Euler
ReferenceThis file contains main topics for the help window in EMT Math Toolbox (EMT). The list of topics opens when the help window is opened and if the search string is empty.
In the help window, you can double click on any of the topics below. Double click the breadcrumbs in the first line to go back to a previous topic.
Overview - The ideas behind EMT
User Interface - About the Windows and Menus
Help - Help for the Help Window
Keyboard - Keys for the Command line
Comments - Syntax for Comment Sections in Notebooks
Variables - Variables and Data Types in EMT
Data Types - Syntax for Data Types in EMT
Formats - Formatting the Output in EMT
Symbolic Variables - Symbolic Variables
Expressions - Expressions in EMT and Maxima
Matrix Language - The matrix language oF EMT
Calculus - Derivatives, Integrals, Differential Equations
Solve Equations - Solve Equations and Systems of Equations
Plots - Plots in 2D and 3D
Algebra - Symbolic Manipulation of Expressions
Linear Algebra - Vectors and Matrices in Linear Algebra
Statistics - Statistics in EMT
Optimization - Linear and non-linear Optimization
Regression - Linear Regression
Polynomials and Splines - Polynomials and Splines in EMT
Programming Language - Functions in EMT
Scripts - External *.e files for EMT
Installation - Installation of EMT and additional Modules
Notebooks - Commands and Comments in Notebooks
Menu - The Menu of of the Notebook Window of EMT
Graphics - Save and Export Graphics of EMT
EMT (Euler Math Toolbox) is a program to do interactive mathematics on all levels. It is named after the mathematician Leonhard Euler (1707-1783) to honor his achievements for modern mathematics.
For starters it is strongly recommended to open the tutorial about first steps in EMT via the menu entry
File/Open Tutorial or Examples
The core of EMT is its notebook interface which can contain
It can be saved in HTML or PDF format (using Latex).
From the notebook, the user can access the numerical kernel of EMT with its matrix language, the programming language of EMT, and the symbolic Algebra (using the open CAS Maxima which is installed with EMT), as well as many other programs and features like
The underlying numerical kernel comes with
and numerous built-in functions for all kinds of applications, mostly written in efficient C or in the programming language of EMT. The matrix language is implemented in a very systematic and easy to grasp way.
The programming language stores its code in Euler files (*.e). It is an advanced Basic dialect. An Euler file can contain any code and several functions. It is loaded using the "load" command. The functions can be listed in the help window using the filename (Try "sets.e").
The graphics interface can generate 2D and 3D graphics with an anti-aliased view. The graphics can be contained in the notebook or exported to files as PNG, SVG, Bitmap or Windows Metafile.
For a start, the best suggestion is to go through one of the tutorials that come with Euler Math Toolbox.
For more information see:
User Interface - The UI in more details Notebooks - All about notebooks in EMT
In the center of the simple to use user interface is the notebook window which contains
EMT is command line oriented. The cursor (up or down, page up or down) moves between the command lines. A command line can be a multi-line. Pressing return executes the command. The position of the cursor does not matter.
To ease editing, EMT uses dialogs. There are
Functions can also be edited in the notebook directly.
The graphics window of EMT can be hidden using Ctrl-G. In this case, the graphics appears in the notebook window using the TAB key. In any case, the graphics can be inserted in the notebook by ending a command with ":".
EMT can do copy and paste (Ctrl-C, Ctrl-X, Ctrl-V). Alternatively, Ctrl-Backspace deletes lines and Ctrl-U inserts all deleted lines at once. Ctrl-Up will recall the previous command into the current command line. Return executes the line or multi-line and Shift-Return executes all commands in the current section (between headings or empty lines).
F1 or double clicks open this help window. Also note the status line with a short help of the current line and a possible suggestion while typing.
Notebooks can be saved and loaded (which starts a new EMT session by default). They can also be exported as HTML pages using a CSS file which can be modified by the user. It is possible to open one second notebook in a separate session. EMT then shows a boundary which indicates that this is not the primary session of EMT.
See:
Exports (Overview),
Menu (Overview)
To store a value into a numerical variable, use := or =.
>a := 3.5 >b = 5 // alternative >v = 1:5 // variable containing a vector
Names of variables can contain
For more Information see
Symbolic Variables - Variables in Maxima
Data Types - data types of EMT
Variables in Functions - about local and global Variables
Matrices - about matrices and sub-matrices
From the viewpoint of the user, EMT has the following data types.
Numerical vectors are just matrices with one row or column. Column and row vectors make a difference. Collections are just immutable collections of other data. There are also expandable lists via glist().
Here are some typical examples of values in EMT.
>-1.234 // decimal dot, not comma!
-1.234
>1.45e22 // 1.45*10^22
1.45e+22
>[1,2,-5,6] // 1x4 vector
[1, 2, -5, 6]
>[1,0;0,1] // identity matrix id(2)
1 0
0 1
>5+6i // complex number 5+6*I
5+6i
>[I,2*I,3+4*I] // complex vector
[ 0+1i, 0+2i, 3+4i ]
>Pi, E, I // constant functions
3.1416
2.7183
0+1i
>"This is a string"
This is a string
>''Say "Hello!"'' // string containing "
Say "Hello!"
>["Vector","of","strings"]
Vector
of
strings
>~1.5,1.7~ // interval
~1.5,1.7~
>1 0.1 // interval, use F8 for the plus/minus
~0.9,1.1~
>~Pi~ // small interval around pi
~3.141592653589792,3.141592653589794~
The numerical data types of EMT are based on the IEEE standard with an accuracy of 16 decimal digits. The default output does not show all significant places.
>1/3 // default output of a single number
0.333333333333
>longest 1/3 // all digits
0.3333333333333333
>fraction 1/3 // output as fraction
1/3
>shortest pi // very short output
3.1
>long exp(1+2*I) // complex number, long output
-1.13120438376+2.471726672i
>sin(pi), longest sin(pi) // rounding towards 0
0
1.224646799147353e-16
There are more formats for the output and input.
>print(1000*1.03^5,2,15," Euro") // print function
1159.27 Euro
>degprint(21.567 )
21 34'1.20''
>printdate(date(day(2011,9,11,12)+1000))
2014-06-07 12:00
>printhex(sqrt(2))
1.6A09E667F3BCD*16^0
To enter vectors and matrices use [].
>shortformat;
>A=[1,2,3;4,5,6] // 2x3 matrix
1 2 3
4 5 6
>A[1] // first row
[1, 2, 3]
>A[,2] // second column
2
5
>A[1,2:3] // first row, column 2 to 3
[2, 3]
>[1,2,3;4] // second row auto-filled with 0
1 2 3
4 0 0
>[1,2;3,I]; // complex matrix
>v=1:10 // row vector 1 to 10
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
>v[5:7] // elements 5 to 7
[5, 6, 7]
EMT can also use strings and vectors of string.
>"Just a string!"
Just a string!
>"Strange character: "+char(156) // number to character
Strange character:
>strtochar("Help") // get numbers
[72, 101, 108, 112]
>r=6.7; "Area of circle with radius "+r+" = "+(pi*r^2)
Area of circle with radius 6.7 = 141.02609422
There are also intervals. The rule of operators and functions is that all possible points in the intervals are used.
>~-1,2~*~-1,2~ // all x*y with -1 <= x,y <=2
~-2,4~
>~-1,1~^2 // all x^2 with -1 <= x <= 1
~0,1~
>exp(~-1,1~) // all exp(x) with -1 <= x <= 1
~0.36,2.8~
For more information see
Interval Arithmetic - Guaranteed Inclusions
Matrix Language - Matrix Language of EMT
Expressions - Expressions in EMT
We have the following assignment operators in EMT.
Symbolic variables, created with "&=" are defined in the Maxima subsystem as expressions and in EMT as strings (with a symbolic flag).
>f &= x^3-x
3
x - x
>df &= diff(f,x) // evaluates, then assigns to df
2
3 x - 1
>df // prints as display formula
2
3 x - 1
>df(5) // evaluates with x=5 like a string expression
74
Symbolic expressions of Maxima (defined by "&=") can be used in EMT as expressions for numerical algorithms, since there is a symbolic string variable in EMT with the same name.
>f &= exp(x)*(x^2+1);
>df &= diff(f,x);
>solve(df,1,y=2) // solve f'(x)=2 numerically
0.248792696686
>plot2d(df,-3,1): // plot f'(x)
In EMT, a symbolic variable (set with "&=") is simply a string.
>f &= x^3-x;
>"String: "+f
String: x^3-x
If you need a value in EMT and Maxima, use "&:=". Then the variable on the numerical side of EMT is no longer a symbolic string, but a value.
>A &:= [1,2;3,4] // A in both worlds
1 2
3 4
>eigenvalues(A) // can be used numerically
[ -0.372281+0i , 5.37228+0i ]
>&eigenvalues(A) // can be used symbolically
5 - sqrt(33) sqrt(33) + 5
[[------------, ------------], [1, 1]]
2 2
>A.A // numerical result
7 10
15 22
>&A.A // symbolic result
[ 7 10 ]
[ ]
[ 15 22 ]
A variable can also be defined just for Maxima with &&= (purely symbolic variable). Note that "&&=" does not evaluate the right hand side symbolically before the definition. It evaluates when it is called at run-time.
>expr &&= integrate(x^3,x);
>&expr
4
x
--
4
For more information, see
Symbolic Functions - programming Maxima
Maxima at Compile Time - create advanced functions
Variables - non-symbolic variables
Appending a unit converts to the IS system.
>3.56yards // three light years in meters
3.255264
>5.689km/h // int m/sec
1.58027777778
The -> operator converts units. If the conversion goes to a string, the units in this string will be used, and the units will be printed after the result.
>2km/h -> miles/min
0.0207123730746
>4.56km/h -> " in/sec"
49.8687664042 in/sec
For a list of all units see the file units.e.
See:
units (Euler Core),
units (Maxima Documentation)
EMT has vectors and matrices. Vectors are just 1xn (row vectors) or nx1 (column vector) matrices. To enter a matrix, separate the elements by a comma and the rows by a semicolon.
>A = [1,2;3,4]
1 2
3 4
>v = [5,6] // row vector
[5, 6]
>w = [5;6] // column vector
5
6
>v.A // A.v does not work !!!
