Euler Math Toolbox 2022-05-18

Fixed a small error in the Gamma function.

>x=-4:0.05:4; y=x';
>h=min(abs(gamma(x+I*y)),5);
>plot3d(x,y,h,>spectral,angle=0°,zoom=3):

Fixed the "limits" parameter of plot3d for scaled output, and added more documentation for this parameter. "limits" should be 1x2 vector [zmin,zmax], and will cut off exceeding z-values. It works only for functions with hue.

>plot3d("min(1/(x^2+y^2),3)",>zhue,>spectral,n=200,limits=[0,2.95]):

Need string f, vector v, real d, real eps, args for f for nelder
Error in:
nelder ...
Improved the documentation of the elliptical functions.
>help ellrf
ellrf is an Euler function.
function map ellrf (x, y, z, eps)
Default for eps : none
Function in file : functions
Carlson's elliptic integral of the first kind RF (x; y; z).
The iteration is implemented in the Euler language.
x, y, and z must be non-negative, and at most one can be zero.
Computes
1/2 integrate(1/sqrt((t+x)*(t+y)*(t+z)),t=0..inf)
See: ellrd, ellf

Fixed a small glitch.

Now the right mouse button works in the following ways:
Clicking into a comment opens the comment window.
Clicking into a program opens the internal editor.

The official forum for Euler Math Toolbox is now at SourceForge at

https://sourceforge.net/p/eumat/discussion/

I will also try to keep the files section in SourceForge up to date with the most recent version. The HTML files remain on my server at Strato, as well as the downloads.

The 32-bit version will no longer be updated, or only at very special request. The version of May 2017 will remain on the server.

Now the right mouse button works in the following ways:

(1) Clicking into a comment opens the comment window.
(2) Clicking into a program opens the internal editor.

The official forum for Euler Math Toolbox is now at SourceForge at

https://sourceforge.net/p/eumat/discussion/

I will also try to keep the files section in SourceForge up to date with the most recent version. The HTML files remain on my server at Strato, as well as the downloads.

The 32-bit version will no longer be updated, or only at very special request. The version of May 2017 will remain on the server.

Now the right mouse button works in the following ways:

(1) Clicking into a comment opens the comment window.
(2) Clicking into a program opens the internal editor.

The help window now searches for strings automatically if no topic is found.

The function mset() is now fixed and works, even if there is nothing to set.

>v=dup(1:3,2); mset(v,mnonzeros(v<0),0)
1 2 3
1 2 3
There is no v[i]<0. But in the following v, there are v[i]<0.

>v=dup(-1:1,2); mset(v,mnonzeros(v<0),0)
0 0 1
0 0 1
Note how mnonzeros(v<0) works. It returns the indices, where this is true.

>mnonzeros(v<0)
1 1
2 1
For vectors, you can also use the following.

>v=-5:5; v[nonzeros(v<0)]=0
[0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5]
There is a new modifier "usenan". It evaluates the command with "errors off" and sets "errors on" afterwards.

>usenan log(-3:3)
[NAN, NAN, NAN, NAN, 0, 0.693147, 1.09861]
E.g., let us compute sum(1/v), where v is not 0.

>v=0:5; usenan w=1/v
[NAN, 1, 0.5, 0.333333, 0.25, 0.2]
Now, we eliminate the NANs before we sum up.

>sum(w[nonzeros(!isNAN(w))])
2.28333333333
The plot3d(x,y,z) routine for plotting surfaces now respects >zscale. In this case, z is assumed to be a function value. The function is scaled so that everything fits into a unit cube. But the function values are scaled with respect to x and y first.

>x=-10:10; y=x'; z=x*y^2; plot3d(x,y,z,>zscale,angle=-40°,zoom=2):
Changes

The value of zscale is used to scale into that direction.

>plot3d(x,y,z,zscale=0.6,angle=-40°,zoom=2):
Changes

Additionally, scale=[sx,sy,sz] can be used.

>plot3d(x,y,z,>zscale,scale=[1,1.5,0.5],angle=-40°,zoom=2):
Changes

The help window now searches for strings automatically if no topic is found.

The function mset() is now fixed and works, even if there is nothing to set.

>v=dup(1:3,2); mset(v,mnonzeros(v<0),0)
1 2 3
1 2 3
There is no v[i]<0. But in the following v, there are v[i]<0.

>v=dup(-1:1,2); mset(v,mnonzeros(v<0),0)
0 0 1
0 0 1
Note how mnonzeros(v<0) works. It returns the indices, where this is true.

