iconEuler Reference

Euler Core Functions

Content

Introduction

This file contains the documentation for the functions, the operators and the syntax elements, which are built into the C/C++ source code of Euler Math Toolbox (EMT). The file does also contain technical documentation for some built-in features of EMT. The documentation of EMT refers to these core features whenever necessary. This page is a HTML version of "help.txt".

There are many functions in other packages, replacing the built-in functions of this file, or adding additional features. Follow the links provided in this document.

Many built-in functions have "comment" functions in the utility files. This helps to keep groups of related functions in one place. The utility functions are documented in separate HTML pages, which are extracted from the utility files.

Help in EMT

The best way to get help is to use the HTML documentation in form of introduction notebooks and examples or the HTML reference. Also observe the status line in the notebook window for information on the current command or option.

To open the context help for a function in EMT, double click the function name or press F1.

help, mxmhelp, helpwindow
  To find help on a specific function, you can
  
   - Look at the status line, while you type.
   - Press F1, open the help window, and enter the function name.
   - Open the complete reference in the browser.
  
  There are some commands in the notebook window, which print help
  directly to the notebook window.
  
    help topic : Displays help for a user function, a built-in function
      and other Euler commands, programming keywords and
      other items.
  
    mxmhelp topic: Display help for a Maxima command.
  
  See: 
Help (Overview),
Keyboard (Overview)
quit
  quit: Quits Euler immediately.

Operators and Functions

Euler uses the well known mathematical operators and functions. For operators, take care of precedence, or set round brackets in case of doubt. Another point to keep in mind is that trigonometric functions work in radians, not in degrees.

To get an introduction, look at the following notebooks.

Special functions are listed in the following page.

+, -, *, ^, /
  The usual arithmetic operations. Note that these operators, as well as
  most functions in Euler, obey the Euler matrix language.
  
  These operators obey to the rules of the Euler matrix language.
  
  See: 
  
  Division and powers generate runtime errors, if they are not defined
  for the arguments.
  
  Note that the dot . has to be used for matrix multiplication.
  
  See: 
. (Euler Core),
. (Maxima Documentation) The underflows that ^ produces are caught to 0, if underflows is off (the default). Note that ^ in Euler uses left binding, so a^b^c is equal to (a^b)^c. >2*(3+4)^2 >2^2^3-2^8 >(2^2)^3-4^3 >(1e-200)^2 >errors off; 1/0; errors on; >A=[1,2;4,5]; A.A See:
underflows (Euler Core)
abs, exp, log, sign, sqrt
  The usual mathematical functions, for real, complex and intervals
  scalars, vectors and matrices. The functions work element-wise for
  matrices, and obey to the rules of the Euler matrix language.
  
    abs(x) : absolute value
    exp(x) : exponential function
    log(x) : natural logarithm, main branch for complex
    sign(x) : sign of reals
    sqrt(x) : square root function, with main branch for complex values
  
  To compute the exponential etc. for a matrix, use "matrixfunction".
  
  The underflows that exp produces are caught to 0, if underflows is off
  (the default). 
  
  >log(E)
  >abs(1+I);
  >sqrt(complex(-1))^2
  >plot2d("sign",-1,1);
  
  See: 
log10 (Mathematical Functions),
logbase (Mathematical Functions)
sin, cos, tan, asin, acos, atan, arctan2, atan2, arcsin, arccos, arctan
  The usual trigonometric mathematical functions, for real, complex and
  intervals.
  
    sin(x) : sine function
    cos(x) : cosine function
    tan(x) : tangent function
    asin(x) : inverse sine, maps [-1,1] to [-pi/2,pi/2]
    acos(x) : inverse cosine, maps [-1,1] to [0,pi]
    atan(x) : inverse tangent, maps R to [-pi/2,pi/2]
    arcsin(x) : inverse sine, maps [-1,1] to [-pi/2,pi/2]
    arccos(x) : inverse cosine, maps [-1,1] to [0,pi]
    arctan(x) : inverse tangent, maps R to [-pi/2,pi/2]
    arctan2(x,y) : Angle of the polar coordinates of [x,y] in ]-pi,pi].
    atan2(x,y) : Angle of the polar coordinates of [x,y] in ]-pi,pi].
  
  The functions work for radians. To convert from degree to radians use
  x° or rad(x). Use shift-F7 to enter the degree symbol in an Euler
  notebook. To convert back, use rad(x).
  
  The degree symbol ° works in Maxima too.
  
  >sin(90°)
  >cos(pi)
  >deg(atan(1))
  >deg(atan2(1,1))
  >degprint(rad(10,30,20))
  
  See: 
polar (Mathematical Functions),
polar (Maxima Documentation),
rect (Mathematical Functions),
deg (Basic Utilities),
rad (Basic Utilities),
degprint (Basic Utilities)
bin, fac, !, logbin, logfac
    bin(n,m) : Binomial coefficient "m chosen from n".
  
    logbin(n,m) : Logarithm of the bin(n,m).
      This is useful for large n, m.
  
    fac(n), n! : Factorial function of n.
  
    logfac(n) : Logarithm of the factorial function
      Useful for large n.
  
  >bin(10,0:10)
  >bin(49,6)-49!/(6!*43!)
  >logfac(100)-sum(log(2:100))
  
  See: 
binsum (Statistics with Euler Math Toolbox),
gamma (Mathematical Functions),
gamma (Maxima Documentation)
ceil, floor, round, mod
    ceil(x) : The smallest interger above x.
    floor(x) : The largest integer below x.
  
    round(x,[n]) : x rounded to n digits.
      Default is n=0.
  
    mod(x,y) : x modulo y.
      This works for non-integer y too.
  
  >x-floor(x)
  >floor(1.9)
  >round(1.9)
  >round(-1.9)
  
  See: 
print (Basic Utilities),
print (Maxima Documentation),
frac (Basic Utilities)
complex, re, im, arg, conj, real
    re(x) : The real part of x.
    im(x) : The imaginary part of x.
    complex(x) : Makes the real argument x complex.
    conj(x) : Conjugate of the complex value x.
    arg(x) : The argument of the complex value x.
    
    real(x) : makes x real.
      If the imaginary part of x is too large, an error will be
      issued.
  
    >sqrt(-1) // Error!
    >sqrt(complex(-1))
    >A=random(10,10); sort(real(eigenvalues(A.A')))
    >deg(arg(I)) // 90
  
  See: 
carg (Maxima Documentation),
conjugate (Maxima Documentation)
I, E, Pi, pi, %e, %pi
    I : The imaginary unit (also %i and 1i)
    E : exp(1) (also %e)
    Pi : pi() (also pi, %pi)
  
  Do not use e for E. 
  
true, false
    true: The constant 1.
    false: the constant 0.
  
  All boolean functions and operators return 1 or 0.
  
<, >, <=, >=, <>, ~=
  Comparison operators, returning 0 or 1. 0 means false, and 1 means
  true. These operators obey to the rules of the Euler matrix language.
  To assert that the result is true for any or all elements of a matrix,
  use any or all.
  
  a~=b compares for equality taking into account the internal epsilon.
  This operator works for intervals too.
  
    >sum(0.1*ones(1,10))==1 // inevitably false
    >sum(0.1*ones(1,10))~=1 // true
    >(pi>3)&&(pi<4) // true
  
  See: 
any (Euler Core),
all (Euler Core),
epsilon (Euler Core)
&&, ||, !
    a&&b : returns a and-connected with b in case of reals. 
      I.e., the result will be 1, if both a and b are non-zeros.
    
    I&&J : The interval intersection of I and J.
    
    a||b : returns a or-connected with b in case of reals. 
      I.e., the result will be 1, if both a and b are non-zeros.
    
    I||J : The interval union of I and J.
  
    !a : return 0 if a is non-zero, or 1, if a is zero.
  
  These operators work for matrix input like all operators in Euler.
  They produce a matrix of 1 or 0 in this case.
  
  See: 
and (Euler Core),
and (Maxima Documentation),
or (Euler Core),
or (Maxima Documentation),
not (Euler Core),
not (Maxima Documentation),
if (Euler Core),
if (Maxima Documentation),
bitand (Euler Core),
bitor (Euler Core),
bitnot (Euler Core)
and, or, not
    and,or : Shortcut logical operators. 
  
  Only the necessary parts of the conditions are evaluated. "and" binds
  stronger than "or". But it might be more clear to use round brackets.
  
    not : Binds stronger than "and" or "or".
  
  These operators accept only reals. They are good for if-conditions or
  while- and until-statements. For logical operations, use "&&", "||",
  "!".
  
    >v=random(1,10);
    >i=12; if i<=cols(v) && v[i]>0.5 then "yes", else "no" endif;
     Index 12 out of bounds! 
     ...
    >i=12; if i<=cols(v) and v[i]>0.5 then "yes", else "no" endif;
     no
  
  See: 
|| (Euler Core),
! (Euler Core),
! (Maxima Documentation),
bitand (Euler Core),
bitor (Euler Core),
bitxor (Euler Core),
bitnot (Euler Core)
iscomplex, isfunction, isinterval, isreal, typeof, printtype, issymbolic, 
isempty
    isreal(x) : tests, if x is real.
    iscomplex(x) : tests, if x is complex.
    isinterval(x) : tests, if x is an interval scalar or matrix.
    isfunction(x) : tests, if x is a function.
    isstring(x) : tests, if x is a string.
    issymbolic(x) : tests, if x is a symbolic expression.
    isempty(x) : tests, if x is none or an empty vector.
  
    typeof(var) : The type of the expression.
  
  The types are: 0=real, 1=complex, 2=matrix, 3=comples matrix,
  4=reference, 5=command, 6=submatrix, 7=complex submatrix, 8=string,
  9=function, 10=interval, 11=interval matrix, 12=interval submatrix,
  13=string matrix, 4=compresses sparse matrix
  
    printtype(x) : String containing the type of x.
  
bitand, bitor, bitxor, bitnot
  Connect two integer values bit by bit. The double values are converted
  to 32 bit integers first.
  
    bitand(x,y) : bitwise and
    bitor(x,y) : bitwise or
    bitxor(x,y) : bitwise exclusive or
    bitnot(x,y) : bitwise not
  
  See: 
|| (Euler Core),
! (Euler Core),
! (Maxima Documentation)
=, :=, ::=, assignment
    variable = value : Assigns the value to the variable.
      The variable may be a variable name or a
      submatrix of a matrix.
  
    variable := value : More strikt form of the asignment. Recommended.
    variable ::= value : Sets a variable in Euler and in Maxima.
  
  See: 
submatrix (Euler Core),
submatrix (Maxima Documentation),
multiple (Euler Core)
submatrix, [, ]
    [...] : Defines a matrix row by row. The columns must be separated
      by commas, the rows by semicolons. In compatibility, this
      works for symbolic matrices too. [] is an empty 0x0 matrix.
      It can be used to attach rows or columns to it.
  
    ["string",...] : Defines a string vector. 
      [none] is the empty string vector. It can be used
      to append strings or string vectors to it.
  
    A[i,j] : Defines a single element of a matrix or a submatrix
      If an index in i or j is out of range, it is neglected, or
      there will be an error message (the default). Negative
      indices count from the end. If i and j are vectors of
      indices, the submatrix with rows in i, and columns in j is
      used.
  