[23, 34]
>A.v' // v' is the transpose of v
17
39
>A.w
17
39
>w'.w // scalar product
61
Note that . is the matrix product. The usual operators and functions work element for element. This is the matrix language of EMT.
>sqrt(A) // takes the square root of all elements of A
1 1.41421
1.73205 2
>A*A // squares all elements of A
1 4
9 16
Matrices can also be formed of vectors and other matrices.
>A=[1,2;3,4]; // 2x2 matrix
>b=[1;2]; // column vector
>A|b // horizontal concatenation
1 2 1
3 4 2
>A_b' // vertical concatenation
1 2
3 4
1 2
>[b,b^2,b^3] // matrix of columns
1 1 1
2 4 8
>[b';b'^2] // matrix of rows
1 2
1 4
Matrix elements or sub-matrices are referred to with square brackets. For vectors, only one index is necessary, of course. For matrices, one index refers to the complete line.
>A=[1,2;3,4]; // 2x2 matrix
>b=[1;2]; // column vector
>A|b // horizontal concatenation
1 2 1
3 4 2
>A_b' // vertical concatenation
1 2
3 4
1 2
>[b,b^2,b^3] // matrix of columns
1 1 1
2 4 8
>[b';b'^2] // matrix of rows
1 2
1 4
Matrices can be assigned to sub-matrices, if they fit. Values can be assigned to sub-matrices.
>A=[1,2,3;4,5,6;7,8,9]
1 2 3
4 5 6
7 8 9
>A[1:2,1:2]=[3,4;5,6]
3 4 3
5 6 6
7 8 9
>A[1]=A[1]+A[2]
8 10 9
5 6 6
7 8 9
>A[2]=-1
8 10 9
-1 -1 -1
7 8 9
There are numerous functions to generate or handle matrices and vectors. Here are just a few examples.
>setdiag(setdiag(2*id(4),-1,1),1,1) // tri-diagonal matrix
2 1 0 0
1 2 1 0
0 1 2 1
0 0 1 2
>diagmatrix(1:4)
1 0 0 0
0 2 0 0
0 0 3 0
0 0 0 4
>A=normal(3,3); A=A/getdiag(A) // normalize diagonal to 1
1 -7.37532 1.1013
2.80451 1 1.56574
0.461699 -0.0330072 1
>nonzeros(isprime(1:20)) // nonzero indices of vector
[2, 3, 5, 7, 11, 13, 17, 19]
>A=random(3,3); k=mnonzeros(A>0.5) // nonzero indices of matrix
1 2
1 3
2 2
2 3
3 1
3 3
>A
0.262253 0.866587 0.536137
0.493453 0.601344 0.659461
0.967468 0.193151 0.935921
>mset(A,k,0) // set indices to value or matrix
0.262253 0 0
0.493453 0 0
0 0.193151 0
See:
Basic Rule: Operators and functions are applied element-wise to their operands (vectors or matrices), and the result is a matrix of the same size.
In case, the operands or parameters do not fit in size, EMT tries to expand the smaller one to the size of the larger one by duplication.
>sqrt(1:5)
[1, 1.41421, 1.73205, 2, 2.23607]
>(1:5)*4
[4, 8, 12, 16, 20]
>(1:5)^2
[1, 4, 9, 16, 25]
>short (1:5)*(2:6)'
2 4 6 8 10
3 6 9 12 15
4 8 12 16 20
5 10 15 20 25
6 12 18 24 30
If one of the operands is only a vector, the other a matrix, the operators work, as if the vector was duplicated to the size of the matrix.
E.g., if a column vector v and a matrix A are combined, the result is a matrix R[i,j], applying the operator to v[i] and A[i,j]. If a row vector v and a column vector w are combined the result combines v[i] with v[j].
All matrices must have the same size. Row vectors must have the same number of rows as any parameter matrix, and column vectors the same number of columns. If the operands do not fit together, there will be an error message.
Scalar values can be combined with any matrix or vector. In this case, the scalar is combined with all elements of the matrix or vector.
EMT can do calculus symbolically and numerically. Have a look at the following sections to learn more.
Derivatives - Compute Derivatives
Integrals - Compute Integrals
Differential Equations - Solve Differential Equations
For symbolic methods, you can also refer to
For general information about numerical methods in EMT, there are tutorials, examples and references.
Symbolic differentiation is done with &diff by Maxima. Here are some examples.
>expr&=x^2*cos(x*y);
>&diff(expr,x) // derivative at s
2
2 x cos(x y) - x y sin(x y)
>&diff(expr,x,n) // n-th derivative at x
n
d 2
--- (x cos(x y))
n
dx
>&diff(expr,x,1,y,1) // derivative to x and y
2 3
- 3 x sin(x y) - x y cos(x y)
>&diffat(expr,x=pi) // derivative to x at a
2
2 pi cos(pi y) - pi y sin(pi y)
>&diff(expr,x) with [x=pi,y=5] // alternative
- 2 pi
>&diffat(expr,x=pi,5) // n-th derivative to x at a
2 5 3 4
- pi y sin(pi y) + 20 y sin(pi y) + 10 pi y cos(pi y)
There are functions for the gradient, the Jacobian and the Hessian matrix.
>expr &= x^2-x*y+y
2
- x y + y + x
>&gradient(expr,[x,y])
[2 x - y, 1 - x]
>sol &= solve(gradient(expr,[x,y]))
[[y = 2, x = 1]]
>&hessian(expr,[x,y]) with sol[1]
[ 2 - 1 ]
[ ]
[ - 1 0 ]
Numerical differentiation is done with "diff". Note that numerical differentiation is numerically inaccurate or very expensive.
>diff("x^x",1)
1
>function f(x,a) := x^x*sin(a*x)
>diff("f",1;5) // e.g. with a=5
0.459386652651
>diff("x^x",1,2) // n-th derivative at a
1.99999999971
See:
Integrals (Overview)
Integration is available symbolically and numerically with a variety of methods.
For symbolic integration, use &integrate.
>&integrate(expr,x) // anti-derivative
2 3
x y x
- ---- + x y + --
2 3
>&integrate(expr,x,a,b) // definite integral
2 3 2 3
(3 a - 6 a) y - 2 a (3 b - 6 b) y - 2 b
--------------------- - ---------------------
6 6
>&assume(y>0); &integrate(x^y,x) // necessary sometimes
y + 1
x
------
y + 1
Or you can define an integral first, then evaluate it.
>function f(x) &= x*cos(x)
x cos(x)
>function F(x) &= integrate(f(x),x)
x sin(x) + cos(x)
>F(pi)-F(0) // works better sometimes
-2
>F &= integrate(f(x),x) // alternative way using an expression
x sin(x) + cos(x)
>&(F with x=b)-(F with x=a) // evaluate expressions
b sin(b) + cos(b) - a sin(a) - cos(a)
For numerical integration, the default integrate() uses an adaptive Gauss scheme.
>integrate("exp(-x^2)",0,1) // e.g.
0.746824132812
>function f(x,a) := exp(-x^2*a)
>integrate("f",0,1;5) // e.g. with a=5
0.395712309611
Some more methods for integration are available. (gauss, gauss5, romberg, simpson, adaptiveint). Some are from the AlgLib library (alintegrate, alsingular). The fastest method is gauss5.
>gauss5("exp(-x^2)",0,1) // 5 evaluations of the function
0.746824468131
>gauss5("exp(-x^2)",0,1,20) // 20 subintervals, 100 evaluations
0.746824132812
There are also advanced methods using intervals for guaranteed inclusions.
>mxmiint("exp(-x^2)",0,1)
~0.746824132812408,0.746824132812446~
The results are very narrow intervals. But the effort is big. igauss() is another function which gets guaranteed bounds.
>expr &= exp(-x^2);
>igauss(expr,0,1,10,&diff(expr,x,20)) // compute 20-th derivative
~0.7468241328124243,0.7468241328124299~
>igauss(expr,0,1,10,dexpr20est=1e12) // provide estimate
~0.7468241328124243,0.7468241328124299~
Differential Equations can be solved by Maxima symbolically up to degree 2 with ode2.
>&ode2('diff(y,x,2)+'diff(y,x)+y=x^2,y,x)
- x/2 sqrt(3) x sqrt(3) x 2
y = E (%k1 sin(---------) + %k2 cos(---------)) + x - 2 x
2 2
To solve for initial values, there are ic1 (for one initial value) and ic2 (for two).
>sol &= ode2('diff(y,x,2)+y=x^2,y,x) // general solution
2
y = %k1 sin(x) + %k2 cos(x) + x - 2
>ysol &= ic2(sol,x=0,y=0,'diff(y,x)=1) // set initial values
2
y = sin(x) + 2 cos(x) + x - 2
>function f(x) &= y with ysol // define a function from ysol
2
sin(x) + 2 cos(x) + x - 2
Maxima has some more advanced functions for differential equations, including systems of differential equations (&desolve, &ode2, &ic1, &ic2, &bc2).
Numerical solutions can be obtained with many functions in EMT. One is ode(), which uses the LSODA algorithm.
>t=0:0.01:5; plot2d(t,ode("-x*y",t,1)):
Second order differential equations need to be translated to systems of equations.
>function f(x,y) := [y[2],sin(y[1])*x]
>t=0:0.01:5; y=ode("f",t,[1,0]);
>plot2d(t,y[1]):
EMT has more functions for differential equations, including interval functions with guaranteed inclusions (runge, adaptiverunge, heun, lsoda, mxmidgl). For the adaptive versions, you do not need intermediate points.
>lsoda("-x*y",[0,1],1)[-1] // last value of solution, i.e. y(1)
0.606530659726
>adaptiverunge("-x*y",[0,1],1)[-1] // adaptive Runge method
0.606530659715
The function mxmidgl() calls Maxima for a Taylor development of high degree.
>mxmidgl("-x*y",0:0.01:1,1)[-1]
~0.60653065971257,0.6065306597127~
In the numerical kernel of EMT, expressions are strings. If they are marked as symbolic they print via Maxima, otherwise via EMT. Expressions in strings are used for plotting and many numerical functions. For this, the variable in the expression has to be "x".