>mnonzeros(v<0)
1 1
2 1
For vectors, you can also use the following.

>v=-5:5; v[nonzeros(v<0)]=0
[0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5]
There is a new modifier "usenan". It evaluates the command with "errors off" and sets "errors on" afterwards.

>usenan log(-3:3)
[NAN, NAN, NAN, NAN, 0, 0.693147, 1.09861]
E.g., let us compute sum(1/v), where v is not 0.

>v=0:5; usenan w=1/v
[NAN, 1, 0.5, 0.333333, 0.25, 0.2]
Now, we eliminate the NANs before we sum up.

>sum(w[nonzeros(!isNAN(w))])
2.28333333333
The plot3d(x,y,z) routine for plotting surfaces now respects >zscale. In this case, z is assumed to be a function value. The function is scaled so that everything fits into a unit cube. But the function values are scaled with respect to x and y first.

>x=-10:10; y=x'; z=x*y^2; plot3d(x,y,z,>zscale,angle=-40°,zoom=2):
Changes

The value of zscale is used to scale into that direction.

>plot3d(x,y,z,zscale=0.6,angle=-40°,zoom=2):
Changes

Additionally, scale=[sx,sy,sz] can be used.

>plot3d(x,y,z,>zscale,scale=[1,1.5,0.5],angle=-40°,zoom=2):

The help window now searches for strings automatically if no topic is found.

The function mset() is now fixed and works, even if there is nothing to set.

>v=dup(1:3,2); mset(v,mnonzeros(v<0),0)
1 2 3
1 2 3
There is no v[i]<0. But in the following v, there are v[i]<0.

>v=dup(-1:1,2); mset(v,mnonzeros(v<0),0)
0 0 1
0 0 1
Note how mnonzeros(v<0) works. It returns the indices, where this is true.

>mnonzeros(v<0)
1 1
2 1
For vectors, you can also use the following.

>v=-5:5; v[nonzeros(v<0)]=0
[0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5]
There is a new modifier "usenan". It evaluates the command with "errors off" and sets "errors on" afterwards.

>usenan log(-3:3)
[NAN, NAN, NAN, NAN, 0, 0.693147, 1.09861]
E.g., let us compute sum(1/v), where v is not 0.

>v=0:5; usenan w=1/v
[NAN, 1, 0.5, 0.333333, 0.25, 0.2]
Now, we eliminate the NANs before we sum up.

>sum(w[nonzeros(!isNAN(w))])
2.28333333333
The plot3d(x,y,z) routine for plotting surfaces now respects >zscale. In this case, z is assumed to be a function value. The function is scaled so that everything fits into a unit cube. But the function values are scaled with respect to x and y first.

>x=-10:10; y=x'; z=x*y^2; plot3d(x,y,z,>zscale,angle=-40°,zoom=2):
Changes

Additionally, scale=[sx,sy,sz] can be used.

>plot3d(x,y,z,>zscale,scale=[1,1,0.5],angle=-40°,zoom=3):

I have removed the ancient function scale() and replaced it with something more useful. Now scale(A) scales the matrix A into [epsilon,1].

>sort(scale(random(10)))
[3.20579e-012, 0.116789, 0.117949, 0.142182, 0.163337, 0.351734,
0.467792, 0.656083, 0.984992, 1]
This is useful for plots. E.g., we disturb the matrix (i*j) randomly. To get spectral colors, we scale the result to [0,1]. This will be a noisy plot of x*y.

>n=50; A=(1:n)*(1:n)'; clg; plotrgb(getspectral(scale(A+n*normal(n,n)))):
Changes

If you plot the same with plot3d, you do not need to scale, because plot3d() has its own scaling parameter. It needs to print the labels correctly.

>plot3d(A+n*normal(n,n), ...
scale=[80,80,1],angle=-70°,center=[0,0,0.5],zoom=1.7):
Changes

I have also expanded the function crossproduct(). It will work now for matrices with three rows, and it will return the results in such a matrix.

>v=[2,3,4]'; crossproduct(v,v|(v+1))
0 -1
0 2
0 -1
I also added the Gram determinant. It computes the volume of the parallelotope spanned by the columns of a matrix. For two vectors in three dimensions, this happens to be the same as the cross product.

>w=[3,4,-2]'; gramdet(v|w), norm(crossproduct(v,w))
27.2213151776
27.2213151776
The generalized cross product is added too. It returns a vector perpendicular to all n-1 columns of a matrix with n rows. The length of the vector is equal to the area of the parallelotope spanned by the columns.

