627564273003;0,3084110353211584

There is a new output format with separated groups of numbers. For this, the function print() got some more parameters. One of them is "sep", which defines the separator string.

>print(123456789.1234,2,sep="'")
123'456'789.12
It is possible to use a hard space for this with sep=true (>sep).

Let us define an operator, which prints an amount of money with a decimal comma. I am not using the Euro character euro$ here, since it will not be present on all machines.

>function prefix money (x) := print(x,2,12,>sep,dot=",",unit=" Euro"); ...
money 12345.678
12 345,68 Euro
We can set this output format for all further outputs.

>userformat("money");
Then we can compute and print a 3% growth of a seed capital.

>1000*1.03^(0:10)'
1 000,00 Euro
1 030,00 Euro
1 060,90 Euro
1 092,73 Euro
1 125,51 Euro
1 159,27 Euro
1 194,05 Euro
1 229,87 Euro
1 266,77 Euro
1 304,77 Euro
1 343,92 Euro
To reset the format, use reset().

>reset;

Plots

Now, the anti-aliasing for graphics now uses the same algorithm as the anti-aliasing for images in the notebook by default. This algorithm now has an improved smoothing, which can be turned down in the Graphics Options.

>plot3d("x*y",>contour):
Changes

There is a different frame type for 3D plots now.

>plot3d("exp(-x^2-y^2)",r=3,>spectral,frame=3,n=100,grid=12):
Changes

The default space on the left of 2D plots has been increased a bit. Moreover, there is a "smaller" parameter in plot2d() now.

>plot2d("exp(x)",1,10,yl="exp(x)",xl="x",smalle
r=3):
Changes

Control Structures

It is now possible to span loops over multi-line commands. One should not print anything inside the loop, however.

>x=1; repeat; ...
x=(x+2/x)/2; until x^2~=2; ...
end; x,
1.41421356237
It is better and advisable to use a function instead. Command lines are meant to be short, and are not for programming.

>function it ...
useglobal;
repeat; x=(x+2/x)/2; until x^2~=2;
end;
endfunction
>x=1; it; x,
1.41421356237
Even better is the following version.

>function it(start,a) ...
x=start;
repeat xnew=(x+a/x)/2;
until xnew~=x;
x=xnew;
end;
return x;
endfunction
>it(1,2)
1.41421356237

There was a bug in the graphics display for inserted images if enlarged display was chosen or the aspect was not integer.

>aspect(1.8); plot2d("x^3-x"); insimg(); aspect();

EMT got an improved PDF output. It is now copying a style file to the current directory, which you are invited to edit. The default output is now for a 4:3 beamer. You can easily change this in the eumat.sty file.

Test File

The magnification for a beamer now got a shortcut (Ctrl-F11), and will automatically generate large images ("Keep Large Images"). There is a new full screen mode (F11), which turns off the menu and the border. This will also active the one-window interface.

In times of better and larger screens and of faster computers with more memory, I found it appropriate to increase the size of the embedded graphics. It is possible to set an own size in the menu, with the insimglines() function, or by using the insimg() command instead of the colon ":" after the plot command.

>plot2d("x^3-x"); insimg(20);
Changes

Moreover, I darkened the colors of the frame and the text for a more professional appearance. The colors can be set with textcolor(), framecolor() and frame3dcolor(). The function reset() defaults everything as usual.

By default, EMT does no longer use fatter lines. You can change that in the graphics menu.

By default, EMT is now using another algorithm for anti-aliasing. It the same algorithm that I had to program for the Linux version. It looks a bit smoother and better.

>plot3d("x^3*y+x^2",>contour,angle=30°,zoom=3,n=100):

In times of better and larger screens and of faster computers with more memory, I found it appropriate to increase the size of the embedded graphics. It is possible to set an own size in the menu, with the insimglines() function, or by using the insimg() command instead of the colon ":" after the plot command.

>plot2d("x^3-x"); insimg(20);
Changes

Moreover, I darkened the colors of the frame and the text for a more professional appearance. The colors can be set with textcolor(), framecolor() and frame3dcolor(). The function reset() defaults everything as usual.

By default, EMT does no longer use fatter lines. You can change that in the graphics menu.

By default, EMT is now using another algorithm for anti-aliasing. It the same algorithm that I had to program for the Linux version. It looks a bit smoother and better.