  Note that matrix indices start with 1 in Euler. -1 is the last element
  of the matrix.
  
    A[i] : The i-th element of a vector, or the i-th row of a matrix 
      with more than one row. A[-1] is the last element (resp.
      row). Note that A[i] returns different things if, A is a 1xm
      or a nxm (n>1) matrix. To make sure to get the i-th row, use
      A[i,].
  
    s[i] : The i-th element of the string vector s, or a string vector
      with the indices from the row vector i.
  
    A[:,j] : Denotes all rows here. This is the j-th column of A, or
      the columns with incides in the row vector j.
  
    A[,j] : Short form for A[:,j].
  
    A[i,:] : The i-th row of A
      or the rows with indices in the row vector i.
  
    A[i,] : Short form of A[i,:].
  
  See: 
= (Euler Core),
= (Maxima Documentation),
{ (Euler Core)
(, )
  Round brackets are used to group operations as in (3+4)*5, to call
  functions as in sin(pi), or to evaluate expression.
  
  When expr is a string expression, expr(...) evaluates it in Euler. 
  The parameters are automatically assigned to x, y, z. Other
  parameters may be added as assigned parameters as in expr(2,a=1).
  The evaluation sees global variables too, and local variables in the
  function, which contains the expression evaluation.
  
  Round brackets may be used instead of v[5], if there is no function
  with the name v (else v(5) is a function call with parameter 5).
  However, this is not recommended, and is disabled in strict mode.
  

Functions for Strings and Characters

Strings in Euler are mainly used to hold expressions. An expression can be evaluated by the Euler interpreter using parameters. For many examples and a full reference refer to the documentation of Euler.

Strings are also used for file names, error texts and other text information.

ascii, char, substring, strfind, strrepl, strlen, tolower, toupper, strtochar, 
chartostr, strtokens, hardspace
    ascii(string) : The ascii code of the first letter in the string.
    char(n) : String containing only the character with ascii code n.
    
    substring(string,n,m) : The substring from n to m (including).
      Negative indices count from the back.
    
    strfind(string,sub[,n]) : Position of sub in the string. 
      The search starts after the n-th position.
      pos=0 means that sub was not found.
  
    strrepl(s,s1,s2) : Replace s1 with s2 in s
    
    strlen(string) : The lengths of the string.
    tolower(string) : The string in lower case.
    toupper(string) : The string in upper case.
    
    strtochar(string) : Vector of letters in the string.
    chartostr(vector) : String with characters in the vector.
    
    strtokens(string,{seperator}) : Vector of words in this string
      
    hardspace() : ASCII number of hard space character
  
  The default separator string for strtokens() contains the characters
  blank, tabulator, comma and semicolon.
  
  >s="me and you", substring(s,strfind(s,"and",0),-1)
   me and you
   and you
  >strtokens(s)
   me
   and
   you
  >strtokens("A-B-C=D","-=")
   A
   B
   C
   D
  >chartostr(flipx(strtochar("the brand new car")))
   rac wen dnarb eht
  >s=12+char(hardspace())+13, s()
   12 13
   1213
  
  See: 
key (Euler Core),
key (Maxima Documentation),
chartoutf (Euler Core),
utf (Euler Core),
strxfind (Euler Core),
strxrepl (Euler Core)
strxfind, strxrepl
  The functions strxfind() and strxrepl() search with a regular
  expressions. Regular expressions are a somewhat complicated tool to
  match strings to pattern. The function strxfind() will return the
  matched region and a list of submatches if the regular expressions
  contained patterns in round brackets. The function strxrepl() can
  use these submatches in the replacement string at any place.
  
    strxfind(string,expr[,start]) : Match with regular expression
      The function returns the position of the match, the matched
      string, and a string vector of submatches. If the match is not
      successful the function returns 0.
      
    strxrepl(string,expr,repl) : Replace with regular expression
      The function returns the string with replacements.
  
  The most often used patterns are the following.
  
    . : any character
    [0-9] : range of characters
    [[:alpha:]] : character class (to be used in range)
    [^0-9 ] : excluded characters in range (digits and blank)
    .+ : repetition of any character (at least one)
    [0-9]* : repetition (any or no digit)
    \[, \\, \* : escaped format characters
    ^ : start of string
    $ : end of string
    (.+) : submatch region (looking for any character)
    (a|an) : alternative
    
  In the replacement of strxrepl(), $0 is the complete matched string,
  and $1, $2 etc. denote the submatches.
  
  >s="Albert Einstein 14.03.1879 Ulm.";
  >{pos,found}=strxfind(s,"[0-9\.]+"); found,
   14.03.1879
  >{pos,found,v}=strxfind(s,"([0-9]+)\.([0-9]+)\.([0-9]+)");
  >v
   14
   03
   1879
  >strxrepl(s,"([0-9]+)\.([0-9]+)\.([0-9]+)","$3-$2-$1")
   Albert Einstein 1879-03-14 Ulm.
  
  See: 
strfind (Euler Core),
strrepl (Euler Core)
utf, chartoutf, fromutf, toutf, Unicode
  EMT has some support for Unicode. Strings can contain UTF-8 encoded
  Unicode characters. These strings print correctly on the output, and
  text in plots with such strings display the Unicode characters. 
  
  The easiest way to generate a Unicode string is a string constant of
  the form u"...". The string can contain HTML entities, which are
  translated to unicode characters. You can find a list of entities in
  the documentation of EMT and in the Internet. An example is
  u"&alpha;". Unicode strings can also contain Unicode characters by
  their number as in u"&296; is a German character!".
  
  In fact any local characters will be translated into Unicode from
  u"..." (or from the utf() function below).
  
    utf(string) : Translate entities to Unicode.
  
    chartoutf(vector) : Translate characters to a Unicode string.
      The characters are given by Unicode numbers. The
      converse function is strtochar(string), which
      recognizes a Unicode string.
  
    fromutf(string) : Translates a Unicode (UTF8) string
    toutf(string : Translates a string to Unicode (UTF8)
  
  See: 
strtochar (Euler Core)
key, testkey
    key() or key(text): Waits for a key.
      The optional text is displayed in the status
      line. The function returns the ascii key code,
      or the internal key code, mapped to codes from 0
      to 15.
  
    testkey(): Tests, if a key has been pressed.
  
  See: 
ascii (Euler Core),
ascii (Maxima Documentation),
wait (Euler Core)
input, lineinput
    input(prompt) : Prompts the user for an expression. 
      Returns the value of this expression. In case of an
      error, the prompt will be repeated. The user can
      break out with escape.
  
    lineinput(prompt) : Prompts for a string, which is not interpreted.
  
  See: 
evaluate (Euler Core)

The Matrix Language

The matrix language of Euler is the core of the functionality in Euler, which helps to avoid loops in many cases. Using the matrix languages allows for easy interactive computations.

Most operators and functions in Euler work for real, complex and interval scalar, vector and matrix input, whenever that makes sense. The operators and functions obey to the rules of the Euler matrix language.

The following are functions that create or modify matrices and vectors.

More functions are contained in Basic Utilities or in Linear Algebra. For an introduction to matrices in Euler, read

largematrices
  Large matrices or vectors might not print in full size. To change this
  toggle the behavior with
  
  >largematrices on
  >largematrices off
  
  or use the operator showlarge.
  
  >showlarge random(10,10)
  
  See: 
showlarge (Basic Utilities)
zeros, ones
    zeros([n,m]) zeros(n,m) zeros(n) : Matrix filled with with x=0.
    ones([n,m]) ones(n,m) ones(n) : Matrix filled with with x=1. 
  
  >shortformat; A=random(2,3)
      0.44462     0.29947     0.28269 
      0.88323     0.27091     0.70442 
  >B=ones(size(A))
      1           1           1 
      1           1           1 
  
  See: 
size (Euler Core)

Functions for Vectors and Matrices

More functions for matrices and vectors can be found in the reference for Linear Algebra.

:, linspace
    i:n : The vector [i,(i+1),...,n] for integers i,n.
  
    x1:d:x2 : The vector [x1,x1+d,...,x1+k*d], 
      If d is negative, we get the vector [x1,x1+d,...,x1+k*d], where k
      is maximal such that x1+kd>=x1. For the comparison with the bound,
      the internal epsilon is used.
  
    linspace(a,b,n) : n+1 linear spaces values in [a,b], 
    	This includes a and b.
  
  The colon : is also used for direct Maxima commands and for indices in
  sub-matrices.
  
  See: 
epsilon (Euler Core),
equispace (Basic Utilities),
chebzeros (Basic Utilities),
::: (Euler Core),
submatrix (Euler Core),
submatrix (Maxima Documentation)
cols, rows, size, redim
    cols(A) : The number of columns of the matrix A.
    rows(A) : The number of rows of the matrix A.
    
    size(A,B,C,...) : The maximal size of A,B,C,... 
      This is used to define a matrix, which can hold
      the result of any operation between the compatible
      matrices A,B,C,... This is necessary, if map or
      the matrix language cannot be used, and a loop
      must be used instead. To access the k-th element
      of a matrix A flattened to a row matrix, use
      A{k}.
  
    redim(A,[n,m]), redim(A,n,m) : returns a matrix with the same content
      but different dimensions n times m.
      Fills with 0, if necessary.
   
  >A=zeros(3,4); size(A),
     [ 3  4 ]
  >rows(A), cols(A),
     3
     4
  >for k=1 to prod(size(A)); A{k}=k; end; short A
      1           2           3           4 
      5           6           7           8 
      9          10          11          12 
  >s=size(1:3,(1:3)')
     [ 3  3 ]
  
  See: 
{ (Euler Core)
., bandmult
  A.B: Matrix multiplication.
  
    bandmult(A,B) : computes A.B and is a bit faster for sparse
      matrices.
  
  This is just the ordinary multiplication of matrices. For sparse
  matrices use cpxmult.
  
  For large, sparse matrices, compress the matrices and use cpxmult.
  
  See: 
. (Euler Core),
. (Maxima Documentation),
cpxmult (Numerical Algorithms)
diag, setdiag, getdiag, band
    diag(A,k) : k-th diagonal of A (deprecated)
    getdiag(A,k) : k-th diagonal of A
    getdiag(A) : diagonal of A
  
    diag([n,m],k,v),diag(n,m,k,v) : nxm matrix with v on its k-th
      diagonal. (See diagmatrix(v,k)).
    setdiag(A,k,x) : A with the k-th diagonal set to x.
  
  In these functions, k=0 is the main diagonal, k=-1 the diagonal below,
  and k=1 the diagonal above. Note that setdiag() does not change the
  matrix A, but returns a new matrix.
  
    band(A,n1,n2) : A, with A(i,j) set to 0, if i+j<n1 or i+j>n2.
  
  See: 
diagmatrix (Linear Algebra),
diagmatrix (Maxima Documentation)
prod, sum, colsum, cumprod, cumsum
  These functions work for row vectors. In the case of matrices, they
  work for each row of the matrix separately. E.g., sum(A) returns a
  column matrix, with entries equal to the sums of the rows of A.
  
    prod(A) : A vector containing the products of the rows of A.
    sum(A) : A column vector containing the sums of each row of A.
    colsum(A) : A row vector containing the sums of each column of A.
    
    cumprod(A) : Matrix with the cumulative products in the rows of A.
      The result R has the same size as A. R[i,k] is the
      product of A[i,1] to A[i,k-1].
    cumsum(A) : Matrix with the cumulative sums in the rows of A.
    
    cummax(A) : Matrix with the cumulative maxima in the rows of A.
    cummin(A) : Matrix with the cumulative minima in the rows of A.
    