>M=normal(4,3); v=vectorproduct(M); M'.v, norm(v), gramdet(M)
0
0
0
0.906895530849
0.906895530849

I have removed the ancient function scale() and replaced it with something more useful. Now scale(A) scales the matrix A into [epsilon,1].

>sort(scale(random(10)))
[3.20579e-012, 0.116789, 0.117949, 0.142182, 0.163337, 0.351734,
0.467792, 0.656083, 0.984992, 1]
This is useful for plots. E.g., we disturb the matrix (i*j) randomly. To get spectral colors, we scale the result to [0,1]. This will be a noisy plot of x*y.

>n=50; A=(1:n)*(1:n)'; clg; plotrgb(getspectral(scale(A+n*normal(n,n)))):
Changes

If you plot the same with plot3d, you do not need to scale, because plot3d() has its own scaling parameter. It needs to print the labels correctly.

>plot3d(A+n*normal(n,n), ...
scale=[80,80,1],angle=-70°,center=[0,0,0.5],zoom=1.7):
Changes

I have also expanded the function crossproduct(). It will work now for matrices with three rows, and it will return the results in such a matrix.

>v=[2,3,4]'; crossproduct(v,v|(v+1))
0 -1
0 2
0 -1
I also added the Gram determinant. It computes the volume of the parallelotope spanned by the columns of a matrix. For two vectors in three dimensions, this happens to be the same as the cross product.

>w=[3,4,-2]'; gramdet(v|w), norm(crossproduct(v,w))
27.2213151776
27.2213151776
The generalized cross product is added too. It returns a vector perpendicular to all n-1 columns of a matrix with n rows. The length of the vector is equal to the area of the parallelotope spanned by the columns.

>M=normal(4,3); v=vectorproduct(M); M'.v, norm(v), gramdet(M)
0
0
0
1.97391989727
1.97391989727

I added a function fzeros() which finds all zeros of another function in an interval. It makes use of fextrema() and bisectin().

>fx := "cheb(x,10)+2*x^2"; ...
plot2d(fx,-1,1); w=fzeros(fx,-1,1); plot2d(w,fx(w),>points,>add):
Changes

Due to user demand, there is no a Python mode.

>pythonmode on
Python mode is on
From now on every command, is interpreted as a Python command.

>print "This is Python"
This is Python
Multi-lines work as expected. For Python control structures, you must use multi-lines and indentation.

>n=1 ...
for k in range(2,40): ...
n=n*k ...
print n
20397882081197443358640281739902897356800000000
You can use "function python" for multi-line commands too.

>function python ...
n=1
for k in range(2,40):
n=n*k
print n
endfunction
20397882081197443358640281739902897356800000000
You can also define functions with "def" in a multi-line command, or as a "python" function.

>function python fak(m) ...
n=1
for k in range(2,m):
n=n*k
return n
endfunction
>print fak(40)
20397882081197443358640281739902897356800000000
Maxima lines work too.

>:: 40!
815915283247897734345611269596115894272000000000

EMT works in this mode with "euler".

>euler 40!
8.15915283248e+047
>pythonmode off
Python mode is off

Some Maxima replacements got fixes to make things more logical. To escape any character in a symbolic expression you can now use \c.

The following defines an infix operator | in Maxima which simply multiplies to values. We need to escape the | and the := in symbolic expressions.

>&infix("\|",115); &a\|b \:= a*b; &3\|4
12

It might be easier to use the direct mode for this as written in the Maxima documentation.

>::: infix("|",115)$ a|b := a*b$ 3|4
12

Furthermore, &!... will prevent any replacements in the symbolic expression. (If you need an expression to start with ! enter &!!...)

>&!(a+b)|(a+b)
2
(b + a)

There are some changes in the web pages that are installed with EMT. Now, links on these pages show the installed documentation. There is a link to the web site in the link box. The PDF files are also linked to the web site, as well as the search results.

There have been some fixed in the Maxima interface, nothing the ordinary user would notice.

In purely symbolic functions := is now replaced by : to make those mre readable. If you do not want this use the direct mode ":::".

>function cfx(n) &&= block([x], ...
for i:=1 thru n do x:=1+1/x, x);
>&cfx(3)
1
--------- + 1
1
----- + 1
1
- + 1
x

It is now possible to define purely symbolic strings with the following syntax. Note that s in the following definition is a string, not an expression.

>s &&= ""sin(pi/5)""; &s
sin(pi/5)

The function frac() got a a local epsilon.

>frac(1/7+1e-9,eps=1e-4), frac(1/7+1e-9)
1/7
20408163/142857140