Euler Math Toolbox 2022-05-18

I had to delete an old compilation of listc.c from the distribution. A few virus checkers raised a false alarm on this file. Other compilations of Tiny C do not have this problem.

I removed lists.e from the list of Euler utilities. The next version of Euler will contain some sort of lists.

The version numbers will now reflect the date of the version rather than a number. This looks more natural for programs that are updated as often as Euler Math Toolbox.

There is finally a 64-bit version of Euler. It is in beta state, but everything so far looked okay.

Output Formats

I updated the commands to set the output formats.

The command format(n) now works like goodformat(n). To set a fixed format use format(n,m).

>format(5); [1,2,pi]
[1, 2, 3.1416]
Use reset() or defformat() to revert to the default format.

>defformat;
The scalar format now prints large numbers in a more friendly way.

>12!
479001600
The format(n,m) works almost as before. But it will now display 0.00, if the number is too small.

>format(5,2); random(10,10)/10
0.07 0.02 0.09 0.03 0.05 0.03 0.04 0.03 0.03 0.09
0.03 0.07 0.02 0.04 0.03 0.09 0.02 0.05 0.01 0.06
0.04 0.07 0.05 0.03 0.05 0.05 0.02 0.03 0.09 0.05
0.05 0.06 0.07 0.10 0.02 0.09 0.01 0.10 0.00 0.04
0.05 0.04 0.02 0.02 0.03 0.08 0.05 0.03 0.03 0.02
0.04 0.02 0.06 0.07 0.10 0.08 0.04 0.05 0.09 0.04
0.07 0.01 0.09 0.01 0.02 0.05 0.00 0.06 0.01 0.06
0.02 0.10 0.08 0.03 0.06 0.09 0.10 0.05 0.02 0.04
0.07 0.02 0.08 0.04 0.04 0.08 0.04 0.02 0.00 0.02
0.04 0.04 0.04 0.03 0.02 0.06 0.08 0.09 0.09 0.01
The format prefixes now return to the previous settings not to goodformat as before.

>longest pi, pi,
3.141592653589793
3.14
The default format is goodformat(6) with zero rounding and dense output on.

>defformat; 10^(-10:10)
[1e-010, 1e-009, 1e-008, 1e-007, 1e-006, 1e-005, 0.0001, 0.001,
0.01, 0.1, 1, 10, 100, 1000, 10000, 100000, 1e+006, 1e+007,
1e+008, 1e+009, 1e+010]
By default, scalars will print with goodformat(12).

>pi
3.14159265359
Manual Axes

I added some more functions for manual axes and grid lines, and documented those in the tutorials.

First of all, you could change the default style of the axes globally. The function gridstyle() now lets you set styles and colors of the grid lines and labels, and the color of the frame.

The settitle function now lets you select a color. Previously, you had to change the global text color for this.

To revert to the defaults, always use reset().

>gridstyle("->",color=gray,textcolor=gray,framecolor=gray); ...
plot2d("x^3-x",grid=1); ...
settitle("y=x^3-x",color=black); ...
label("x",2,0,pos="bc",color=gray); ...
label("y",0,6,pos="cl",color=gray); ...
reset():
Changes

Sometimes, you wish to build a plot with axes step by step. There new functions xaxis() and yaxis() now.

>fullwindow; ...
gridstyle(color=darkgray,textcolor=darkgray); ...
plot2d(["2^x","1","2^(-x)"],a=-2,b=2,c=0,d=4,grid=
0,color=4:6); ...
xaxis(0,-2:1,style="->"); xaxis(0,2,"x",<axis); ...
yaxis(0,4,"y",style="->"); ...
yaxis(-2,1:4,>left); ...
yaxis(2,2^(-2:2),style="."); ...
labelbox(["2^x","1","2^-x"],colors=4:6,x=0.8,y=0.2
); ...
reset:
Changes

Python >>>

This version allows Python commands to start with >>> ... This applies to multiple lines too. Note that >> ... for Yacas has been disabled.

>>> ...
print 'Hello World!' ...
print 'Did you know that 111111111^2 =',111111111**2,'?'
Hello World!
Did you know that 111111111^2 = 12345678987654321 ?
RGB Plots

I improved the RGB plots by introducing the function getwindowfromplot. This function returns a screen window from plot coordintates. Without parameters, it returns the full area of the current plot.