  >sum([1,2;4,3])
                3  
                7  
  >colsum([1,2;3,4])
    [4,  6]]
  >plot2d(cumsum(normal(1,1000)));
  >cumprod(1:5), (1:5)!
    [1,  2,  6,  24,  120]]
    [1,  2,  6,  24,  120]]
   
dup, _, |
    dup(v,n) : duplicates the vector v n times 
      The duplication takes place vertically or horizontally
      depending on v. Row vectors are duplicated vertically,
      column vectors are duplicated horizontally.
  
    v_w : sets v atop of w.
      If one matrix has less columns than the other, it is filled
      with zeros. However, if v or w is a scalar value, this value
      is used for the fill. So v_5 adds one row filled with 5 to v.
      In any case, the number of rows is the sum of the rows of v
      and w, the number of columns is the maximum of the number of
      columns of v and w. This applies also, if v or w has zero
      rows.
  
  _ is also used to access the built-in version of a function, if an
  overwrite function exists. E.g., "_insimg" uses the built-in variant of
  "insimg".
  
    v|w : sets v aside of w.
      Works analogously to v_w. So, v_5 will enter a column filled
      with 5 to v. The number of columns is the sum of the numbers
      of columns, and the number of rows is the maximum of the
      number of rows of v and w.
  
  | : is also uses to append flags in Maxima compatibility mode and
  symbolic expressions.
  
    s|t : does append strings, or strings and numbers. 
      The numbers will be formatted according to the current format.
      Note, that s+t will also append strings, or strings and
      numbers. Then s must be a string.
  
  >shortformat;
  >(1:3)_2
      1           2           3 
      2           2           2 
  >(1:3)'|2
      1           2 
      2           2 
      3           2 
  >random(3,3)_random(2,2)
      0.02744     0.70953     0.72939 
      0.58709     0.47649     0.58604 
      0.76816     0.54169     0.47627 
      0.57003     0.40318           0 
      0.13538     0.43876           0 
  >"me"|" and "|"you"
   me and you
  >"Result : "+1.5^2*pi
   Result : 7.06858347058
   
any, all, nonzeros, mnonzeros, mget, mset, firstnonzero, lastnonzero
    any(A) : 1, if any entry in A is different from 0.
    all(A) : 1, if all entries in A are different from 0.
  
  all(A) is equivalent to !any(A==0). For numerical comparison
  respecting the internal epsilon, use A~=0.
  
  In "if" clauses you must use any(v>2) to check, if any element of v is
  greater than 2.
  
    nonzeros(v) : The indices of the non-zero elements of v.
      v must be a vector. The result is a row vector w, of
      at most the size of v.
  
    mnonzeros(M) : The indices of the non-zero elements of M.
      The result is a nx2 vector with rows of indices i,j.
      
    firstnonzero(M): Index of first non-zero element in each row of M
      If there is none, the result is 0. For a matrix, the result is
      a column vector.
     
    lastnonzero(M): Index of last non-zero element in each row of M
      Like firstnonzero.
     
    mget(M,k) : Get the elements of M with indices k.
      k is a nx2 vector of rows of indices i,j,
      
    mset(M,k,A) : Sets the indices of M in k to A.
      k is a nx2 vector of rows of indices i,j. Then M[i,j] is set
      to A[i,j].
  
  See: 
~= (Euler Core)

Polynomials

A Polynomial is stored in Euler in a vector of coefficients, starting with the lowest coefficient. Euler can solve complex polynomials, find polynomial fits and interpolations, and do some basic calculations of polynomials.

There are some utility functions to create and handle special orthogonal polynomials, like Chebyshev or Legendre polynomials. See @"gauss", @"functions".

interp, divdif, interpval, polytrans, interpolation
  Interpolation with polynomials. Euler uses the divided difference form
  of the interpolation polynomial. This form requires that the
  interpolation points are passed to the evaluation function of the
  interpolating polynomial.
  
    interp(x,y) : divided differences of the polynomial interpolation.
      Interpolates (x(i),y(i)). Hermite interpolation is
      available with the function hermitinterp. For
      polynomial fit, use polyfit. 
    divdif(x,y) : Alias to interp.
  
    interpval(x,d,t) : evaluates the interpolation in x(i)  in t.
    divdifval(x,d,t) : Alias to interpval.
  
  Both functions work with divided differences d in a vector [x0],
  [x0,x1] etc. The functions work for complex interpolation too.
  
  The function evaldivdif is an alias to divdifval with the t-argument
  at the first place.
  
    polytrans(x,d) : transfer the divided differences into a polynomial.
  
  >x=[0,1]; d=interp(x,[1,1/2])
   [ 1  -0.5 ]
  >frac(interpval(x,d,[0,1])) // evalute in 0,1
   [1,1/2]
  >polytrans(x,d) // this is 1-0.5x
   [ 1  -0.5 ]
  >d=interp(1:5,[-1,1,-1,1,-1])
   [ -1  2  -2  1.33333333333  -0.666666666667 ]
  >plot2d("interpval(1:5,d,x)",1,5);
  
  See: 
evaldivdif (Numerical Algorithms),
interpolate (Numerical Algorithms),
hermiteinterp (Numerical Algorithms),
polycons (Euler Core),
polyfit (Linear Algebra),
remez (Numerical Algorithms)
polyadd, polycons, polydiv, polymult, polytrunc, polyval
  Polynomials in Euler are stored in vectors starting with the constant
  coefficient.
  
    polyval(p,x) : Evaluates p at x.
    polycons(v) : A polynomial with zeros v(i).
    polydiv(p,q) : {h,r}, where h is the result p/q and r the remainder.
    polymult(p,q) : Computes p*q.
    polyadd(p,q) : Computes p+q.
    polytrunc(p) : Truncates zero coefficients.
  
  >p=polycons(1:5), real(polysolve(p))
   [ -120  274  -225  85  -15  1 ]
   [ 1  3  4  2  5 ]
  >polyval(p,1:5)
   [ 0  0  0  0  0 ]
  
  See: 
polysolve (Mathematical Functions),
polyroot (Euler Core),
cheb (Mathematical Functions)
polyroot
    polyroot(p,x) : Computes the root near x.
  
  These functions use a Newton method close to the 0, and a descent
  method away from a zero.
  
  See: 
polyval (Euler Core),
polycons (Euler Core),
polysolve (Mathematical Functions)

File Input and Output

With the following routines, and some utility functions described in other files, it is possible to write to files, and read from files.

For more advanced functions to read and write matrices or other values, see Elementary File Input and Output.

changedir, path, load, home, homedir, userdir, start, eulerhome
  Euler files (*.e) contain sequences of commands or definitions of
  functions. They can be loaded with "load filename" or with the
  load(filename) function. It is not possible to load files inside
  functions. One loaded file can load another file. To skip parts of the
  file use lines containing "comment" and "endcomment" or "stopload".
  
    load(string) : loads an Euler file (*.e)
      Interprets the file until it ends or until "stopload"
      is found. Inside a loaded file, another file can be
      loaded.
  
    changedir(string) : Changes the current directory.
      Returns the new directory. If the string is
      empty, it only returns the current directory.
      Note that the directory is changed
      automatically, if a new notebook is loaded or
      saved. 
  
    cd name : Change the directory to this file
      You can also use cd(expression), or cd "name". "cd" prints
      the current directory.
  
    home() : The home directory of the current notebook
    userhome() : The home directory of the user
  
    eulerhome() : The subdirectory Euler uf userhome().
      If the directory does not yet exist, it will be
      created. You should use this for temporary files.
  
    userdir() : The "Euler Files" directory.
    start() : The directory, where Euler started.
  
    path(string) : Sets a path to look for Euler files. 
      The load command will look through this path and
      choose the first available file. The path items must
      be separated by a semicolon. An example is
      ".;myspecial". This should not be changed by the
      user.
  
    load filename : loads an Euler file. 
      The name may be included in double quotes, or in
      round brackets (if it is a string expression). The
      command can also be used in a file to load another
      file.
  
  >filename=eulerhome()+"test.txt"
   C:\Users\Rene\Euler\test.txt
  >open(filename,"w"); writeln("First Line"); writeln("Second Line"); close();
  >printfile(filename)
   First Line
   Second Line
  >fileremove(filename)
  
  >changedir(eulerhome());
  >changedir(home());
  >cd
  
  See: 
comment (Euler Core),
endcomment (Euler Core),
printfile (Elementary File Input and Output),
printfile (Maxima Documentation),
open (Euler Core)
searchfile, dir, fileremove, filecopy, fileexists
    searchfile(pattern) : Searches for the file pattern.
    searchfile() : Searches the next file with this pattern.
    fileremove(file) : Removes a file
  
    filecopy(name1,name2) : Copies a file.
    fileexists(name) : File exists?
  
  See: 
dir (Euler Core),
dir (Elementary File Input and Output)
open, close, eof
  Euler can keep one file open for input and another file for output.
  
    open(filename[,type]) : opens a file with type="r","w","rb","wb"
      Opens for reading (type is "r") or writing
      (type is "w"). Binary mode can be achieved
      with "rb" or "wb". Default type is "r".
  
    close() : closes all opened files.
    eof() : returns 1, if the file is read completely.
  
  See: 
putchar (Euler Core),
putword (Euler Core)
putchar, putword, putlongword, putuchar, putuword, putulong
    putchar(c) : puts a character to a previous opened file.
      c is the ascii code of the character.
  
    putuchar(c) put c as unsigned character.
    putword(x) : puts x a s two bytes to the file.
    putuword(x) : put x as unsigned word.
    putlongword(x) : puts x as four bytes to the file. 
    putulongword(x) : The unsigned thing.
  
  These functions use the internal format. They are useful for binary
  output. For writing text, use write() and writeln().
  
  See: 
write (Euler Core),
getstring (Euler Core),
open (Euler Core),
close (Euler Core),
close (Maxima Documentation)
getword, getlongword, getuchar, getuword, getulongword, getline
    getchar() : reads one character from the file.
    getuchar() : for unsigned characters.
    getchar(n) : reads n characters, and returns in a 1xn vector.
    getword() : reads a word.
    getuword() : reads an unsigned word.
    getword(n) : reads n words.
    getlongword() : reads a long word.
    getulongword() : reads an unsigned long word.
    getlongword(n) : reads n long words.
    getline() : returns a line from the file.
  
  See: 
write (Euler Core),
getstring (Euler Core),
open (Euler Core),
close (Euler Core),
close (Maxima Documentation)
getstring, write, writeln
    getstring(n) : reads a string of length from an opened file.
    write(string) : writes a string to the open file.
    writeln(string) : writes a string to a line in the open file.
  
  See: 
open (Euler Core),
close (Euler Core),
close (Maxima Documentation)
urlopen, urlclose, urlgetline, urleof
  EMT can get a connection to an internet URL for reading. It can read
  line by line from this URL.
  
    urlopen(string) : Opens the connection
    urlgetleine() : Read a line from the URL.
    urlclose() : Closes the connection
    urleof() : End of file for reached?
  