>expr := "x^3-x" // expression in string
x^3-x
>plot2d(expr,-1,1): // plot expression
>expr &= x^3-x // symbolic expression
3
x - x
>expr // prints via Maxima
3
x - x
>plot2d(expr,-1,1): // plot it
Expressions can be evaluated numerically. The variables x,y,z are assigned automatically. Other variables can be assigned by assigned parameters or via global variables.
>expr &= a*x^2
2
a x
>a=4; expr(2.5) // use global variable
25
>expr(2.5,a=5) // use assigned variable
31.25
>"x*y^2"(3,4) // direct evaluation
48
Thus many algorithms of EMT can use expressions in x instead of functions. But as soon as additional parameters are involved, which are not global, functions should be preferred.
>a=5; plot2d("a*x^3-x",0,1): // use global "a"
>function f(x,a) := a*x^3-x
>plot2d("f",0,1;4): // "use a=4" as semicolon parameter
>plot2d({{"f",4}},0,1): // Alternative
>plot2d("f(x,4)",0,1): // Alternative
Symbolic expression can be evaluated by Maxima with "with". This is a nice way in EMT instead of the use of "at".
>expr &= exp(-x^2)
2
- x
E
>&expr with x=a
2
- a
E
>&expr with x=5.67
1.09121450630144e-14
>&expr with x=sqrt(log(y))
1
-
y
>&x^2+y^2 with [x=5,y=4]
41
>&at(x^2+y^2,[x=5,y=4]) // original Maxima syntax
41
"with" is a simple way to use results of "solve".
>&solve(x^3-2*x+1,x), &x with %[1]
- sqrt(5) - 1 sqrt(5) - 1
[x = -------------, x = -----------, x = 1]
2 2
- sqrt(5) - 1
-------------
2
>&solve([x+y=5,x-2*y=4]); &[x,y] with %[1]
14 1
[--, -]
3 3
A special problem worth mentioning here is the conflict between an expression and a function with the same name. To avoid this, use names like fx, fxy or expr, which are not used for functions. If a function and an expression with the same name exists, the expression will be used.
For more information about differential equations refer to the tutorials or the reference.
EMT Math Toolbox has plots in 2D and 3D, both for data and functions. Click on the sections below.
Plots in 2D - Plots in the 2D plane
Plots in 3D - Plots in the 3D space
Implicit Plots - Plots of level lines of functions
Logarithmic Plots - Plots with logarithmic scales
Styles of 2D Plots - Setting own plot labels, grid styles or axis ticks
Statistical Plots - Plots for statistical data
Plot Aspect - Aspect Ratio of the plot window
Basic Plots - Very basic functions for plots
It is possible to plot in Maxima using Gnuplot. or in Python using Math Plot Lib. But we do not cover these subjects here.
For more information on plots, refer to
For planar plots, EMT uses the function plot2d. This function can plot functions and data.
A single expression in "x" (e.g. "4*x^2") or the name of a function (e.g. "f") produces a graph of the function.
>a:=5.6; plot2d("exp(-a*x^2)/a");
>p=chebpoly(10); plot2d("polyval(p,x)",-1,1): // plot polynomial
>plot2d(''integrate("sin(x)*exp(-x^2)",0,x)'',0,2): // plot integral
Functions like "plot2d" that expect other functions or expressions as an argument can pass additional parameters to the argument function with semicolon parameters or with a call collection.
>function f(x,a) := x^2/a+a*x^2-x; // define a function
>a=0.3; plot2d("f",0,1): // plot with a=0.3
>plot2d("f",0,1;0.4): // plot with a=0.4
>plot2d({{"f",0.2}},0,1): // Alternative: plot with a=0.2
>plot2d({{"f(x,b)",b=0.1}},0,1): // plot with 0.1
Symbolic expressions can also be used, since they are stored as simple string expressions if they are defined with "&=".
>a:=5.6; expr &= exp(-a*x^2)/a // define expression
2
- a x
E
-------
a
>plot2d(expr,-2,2): // plot from -2 to 2
>plot2d(expr,r=1,thickness=2): // plot in a square around (0,0)
>plot2d(&diff(expr,x),>add,style="--",color=red): // add another plot
>plot2d(&diff(expr,x,2),a=-2,b=2,c=-2,d=1): // plot in rectangle
>plot2d(&diff(expr,x),a=-2,b=2,>square): // keep plot square
For data plots, you need one or more rows of x-values, and one or more rows of y-values. For point plots use ">points", for mixed lines and points use ">addpoints".
>xdata=[1,1.5,2.5,3,4]; ydata=[3,3.1,2.8,2.9,2.7]; // data
>plot2d(xdata,ydata,a=0.5,b=4.5,c=2.5,d=3.5,style="."); // lines
>plot2d(xdata,ydata,>points,>add,style="o"): // add points
>p=polyfit(xdata,ydata,1); // get regression line
>plot2d("polyval(p,x)",>add,color=red): // add plot of line
Thus, we can also plot curves or filled curves.
>t=linspace(0,2pi,1000); // parameter for curve >x=sin(t)*exp(t/pi); y=cos(t)*exp(t/pi); // x(t) and y(t) >plot2d(x,y,r=10): // plot curve >plot2d(x,y,r=10,>filled,style="/",fillcolor=red): // fill curve
The plot of multiple functions can be done with >add, possibly using a loop.
>plot2d("(1-x)^10",0,1); // plot first function
>for i=1 to 10; plot2d("bin(10,i)*x^i*(1-x)^(10-i)",>add); end;
>insimg;
But it can also be done with a matrix of x-values, and a matrix of y-values.
>x=linspace(0,1,500); >n=10; k=(0:n)'; // n is row vector, k is column vector >y=bin(n,k)*x^k*(1-x)^(n-k); // y is a matrix then >plot2d(x,y):
Another method is a vector of expressions.
>plot2d(["sin(x)","cos(x)"],0,2pi): // plot vector of expressions We can get a vector with symbolic expressions from Maxima using makelist() and mxm2str(). >v &= makelist(binomial(10,i)*x^i*(1-x)^(10-i),i,0,10) // make list >mxm2str(v) // get a vector of strings from the symbolic vector >plot2d(mxm2str(v),0,1): // plot functions
If the parameter >points or >addpoints is present, the plot will show points with a style given in the parameter pointstyle.
>k=0:10; x=bin(10,k); plot2d(k,x,a=0,b=10,c=0,d=300,>points,,style="[]"): >x=-1:0.1:1; y=x^2; plot2d(x,y,>addpoints,style="--",pointstyle="x"):
It is possible to plot bars using >bar. There is also the statistical plot function plotbars() for the same purpose. There are also automatic histograms.
>plot2d(normal(10000),distribution=30);
>plot2d("qnormal(x)",thickness=5,color=red,>add):
>plot2d(random(100),>histogram,>even):
>x=intrandom(100,10);
>columnsplot(getmultiplicities(1:10,x),1:10):
For more information, refer to the tutorial about 2D plots, or the documentation about plot2d().
plot2d - Reference to the main function for 2D Plots
Plot Aspect - Change the aspect ratio of the plot window
Plots - Overview of Plots
EMT uses the "logplot" parameter for logarithmic scales. 1 is for the x-axis, 2 for both axes, and 3 for the x-axis only.
>plot2d("exp(x+sin(x))",0,100,logplot=1):
>plot2d("exp(x+sin(x))",10,100,logplot=2):
>plot2d("log(x*(2+sin(x/100)))",10,1000,logplot=3):
This does also work with data plots.
>x=10^(1:20); y=x^2-x; >plot2d(x,y,logplot=2): Plots - Overview of Plots
By default, EMT computes automatic axis ticks and adds labels to each tick. This can be changed with the grid parameter.
>plot2d("x^3-x",grid=0) // no grid, frame or axis
>plot2d("x^3-x",grid=1) // x-y-axis
>plot2d("x^3-x",grid=2) // default ticks
>plot2d("x^3-x",grid=3) // x-y- axis with labels inside
>plot2d("x^3-x",grid=4) // no ticks, only labels
>plot2d("x^3-x",grid=5) // default, but no margin
>plot2d("x^3-x",grid=6) // axes only
>plot2d("x^3-x",grid=7) // axes only, ticks at axis
>plot2d("x^3-x",grid=8) // axes only, finer ticks at axis
>plot2d("x^3-x",grid=9) // default, small ticks inside
>plot2d("x^3-x",grid=10) // no ticks, axes only
The parameter <frame switches off the frame, and framecolor=blue sets the frame to a blue color.
If you want your own ticks, you can use style=0, and add everything later.
>plot2d("x^3-x",grid=0); // plot
>frame; xgrid([-1,0,1]); ygrid(0): // add frame and grid
For the plot title and labels of the axes, see the following example.
>plot2d("exp(x)",-1,1);
>textcolor(black); // set the text color to black
>title("y=exp(x)"); // title above the plot
>xlabel("x"); // "x" for x-axis
>ylabel("y",>vertical); // vertical "y" for y-axis
>label("(0,1)",0,1,color=blue); // label a point
>reset: // to revert the text color
The axis can be drawn separately with xaxis() and yaxis().
>plot2d("x^3-x",<grid,<frame);
>xaxis(0,xx=-2:1,style="->"); yaxis(0,yy=-5:5,style="->"):
Text on the plot can be set with label(). In the following example, "lc" means lower center. It sets the position of the label relative to the plot coordinates.
>function f(x) &= x^3-x
>plot2d(f,-1,1,>square);
>x0=fmin(f,0,1); // compute point of minimum
>label("Rel. Min.",x0,f(x0),pos="lc"): // add a label there
There are also text boxes.
>plot2d(&f(x),-1,1,-2,2); // function >plot2d(&diff(f(x),x),>add,style="--",color=red): // derivative >labelbox(["f","f'"],["-","--"],[black,red]): // label box
There are interactive sliders, which can set values by the user. The function dragvalues() provides this.
>function plotf([a,b]) := plot2d("exp(a*x)*cos(2pi*b*x)",0,2pi;a,b);
>dragvalues("plotf",["a","b"],[-1,2],[[-2,2];[1,10]], ...
> heading="Drag these values:",hcolor=black):
Plots - Overview of Plots
To draw the set f(x,y)=c for one or more constants c you can use plot2d() with its implicit plots in the plane. The parameter for c is level=c, where c can be vector of level lines. Additionally, a color scheme can be drawn in the background to indicate the value of the function for each point in the plot. The parameter "n" determines the fineness of the plot.