>plot3d("x^3*y+x^2",>contour,angle=30°,zoom=3,n=100):

It is now possible to use Unicode characters in plots and strings. For this, EMT has a new type of strings. The easiest way is to use u"...". Then inside the string, HTML entities can be used.

Unicode strings get a special flag and print as Unicode.

>sa = u"α"
?
Concatenating a Unicode string to another string makes the result a Unicode string.

>sa + " = " + 45 + u"°"
? = 45°
This output will export to HTML and can be saved with the notebook, unless HTML export or saving is disabled by the user.

Note that copy and paste does not work with Unicode strings. The kernel of Euler still works with local character sets. HTML entities are the only save way to port special characters to other encodings.

>u"Ä is a German character!"
? is a German character!
Special characters should not be used in variable names. The following assignment will work here, but may fail in other encodings.

>? = 5
5
In plots, Unicode strings can be used.

>plot2d("x^3*sin(x)",-4pi,4pi, ...
title=sa+u"³ - "+sa,xl=sa, ...
>vertical,yl=u"y = f(α)"); ...
x=3/2*pi; plot2d(x,x^3*sin(x),>points,>add); ...
label(sa+u"=π",x,x^3*sin(x),pos="lc"):
Changes

Moreover, it is possible to use HTML entities in comments. This allows ?, ? etc. The characters will be exported in the notebook.

Copy and paste into another program will copy the original input with HTML entities. To get a nice copy for documents, use "Copy as formatted Output" (Ctrl-Shift-F).

Plots

I improved the vertical styles in plots.

>plot2d("log(x)",0,2,yl="log(x)",xl="x",>ver
tical):
Changes

This does also apply to special axis styles. Moreover, a transparent color will hide the text around the text box or label box.

>plot2d("x^3-x",<frame,grid=0); ...
xaxis(0,xx=[-2,-1,1,2],style="->"); ...
yaxis(0,yy=[-5,5],>vertical,>left,style="->"); ...
textbox(["Function",u"x³ - x"],color=transparent, ...
tcolor=black,x=0.3,y=0.2):

The nlfit() function (which uses the Levenberg Marquardt algorithm to minimize |f(v)| for a non-linear fit) is now more sophisticated.

>function f([x,y]) &= [x,y,x^2+2*y^2+1]
2 2
[x, y, 2 y + x + 1]

>function Df([x,y]) &= jacobian(f(x,y),[x,y])
[ 1 0 ]
[ ]
[ 0 1 ]
[ ]
[ 2 x 4 y ]

>nlfit("f","Df",[1,2])
[1.23914e-010, 2.62476e-010]
The Nelder Mead method is more stable. The speed of the algorithms compare for small problems. For large problems, nlfit() should be much better, since it is locally a Newton method.

>neldermin("norm(f(x))",[1,2])
[-3.27698e-008, 2.87569e-007]
This is another alternative.

>powellmin("norm(f(x))",[1,2])
[1.65478e-012, 7.1513e-009]
Fixes

There is the famous typo

>[1,1.2.4]

instead of

>[1,1.2,4]

This will now issue an error message.

If you need a scalar product after or before a number, you need to put the number in brackets.

>(1).(5)
5
Of course, this will not be necessary for variables containing numbers.

>a=4.5; a.a
20.25
Model Fit

The modelfit() function now has parameters for neldermin(). The defaults will do in most cases.

The nlfit() function (which uses the Levenberg Marquardt algorithm to minimize |f(v)| for a non-linear fit) is now more sophisticated.

>function f([x,y]) &= [x,y,x^2+2*y^2+1]
2 2
[x, y, 2 y + x + 1]

>function Df([x,y]) &= jacobian(f(x,y),[x,y])
[ 1 0 ]
[ ]
[ 0 1 ]
[ ]
[ 2 x 4 y ]

>nlfit("f","Df",[1,2])
[1.23914e-010, 2.62476e-010]
The Nelder Mead method is more stable. The speed of the algorithms compare for small problems. For large problems, nlfit() should be much better, since it is locally a Newton method.

>neldermin("norm(f(x))",[1,2])
[-3.27698e-008, 2.87569e-007]
This is another alternative.

>powellmin("norm(f(x))",[1,2])
[1.65478e-012, 7.1513e-009]
Fixes

There is the famous typo

>[1,1.2.4]

instead of

>[1,1.2,4]

This will now issue an error message.