  See : open, toutf, fromutf

Expressions

As mentioned in the section about strings, strings can hold expressions. Expressions can be evaluated using the evaluate function or its short form string(), resp. string(parameters). For examples, read the following introduction.

eval, @(, evaluate, error, errorlevel
    eval("f",v,...) : calls the function f with parameters v,...
  
  This is obsolete, because f(v,...) will work for strings containing
  function names and also for strings containing expressions. 
  
    "expression"(...) : Euler evaluates strings containing expressions.
      Note that expressions in Euler are by default
      expected to be expressions in x, y, and z.
      Functions, of course, can use any variable name.
      The parameters of an expression in expr(...) are
      assigned to x, y, and z in this order. Additional
      parameters can be assigned using the form
      var=...
  
  Evaluating an expression will use global variables, besides the
  arguments passed to the evaluations. Global variables will be used
  even if there is a local variable with the same name in the function
  which contains the evaluation.
  
  Expressions can be contained in call collections. The call collection
  will then be of the form {{expression,parameter,...}}. This is just a
  shortcut to expression(parameter,...).
  
  A string of the form "@(a,b,c,...) expression" is evaluated as the
  expression with parameters a,b,c etc. This works like an anonymous
  function. It should be preferred to store functions in functions, not
  in strings, however.
  
    evaluate(expression) : evaluates the expression string.
      This is obsolete, since expression() will do
      the same, and variables expression(x,y,z)
      will assign local variables x, y and z.
      Additional parameters can be assigned as
      var=... Global variables can be used.
  
    error(string) : issues an error and prints the message string.
  
    errorlevel(string) : evaluate the string
      It returns the result, and 0 if no error, else
      the error number.
  
  >function f(x) := x^3+x
  >f(0.5)
   0.625
  >eval("f",0.5)
   0.625
  >"f"(0.5)
   0.625
  
  >expr="x^3+x";
  >expr(0.5)
   0.625
  >expr:="a*x^3+x";
  >expr(0.5,a=1)
   0.625
  
  >expr="@(a) a^3+a";
  >expr(0.5)
   0.625
  
  >errorlevel("1/0")
   1
  
  See: 
evaluate (Euler Core),
expreval (Basic Utilities),
collection (Euler Core)

Intervals

Euler contains an interval arithmetic. Most built-in mathematical functions and operators can handle intervals.

The basic rule of interval arithmetic is that the result is an interval containing all possible results that can occur, if an input parameter is chosen for each parameter interval. Note, that if the same interval occurs twice in the parameter list, the intervals are independent. See the documentation for more details.

For interval algorithms see @"interval". For an introduction, visit the following notebooks.

diameter, middle, left, right
    right(w) : right end of the interval w.
    left(w) : left end of the interval w.
    diameter(w) : right(w)-left(w)
  
  These functions are functions for intervals, but they works for real
  numbers too.
  
  See: 
~ (Euler Core),
~ (Maxima Documentation)
interval, ~, ±
    interval(a,b) : the interval [a,b].
    ~a,b~ : the interval [a,b].
    a±d : is like ~a-d,a+d~ (use shift-F8 for ?)
    ~a~ : returns an interval [a-eps,a+eps].
  
  >1±0.05
   ~0.95,1.1~
  >~-1,1~
   ~-1,1~
  >~4~
   ~3.999999999999999,4.000000000000002~
  
  See: 
middle (Euler Core),
left (Euler Core),
right (Euler Core),
expand (Euler Core),
expand (Maxima Documentation),
intersects (Euler Core),
|| (Euler Core)
intersects, <<, <<=, expand
    intersects(a,b) : tests two intervals on non-empty intersection.
    a<<b : true if a is contained in b.
    a<<=b : true if a is contained in or equal to b.
    
    expand(x,d) : Expands a real number or an interval to a larger interval 
      The interval [x-d,x+d], if x is real, and an interval
      of d times the diameter of x, if x is an interval.
  
  >~2~ << expand(2,epsilon)
   1
  >expand(2,epsilon) << ~2~
   0
  
  See: 
interval (Euler Core),
|| (Euler Core)

Functions for Plots

Euler has the primary plot routines plot2d and plot3d. The following functions should only be used for special purposes. For more information, read the following notebooks.

plots
  Euler uses an external window for plots, but it can also copy plots to
  the text window. The preferred way to plot is via the utility
  functions plot2d and plot3d. These functions can plot
  
  - 2D and 3D functions and parametric curves or surfaces
  - implicit function of two or three variables
  - anaglyph, contour, or level line plots
  - user animated 2D and 3D plots
  - complex nets
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
plot3d (Plot Functions),
plot3d (Maxima Documentation)
aspect, getaspect
    aspect(x) : Sets the aspect ratio of the plot window.
      x=0 resets to the default values. This function will
      disable the "Graphics Window" settings in "Set Aspect"
      until aspect(0) or a restart of Euler.
  
    getaspect() : Gets the current aspect ratio of the Euler window.
  
  See: 
aspect (Euler Core),
aspect (Plot Functions)
frame
    frame() : Draws the frame around the plot.
      The frame size is the graphics window minus the boundary
      for labels.
  
    framecolor(color) : Sets the color for the frame. 
      The white color 0 prevents the frame from
      drawing.
  
  See: 
shrinkwindow (Plot Functions),
fullwindow (Plot Functions),
window (Plot Functions)
coordinates
  Euler uses plot coordinates or screen coordinates. 
  
  The screen coordinates always range from 0 to 1024, no matter if the
  actual screen format is square or rectangle. These coordinates are only
  used by elementary plot functions, and should not be used by the
  normal user.
  
  Plot coordinates are set by the recent plot. They refer to an area in
  the plane.
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
setplot (Euler Core),
setplot (Plot Functions)
hold, holding, clg
  Note that these functions should not be used with plot2d and plot3d.
  To add another plot use add=1 for plot2d.
  
    hold : toggles holding of the graphics on or off. 
      This applies to the fact that plot clears the graphics
      normally.
  
    hold on : toggles holding on.
  
    hold off : toggles holding off.
  
    holding() : The holding state.
  
    holding(f) : sets the holding state. 
      f should be 1 or 0. Returns the old state.
  
    clg : clears the graphics screen.
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
plot (Euler Core),
plot (Plot Functions)
huegrid, solidhue, solidhuecontour, subgrid
    huegrid(flag) : Turn the grid on or off.
  
    subgrid([ni,nj]) : Skip ni, nj gridlines in solid ior solidhue plots.
      Only grid lines i,j equal to 0 modulo ni,nj are
      drawn. This is used by plot3d to generate nice
      looking solid and hue plots.
  
  Instead of these functions, use parameters of plot3d.
  
  See: 
plot3d (Plot Functions),
plot3d (Maxima Documentation),
solid (Euler Core),
subgrid (Euler Core),
subgrid (Plot Functions),
huecolor (Euler Core)
keepsquare
    keepsquare(f) : sets a flag to determine the autoscaling. 
      If the flag is on, autoscaling will keep the y
      coordinates the same range as the x coordinates.
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
plot (Euler Core),
plot (Plot Functions)
plot, linestyle, setplot, linewidth, style
  These functions should be replaced by the various options of plot2d.
  
    plot(x,y) : connects the points x(i),y(i) with lines. 
      The coordinates are plot coordinates, which are set
      before the plot from the range of x and y, unless
      setplot() has not been called before, or scaling is turned
      off. y can contain more rows as x. This will plot x,y[r]
      for all rows of y simultaneously.
  
    plot() : The x and y plot range (1x4 vector [x1,x2,y1,y2]).
  
    setplot([x1,x2,y1,y2]) : sets the plot coordinates.
  
    scaling(f) : Sets the scaling flag. 
      Returns the previous value. 
  
    linewidth(n) : Sets the width of the line. 
      Returns the old width.
  
    style(string) : Sets line styles and marker styles.
      Ignores wrong styles or style=none.
  
  These functions should no longer be used and are replaced by plot2d.
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
rgb (Plot Functions)
mark, markwithcolor, markersize, markerstyle
    mark(x,y) : Works like plot, but does not connect the points.
      Use plot2d with >points instead of this function.
  
    markwithcolor(x,y,c) : Uses the color c to mark x and y.
      This is used plot3d to mark points in 3D with
      different colors.
  
    markersize(n) : Sets the marker size in screen coordinates.
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
coordinates (Euler Core),
markerstyle (Euler Core)
bar, barcolor, barRGB, polygon
  These functions should be replaced by the various options of plot2d.
  
    bar([xl,yu,w,h]) : Draws a rectangle 
      Use upper left coordinates (xl,yu), and lower
      right coordinates (xl+w,yu+h) in screen
      coordinates. The fill style is set with barstyle,
      and the color with barcolor. 
  
    barcolor([n]) : Sets the color index for the bar. 
      Returns the old color.
  
    barRGB(r,g,b) : Plots a matrix of red-green-blue values.
      The values r,g,b must of same size with values
      between 0 and 1.
  
    polygon(x,y,f) : Filled curve with optional outline.
      The fill style is set with barstyle, and the fill
      color with barcolor.
  
  See: 
color (Euler Core),
color (Maxima Documentation),
plotbar (Plot Functions),
polygon (Euler Core),
polygon (Maxima Documentation),
barstyle (Euler Core)
style, barstyle, markerstyle, linestyle
    barstyle(string) : Sets the fill style for bars and polygons.
      Available are "#" for solid colors, "#O" or "O#"
      for solid colors with black boundary, "O" for
      boundaries only, "\", "/" "\/", "+", "-", "|" for
      hatched styles.
  
    linestyle(string) : Sets the linestyle.
      Available line styles are "-" solid, "--"
      dashed, "." dotted, "-." for dash-dotted, ".-."
      and "-.-" for dash-dotted, or "->" for arrows.
      There is also an invisible style "i". Returns
      the previous value.
  
    markerstyle(string) : Sets the style for the markers. 
      Available styles are "<>", "[]", "o", ".",
      "..", "...", "..#", "+", "|", "-" or "*", and
      "<>#", "[]#", "o#" for filled markers, and
      "<>w", "[]w", "ow" for markers filled with the
      background color.
  
  These style functions return the old style, so that the plot function
  can reset the previous style. This is necessary, since some plot
  functions do not set the style, but use the global style.
  
  style(string) : Set the style of lines and points.
      Combines line styles like ".", markerstyles like "m.",
      and barstyles like "b/". Does not return the old style.
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
plot (Euler Core),
plot (Plot Functions),
mark (Euler Core),
mark (Plot Functions),
bar (Euler Core)
plotcubes
    plotcubes(M,[color]) : plots lines of M as cubes in 3D. 
      The lines contain [x1,y1,z1,x2,y2,z2] with
      lower and upper limits for the sides of the
      cubes. The cubes are drawn in the order of
      the lines. The utility function plotcubes()
      sorts the cubes properly. 
  
  Each cube can have a different color in the optional color vector.
  
  See: 
plot3d (Plot Functions),
plot3d (Maxima Documentation),
plot3dbars (Plot Functions),
plotcubes (Euler Core),
plotcubes (Plot Functions),
rgb (Plot Functions)
addpage, deletepages, showpage, pages, copytopage, copyfrompage
  Functions for animation of plots in Euler. Pages are hidden graphics.
  The user can add as many pages as the memory permits. The idea is to
  create these pages once, and then play them in an animated sequence.
  It is possible to hide the current page during the creation of the
  sequence by showing some other fixed page.
  