>s=linspace(0,1,100); t=s'; c=rgb(s,t,0); ...
setplot(0,1,0,1); plotrgb(c,getwindowfromplot()); xplot():
Changes

Hard Spaces

You can now enter a variable with a space or a function with a space in its name, if you press Ctrl-Space.

Note that the hard space character should be saved for other users with the same encoding, unless you use unicode for notebooks. Since this versions, I save the default notebooks in Unicode.

>variable with space = 1.45
1.45
>variable with space + 70
71.45
You can also format numbers with a hard space.

>one million = 1 000 000
1000000
Statistics

There are new functions to generate random variables. The function randpint generates numbers from 1 to k based on given probabilities.

>X=randpint(1,1000,[0.2,0.3,0.5]); getmultiplicities(1:3,X)
[177, 321, 502]
The function randmultinomial() generates n events of m throws with given probabilities.

>randmultinomial(10,1000,[0.2,0.3,0.5])
178 321 501
199 287 514
209 287 504
216 279 505
187 288 525
211 324 465
198 317 485
205 317 478
203 298 499
216 306 478
This is used to apply the Monte Carlo method in the chi-square test. Assume, we observe a vector of number of times of 1 to 6 in dice throws. We may ask, if the dice is fair.

>X=[10,12,8,4,12,15]
[10, 12, 8, 4, 12, 15]
The default chitest() compares to the expected times of 1 to 6 using an assumed chi-square distribution.

>chitest(X,ones(1,6)/6*sum(X))
0.20873101314
The Monte Carlo method simulates the dice throws and checks the chi-squared statistics for excess of the observed statistics.

>chitest(X,ones(1,6)/6*sum(X),montecarlo=10000)
0.214
Boxplots

I have changed the boxplot function. It now uses data instead of quartiles. But it can now also display outliers.

>boxplot(randnormal(5,1000,1000,10)):
Changes

I removed the quantile function. There is now a quartile function, which returns the Q1,M,Q2 quartiles, and the minimal and maximal data which are not outliers. By definition, outliers are data which are more than 1.5 times the range less than Q1 or greater then Q2.

>x=normal(100); [min(x),max(x)], quartiles(x), quartiles(x,outliers=none),
[-2.46061, 1.84706]
[-2.46061, -0.539466, 0.135262, 0.813865, 1.84706]
[-2.46061, -0.539466, 0.135262, 0.813865, 1.84706]
Radius in Plots

You can now use a vector of radii for plots. This maybe useful sometimes.

>plot2d("sin(x)",r=[pi,1.2]):
Changes

This does also work in 3D. Together with the scaling you get a flexible approach.

>plot3d("-x^3+y^2-x*y",r=[1,2],scale=[1.8,1,2],>contour):

I collected the changes from 23.0 to 23.5 for EMT 24. Due to high workload in the winter term there will only be fixes for the next months.

EMT now includes the recent version of TinyC.

I updated the commands to set the output formats.

The command format(n) now works like goodformat(n). To set a fixed format use format(n,m).

>format(5); [1,2,pi]
[1, 2, 3.1416]
Use reset() or defformat() to revert to the default format.

>defformat;
The scalar format now prints large numbers in a more friendly way.

>12!
479001600
The format(n,m) works almost as before. But it will now display 0.00, if the number is too small.

>format(5,2); random(10,10)/10
0.07 0.02 0.09 0.03 0.05 0.03 0.04 0.03 0.03 0.09
0.03 0.07 0.02 0.04 0.03 0.09 0.02 0.05 0.01 0.06
0.04 0.07 0.05 0.03 0.05 0.05 0.02 0.03 0.09 0.05
0.05 0.06 0.07 0.10 0.02 0.09 0.01 0.10 0.00 0.04
0.05 0.04 0.02 0.02 0.03 0.08 0.05 0.03 0.03 0.02
0.04 0.02 0.06 0.07 0.10 0.08 0.04 0.05 0.09 0.04
0.07 0.01 0.09 0.01 0.02 0.05 0.00 0.06 0.01 0.06
0.02 0.10 0.08 0.03 0.06 0.09 0.10 0.05 0.02 0.04
0.07 0.02 0.08 0.04 0.04 0.08 0.04 0.02 0.00 0.02
0.04 0.04 0.04 0.03 0.02 0.06 0.08 0.09 0.09 0.01
The format prefixes now return to the previous settings not to goodformat as before.