>expr := "2*x^2+x*y+3*y^4+y"; // define an expression f(x,y) >plot2d(expr,level=0): // Solutions of f(x,y)=0 >plot2d(expr,level=0:0.5:20,>hue,contourcolor=white,n=200): // nice >plot2d(expr,level=0:0.5:20,>hue,>spectral,n=200,grid=4): // nicer
This works for data plots too. But you will have to specify the ranges for the axis labels.
>x=-2:0.05:1; y=x'; z=expr(x,y); >plot2d(z,level=0,a=-1,b=2,c=-2,d=1,>hue): >z=z+normal(size(z))*0.2; >plot2d(z,level=0.5,a=-1,b=2,c=-2,d=1):
It is possible to fill regions of values for a specific function. For this, level must be a 2xn matrix. The first row are the lower bounds and the second row contains the upper bounds.
>plot2d(expr,level=[0;1],style="-",color=blue): // 0 <= f(x,y) <= 1 >plot2d(expr,level=[0:0.2:5;0.05:0.2:5.05],color=lightgray):
The following is the set of the maximal value of four linear conditions less than or equal 3. This is A[k].v<=3 for all rows of A. To get nice corners, we use n relatively large.
>A=[2,1;1,2;-1,0;0,-1];
>function f(x,y) := max([x,y].A');
>plot2d("f",r=4,level=[0;3],color=green,n=111):
3D Plots in EMT are done in central projection. The algorithm used is very quick and usually produces very nice plots. For photo-realistic output, you can install Povray and use it from EMT. EMT can also do anaglyph plots for red/cyan glasses.
The most elementary form shows the graph of a function in x and y and uses an expression in x and y or a function. The plot is scaled to fit into the unit cube, so that the default viewing parameters work well. These parameters can be changed.
>plot3d("x^2+y^2"):
Plots can have contour lines and shading. The contour lines can be thin or show a range of z-values.
>plot3d("x*abs(y)^1.4", ...
> contour,>spectral,angle=-60 ,height=20 ,scale=1.1):
>plot3d("x^2+y^3",>contour,angle=30 ,level=-2:0.2:2):
>plot3d("x^y-y^x",a=0,b=4,c=0,d=4,n=200,level=0,light=[0,1,0]):
>function f(x,y,a) := x^2+y^a
>plot3d("f";4.5,a=0,b=1,c=0,d=1,>contour,angle=20 ): // with a=4.5
>plot3d("x^2+y^2",>spectral,n=100):
>plot3d("x^2+y^2",>zhue,levels=-2:0.2:2,dl=0.01,n=100):
For surfaces which are not the graph of a function, three functions or
expressions can be used. In this case, the user needs to scale the plot
properly to fit into the default view area.
>plot3d("x","y","x*y",xmin=-2,xmax=2,ymin=-2,ymax=2, ...
> scale=[2,2,1],n=10):
>plot3d("x","cos(y)","sin(y)",xmin=-2,xmax=2,ymin=0,ymax=2pi, ...
> scale=[1.5,1.5,1.5],n=100,grid=10,fillcolor=[gray,black]):
Matrices of x-, y-, and z-coordinates can be used to represent the surface. The matrix language of EMT allows that x can be row vector and y a column vector only. The grid determines the number of grid lines and should be a multiple of the number of intervals.
>x=-2:0.05:2; y=x'; plot3d(x,y,x*y,scale=[2,2,1]; grid=10):
For an example with shading, we plot a ball.
>s=linspace(0,2pi,180); t=linspace(-pi/2,pi/2,90)'; >plot3d(cos(s)*cos(t),sin(s)*cos(t),sin(t), ... > >hue,grid=[6,4],<frame,color=blue,scale=1.5,height=45 ,angle=45 ): It is possible to shade with different values, and use different values for the levels. Then the plot can be limited to a range of values.
>x=-1:0.02:1; y=x'; z=x+y+(x-1)^2; >plot3d(x,y,z,scale=[2,2,1],hues=scalematrix(x+y),>spectral, ... > levels=[-1:0.2:1],values=x+y,limits=[-1,1], ... > angle=40 ):
For clouds of points or lines, use >points or >wire.
>A=normal(3,500); plot3d(A[1],A[2],A[3],>points): // cloud of points >W=cumsum(A); plot3d(W[1],W[2],W[3],>wire): // path in 3D
For more information, refer to the tutorial for 3D plots or the documentation of plot3d.
Plot Aspect - Change the aspect ratio of the plot window
plot3d - Main function for 3D plots
Plots - Overview of Plots
The default plot uses a square plot window. The layout of the graphics depend on the aspect ratio and the selected font, because space is needed for the title and the labels. The function fullwindow() will get rid of this extra space if you do not want to use a title or labels.
You can change this with the function aspect() which also refreshes the layout of plots. With a parameter it sets the width/height ratio. Without parameters, aspect() is resetting to the default aspect ratio.
>aspect(2); >plot2d(["sin(x)","cos(x)"],0,2pi): >aspect();
In 3D Plots, we can change the scaling of the z-values. The following is a polar plot, which graphs the function on a disk.
>aspect(2);
>plot3d("x^2+y^4",fscale=0.4,scale=1.6,>contour,>spectral>polar):
>reset;
The reset() function restores plot defaults including the aspect ratio.
It is also possible to change the aspect ration permanently in the Options Menu. This should only be used if you want to start EMT with a new aspect ratio. The ratio can be tied to the size of the plot window. This is only for small changes which do not need another layout of the plots.
There are many functions which are specialized on statistical plots. For a demonstration, refer to the tutorial on statistics.
One of the often used plots is a column plot.
>columnsplot(cumsum(random(10)),style="/",color=blue):
It can also show strings as labels.
>months=["Jan","Feb","Mar","Apr","May","Jun", ...
> "Jul","Aug","Sep","Oct","Nov","Dec"];
>values=[10,12,12,18,22,28,30,26,22,18,12,8];
>columnsplot(values,lab=months,color=red,style="-");
>title("Temperature"):
To plot an experimental statistical distribution, you can use distribution=n with plot2d.
>w=randexponential(1,1000); // exponential distribution >plot2d(w,>distribution): // or distribution=n with n intervals
Or you can compute the distribution from the data and plot the result with >bar in plot3d, or with a column plot.
>w=normal(1000); // 0-1-normal distribution
>{x,y}=histo(w,10,v=[-6,-4,-2,-1,0,1,2,4,6]); // interval bounds v
>plot2d(x,y,>bar):
The statplot() function sets the style with a simple string.
>statplot(1:10,cumsum(random(10)),"b"):
For more statistical plots, refer to the tutorial on statistics, or the following introduction.
Statistics - Introduction to Statistics in EMT
There are also very basic functions of plots. The following is just a brief introduction into these elementary functions.
plots - Core functions for elementary Plots.
There are screen coordinates, which always range from 0 to 1024 in each axis, no matter if the screen is square or not. Ant there are plot coordinates, which can be set with setplot(). The mapping between the coordinates depends on the current plot window. E.g., the default shrinkwindow() leaves space for axis labels and a plot title.
In the example, we just draw a few random lines in various colors. For details on these functions, study the core functions of EMT.
>clg; // clear screen >window(0,0,1024,1024); // use all of the window >setplot(0,1,0,1); // set plot coordinates >hold on; // start overwrite mode >n=100; X=random(n,2); Y=random(n,2); // get random points >colors=rgb(random(n),random(n),random(n)); // get random colors >loop 1 to n; color(colors[#]); plot(X[#],Y[#]); end; // plot >hold off; // end overwrite mode >insimg; // insert to notebook
It is necessary to hold the graphics, since the plot() command would clear the plot window.
To clear everything we did, we use
>reset;
For another example, we draw a plot as an inset in another plot. This is done by defining a smaller plot window. Note that this window does not provide room for the axis labels outside the plot window. We have to add some margin for this as needed. Note that we save and restore the full window, and hold the current plot while we plot the inset.
>plot2d("x^3-x");
>xw=200; yw=100; ww=300; hw=300;
>ow=window();
>window(xw,yw,xw+ww,yw+hw);
>hold on;
>barclear(xw-50,yw-10,ww+60,ww+60);
>plot2d("x^4-x",grid=6);
>hold off;
>window(ow);
A plot with multiple figures is achieved in the same way. There is the utility figure() function for this.
The Tutorial about plots contains another example of a function made of basic plot routines.
See:
window (Plot Functions),
figure (Plot Functions)
EMT can generate random numbers for various distributions. They are usually named rand... with a few exceptions. E.g.
>mean(normal(100000)) // 0-1-normal distributed values
0.000592833227266
>dev(random(100000)) // uniformly distributed values in ]0,1[
0.288713900972
>w=randnormal(1,100000,1000,10);
> // 1x100000 matrix of 1000-10-normal distr. values
>plot2d(w,>distribution): // plot the distribution
>w=intrandom(1,600,6); // 1x600 matrix of dice throws
>getmultiplicities(1:6,w) // count multiplicities
[97, 109, 95, 104, 112, 83]
>mean(randexponential(1,100000,2)) // exponential distribution
2.00089081545
EMT has many distribution functions and their inverses.
>w=intrandom(1,600,6); // 1x600 matrix of dice throws
>getmultiplicities(1:6,w) // count multiplicities
[97, 109, 95, 104, 112, 83]
>mean(randexponential(1,100000,2)) // exponential distribution
2.00089081545
>normaldis(3) // 0-1-normal distribution function
0.998650101968
>[invnormaldis(2.5%,1000,10),invnormaldis(97.5%,1000,10)]
[980.4, 1019.6]
> // 90% confidence interval of 1000-10-normal distribution
>invchicdis(chicdis(30,10),10) // inverse complimentary chi-square
30
Most density functions are also present. They are named like qnormal().
>plot2d(normal(1000),>distribution); // plot empirical data
>plot2d("qnormal(x,0,1)",>add,color=red,thickness=2):
> // add expected density
Here are some examples of statistical indicators, derived from empirical data.