  Pages 1 and 2 can be combined for an anaglyph view. The graphics
  function (usually plot3d) will plot on both pages with a slightly
  different view. The combination merges both graphics in such a way,
  that the use of red/cyan glasses is possible.
  
    addpage() : Copies the graphics to a new page.
    deletepages() : Deletes all pages.
  
    showpage(n) : Shows the page number n (1 to number of pages). 
      If n is 0, it shows the normal graphics. You can show
      page 1, and draw on page 0.
  
    showpage(-1) : combines page 1 and 2 to an anaglyph image, 
      and sets the graphics to anaglyph mode. This is
      cancelled with a new plot or clg().
  
    pages() : The number of pages.
  
    copytopage(n), copyfrompage(n) : Copies to or from the page number n
      to the current graphics
  
  See: 
animate (Plot Functions),
anaglyph (Euler Core),
plot3d (Plot Functions),
plot3d (Maxima Documentation)
anaglyph, setanaglyph
  Anaglyph plot in Euler are realized by combining the pages 1 and 2 to
  a single rec/cyan image with the command "showpage(-1)". Both pages
  should contain two views of the same 3D scene with a slight different
  view.
  
    setanaglyph(a) : Tilts the current view with the angle a to the right.
  
  See: 
pages (Euler Core)
mesh, meshbar, meshfactor
  Obsolete. Use plot3d instead.
  
    mesh(A) : Plots a simple 3D plot of a(i,j) over a grid of points (i,j).
  
    meshbar(A) : Works like mesh(A) with bars.
      The plot consists of columns of height a(i,j). Works
      for 1xn vectors too.
  
    meshfactor(f) : Dumps the mesh with the factor (0 to 1) in height.
  
  See: 
plot3d (Plot Functions),
plot3d (Maxima Documentation)
solid, wire, fillcolor, project, view, twosides
  These are elementary functions for plotting. In most cases, they
  should be regarded as obsolete. Use plot3d instead.
  
    solid(x,y,z) : Plots a solid 3D plots of points x(i,j), y(i,j), z(i,j). 
      I.e., a rectangular grid (x,y) is mapped to these
      points. The two sided may have different colors,
      unless the twosides flag is off. Take care that the
      mapping is one-to-one and the plot does not
      self-intersect.
  
    solid(x,y,z,i) : Does not connect the i-th row to the i+1-th row. 
      This produces disconnected solid plots.
  
    solid(x,y,z,i,h,w,v,l) : Solid plot with shading and levels.
      
  This is the basic function for plot3d, which should be used instead. h
  is a matrix of hue values (between 0 and 1) for the shading. w is a
  matrix of values for each matrix point. v is either a row vector of
  level lines, which should be drawn, or a 2xn matrix of ranges of level
  lines. In case of ranges, the areas in each triangle between two
  bounds of each range are drawn with the contour color. v is ignored,
  if it is none. l is a 1x2 vector of limits. If limits is none or not a
  valid range, it is ignored. Triangles are clipped to these limits.
  
    wire(x,y,z) : Like solid, but does not fill the plot.
    fillcolor([n,m]) : sets the fill colors for both sides. 
      n,m are color indices (0..15) or rgb colors.
  
    project(x,y,z) : Projects the coordinates to the screen, 
      Just as in a solid or wired plot. Returns
      {vx,vy,vd}, where vx and vy are the screen
      coordinates, and vd is the distance to the eye. To
      mark a point in 3D, use this function, then
      fromscreen() and finally mark(), since mark works
      with plot coordinates.
  
    view([d,f,a,b]) : Sets the viewpoint of the camera. 
      d the camera distance to 0, f is the zoom factor,
      a the angle from the negative x axis and b the
      angle in height (measured in radial coordinates).
  
    twosides(f) : Turns the different colors for solid plots on or off.
  
  See: 
plot3d (Plot Functions),
plot3d (Maxima Documentation),
contourcolor (Euler Core),
color (Euler Core),
color (Maxima Documentation),
fromscreen (Plot Functions),
mark (Euler Core),
mark (Plot Functions),
solidhue (Euler Core),
solidhue (Plot Functions)
color, wirecolor, huecolor, contourcolor, setcolor, resetcolors
    color(n) : Sets the color for the plot.
      Ignores n=none. It returns the current color if n is
      missing. The color can either be an index (0..15), or a
      RGB color, defined by the "rgb" function. Returns the old
      value. 
  
  The predefined colors are: white=0,black=1,red=2,green=3,blue=4,
  cyan=5, olive=6, lightgray=7, gray=8, darkgray=9, orange=10,
  lightgreen=11, turquoise=12, lightblue=13, lightorange=14, yellow=15.
  These names are defined as constants.
  
  If the color is a vector of colors, each element will be used for one
  row of a plot of a matrix.
  
    wirecolor(n) : Sets the color of the wires. 
      Ignores n=none.
  
    huecolor(n) : Sets the color for the shading. 
      Ignores n=none. If the color=-1, then the hue
      determines the color, and if color=-2 the hue is
      translated into a spectral shading.
  
    contourcolor(n) : Set the color for contours. 
      Ignores n=none.
  
    setcolor(i,red,green,blue) : Sets the color number i 
      Red, green, blue are for all future
      plots. The values must be in in [0,1].
      The colors can also be set in the Euler
      menu. The colors are reset at each
      start of Euler.
  
  This function is not necessary for single plots, since colors can be
  set individually, using e.g. color=rgb(r,g,b). The colors set in
  this way are not saved by Euler globally.
  
    resetcolors() : resets all colors to the default values.
  
  See: 
rgb (Plot Functions),
solid (Euler Core)
hatchgridsize, pixel
    hatchgridsize(pixels) : Sets the grid size for hatched fills.
      The grid size will be set to a multiple of
      8.
  
    pixel() : The width and height of a screen pixel in plot coordinates. 
      This helps avoiding unnecessary computations in the plot
      routines, especially for adaptive plots.
  
  See: 
toscreen (Plot Functions),
fromscreen (Plot Functions)
mouse, mousedrag, mousepos, mousestate
    mouse() : mouse() or mouse(string). Waits for mouse click or key.
      Waits until the user has clicked into the plot window.
      Returns the x,y-coordinates in plot coordinates. If the
      user pressed a key, the function will return the key, not
      an 1x2 vector.
  
    mousedrag() : Waits for mouse or keyboard events.
      The function returns {flag,m,time}, where flag is 0
      for keyboard events, 1 for mouse down, 2 for dragging,
      3 for mouse up. The result m contains the mouse
      position or the keyboard code. The time is the time of
      the event in milliseconds.
  
    mousestate() : 0 for mouse down and 1 for mouse up.
    mousepos() : The current mouse position in plot coordinates.
  
  See: 
plot2d (Plot Functions),
plot2d (Maxima Documentation),
setplotm (Plot Functions),
fromscreen (Plot Functions),
toscreen (Plot Functions),
dragvalues (Plot Functions)
ctext, rtext, text, vtext, vctext, vutext, vcutext, textheight, textwidth, 
textcolor
  The text functions draw text to the graphics window. The font is
  fixed, and can be changed globally. These functions use screen
  coordinates. To label anything in a 2D plot, use "label".
  
    text(string,[n,m]) : Draws the string to screen coordinates n,m
      The screen is aligned at the upper left edge.
      All text functions work in the form
      text(string,n,m) too.
  
    ctext(string,[n,m]) : Draws the string horizontally centered.
    vtext(string,[n,m]) : Draws a string downward.
    vctext(string,[n,m]) : Draws a string downward centered.
    vutext(string,[n,m]) : Draws a string upwards.
    vcutext(string,[n,m]) : Draws a string upwards centered.
    rtext(string,[n,m]) : Draws it right aligned.
    textheight() : The maximal text height.
    textwidth() : The average text character width.
  
    textcolor([n]) : sets the color index for text (0..15), 
      or returns the current color only.
  
  See: 
label (Plot Functions),
label (Maxima Documentation),
coordinates (Euler Core)
setfont
    setfont(lines,name) : Sets the graphics font.
      This sets a font with size and name for one
      session. The font is automatically applied to
      the current graphics. Euler can currently use
      only one font. The default is "Courier New"
      and 40 lines per screen. If the name is blank,
      the font is not changed. Use this for special
      purposes or for small prints, where larger
      font sizes are important.
  
  The utility function setfont(fontsize,size,name) provides a better
  interface.
  
    setfont() : Resets the old font size and name.
  
  See: 
setfont (Euler Core),
setfont (Basic Utilities)
density
    density(A) : Draws the elements a(i,j) 
      it uses darker and lighter values in a square grid seen
      from above. You should scale the matrix to [0,1) since
      only the fractional part is used.
  
  Use plot2d instead.
  
  See: 
contour (Euler Core),
contour (Maxima Documentation)
getred, getgreen, getblue, putred, putgreen, putblue
  These functions transfer one RGB encrypted pixel color to red, green,
  blue values in the interval [0,1], and reverse. Used by loadpixels(),
  savepixels(), and by rgb(r,g,b), getrgb(x).
  
    getred(x) : Gets the red part of an RGB color
    getgreen(x) : Gets the green part of an RGB color
    getblue(x) : Gets the blue part of an RGB color
    
    putred(x) : RGB color with this red part
    putgreen(x) : RGB color with this green part
    putblue(x) : RGB color with this blue part
  
  See: 
rgb (Plot Functions),
getrgb (Plot Functions),
loadrgb (Euler Core),
savergb (Euler Core)
loadrgb, savergb
  Load and save an array of pixels into an image file of PNG format.
  
    loadrgb(filename) : loads a PNG into an RGB array
    savergb(x,filename) : saves an RGB array into a PNG file
  
  To convert the RGB format to its channels, use getred, getgreen, getblue,
  or getrgb. To convert the channels into an RGB format use rgb.
  
  See: 
rgb (Plot Functions),
getred (Euler Core),
getgreen (Euler Core),
getblue (Euler Core),
putred (Euler Core),
putgreen (Euler Core),
putblue (Euler Core),
getrgb (Plot Functions)
contour
  Use plot2d or plot3d instead.
   
    contour(A,v) : Draws contour lines of a(i,j) at heights v(k). 
      a(i,j) is interpreted as the values of a function f
      at (i,j). Note, that i and j are always scaled to a
      square grid, unless you specify a non-square plot
      window. The function does not draw the plot
      coordinates, so they must be set manually using e.g.
      xplot.
  
  The advanced utility function plot2d should be used instead of this
  elementary function.
  
  See: 
density (Euler Core),
density (Plot Functions),
fcontour (Plot Functions),
contourcolor (Euler Core)
plotarea, fullplotarea, margin
    plotarea(x,y) : Determines the necessary plot range
  
    fullplotarea() : Returns the full plot area including margin.
      This works like plot(), but takes care of margins.
  
    margin(x) : Sets a margin between 0 and 1 around the plot.
  
  Use the parameters of plot2d instead.
  