>longest pi, pi,
3.141592653589793
3.14
The default format is goodformat(6) with zero rounding and dense output on.

>defformat; 10^(-10:10)
[1e-010, 1e-009, 1e-008, 1e-007, 1e-006, 1e-005, 0.0001, 0.001,
0.01, 0.1, 1, 10, 100, 1000, 10000, 100000, 1e+006, 1e+007,
1e+008, 1e+009, 1e+010]
By default, scalars will print with goodformat(12).

>pi
3.14159265359

I added some more functions for manual axes and grid lines, and documented those in the tutorials.

First of all, you could change the default style of the axes globally. The function gridstyle() now lets you set styles and colors of the grid lines and labels, and the color of the frame.

The settitle function now lets you select a color. Previously, you had to change the global text color for this.

To revert to the defaults, always use reset().

>gridstyle("->",color=gray,textcolor=gray,framecolor=gray); ...
plot2d("x^3-x",grid=1); ...
settitle("y=x^3-x",color=black); ...
label("x",2,0,pos="bc",color=gray); ...
label("y",0,6,pos="cl",color=gray); ...
reset():
Changes

Sometimes, you wish to build a plot with axes step by step. There new functions xaxis() and yaxis() now.

>fullwindow; ...
gridstyle(color=darkgray,textcolor=darkgray); ...
plot2d(["2^x","1","2^(-x)"],a=-2,b=2,c=0,d=4,grid=
0,color=4:6); ...
xaxis(0,-2:1,style="->"); xaxis(0,2,"x",<axis); ...
yaxis(0,4,"y",style="->"); ...
yaxis(-2,1:4,>left); ...
yaxis(2,2^(-2:2),style="."); ...
labelbox(["2^x","1","2^-x"],colors=4:6,x=0.8,y=0.2
); ...
reset:

For this version, I updated the Help Window. It does not display the content of the file "overview.txt", starting with a list of topics, so that it is easy to find information about how to do some task. I think this is far more convenient for the new user.

The help window does also display breadcrumbs with a list of recently visited pages.

This version allows Python commands to start with >>> ... This applies to multiple lines too. Note that >> ... for Yacas has been disabled.

>>> ...
print 'Hello World!' ...
print 'Did you know that 111111111^2 =',111111111**2,'?'
Hello World!
Did you know that 111111111^2 = 12345678987654321 ?
RGB Plots

I improved the RGB plots by introducing the function getwindowfromplot. This function returns a screen window from plot coordintates. Without parameters, it returns the full area of the current plot.

>s=linspace(0,1,100); t=s'; c=rgb(s,t,0); ...
setplot(0,1,0,1); plotrgb(c,getwindowfromplot()); xplot():
Changes

Hard Spaces

You can now enter a variable with a space or a function with a space in its name, if you press Ctrl-Space.

Note that the hard space character should be saved for other users with the same encoding, unless you use unicode for notebooks. Since this versions, I save the default notebooks in Unicode.

>variable with space = 1.45
1.45
>variable with space + 70
71.45
You can also format numbers with a hard space.

>one million = 1 000 000
1e+006
Statistics

There are new functions to generate random variables. The function randpint generates numbers from 1 to k based on given probabilities.

>X=randpint(1,1000,[0.2,0.3,0.5]); getmultiplicities(1:3,X)
[177, 330, 493]
The function randmultinomial() generates n events of m throws with given probabilities.

>randmultinomial(10,1000,[0.2,0.3,0.5])
172 322 506
208 296 496
210 275 515
204 290 506
199 282 519
200 332 468
206 305 489
194 324 482
202 300 498
219 303 478
This is used to apply the Monte Carlo method in the chi-square test. Assume, we observe a vector of number of times of 1 to 6 in dice throws. We may ask, if the dice is fair.

>X=[10,12,8,4,12,15]
[10, 12, 8, 4, 12, 15]
The default chitest() compares to the expected times of 1 to 6 using an assumed chi-square distribution.