>x=normal(1000);
>boxplot(x):
>mean(x), dev(x)
-0.0284789448205
1.01524089997
>median(x)
0.00772826335585
>getfrequencies(x,-4:4) // count number of items in intervals
[4, 27, 139, 328, 345, 139, 18, 0]
>covar(x[1:2:999],x[2:2:1000]) // empirical covariance
-0.0408698723263
>quartiles(x) // quartiles, neglecting outliers
[-2.658, -0.696358, 0.00772826, 0.640064, 2.64099]
>quantile(x,10%) // 10% quantile
-1.32974164884
>plot2d(-4:4,empdist(-4:4,sort(x))): // empirical distribution
There are statistical tests too.
>X=normal(3,10); // get 3 rows of data to test for same mean
>varanalysis(X[1],X[2],X[3]+2) // will be rejected
3.66187622312e-05
>mediantest(X[1],X[2]+2) // will be rejected
0.0115070687826
>signtest(X[1],X[2]+2) // will be rejected
0.9990234375
>v=randexponential(1,1000,0.5);
>x=getfrequencies(v,0:0.1:1) // frequencies of exponential data
[171, 169, 123, 85, 81, 66, 51, 41, 47, 37]
>y=ones(1,10)*sum(x)/10 // assumed uniform distribution
[87.1, 87.1, 87.1, 87.1, 87.1, 87.1, 87.1, 87.1, 87.1, 87.1]
>chitest(x,y) // will be rejected
0
For statistical plots refer to
Statistical Plots - Special plots for statistics
For more information about statistics in EMT, refer to
To solve a an equation or a system of equations symbolically, use &solve(). The function returns a vector of solutions. To insert one of the solutions into an expression, use with.
>&solve(diff(x^3-x,x)) // all critical values (assume expr=0)
1 1
[x = - -------, x = -------]
sqrt(3) sqrt(3)
>expr &= x^3-x
3
x - x
>&solve(diff(expr,x)=0,x), &expr with %[1] // value at maximum
1 1
[x = - -------, x = -------]
sqrt(3) sqrt(3)
2
----
3/2
3
Systems are solved in a similar way. The solution is a vector of solutions, which are in turn vector of equations [x=3,y=4] etc.
>sol &= solve([x^2+y=4,x-2*y=a],[x,y]) // system of 2 equations
- sqrt(8 a + 65) - 1 - sqrt(8 a + 65) - 4 a - 1
[[x = --------------------, y = --------------------------],
4 8
sqrt(8 a + 65) - 1 sqrt(8 a + 65) - 4 a - 1
[x = ------------------, y = ------------------------]]
4 8
>&[x,y] with sol[1] with a=4 // get first solution with a=4
- sqrt(97) - 1 - sqrt(97) - 17
[--------------, ---------------]
4 8
To solve an equation numerically, the default function is solve(), which uses a secant algorithm.
>solve("x^x-3",1) // solve near 1
1.82545502292
>solve(&diff(x^x,x),0.5) // with symbolic expression
0.367879441171
>solve(&diff(x^x,x),1,y=2) // target value y=2 near 1
1.36367018703
>function f(x,a) := x^x-a*x
>x1 := solve("f",1,2;0.2,y=1) // with a=0.2 between 1 and 2
1.19639572649
There are more algorithms, some of them with guaranteed inclusions.
>bisect("x^3",1,2,y=2,eps=0.01) // reduced accuracy
1.26171875
>ibisect("x^3",1,2,y=2) // guaranteed inclusion
~1.259921049894,1.259921049897~
>secantin("x^x",0,0.2,y=0.9) // keep solution in interval
0.0300653684896
>expr &= diff(x^x,x)
x
x (log(x) + 1)
>newton(expr,&diff(expr,x),0.5) // Newton for expr=0
0.367879441171
>inewton(expr,&diff(expr,x),0.5) // guaranteed and fast solution
~0.36787944117144222,0.36787944117144245~
To solve a system of equations, Broyden's algorithm is fast and converges for close starting points.
>function f(x) := [x[1]^2+x[2]^2-10,x[1]+2*x[2]-4]
>broyden("f",[0,3]), f(%) // solve and test result
[-1.53238, 2.76619]
[0, 0]
>function f([x,y]) := [x^2+y^2-10,x+2*y-4] // easier definition of f
For the Newton algorithm, we need the Jacobian.
>function f([x,y]) &= [x^2+y^2-10,x+2*y-4] // f
2 2
[y + x - 10, 2 y + x - 4]
>function Df([x,y]) &= jacobian(f(x,y),[x,y]) // Jacobian of f
[ 2 x 2 y ]
[ ]
[ 1 2 ]
>newton2("f","Df",[0,3]) // Newton algorithm
[-1.53238, 2.76619]
>inewton2("f","Df",[0,3]) // guaranteed inclusion
[ ~-1.532380757938123,-1.532380757938117~,
~2.766190378969058,2.766190378969062~ ]
Expanding and factoring of symbolic expressions can be done with very large expressions.
>&factor(diff(expand((1+x)^20),x))
19
20 (x + 1)
>&factor(x^6+x^2+1)
6 2
x + x + 1
>&factor(x^6+x^2+1) | modulus:=13 // factor in Z modulo 13
2 4 2
(x + 6) (x - 6 x - 2)
>&expand((1+x)^4+(1+x)^2+1/(1+x)^2+1/(1+x)^4,2,2) // limit expansion
1 4 1 2
------------ + (x + 1) + -------- + x + 2 x + 1
2 4
x + 2 x + 1 (x + 1)
To factor numbers, use a direct mode.
>:: factor(100!)
97 48 24 16 9 7 5 5 4 3 3 2 2 2 2
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
59 61 67 71 73 79 83 89 97
There are various to extract parts of an expression.
>expr &= expand((a*x^3+a^2*x+b*x+y)*(a*x+y))
2 3 2 2 4 2 3 2
y + a x y + b x y + a x y + a x y + a x + a b x + a x
>&part(expr,2) // second term in expression
3
a x y
>&collectterms(expr,x) // make a polynomial of x
2 2 3 2 4 3 2
y + x (b y + a y + a y) + a x y + a x + (a b + a ) x
>&coeff(expr,x,2) // coefficient of x^2
3
a b + a
>&combine(a/b+c/d+e/b) // combine fractions
e + a c
----- + -
b d
For simplification, there are many methods.
>&ratsimp(1/x+1/x^2+(x+1)/x) // try to simplify a fraction
2
x + 2 x + 1
------------
2
x
>&partfrac((x^2+1)/(x^2-1),x) // sum as partial fractions
1 1
- ----- + ----- + 1
x + 1 x - 1
>&trigsimp((sin(x)+cos(x))*(2*sin(x)+cos(x))) // apply rules for trig
2
sin (x) + 3 cos(x) sin(x) + 1
>&radcan((log(1+2*a^x+a^(2*x))/log(1+a^x))) // apply rules for radicals
2
There are many options for expansions and reductions.
>&trigexpand(sin(5*x)) // make it a polynomial in sin(x), cos(x)
5 2 3 4
sin (x) - 10 cos (x) sin (x) + 5 cos (x) sin(x)
>&trigreduce(sin(x)^5) // make it a trigonometric series
sin(5 x) - 5 sin(3 x) + 10 sin(x)
---------------------------------
16
>&log(a*b) | logexpand:=all // apply rules of log
log(b) + log(a)
>&logcontract(log(x)+log(y)) // collect sums of log
log(x y)
>&sqrt(x^y) | radexpand:=all // apply rules for sqrt
y/2
x
>&multthru(a*x+1,b+c) // multiply into sum
c (a x + 1) + b (a x + 1)
>&distrib((1+x*(2+x))*(1+y)) // apply distrib. rule
x (x + 2) y + y + x (x + 2) + 1
>&rootscontract(sqrt(x)*sqrt(y)) // contract roots
sqrt(x y)
>&scsimp(a*b+b*c,a*b+d=1,b*c=d) // simplify with rules
Do not forget, that you can map a function to each argument.
>&map(sqrt,x+y)
sqrt(y) + sqrt(x)
>&map(ratsimp,(1+x)/(1-x^2)+(x^2+2*x+1)/(x^2-1))
x + 1 1
----- - -----
x - 1 x - 1
You can get examples for manipulations.
>:: example(radcan)
Functions of one variable are minimized with fmin(). With several variables, use the Nelder Mead algorithm or Powell's algorithm
>fmin("x^3-x",0,1), "x^3-x"(%)
0.57735026525
-0.38490017946
>function f([x,y,z]) &= x^2+y^2+z^2+exp(-x*y*z)+x
- x y z 2 2 2
E + z + y + x + x
>nelder("f",[0,0,0]) // start from x=y=z=0
[-0.5, 3.58599e-07, -5.04735e-07]
>powellmin("f",[0,0,0]) // start from x=y=z=0
[-0.5, 1.05376e-08, 1.05376e-08]
>&grad(x^2+y^2+z^2+exp(-x*y*z)+x) with [x=-1/2,y=0,z=0]
[0, 0, 0]
Alternatively, the Broyden algorithm could be used for grad f=0.
>function f([x,y,z]) &= x^2+y^2+z^2+exp(-x*y*z)+x
- x y z 2 2 2
E + z + y + x + x
>function df([x,y,z]) &= grad(f(x,y,z));
>broyden("df",[0,0,0])
[-0.5, 0, 0]
The simplex algorithm can be called via simplex() or the LPSOLVE library.
>A=[1,2;3,4;4,2] // mxn matrix A
1 2
3 4
4 2
>b=[10,8,7]' // column vector b
10
8
7
>c=[1,1] // row vector c
[1, 1]
>simplex(A,b,c,>max) // solve cx->max with Ax<=b, x>=0
1.2
1.1
Here is an example with equalities
>A=random(5,10); b=sum(A);
>c=ones(1,10);
>simplex(A,b,c,eq=0,>max)
0
3.04962
2.00196
0
0.59614
2.43778
0
0
4.16962
0
Another example with a mix of >= and <=.
>A=[1,1;1,1;1,-1;1,-1] // 4 inequalities -1<=x+y<=1, -1<=x-y<=1
1 1
1 1
1 -1
1 -1
>b=[-1,1,-1,1]' // right hand side
-1
1
-1
1
>eq=[1,-1,1,-1]' // equations >=, <=, >=, <=
1
-1
1
-1
>restr=[0,0] // no restriction x>=0
[0, 0]
>c=[1,1.2]; // target function
>simplex(A,b,c,eq,restr,>max) // solve
0
1
For integer problems, use intsimplex() or the efficient LPSOLVE.