  See: 
plot (Euler Core),
plot (Plot Functions),
plot2d (Plot Functions),
plot2d (Maxima Documentation)
insimg, loadimg, loadanaglyph
    insimg(n,"name",flag,crop) : Inserts the graphics into the notebook.
      The graphics is scaled to n lines.
      "name" is optional and appended to the
      file name on export. flag is for
      anti-aliasing, and crop for the crop
      (1x4 vector). This function is
      overloaded with a more comfortable
      function.
  
    loadimg(n,"filename",flag) : Loads an image into the notebook.
      Loads a graphics from a file with maximal
      height n. If larger, the graphics scales
      down.
  
    loadanaglyph(n,"left,"right") : Generates an anaglyph.
      Loads two graphic files and combines
      them to one anaglyph image. The files
      must have the same size, and be at
      least 16x16. The image is inserted
      into the notebook with height at most
      n.
  
  For these three functions there are service functions with more
  flexible parameters.
  
  See: 
insimg (Euler Core),
insimg (Plot Functions),
loadimg (Euler Core),
loadimg (Plot Functions),
loadanaglyph (Euler Core),
loadanaglyph (Plot Functions)
postscript, pswindow
  postscript "file.ps" dumps the graphics to the file in postscript
  format. To set the size of the output in cm, use pswindow([w,h]).
  w=h=15 is the default.
  

Programming Euler

The most part of Euler is programmed in the Euler language itself. These programs are interpreted, which makes them slower than the built-in functions. For practical purposes, the speed difference is very often negligible, especially, if the programs use the matrix language of Euler.

For an introduction to the programming see the following notebook.

function, endfunction, return, parameter, none, alias
  A function in Euler has the following form.
  
    function name (parameter names)
    ...
    return ...
    ...
    endfunction
  
  A function can contain more than one return statement. If the function
  does not contain a return statement, it returns "none", a string with
  one character equal to 1. This result will yield no output.
  
  If the function line in the notebook ends with three dots ..., the
  function can be interpreted with one stroke of the return key. To edit
  such a function use the internal editor with F9, or click into the
  body of the function to edit the function.
  
  See: 
Keyboard (Overview) >function f(x) ... $ if x>0 then return x^3 $ else return x^3 $ endif; $endfunction One-line functions return only the result of a simple expression. They are defined in the following simplified form. >function f(x) := x^x >f(1:5) [ 1 4 27 256 3125 ] A function can return multiple values. The syntax uses {...}. The return values can be assigned to variables. >function sort2 (a,b) ... $ if a<b then return {a,b} $ else return {b,a} $endfunction >{c,d}=sort2(5,4); [c,d] [ 4 5 ] If the return values are used as arguments to other functions, only the first one is used by default. This can be changed with the args modifier. There are functions, which cannot be called. The purpose of these functions is to provide help in the usual form for DLL functions, or for Maxima functions. Use the keyword comment to define such functions. function comment name (parameters) ## Comment ... endfunction You can define an alias name for a function with alias original aliasname See:
return (Euler Core),
return (Maxima Documentation),
parameters (Euler Core),
## (Euler Core),
protect (Euler Core),
overwrite (Euler Core),
maxima (Euler Core),
symbolic (Euler Core),
dll (Euler Core)
repeat, break, continue, while, until, end, for, loop, #, index
  repeat; ...; end;
  
  This is an eternal loop. 
  
    until condition;
    while condition;
  
  These statements must be inside a repeat loop. The until statement
  finishes the loop if the condition is true. It jumps after the
  end of the loop. The while statement finishes the loop if the
  condition is false.
  
    for i=n to m; ...; end;
    for i=n to m step i; ...; end;
    v=...; for x=v; ...; end;
  
  This is the for loop. It can also be used to iterate through a row or
  column vector v as above. Changing v, or the loop step, or the end
  values inside the for loop has no effect on the loop. The vector v is
  copied to an allocated memory area. Again, return or break end the
  loop. 
  
    loop 1 to n; ...; end;
  
  This is a faster loop. The loop index can be called with index() or
  with the special symbol #. Again, return or break end the loop.
  
    break;
    continue;
  
  Each loop can be finished with return or break (usually in an if statement),
  where break jumps to the end of the loop. The continue command jumps
  to the start of the loop. 
  
  >function test ...
  $  s=0;
  $  n=1;
  $  repeat
  $    s=s+1/n;
  $    while s<10;
  $    n=n+1;
  $  end;
  $  return n;
  $endfunction
  >test
   12367
  
  Loops can also be used in the command line, as long as they fit into
  one line.
  
  >s=0; loop 1 to 10; s=s+#; end; s,
   55
  >s=0; for i=1:10; s=s+i; end; s,
   55
  >s=0; for i=1 to 10; s=s+i; end; s,
   55
  >s=0; i=1; repeat; s=s+i; i=i+1; until i>10; end; s,
   55
  >s=0; i=1; repeat; s=s+i; i=i+1; while i<=10; end; s,
   55
  
  See: 
not (Euler Core),
not (Maxima Documentation),
and (Euler Core),
and (Maxima Documentation),
or (Euler Core),
or (Maxima Documentation)
if, elseif, endif, else, then
    if condition then ...; endif;
    if condition then ...; else; ...; endif;
    if condition then ...; elseif condition; ...; endif;
    if condition then ...; elseif condition; ...; else; ...; endif;
  
  The command "then" can be omitted, though this is not encouraged. The
  commands can be used in the command line, as long as they fit into a
  single line.
  
  Note that loops and ifs cannot be used in one-line functions.
  
  >function signumstr (x)
  $  if x<0 then return "negative"
  $  elseif x==0 then return "zero"
  $  else return "positive"
  $endfunction
  >signumstr(-1), signumstr(0), signumstr(1)
   negative
   zero
   positive
  
  See: 
not (Euler Core),
not (Maxima Documentation),
and (Euler Core),
and (Maxima Documentation),
or (Euler Core),
or (Maxima Documentation)
argn, totalargn, args, getarg, arg1, arg2, arg3
  The arg... group of functions gives you access to unnamed arguments of
  functions. Unnamed arguments get the default names arg1, arg2, arg3,
  ... and can be accessed by this name or by the argument number.
  
    argn() : The number of arguments, given to the function.
    args(n) : returns all arguments from the n-th argument on, 
    getarg(n) : get the n-th argument to a function call.
  
    args() : ll parameters from the first unnamed argument 
      or the first argument after the semicolon ";" (semicolon
      parameters).
  
  Note that args() can deliver multiple returns which have the args flag
  (see the documentation for "return"). They can all be used as
  arguments to another function. In the following example, we pass a
  function as a parameter to another function, which calls it with
  args.
  
  >function f(a,b,c) := a*b*c
  >function g(h) := h(args())
  >g("f";5,6,7)
   210
  
  See: 
function (Euler Core),
return (Euler Core),
return (Maxima Documentation)
comment, endcomment, skipcomment, //, /*, */, ##, .., ...
  comment
  ....
  endcomment
  
  Brackets for a comment in a file. The load command will show the
  comment, unless this is turned off. comment and endcomment must be on
  a line of their own.
  
    comments on : turns comments on.
    comments off : turns comments off.
  
    skipcomment
    ....
    endcomment
  
  Like comment, but will never print.
  
    /*
    ....
    */
  
  Shortcut for skipcomment. In functions /* ... */ can also be used.
  
    // is a one line comment, which is removed from a function definition.
  
  In functions, lines starting with ## are the help text for the
  function, if they follow the function header immediately. Otherwise,
  they are ordinary comment, but will not be removed from the function
  body. So they are visible with "type function".
  
    .. skips rest of this line and continues with next line.
  
    ... works like .., but sends all lines as a multi-line command.
  
  See: 
load (Euler Core),
load (Maxima Documentation),
function (Euler Core)
parameters, arguments
  Parameters of functions can have default values. If the parameter has
  a default value in the function definition as in f(x,n=3), this value
  is used, unless two arguments are passed to f. If a parameter is
  missing as in h(x,,4), the default value for the second parameter is
  used.
  
  >function f(x,a=5) := a*x
  >f(2)
   10
  >f(2,3)
   6
  
  Parameters with default values can be set with assigned arguments in
  the form "f(10,a=5)". This allows very flexible functions, where the
  order of the assigned arguments does not matter. A shortcut for a
  boolean assigned argument is ">a" (for a=1) or "<a" for (a=0).
  Moreover, "=a" will set an assigned parameter with the same value as
  the variable "a" in the calling program.
  
  By default, it is only possible to pass arguments with values if the
  corresponding parameter with default value does exist. But the
  modifier "allowassigned" in the definition of the functions allows
  additional assigned arguments. These arguments work like variables in
  the function.
  
  Parameters of functions are passed by value, unless the parameter
  name starts with %... (See the "global" command for global variables).
  Even, if the parameter is passed by reference, it can only change its
  value, not its type. If it is a matrix, it cannot change its size.
  
  >a=4;
  >function change$x (x) ...
  $  x=5;
  $endfunction
  >change$x(a); a,
   4
  >function change$x (%x) ...
  $  %x=5;
  $endfunction
  >change$x(a); a,
   5
  
  Vectors and matrices are passed by reference. So if you change elements
  those elements will be changed in the original matrix. The matrix
  cannot be replaced completely, however. Even "v=v" in the function
  already copies the matrix argument "v" to a local copy.
  
  >v=1:5
   [ 1  2  3  4  5 ]
  >function change$w (w) ...
  $  w[1]=2;
  $endfunction
  >change$w(v);
  >v
   [ 2  2  3  4  5 ]
  
  If the call contains a semicolon instead of the colon to separate
  the parameters as in f(x,y;4,5), the parameters 4,5 are extra
  parameters, which can be passed from f to other functions using the
  args() function. These parameters are called semicolon parameters. For
  an example, see the "args" function.
  
  Semicolon parameters can not be used inside a function, and can only
  be uses via the args() function as parameters for other functions.
  References to variables, which are given as a semicolon parameter, are
  visible however. This does even work in evaluations of expressions,
  when args() is used in the parameter list of the evaluation.
  
  >expr="a*x";
  >function f(expr) := expr(2,args());
  >function g(expr,a) := f(expr;a)
  >g(expr,4)
   8
  
  Parameters can be restricted in the types of the values they can be
  assigned to, when the function is called. Possible restrictions are
  
    real, complex, interval, integer: include scalars, vectors or matrices
    scalar: no vector, no matrix
    vector: row vector, includes scalar
    column: column vector, includes scalar
    numerical: not a string or compressed matrix
    string: excludes vector of strings
    cpx: compressed matrix
    positive, nonnegative: additional hint
  
  These types can be combined. E.g., "complex vector" will accept all
  row vectors of type complex or real. 
  
  There are the shortcuts 
  
    number = real scalar
    natural = non-negative integer
    index = positive integer scalar
    indices = positive integer vector
  
  These parameter types are restrictions. There is one parameter type,
  which is not a restriction.
  
  none: Allows the special string none for this parameter.
  This is automatically assigned to a parameter with default value none.
  
  >function f(x:number) := x^2
  >f(1:5)
   Function f needs a scalar for x
   
   Error in :
   f(1:5)
      ^
  >f(5)
   25
  
  The scalar type is often necessary. To allow for a vector, use "map".
  
  Assigned parameters are parameters with a default value as in
  f(x,y,c=5). These parameters must be the last parameters in the
  parameter list, even after semicolon parameters. As a special
  convenience >c is equivalent to c=true, and <c is equivalent to
  c=false. Boolean parameters are often used in plot2d and plot3d.
  