>chitest(X,ones(1,6)/6*sum(X))
0.20873101314
The Monte Carlo method simulates the dice throws and checks the chi-squared statistics for excess of the observed statistics.

>chitest(X,ones(1,6)/6*sum(X),montecarlo=10000)
0.2137

I have changed the boxplot function. It now uses data instead of quartiles. But it can now also display outliers.

>boxplot(randnormal(5,1000,1000,10)):
Changes

I removed the quantile function. There is now a quartile function, which returns the Q1,M,Q2 quartiles, and the minimal and maxima data which are not outliers.

By definition, outliers are data which are more than 1.5 times the range less than Q1 or greater then Q2.

>x=normal(100); [min(x),max(x)], quartiles(x), quartiles(x,outliers=none),
[-2.97118, 3.15595]
[-2.97118, -0.90094, -0.229074, 0.485019, 2.55719]
[-2.97118, -0.90094, -0.229074, 0.485019, 3.15595]
Radius in Plots

You can now use a vector of radii for plots. This maybe useful sometimes.

>plot2d("sin(x)",r=[pi,1.2]):
Changes

This does also work in 3D. Together with the scaling you get a flexible approach.

>plot3d("-x^3+y^2-x*y",r=[1,2],scale=[1.8,1,2],>contour):

I added an adaptive method from AlgLib for integrals. It is a very similar method as the adaptive Gauss method, which is the default for integrate.

>alintegrate("exp(-x^2)",0,10), integrate("exp(-x^2)",0,10)
0.886226925453
0.886226925453
>function f(x) := exp(-x^2);
>tic; loop 1 to 1000; _alintegrate("f",0,10); end; toc;
Used 0.39 seconds
In fact, for this particular function, the adaptive Gauss method as implemented in Euler takes the same time.

>tic; loop 1 to 1000; adaptivegauss("f",0,10); end; toc;
Used 0.39 seconds
AlgLib does also contain a special algorithm for functions with singularities at the ends of the interval. The function should be of the type

Changes

with a smooth function g. You have to provide alpha and beta.

>alsingular("1/sqrt(x)",0,1,-0.5,0)
2
Fixes

The strrepl() function was not documented well. Moreover, it did not work in some cases.

>strrepl("test","t","")
es

http://euler.rene-grothmann.de/Programs/Changes.html

· Extended the Bessel functions besseli() and besselj() to 0, and to x smaller than 0 if the order is integer.
· Note that Euler is different from other packages including Maxima. The main argument x comes in front of the order. This style allows Euler to pass the order as an additional argument. Therefore, it is the default style of Euler. It is also the most elegant style. Many functions like evalpoly follow the same convention. The interpolation can be written in the same way.

diff Function:
· The diff function got a fix. It did not work in another function for a function, which used semicolon parameters.

Matplotlib:
· If this library was imported and Euler was restarted, the next import of this library crashed Euler. Fixed by not finalizing Python.

Sound:
· Updated the sound library. The default number of points for analyzesound is now 2^16. It works, if the sound is shorter taking a power of 2 less than the sound. You can also set the parameter points to the number of points in your sample.
· Due to the higher number of samples, the frequency is now very accurate.

I fixed hatched fill styles. The SVG output did not work correctly, and in the screen output there were tiny inaccuracies in slashed bar hatch fills.

Moreover, the hatch size did not scale with the graphics window. It does now, if the scaling of the line width is on (the default). Consequently, some hatched graphics may look different now.

>t=linspace(0,2pi,5); plot2d(cos(t),sin(t),r=1,>filled,style="/"):


It is possible to make the hatch grid finer or more coarse. The default hatch size is 12. You can use 8, if you like.

>hatchgridsize(8); plot2d(normal(1,1000),>distribution,style="/"):


>hatchgridsize(16); plot2d("x^3-x",r=1.2,>filled,style="/"):

It is now possible to save the Euler notebook during a computation. This is necessary, since Python scripts cannot be interrupted. Now you can at least save the notebook and exit the program.

I switched to Visual Studio 2012 Professional. Euler got compiled with the default settings. I had to fix a bit of the code, since the order of the initializations was important. Now this should work, no matter in which order the modules are linked.