>A=random(10,2); b=ones(10,1)*100;
>c=[1,1];
>simplex(A,b,c,>max)
0
121.117
>intsimplex(A,b,c,>max)
0
121
>ilpsolve(A,b,c,>max)
0
121
For non-linear problems, you can use the Newton-Barrier method.
>A=1+random(1000,2); // 1000 linear conditions
>b=100+random(1000,1)*100; // right hand side
>A=A_-id(2); b=b_zeros(2,1); // add x>=0
>function f([x,y]) &= 1/(x*y); // target function
>function df([x,y]) &= gradient(f(x,y),[x,y]); // gradient
>function Hf([x,y]) &= hessian(f(x,y),[x,y]); // Hessian
>newtonbarrier("f","df","Hf",A,b,[1,1])
[29.6815, 25.91]
See:
simplex (Linear Programming)
For more examples and the reference see
Linear regression with polynomials can be done with polyfit().
>function f(x) := 0.8*x+0.1; // correct function
>x=0:0.1:1; y=f(x)+random(cols(x))*0.4; // random values
>plot2d(x,y,>points); // plot values
>p=polyfit(x,y,1) // fit with polynomial of degree 1
[0.374996, 0.691596]
>plot2d("polyval(p,x)",>add,color=red): // add fit to plot
General linear fits can be done with fit(A,y). This minimizes |Ax-y|.
>function f(x) := exp(-0.2*x)+0.5*exp(-0.4*x)+0.3
> // of type a*exp(-0.2*x)+b*exp(-0.4*x)+c
>x=0:5; y=f(x)+random(cols(x))*0.4;
>plot2d(x,y,>points); // plot random data
>A=exp(-0.2*x')|exp(-0.4*x')|1
1 1 1
0.818731 0.67032 1
0.67032 0.449329 1
0.548812 0.301194 1
0.449329 0.201897 1
0.367879 0.135335 1
>a=fit(A,y')
3.25472
-1.47332
0.0104999
>function ff(x) := a[1]*exp(-0.2*x)+a[2]*exp(-0.4*x)+a[3]
>plot2d("ff(x)",>add,color=red):
>mean(ff(x)-y), dev(ff(x)-y) // 0 and norm of difference
0
0.0232198683434
For non-linear fits of all kinds, one choice is Nelder-Mead.
>function f(x) := exp(-0.2*x)+0.5*exp(-0.4*x)
> // of type exp(c*x)+0.5*exp(d*x)
>x=0:5; y=f(x)+random(cols(x))*0.1;
>plot2d(x,y,>points); // plot random data
>function ff(x,[c,d]) := exp(c*x)+0.5*exp(d*x);
>function ferr(v) := sqrt(sum(ff(x,v)-y)^2)
>v=nelder("ferr",[-1,-1])
[-0.119043, -0.768554]
>plot2d("ff(x,v)",>add,color=red):
There are many methods to solve a linear system in EMT.
>A=[1,2;3,4]
1 2
3 4
>b=[1;4]
1
4
>A\b // Gauss algorithm
2
-0.5
>A=redim(1:9,3,3) // singular matrix !
1 2 3
4 5 6
7 8 9
>b=sum(A)
6
15
24
>fit(A+normal(size(A)),b) // fit with orthogonal methods
1.00405
0.618711
1.39271
>svdsolve(A,b) // fit using singular values
1
1
1
For hand work there is a pivotize() function.
>shortformat; A=[0,1,2;3,4,5;1,2,0]; b=sum(A);
>M=A|b // produce the Gauss scheme
0 1 2 3
3 4 5 12
1 2 0 3
>fraction pivotize(M,2,1)
0 1 2 3
1 4/3 5/3 4
0 2/3 -5/3 -1
>fraction pivotize(M,1,2)
0 1 2 3
1 0 -1 0
0 0 -3 -3
>fraction pivotize(M,3,3)
0 1 0 1
1 0 0 1
0 0 1 1
>A=[0,1,2;3,4,5;1,2,0]; M=A|sum(A)
0 1 2 3
3 4 5 12
1 2 0 3
>echelon(M) // all in one step
1 0 0 1
0 1 0 1
0 0 1 1
There are also function to handle a Gauss-Jordan scheme. See the tutorial on Linear Algebra for an example.
There are a lot more algorithms for Linear algebra.
>A=redim(1:9,3,3)
1 2 3
4 5 6
7 8 9
>kernel(A)
1
-2
1
>svdkernel(A) // orthogonal basis of kernel
0.408248
-0.816497
0.408248
>eigenvalues(A), real(%)
[ 16.1168+0i , -1.11684+0i , 0+0i ]
[16.1168, -1.11684, 0]
>{v,X}=eigen(A); v, // LAPACK algorithm for eigenvalues
[ 16.1168+0i , -1.11684+0i , 0+0i ]
>real(inv(X).A.X)
16.1168 0 0
0 -1.11684 0
0 0 0
>{U,v,W}=svd(A);
>v // singular values
[16.8481, 1.06837, 0]
>{R,Q,c}=qrdecomp(A); // QR decomposition
>Q'.R // is A
1 2 3
4 5 6
7 8 9
For symbolic linear algebra, you can define symbolic matrices or transport a numerical matrix to Maxima.
>A &= [1,2,3;4,5,6;7,8,x]
[ 1 2 3 ]
[ ]
[ 4 5 6 ]
[ ]
[ 7 8 x ]
>&determinant(A), &solve(%=0)
- 2 (4 x - 42) + 5 x - 57
[x = 9]
>A=[1,2;3,4]; // numerical matrix
>&invert(@A) // use direct in symbolic expression
[ - 2 1 ]
[ ]
[ 3 1 ]
- - - ]
[ 2 2 ]
>&eigenvalues(@A)
5 - sqrt(33) sqrt(33) + 5
[[------------, ------------], [1, 1]]
2 2
>mxmset("A",random(2,2)) // define a matrix A in Maxima
>&A
[ 0.2003215849526854 0.4025401003732851 ]
[ ]
[ 0.6524835123965044 0.9308647905120194 ]
The rule for interval arithmetic is that operators combine all elements of the parameter intervals and return an interval containing all these results. Functions are applied to all elements of the parameter interval similarly.
>~1,2~ * ~7,8~ // result: ~1*7,2*8~ >~-1,1~ * ~1,1~ // result: ~-1,1~ !!! >~-1,1~^2 // result: ~0,1~ !!! >sin(~-1,1~) // result: ~sin(-1),sin(1)~
Often, we express a function in terms of an expression. Then the rules for expressions overestimate the result of the function. There are clever methods to get closer inclusions.
>expr &= x^4-x^3+x^2+x-1 >x=1 0.02 // another way to enter an interval, use F8 >expr(x) // simple evaluation >expr(fmin(expr,left(x),right(x))) // best lower estimate >expr(fmax(expr,left(x),right(x))) // best upper estimate >ieval(expr,x) // using sub-division of interval
Integrals are good for obtaining guaranteed solutions.
>ibisect("x^3-x",1,2,y=1.4) // interval bisection
>expr &= x^x
>inewton(expr,&diff(expr,x),1,y=1.4) // interval Newton
>mxminewton(expr,1,2,y=1.4) // automatic call of Maxima
>mxmiint("x^x",1,2) // elaborate inclusion of integral
>est20=ieval(&diff(exp(-x^2),x,20),0,2)
> // estimate of 20-th derivative
>igauss("exp(-x^2)",0,2,dexpr20est=est20) // close approximation
>longest integrate("exp(-x^2)",0,2) // for comparison
There are also interval methods for linear system and for non-linear systems of equations.
Numerical polynomials are stored as vectors v of coefficients with the constant coefficient in v[1]. They can be evaluated and manipulated to some degree.
>p=[1,2,3]; // 1+2x+3x^2
>plot2d("evalpoly(x,p)",-1,1):
>plot2d("evalpoly",-1,1;polydif(p)): // derivative of p
>polymult(p,p) // coefficients of p^2
There are some special polynomials.
>chebpoly(10) // coefficients of 10-th Chebychev polynomial >cheb(0.6,10) // evaluate it in 0.6 >legendre(8) // Legendre polynomial
Interpolation can be done with the Newton scheme.
>p=[1,-1,2,-1] // a polynomial
>xdata=1:4; ydata=evalpoly(xdata,p) // evaluated in 1,2,3,4
>dd=divdif(xdata,ydata) // divided differences
>plot2d(xdata,ydata,>points); // plot points
>plot2d("evaldivdif(x,dd,xdata)",>add): // plot interpolation
>polytrans(xdata,dd) // get back the polynomial
>q=polyfit(xdata,ydata,2) // fit polynomial of degree 2
>plot2d("evalpoly(x,q)",color=green,>add): // add to plot
Symbolic Polynomials can be derived from the vector with "sum". The converse can be done with "makelist".
>p &:= chebpoly(10) // vector of coefficients in Maxima and EMT >pol &= sum(p[k]*x^(k-1),k,1,11) // make it a polynomial >&makelist(coeff(pol,x,k),k,0,10), vpol = %() // get the vector back
Furthermore, EMT can compute the zeros of polynomials numerically.
>p=1|zeros(1,10)|1 // z^11 + 1 >plot2d(polysolve(p),>points): // Baurhuber iteration, plot zeros >polysolve(p,>usematrix) // eigenvalues of companion matrix
Interpolation with cubic splines are can be done with "spline".
>xdata=-5:5; ydata=zeros(size(xdata)); ydata[6]=1; // data points
>plot2d(xdata,ydata,a=-5,b=5,c=-2,d=2,>points);
>s=spline(xdata,ydata); // compute spline
>plot2d("evalspline(x,xdata,ydata,s)",>add): // evaluate
>t=linspace(-5,5,1000);
>max(abs(evallinspline(t,xdata,ydata)-evalspline(t,xdata,ydata,s)))
> // maximal distance between lineare and cubic spline
For B-Splines, use nspline().
>plot2d("nspline";(1:4)',2,1:7,a=0,b=8):
The internal accuracy of a number in EMT is the IEEE standard of about 16 decimal digits. EMT does not print all digits by default. This saves space and looks better. For one number, the following operators can be used.