  Arguments can be passed to a function with no parameters. The
  function names these parameters as "argX", where X is the position
  number of the parameter. Additional arguments are not possible for
  functions with parameters. In this case the parameter and argument
  counts must agree.
  
  See: 
args (Euler Core),
args (Maxima Documentation),
argn (Euler Core),
useglobal (Euler Core),
global (Euler Core)
useglobal, global, setglobal
  useglobal : Allows reference to global variables in functions.
  global variable: Allows reference to a global variable in functions.
  setglobal variable: Sets the variable global in all functions.
  
  Note that useglobal is set in all one-line functions automatically.
  
  >function f(x)
  $  useglobal; // alternatively: global a;
  $  return a*x;
  $endfunction
  >a=4; f(2)
    8
  
  See: 
function (Euler Core)
map
  Vectorization
  
  Mathematical functions implemented in Euler functions should be ready
  to handle vector input. Often, this is automatic. But if the function
  contains conditional branches, or other more complicated routines,
  that might not be the case. To make sure, you can plot the functions
  and use them in numerical routines, you can define the function as a
  map function at compile time.
  
    function map f (x:scalar; p:vector)
    ...
  
  In this case, f will map to the elements of a vector x. I.e., it will
  compute the value for each element of x one by one. All parameters
  after a semicolon are excluded from the mapping.
  
  It is also possible to map a function at run time. Use either of the
  following commands.
  
    map("f",X,Y,...) : maps the function f to the elements of X,Y. 
    fmap(X,Y,...) : does the same.
  
  The mapping obeys the rules of the Euler matrix language. The result
  of f must be a scalar numerical value. Parameters after a semicolon
  instead of a comma are protected from the mapping, and are used as
  they are.
  
  You can map to a function with no result (return missing). The map
  function will return no result too. No result is simulated in Euler
  with the value none.
  
  The following function restricts to a number, but since "map" is used,
  it can map itself to vectors or matrices. It will still not work for
  complex scalars or matrices.
  
  >function map test (x:number) ...
  $  if x<0 then return 0
  $  else return x^2
  $endfunction
  >test(-1:1)
   [ 0  0  1 ]
  
  See: 
function (Euler Core),
none (Euler Core)
protect, overwrite
  protect: protect all current function definitions and variables 
  from overwriting.
  
  Variables starting with default... are not protected. The protection
  can be ignored with a global flag in the Euler menu.
  
  overwrite: keyword to enable the redefinition of a protected function.
  
  Functions can be protected just as built-in functions. By default,
  protected functions can not be redefined. But the overwrite keyword
  unlocks the protection.
  
  function overwrite f (...)
  ...
  
  See: 
function (Euler Core)
trace, traceif
  Traces Euler functions.
  
    trace : toggle trace mode
    trace on,off : set trace mode
    trace function : toggles tracing the function
  
  While tracing is on, the following keys are available.
  
    cursor down : single step
    cursor right : step over subroutines
    cursor up : go until return
    cursor left : end trace
    insert : evaluate expression
    escape : abort execution
  
  In functions, it is useful to start tracing conditionally.
  
    traceif condition : start tracing, if the condition is true
  
  You can then inspect local values at this point, and continue execution
  with the cursor left key.
  
    trace errors : trace on errors
  
  This is useful to start tracing on errors. The user can inspect
  variables. In a function, only local variables are available. An empty
  input aborts the execution.
  
{, }, multiple
    return {a,b,c} : Returns multiple values from a function.
  
    {a,b,c}=f(...) : Assigns these values to variables.
  
    A{k} : The k-th element of the matrix A, as if A was flattened 
      to a row matrix. Equivalent to redim(A,1,prod(size(A)))[k],
      but faster. Indices out of bounds produce an error. This
      works for string matrices too.
  
    A{i,j} : The i,j-th element of the matrix A, often like A[i,j].
      But for vectors, the index exceeding 1 will be set to 1.
      This allows functions obeying the matrix language of Euler,
      since f(A,B) will work for a row vector A and B, combining
      the proper elements, if A{i,j} and B{i,j} are combined.
      A{i,j} cannot be used for assignments. Indices out of
      bounds produce an error.
  
  See: 
= (Euler Core),
= (Maxima Documentation),
{{ (Euler Core),
}} (Euler Core)
{{, }}, collection, call
  Collections in EMT can contain any other data type. They are behave
  like immutable lists. Use collections to return multiple values from a
  function in one EMT object (in contrast to multiple returns).
  
    {{a,b,c,...}} : Creates a collection of objects
  
  Collections can also catch multiple returns of a function.
  
  >c={{sort(y)}}
  
  Moreover, collections can be executed, if the first element is the
  name of a function or an expression. We call this a "call collection".
  The rest of the list is passed to the function as extra parameters.
  This is an alternative to semicolon parameters in calls to plot
  functions or numerical algorithms.
  
  >function f(x,a) := a*x^2
  >plot2d({{"f",5}},-1,1):
  
  In expressions, the extra parameters (besides the default x,y,...)
  must be named elements of the collection.
  
  >L={{"sin(a*x)",a=6}}; L(4)
  
  See:{,},glist,;,evaluate
operator, postfix, prefix, strong
    function operator op (x,y) := ...
    function postfix f (x) := ...
    function prefix g (x) := ...
  
  Define operators and postfix functions. Usage:
  
  4 op 5
  5 f
  g "test"
  
  The operators bind weakly. 
  
  1 + 5 op 6 + 1 -> (1+5) op (6+1)
  
  The modifier "strong" makes them bind strongly.
  
  >function strong operator op (x,y) := ...
  >1 + 5 op 6 + 1 // is like: 1 + (5 op 6) + 1
  
  There is no strong version of prefix functions.

Numerical Algorithms

There are more numerical algorithms in Euler than these core functions. You find an overview on the following pages.

and in some other introduction notebooks.
nelder, brent
    brent("f",a,d,eps) : returns a minimum close to a. 
      The function goes away from a with step size d
      until it finds a good interval to start a fast
      iteration. Additional parameters are passed to
      f.
  
  The function fmin is recommended instead of this function.
  
    nelder("f",v,d,eps) : return a minimum close to v.
      For functions f(v), accepting 1xn vectors v.
      The parameter d is the initial simplex size.
      eps is the final accuracy.
  
  See: 
nelder (Euler Core),
nelder (Numerical Algorithms),
neldermin (Numerical Algorithms),
brentmin (Numerical Algorithms),
fmin (Numerical Algorithms),
fmax (Numerical Algorithms)
runge1, runge2
    runge1("f",a,b,n,y) : does n steps of the Runge-Kutta method 
      to solve y'=f(x,y). y may be a 1xn vector or a
      real (the initial value), (a,b) is the
      x-interval.
  
    runge2("f",a,b,y,eps,step) : does the same with adaptive step sizes.
      The initial size is step, and the
      accuracy is eps.
  
  The user should use runge, and adaptiverunge instead.
  
  See; runge,adaptiverunge
toeplitz, toeplitzsolve
    toeplitz(v) : returns the Toeplitz matrix T to the vector v. 
      v must be a 1xn vector with odd n=2m+1. T has the
      property that T[i,j]=v[m+i-1-j]
  
    toeplitzsolve(v,b) : Solves T\b fast, where T is the Toeplitz matrix to v.
  
svd
    svd(A) : computes {U,v,W}, the singular value decomposition of A.
      A must be a real matrix. U,V are orthogonal, and we get
      A=U.diag([cols(A),cols(A)],0,v).W'
  
  A must have more rows than columns.
  
  See: 
svdsolve (Linear Algebra),
fit (Linear Algebra)
accu, accure, accuim, accua, accub
  The long accumulator in Euler is mainly used to compute an exact scalar
  product, necessary for residuum iteration. However, it can be used for
  other purposes as well. Euler cannot add or multiply two accumulators,
  but can add any value, or a product to the accumulator.
  
  The accu... functions gets the content of the long accumulator (or its
  real, imaginary (for complex accus), upper or lower (for interval
  accus) part.
  
  See: 
accuload (Euler Core),
accuadd (Euler Core),
residuum (Euler Core)
accuload, accuadd
    accuload(v) : Loads the sum of the vector v to the accumlutor.
    accuload(v,w) : Loads the scalar product of v and w.
    accuadd(v) : Adds the sum of the vector v to the accumulator.
    
    accuadd(v,w) : Adds the scalar product of v and w. 
      Both must be 1xn vectors.
  
  See: 
accu (Euler Core),
accure (Euler Core),
accua (Euler Core),
accub (Euler Core),
residuum (Euler Core)
residuum
    residuum(A,x,b) : Computes A.x-b exactly to the last digit. 
      A, x and b must be compatible matrices.
  
  See: 
xlgs (Numerical Algorithms)

Functions for Units

Units can be converted in Euler. However, Euler does not check for correct application of units. It simply converts numbers. The final results should be stored in the international system (IS).

units, easyunits, ->
  Units are simple conversion constants in Euler. All units end with $,
  but it is possible to use them without $, if easyunits is set.
  
    easyunits on/off: sets or releases use of units without $.
    easyunits: toggles use of units without $.
  
  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
  >2km/h->" m/s"
   0.555555555556 m/s
   
  See: 
Units (Overview)

Functions for Maxima

The main interface to Maxima should be the symbolic expression and the symbolic function. With the help of these, Maxima integrates seamless into Euler.

More commands are defined in a utility file. Those commands start with mxm..., and use Maxima to provide symbolic results at run time.

There are also commands in Maxima, which extend the Maxima syntax for Euler.

For a start and more information consult the following pages.

maxima, symbolic, &, &=, &:, ::, :::, @:, ::=, &:=, &&=, $
  Maxima is a powerful, open Algebra system contained in Euler. By
  default, the Maxima process starts in the background, whenever Euler
  starts or is restarted. Note that Euler restarts by default when a new
  notebook is loaded.
  
  Maxima can be used directly or via symbolic expressions. The direct
  mode has a compatibility mode, which makes the syntax of Maxima and
  Euler more similar. It is the preferred mode.
  
    :: ... Maxima command in compatibility mode.
    maxima ... is an alternative way to start a Maxima command.
    ::: ... Maxima command in direct mode
    : ... Maxima command in direct mode (disabled by default).
  
    &expression : evaluates a symbolic expression. 
      Returns the string evaluated in Maxima. An alternative
      is ::expression. The expression must be well formed.
      Maxima command flags can be appended using |, and with
      can be used to assign values. The expression can be
      enclosed in quotes.
  
  >&(x+4)^2 | expand
           2
          x  + 8 x + 16
  >&"diff(x^5,x,2) with x=5"
                2500
  
  The result in Euler is a string with the symbolic flag. It can be used
  line any other expression.
  
  >plot2d(&diff(sin(x)*x,x),0,2pi);
  
  function f(x) &= expression defines a function for Maxima and Euler.
  The expression is evaluated before the function is defined.
  
  >expr &= x^4;
  >function f(x) &= diff(expr,x)
             3
          4 x
  
  Symbolic functions assigns a symbolic expression to a variable in
  Maxima and Euler, after the expression has been evaluated by Maxima.
  
  a &&= expression assigns an expression only in symbolic form. The
  expression is evaluated before it is defined. This is useful if the
  expression contains functions, which cannot be evaluated numerically.
  
  >function f(x) &&= expression 
    
  defines a function for Maxima only. The expression is not evaluated in
  the case. It works like a macro. So this definition can be used for
  functions, which need to evaluate at run time with additional
  information.
  