If you need a scalar product after or before a number, you need to put the number in brackets.

>(1).(5)
5
Of course, this will not be necessary for variables containing numbers.

>a=4.5; a.a
20.25

The old LPSOLVE demos by Notebaert are back. There had to be some changes in the interface, if empty matrices are involved.

LPSOLVE Demos

Diagonals of Matrices

Since the diag() function was a bit overloaded, I added two new functions, which are more easy to use and to remember.

getdiag(A,k) now reads the k-th diagonal off A. The default is k=0.

>A=random(3,3); getdiag(A)
[0.655416, 0.525, 0.28269]
Other diagonals are shorter, of course.

>A=random(3,3); getdiag(A,1)
[0.595017, 0.323032]
To get a diagonal matrix, there is the simple function diagmatrix().

>diagmatrix(1:3)
1 0 0
0 2 0
0 0 3
For other diagonals, use shorter vectors.

>diagmatrix(1:2,1)
0 1 0
0 0 2
0 0 0
This is a replacement of the following command.

>diag(3,3,1,1:2)
0 1 0
0 0 2
0 0 0
Remember that there is also band().

>band(A,-1,1)
0.095153 0.595017 0
0.72868 0.465164 0.323032
0 0.502255 0.168603

I improved dragvalues(). It does now accept different stop numbers for each drag. Moreover, it will round to the given number of digits before it calls the plot function.

Here is an example. The example is by Ron Gerber via email.

>function plotstar ([m,e,r]) ...
fullwindow();
x=mod((0:1:360)*r,360)°;
plot2d(cos(m*x)^e*cos(x),cos(m*x)^e*sin(x),grid=0,thickness=0.5);
endfunction
>v=dragvalues("plotstar",["M","E","R"
], ...
[4,1,31],[0,12;1,10;1,50], ...
stops=[12,9,49],digits=0, ...
heading="Drag these values:",cheading=black); ...
plotstar(v):

The documentation of EMT got an update, lots of errors and typos were removed, the English was fixed at many places, and the content of the help window go a nicer format. Documentation is always work in progress.

The source of Euler contains a new version of pipes.cpp, which compiles and runs on the command line. Symbolic expressions and graphics do not work.

The documentation of EMT got an update, lots of errors and typos were removed, the English was fixed at many places, and the content of the help window go a nicer format. Documentation is always work in progress.

The source of Euler contains a new version of pipes.cpp, which compiles and runs on the command line. Symbolic expressions and graphics do not work.

I added modelfit() for non-linear fit of model functions to data. The algorithm uses the Nelder-Mead minimization by default. It defines an error function and minimizes the norm of the error.

>x=0:0.2:2; y=exp(-x)+0.8*exp(-2*x)+0.01*normal(size(x));

We define a model function.

Changes

The function needs to vectorize. But this is automatic in expressions.

>function model(x,p) := p[1]*exp(-x)+p[2]*exp(p[3]*x);
>pbest=modelfit("model",[1,1,-1],x,y)

[0.929599, 0.872998, -1.92642]

>plot2d(x,y,>points); plot2d("model(x,pbest)",>add,color=red):

Changes

>norm(y-model(x,pbest))

0.0275031948945

Of course, you can just as well do this yourself.

>function err (p) := norm(y'-model(x',p))
>neldermin("err",[1,1,-1])

[0.929599, 0.872998, -1.92642]

Fixes

The powellmin() function did not pass semicolon parameters correctly to the function. Consequently, the modelfit() function did not work with Powell's method.

>pbest1=modelfit("model",[1,1,-1],x,y,>powell)

[0.929599, 0.872998, -1.92642]

B-Splines

A function, which computes the normalized B-Spline N(x,k,n,t) for a set of knots x[k:k+n+1] has been added. The function does work for multiple knots too.

>t=[1,2,2,3,4,5,6]; plot2d("nspline(x,2,3,t)",1,6):

Changes

The function vectorizes for row vectors x and column vectors k, or for matrices x and scalar k.

In the following example, we demonstrate that the normalized B-Splines add to 1.

>t=1:10; x=1:0.01:10; y=nspline(x,(1:6)',3,t); ...
plot2d(x,sum(y')'):

Changes