>pi >longest pi >long pi >short pi >shortest pi >fraction 1/3+1/7
To format a number, there are many more functions. The result is always a string. Note that you can append strings with +.
>print(pi,5,10) // with number of digits and total length
>K=1012.56; i=3.75%;
>"E = "+print(K*(1+i),2,unit="$") // with additional unit
>printdollar(K*(1+i)^4) // shortcut
>degprint(pi/4)
>degprint(0.56,<dms) // not with minutes and seconds
>"polar: "+polarprint(1+I)
>frac(1/3)+" + "+frac(1/7)+" = "+frac(1/3+1/7) // convert to frac
>printf("pi = %10.2f",pi) // C style printf
>"x = "+(pi*4+1) // string plus number in default format
You can also set the output format permanently.
>format(5,2); // 5 places with 2 digits after decimal dot >pi >12121.454354 >random(3,3) >1:10 >format(10); // nice format with 10 digits of accuracy >pi
By default, EMT makes a difference between the output of scalars, vectors and matrices. By default, scalars use a longer format, and vectors use a denser format. The function format(n,m) applies to all output, however.
>reset; // reset output >pi // uses the longer scalar format >1:10 // uses a dense vector output >random(3,3) // uses the default matrix format >denseoutput(0); 1:5 // turn off dense output >denseoutput(2); 1:5 // default >setscalarformat(5); pi // use 5 digits of accuracy for scalars
Some commands change the format for the subsequent output.
>shortestformat; random(2,8), pi // only for matrices >shortformat(>all); pi // for all output >longestformat; pi // always for all output >goodformat(12); pi >format(8,5); random(4,4) // for all output >goodformat(8,4); random(4,4), pi // for matrices only >goodformat(4,>all); random(4,4), pi >fracformat(10); intrandom(6,6,6)/intrandom(6,6,6) // fractions >defformat; // reset the default
Large matrices are eclipsed by default. There are several ways to see the full matrix.
>largematrices on; // or off - toggles the eclipsed output >showlarge A // shows A without eclipses >shortest A // may help to see the full matrix A
There are one-line and multi-line programs.
>function f(x) := x^3-x; // one-line function >function map f(x) // multi-line function ... $ if x>0 then return x^2 // if x>0 would not work for vectors x $ else return x^3 $ endif; $endfunction
To start a multi-line function type "function f(x)" and press Ctrl-Return. To edit a multi-line function click into its body. Use Ctrl-Return for line breaks, and Ctrl-Back for joining lines. Alternatively, you can press F9 and edit in a separate editor.
A variety of control structures are available, which work in one-line commands too, but usually spread several lines in programs. The loop with "repeat" is eternal, but can be left with until, while, or break;
>function f(x) ... $repeat $ xnew=(x+2/x)/2; $ until x~=xnew; $ x=xnew; $end; $return x; $endfunction
Other loops start with "for" and "loop". The for loop can loop over a vector.
>s=0; loop 1 to 10; s=s+#; end; s, >s=0; for i=1 to 10; s=s+i; end; s, >s=0; for i=10 to 1 step -1; s=s+i; end; s, >v=1:10; s=0; for i=v; s=s+i; end; s,
The "if" structure can have "else" and "elseif" branches. It needs an "endif". The condition does not work for vectors. So we need to vectorize with "map".
>function map f(x) ... $ if x<0 then return -1; $ elseif x==0 then return 0; $ else return 1 $ endif; $endfunction
Multiple return values are possible. Only the first return value will be used as an argument to another function, unless the modifier "args" is set.
>function f(a,b) := {b,a}
>{x,y}=f(1,2); x, y,
>function args f(a,b) := {b,a}
>{x,y}=f(f(1,2)); x, y, // would not work without "args"
Parameters can be restricted to specific data types.
>function map f(a:real) := integrate("sinc(x)",0,a)
>f(0:0.5:2)
For details on this, see the reference.
parameters - Reference for parameters
function - Reference for functions
trace - How to find errors in functions
alias - Define an alias for a function
There are local and global variables. Inside a function, only local variables and parameters are visible. To access global variables, "useglobal" or "global var" must be present.
>function changeA () ... $ global A; $ A=random(4,4); $ endfunction >A=id(6); >changeA(); A,
To generate a global variable with a given value use "global var=value".
>function newvar (x) ...
$ global nv=x;
$endfunction
>newvar(4); nv,
4
Variables ending with $$ are automatically global variables.
>a$$ = 1:4 // assign to a global variable a$$ >v = a$$ // retrieve value of global variable a$$
Moreover, variables can be set as global.
>v=1:3; >setglobal v; >function changev () ... $ v[2]=5; $ endfunction >changev(); v,
Note that one-line functions set "useglobal" automatically. So all global variables can be used. But parameters have precedence.
>function f(x) := a*x
>a=6; f(5)
30
Note that it is not a good style to use global variables to return values from functions. Instead, multiple returns should be used. Moreover, it is not a good style to access a global variable in a function (with the exception of units). Instead, pass the variable as an argument to the function. But for efficiency and simplicity, it is often nicer to use global variables.
The evaluation of an expression can also see global variables, in fact only global variables and variables given as arguments to the evaluation.
>function ev(fx,x) := fx(x)
>a=6; ev("a*x",2)
12
Arguments take precedence over global variables.
>a=5; f(2,3) 6 >"a*x"(2,a=3) 6 >"a*x"(2) 10
Furthermore, there is a global list type. Lists can grow as long as needed.
>glist("v");
>for i=1 to 1000; glistadd("v",i^2); end;
>s=0; for i=1 to 1000; s=s+glistvar("v",i); end; s,
>sum((1:1000)^2) // check
See:
global (Euler Core),
gvar (Euler Core),
glist (Euler Core)
Symbolic functions are functions which are defined for Maxima and for the numerical kernel of EMT. These functions must be one-line functions defined with &=. The definition can spread over several lines nevertheless.
>function f(x) &= x^x >function df(x) &= diff(f(x),x) >function h(x) &= diff(f(x),x) + ... > diff(f(x),x,2)
The body of the function is evaluated symbolically before it is defined. Thus df() in the example above is a valid function for EMT which can be evaluated numerically quickly and efficiently. Moreover, it can be used in other symbolic functions.
>plot2d("df",0,2);
>&df(x+1)
There are also purely symbolic functions. These functions are known only to Maxima. They are not evaluated at the time they are defined. The following function computes df/dx^2+df/dy^2. It must of course be evaluated, when it is used, not at the time of its definition.
>function L(f) &&= diff(f,x,2)+diff(f,y,2) >u &= ratsimp(realpart((x+I*y)^3/exp(x+I*y))) >&ratsimp(L(u)) // should be 0, since u(x,y) is harmonic) See:
Using a symbolic function inserts a Maxima result at compile time into the numerical EMT function.
>function f(x) &= integrate(x^3*cos(x),x) >type f // inspect the function
Symbolic expressions can also be inserted into multi-line functions. Again, the expression is evaluated at compile time, and the result is inserted into a normal EMT function. Take care to set brackets, since the result of the symbolic evaluation is inserted verbatim.
>expr &= x^x-2;
>function newtoniterator (x)
$ return x-(&:"expr")/(&:"diff(expr,x)");
$ endfunction
>type newtoniterator // look at the definition
>x=2; newtoniterator(x)
>iterate("newtoniterator",x)
Comments can contain some markups.
Images cannot be read from the Web. For "See:", local web links are possible.
Paragraphs consists of one long line in the editor. Line breaks will start a new line. Those lines are formatted into paragraphs in the editor and the EMT window. Paragraphs should be separated with an empty line. It is recommended to press Return twice after each paragraph and heading.
Spell checking is toggled with F12. To enter a word into the private dictionary mark the word and press Ctrl-F12.
See:
Exports (Overview)
Most of the functions of EMT are not in the C core, but written in the programming language of EMT and loaded from an Euler files (*.e) when the program starts. Euler files can be edited with any editor. To start the editing, enter
>load file
into the notebook and press F9 (internal editor) or F10 (external editor, by default JE). The file will be saved in the directory of the current notebook. In the case of the internal editor, EMT will wait for the editing to finish.
Scripts can contain
Make sure, functions have proper comment lines. For core functions or functions in C loaded by a DLL, there are comment functions. For functions defined in Maxima in EMT, there is the maximafunction definition.
:: h(x) := x^3; maximafunction h (x) ## Computes x^3 endfunction
Then
>help &h
will work. And &h will be known in the help window.
Help for Euler files is available by entering filename.e in the help window, or by double clicking on the line consisting of the load command.
Scripts can be run on the command line. Output will be lost however unless files are used. Use the parameter "-script script.e" to start EMT for a script. Details are explained in the web documentation.
You can open the help window at any time with F1. By default, EMT will display the topic explaining the most recent command. Press escape to clear the input line. While you type, the window will display all available topics. To select the first topic, press return. To select any other topic, double click on it.
The following wildcard characters and commands are supported in the search line.
The following key strokes are useful in the help window.
EMT tries to use the keys in the same way as other programs. But there are some special keys.
See:
function (Euler Core),
python (Euler Core)
The full version of EMT installs
If the binaries of Latex are not in your system path you can configure Latex in EMT. If you install Anaconda its Python interpreter will be found by EMT. It is accessed via the Python DLL in Windows. The Povray executable is stored in a string if it not in the system path. See the tutorial about Povray for details. The external editor can be configured inside EMT too.
The simple interface of EMT consists of the following windows.
In the notebook window, command lines can be entered. Each command line consists of up to four elements.
The command line can be a multi-line, with "..." connecting the lines. In this case, all connected commands in the multi-line are executed at once no matter where the cursor is in the multi-line. The connected lines do not show the ">" prompt.
A special case are functions (not one-line functions). They consist of a header line in red, and a body in blue, which ends with a line containing "endfunction". More about this in the section about programming EMT.
To edit a comment, use the comment editor. Press F5 to open this editor. More on this in the section about Comments.
To include the current graphics into the notebook below the command which generates the graphics, use a colon ":" at the end of the command line. An alternative is the command insimg().
Notebooks can be saved in "*.en" files, or exported in HTML, Markdown, and PDF. More on this in the section about Exports.
See:
Keyboard (Overview),
Comments (Overview),
Menu (Overview),
Exports (Overview)
EMT programs can write to files and read files. For this have a look at the programming language of EMT. The function to handle data tables in statistics can even write data in a variety of formats, such as CSV.
It is possible to dump all output of EMT to an external file. This function was useful in old versions of EMT which did not have a notebook interface.
Notebooks can be exported in various ways.
For all of these versions, you need write access to the current directory. For a tutorial notebook, you should save the file externally from the installation. You can do this with other notebooks too. EMT will ask you to keep the old file or delete it after saving. It will copy all images, creating an images directory by default, and also all C files or Euler files that start with the same name as the current notebook.
The notebook file is essentially a text file. It is human readable. By default is an UTF-8 file. For details on the syntax, study a simple Euler notebook file.
The HTML export is optionally using a CSS. You need to copy one of the provided CSS patterns using the Extras menu to the current directory as "Pattern.html". Change the pattern file to your liking. The export will save the Euler notebook first, then generate the HTML export, and open it in the default browser. You can configure the browser to be used.
Note that the HTML export the images in the notebook, including the Latex formulas, unless MathJax is forced in the dialog. It is possible to keep large versions of the graphics and the Latex formulas in the Options menu. In this case, it is possible to link to the large versions in the HTML export.
The PDF export is using "pdflatex". Latex must be installed completely for this. You can configure the path to Latex in the Options menu. A file "eumat.sty" is copied to the current directory. Modify this style file if needed. The PDF is opened in the default PDF browser.
The Markdown file is similar to the HTML export. But the markdown language is much easier to write. EMT will also use HTML entities to format the output by default, since not all Markdown browsers accept escaped characters.
The main menus in EMT are the following.
Let us give some more tips about various menu items and settings.
Load and Save Notebooks
Notebooks are usually saved into "Euler Files" in the user directory. The file dialog remembers the last directory. The directory of the notebook will become the default directory and data files without a path are searched in this file. Note that images are stored into a sub-directory "images" by default. If notebooks are saved into another position EMT will ask to delete the old notebook and its files.
To open tutorials there is a menu entry in the "Help" menu. It is possible to open a second notebook in another copy of the program. This second notebook is marked as secondary and will not remember its settings.
About loading and saving graphics, see the section about Graphics in EMT.
Edit Notebooks
You will usually just enter or edit a command line and execute it with the return key. But it is possible to execute a complete section of command lines at once with shift-return, and all commands from the current one to the end with ctrl-return.
EMT has a command history. See the menu or the section about the Keyboard for more details. You can also use cut and paste as usual with Ctrl-X, Ctrl-C and Ctrl-V.
Command lines can be split into multi-lines (ctrl-return) or in two separate lines (shift-ctrl-return), and joined to one line (ctrl-back).
With F12 a simple spell checker is available in English or in German. To add a word to the checker mark the word and press Ctrl-F12. To search for a word use Ctrl-F and F3.
Options
This is more than we can explain here. Some general keys are included here.
The graphics is displayed in a square window by default. But the aspect can be set permanently here. Alternatively, it can be set for each graphics with the aspect() command.
EMT supports folding of mutli-lines and functions (ctrl-l). They are only fully visible if they have the focus. For a single line this can be achieved with %% or %+ at the start of the line.
User Menu
This will be filled if you load an Euler file containing the "menu" or "submenu" command.
See:
Graphics (Overview),
Keyboard (Overview),
Options (Overview)
Let us go through the Options menu, explaining some details
-Show Graphics (TAB) -Graphics in Text Window (CTRL-G)
This will toggle between display of the graphics window in a separate window and inside the text window. In any case, the TAB key will show the graphics.
-Set Aspect
The aspect ratio (width by height) of the graphics can be set here. There are some special settings and a general width/height ratio. The graphics is square by default. The graphics ratio can also follow the dimensions of the graphics window.
There is also an aspect() command which can set the ratio. It is the preferred way since it will also recompute the borders.
-Fold all Multi-Lines (Ctrl-L)
This will clear the text window by folding functions and multi-lines which are not active.
For information on the rest of the Options menu see:
Beamer Settings
This will make the text and graphics window as large as possible to fit onto the screen. Note, that EMT remembers the sizes of the windows for each screen size separately. In case of F11, this is not saved.
A simple magnification of the content of the text window.
Insert images in the text window at reduced size for small displays.
The graphics of EMT is generated with low level routines, such as drawing a line or a text. EMT contains high level routines to ease the generation of plots, such as plot2d(). In this menu, the user can control the appearance of the plots.
EMT has 16 default colors for graphics and 6 default colors for text in the notebook. You can change the default colors with a color picker here, or opt to use grayscale.
Anti-Aliasing is a trick to get more smooth graphics. By default, the graphics is drawn at three times its width and then downscaled to get a smooth appearance. EMT uses a C coded way to reduce the image. This needs some system resources and thus can be turned off for the graphics window or the graphics in the text window. The default method of Windows can also be used which might be a bit faster. Sharper Anti-Aliasing improves the quality at the cost of performance.
The bigger graphics can either be saved or not. It might be useful to have the three times bigger graphics for later use. But it takes some space on the disk. If you keep the big graphics you can generate links to them in the HTML export.
The thickness of lines depends on the font size, which in turn depends on the graphics size and the user selected font. This may not always be what you need. For large prints, you do not want the lines to get fat.
Selecting the font is important for graphics export. You should use the setfont() method for this. But the font can also be changed permanently. The graphics font is set in relation to the graphics size. For big prints you may want to keep the font smaller. You can use other fonts for text, such as fonts designed for programmers, to improve readability. You need to install those fonts into Windows.
SVG is not the correct way to insert graphics into your documents. I suggest using a properly sized PNG file. Usually 2000x2000 pixel is enough for print. Use setfont() and aspect() to control the output.
Images are saved by default. If not they not to be recreated by running commands in the notebook (Shift-Return).
This makes sense only for images containing Latex formulas. It allows using the graphics in Latex with an overlay of formulas created by Latex.
See:
setfont (Euler Core),
setfont (Basic Utilities)
If you want to use another browser than the system default browser you can set this here. The default documentation directory is the start directory of EMT.
The default external editor is the included Java editor JE. It requires Java. If you want another editor you can set it here. A good recommendation is Notepad++.
EMT can load external libraries and execute external programs. For security reasons, the program will ask before doing this.
This can turn off the help in the status line below the text window, or transfer this help to the window title instead. The latter was necessary for Linux and Wine.
If not you can save your old variables between sessions. This is not recommended if you want to create consistent notebooks. Use Euler files (load *.e) to load commands and variables into a notebook.
EMT can handle Python 3.6 and later. But for backward compatibility, the default is Python 2.7. You can install two versions of Python. Install the default version for your programming needs later.
If not the line width will be as broad as the window. This is not recommended for compatible notebooks.
EMT keeps a junk of memory for its stack size, usually 1GB of RAM. Moreover, it needs at least 256 KB to store the graphics in vector format. You can increase these sizes for more complex tasks.
Notebooks (*.en) are human readable text files and can be saved in the default encoding or UTF-8. EMT saves three bytes to mark files at UTF-8.
This adapts some internal settings for Linux. Usually, Linux is detected automatically. EMT runs very nicely in the recent version of Wine.
This is only useful for those of you that use another Maxima version than the included one.
Maxima can be used with the original Maxima syntax or the adapted EMT syntax. The same can be achieved with ":::" instead of "::". The seamless expressions "&..." use the adapted syntax.
For some functions, Maxima asks questions in an interactive mode. These questions can be answered automatically in some reasonable way.
Maxima output is numbered so that the output can be referred to. EMT removes these numbers for its notebook style. To refer to an output assign it to a variable.
If disabled, the Maxima system will be started on first use.
This will always use the direct mode of Maxima with the ": ..." command.
Yacas is no longer supported.
See:
maxima (Euler Core)
This sets the directory where the Latex commands are located, and the work directory. The default work directory the Euler directory in the user home.
The Latex command window is usually not interesting to see.
This concerns the "mathjax:" command in comments. If disabled the formulas will not be parsed by Latex.
Latex formulas are parsed to images. If anti-aliasing is used the images are three times larger and reduced for a smooth look. The larger images can be kept or discarded.
Usually the PNG files have a white background. This can be changed there. The anti-aliasing will be less effective.
Removes the log files etc.
If disabled EMT relaxes the requirements to use () for functions and [] for indices.
This is more strict than the default "=".
This interprets "3 -4" as one command or one item of a vector. The commands and the elements of a vector have to be separated by command or semicolon.
The functions loaded at the program start are protected. You can use the "overwrite" modifier to redefine these functions, or change this switch.
Relaxing is a method to load old EMT files which use round brackets for indices and other old syntax. There is a command "relax" to switch this on in the Euler file. But you can also use this switch.
If this is switched off empty vectors may be returned. instead of an error message. This creates hard to detect programming errors.
This is off by default and removes another hard to find error in programs. In old versions of EMT, a condition was true if all elements were non-zero. You should instead use "any" and "all" for vectors.
See:
any (Euler Core),
all (Euler Core)
Graphics are stored internally in a meta file format. Thus they can be resized when the graphics window changes its dimensions, and exported to various sizes. The aspect should be kept the same. By default, the graphics window displays a square containing the graphics.
Note that such an internal vector format helps only withing a small range of sizes. To design a graphics for various print sizes requires changing the fonts, the line widths and the complete layout of the graphics. See the tutorial about graphics in EMT for more information, especially the command setfont().
EMT can export graphics in the following ways.
The preferred way to save a graphics is PNG in a proper size for printing. For a 10cm print width you will need about 2000 pixels. But you can also copy and paste to Word or any other Windows program using the Windows Metafile.
Graphics can be inserted into the notebook with the ":" at the end of the command line. For more control over the graphics size, use insimg(). These files are saved along with the notebook as PNG, by default in a subdirectory "images". Optionally, a three times larger file is saved additionally without anti-aliasing. This is useful for print media like PDF which have their own good anti-aliasing.
It is possible to load graphics into EMT notebooks in comments (using image: ...) and below commands (loadimg() command). It is also possible to load graphics into matrices with RGB-values (loadrgb() command)
See:
Comments (Overview),
loadrgb (Euler Core),
insimg (Euler Core),
insimg (Plot Functions)