  >function D2(f,x) &&= diff(f,x,2);
  >&D2(x^5,x) 
             3
         20 x
  
  &:expression returns the string, evaluated in Maxima, and evaluates
  the result in Euler. In functions, the string is inserted into the
  function body at compile time. Finish the command with , or ;, or
  enclose the expression in quotes.
  
  >function f(x) &= x^4;
  >function df(x) := &:diff(expr,x);
  >type df
   function df (x)
   useglobal; return 4*x^3; 
   endfunction
  
  @:expression works like &:expression (deprecated)
  
    a &:= value sets the value for both Euler and Maxima.
    a ::= value does the same (deprecated)
  
    $expression :  a symbolic expression like &expressions. 
      But it prints with Latex, if Latex is installed.
      There might be a short delay, if such a command is
      executed.
  
  See: 
mxmstart (Maxima Functions for Euler),
mxmset (Maxima Functions for Euler),
mxmget (Maxima Functions for Euler),
function (Euler Core)
@, @@
    @expr : Is replaced with the content of the variable expr 
      in symbolic expressions, Maxima commands, and function
      definitions at compile time. It is an alternative to
      symbolic expressions, in case the expression is defined in
      an Euler string only. But it will also work for numerical
      values and even matrices.
  
  >expr:="x^5"; &diff(@expr,x)
      4
      5 x
  >a=solve("x^x",1,y=2), &bfloat(@a^@a)
   1.55961046946
      1.9999999999999988897769753748435b0
  >v=1:5; &map(sqrt,@v)
      [1, sqrt(2), sqrt(3), 2, sqrt(5)]
  
  In function definitions, @expr inside strings is not replaced at
  compile time. Therefore, &"diff(@expr,x)" in functions replaces @expr
  at run time, and computes the derivative then.
  
  >function dtest (expr,var) := &"diff(@expr,@var)"
  >type dtest
   function dtest (expr, var)
   useglobal; return &"diff(@expr,@var)" 
   endfunction
  >dtest("x^4","x") 
            3
         4 x
  
  @@ : In Maxima and symbolic expressions, this allows the input of a
  "@".
  
  See: 
mxm (Euler Core)
mxm
    mxm("diff(x^2,x)") : Evaluates in Maxima. 
      The result is a string. This is deprecated.
      Use &diff(x^x,x) instead.
  
  Note that @a is replaced by the content of the variable a, if @a is
  contained in the Maxima command.
  
  >expr := "x^3-x"
   x^3-x
  >mxm("diff("+expr+",x)")
   3*x^2-1
  >&diff(@expr,x)
            2
         3 x  - 1
  
  See: 
mxmeval (Maxima Functions for Euler),
@ (Euler Core),
@ (Maxima Documentation)
maximamode, euler
  Toggles Maxima mode. If on, all commands are sent to Maxima. In maxima
  mode, 
  
    euler command
  
  will send the command to Euler.
  
    maximamode: Toggles the mode
    maximamode on: Compatibility mode
    maximamode direct: Direct mode
    maximamode off: Maxima mode is off
  
  See: 
::: (Euler Core),
:: (Euler Core),
:: (Maxima Documentation),
: (Euler Core),
: (Maxima Documentation)
latex:, maxima:
  If LaTeX is installed, formulas can be used in comments, in the
  command output and in plots. 
  
  >latex: expression
  
  Produces a Latex formula. Of course, the expression must be a valid
  Latex code.
  
  >maxima: expression
  
  Inserts a formula for this symbolic expression. Any symbolic
  expression or variable can be used.
  
  See: 
texpng (Plot Functions)

Extensions of Maxima in Euler

The following functions are symbolic functions defined in a file, which is loaded every time Maxima starts. These functions are extensions to the Maxima syntax specific to Euler.

grad, hesse, diffat
  grad(expression) symbolic: Gradient of expression
      The variables are sorted alphabetically.
  
  gradient(expression,variable) symbolic: Gradient of expression
  
  hesse(expression) symbolic: Hessian of expression
      The variables are sorted alphabetically.
  
  hessian(expression,[variables]) symbolic: Hessian of expression
  
  diffat(expression,x=value{,n}) symbolic: Evaluated differentiation
      Works like diff(expression,x,n) with    
      x=value
  
crossproduct, scalp
    crossproduct(v,w) symbolic: Cross product in 3-space
  
    scalp(v,w) symbolic: Scalar product
  
getlagrange, solvelagrange
    getlagrange(f,g,[variables]) symbolic: Get Lagrange equations
      for extreme values of f(variables) with the condition
      g(variables)=0.
  
    getlagrange(f,g) symbolic: Get Lagrange equations
      Use the variables of f.
  
    solvelagrange(f,g,[variables]) symbolic: Solve Lagrange equations
      for extreme values of f(variables) with the condition
      g(variables)=0.
  
    solvelagrange(f,g) symbolic: Solve Lagrange equations
      Uses the variables of f.
  
with
    with: expression with [var=value,var=value,...] (symbolic)
    with: expression with var=value (symbolic)
  
  Evaluate the symbolic expressions and assign variable values. Works
  like at(expression,assignments)
  

Global Variables

EMT is based on a variable stack. The usual global variables can be read and their value can change, but neither their type nor their size. Thus EMT needs some functions for global variables which can be changed.

Global variables are cleared when the notebook is reset.

gset, gvar, gremove, gclear
  These functions handle global variables with all supported values
  in EMT.
  
    gset("name",value) : Sets a global variable "name" to a value
    gvar("name") : Reads the global variable "name"
    gremove("name") : Removes the global variable
    gclear() : Removes all global variables
  
  As a shortcut
  
    a$$ = value
    var = a$$
  
  can be used.
  
  See: 
glist (Euler Core)
glist, glistadd, glistinsert, glistvar, glistdelete, glistlength, glistremove
  Global lists can take any supported values in EMT. New elements can be
  added at any position, but appending a new element is the most
  efficient way to add an element. Access to any element is fast.
  
    glist("name") : Create a list "name"
    glistget("name",position) : Get an element from a list
    glistadd("name",value) : Append a value to a list
    glistinsert("name",position,value) : Insert an element to a list
    glistput("name",position,value) : Replace an element in a list
    glistdelete("name",position) : Delete an element
    glistlength("name") : Length of a list
    glistremove("name") : Delete a list
  
    listlists : List of all global lists
    listlists name : all global lists with name containing the string
      Similar to listvars
  
  See: 
gset (Euler Core)

Various Core Functions

matlab
    matlab on
    matlab off
    matlab
  
  Toggle the Matlab mode. In this mode the syntax of Euler is closer
  to Matlab. For a reference read the documentation about Euler and
  Matlab.
  
tic, toc
  tic; ... commands ... toc; measures the time of the commands
  in millisecond resolution.
  
  >tic; inv(normal(100,100)); toc;
   Used 0.047 seconds
   
time, wait
    time() : The time in seconds.
  
    wait(n) or wait(n,text): waits for n seconds (text for status line).
      This functions interrupts, if the user
      presses a key. Returns {time,key} the time
      waited, and a key, if one was pressed.
  
  See: 
key (Euler Core),
key (Maxima Documentation)
epsilon, setepsilon, localepsilon, relerror
    epsilon() : The internal epsilon used for various purposes.
    setepsilon(x) : sets the internal epsilon.
    localepsilon() : sets the epsilon locally for a function.
  
  Some numerical algorithms allow to set the local epsilon using an
  assigned argument eps=...
  
    relerror(x,y) : |x-y|/|y| for y<>0, and x for y~=0.
  
  >bisect("x^2-2",0,2)
   1.41421356237
  >bisect("x^2-2",0,2,eps=0.001)
   1.41455078125
  
  See: 
~= (Euler Core)
free, hexdump, memorydump, list, listvar, listvars, forget, clear, clearall, 
remvalue, type
    free() : The free space on the stack.
  
    list : lists all built-in functions, commands and all functions.
    list string: lists only items containing the string.
  
    listvar: lists all variables, their types and sizes.
    listvar string: lists all variables containing the string.
  
    listvars: lists all variables, their types and sizes.
      Alias to listvar.
    listvars string: lists all variables containing the string.
      Alias to listvar string.
  
    clear var,... : removes the variables.
    clearall : removes all user variables, 
      but not default variables and %variables.
  
    remvalue var,... : removes the variables
  
    forget function,... : removes the functions.
  
    hexdump var : dumps the variable or user defined function var 
      in hexadecimal form.
  
    type function : Type the code of a function
    type var : Type the content of a variable
  
  The list functions can be replaced by the help window and its
  wildcards.
  
  See: 
restart (Euler Core),
restart (Maxima Documentation)
restart
  Restarts Euler and its notebooks. All variables and user functions
  are lost. Usually, this function is called using the menu entry.
  
  See: 
clear (Euler Core),
forget (Euler Core),
forget (Maxima Documentation),
remvalue (Euler Core),
remvalue (Maxima Documentation),
quit (Euler Core),
quit (Maxima Documentation)
name
    name(var) : The name of the variable var.
  
  See: 
%, %%
    %: is the result of the previous command (not assignment). 
      This works in Maxima and Euler. In symbolic expressions, % refers
      to the previous Euler result. If this is another symbolic
      expression, % is replaced by this symbolic expression (in
      round brackets). In Maxima lines, it refers to the previous
      Maxima result. Over several command lines, is safer to use
      variables.
  
    %%: Is used in Maxima block commands to refer to the previous result.
      So &(x^2,%%^2) yields x^4.
  
relax
    relax : Command to relax strict mode, when reading an Euler file.
  
  Strict mode inhibits certain old constructions. It can be switched off
  in the Euler menu. For Euler files some of these regulations are
  relaxed, but not all. To relax all regulations, start the Euler file
  with the relax command.
  
addmenu, submenu
    addmenu command : Adds the command to the user menu.
      The command can contain place holders as ?item.
      Whenever the user places the cursor in front of
      the question mark and starts typing, the place
      holder is removed. If the line contains a place
      holder, the cursor right key positions the cursor
      in front of the next place holder.
  
    submenu command : Generates a new submenu in the user menu.
      The default submenu is the filename of the Euler
      file, or "Other".
  
  The user menu is cleared at each restart.

A dynamic link libraries (DLL) is a Windows feature to add functions to a running program. In Euler, it is possible to compile external functions in DLLs. You can find more information on this in the following notebook.

dll, closedll, tccompile
    dll(name,function,n) : Loads the function f (a string) from the dll
      with the name, assuming it has n arguments.
  
    closedll(name) : Unload the DLL.
  
    tccompile filename : Compile the C file with tccompile.
    
errors, isNAN, NAN, underflows
  Handles generation of NAN (not a number) or error messages.
  
    errors on, off: Turn error messages on or off
  
    NAN : Not a number
    isNAN(x) : Tests, if x is a NAN
  
    underflows on, off: Turns underflows for ^ and exp on and off. 
      The default is off.
  
setkey
    setkey(n,text) : Sets the keyboard function key n to the text.
  
  Some function key do not work without the Ctrl or Shift key.
  
exec, message, execwrite, execwriteln, execread
    exec("command","arguments","directory",print,hidden,wait) : external command.
  
  For details have a look at the utility function exec. This will block
  by default and wait for user confirmation.
  
    execwrite(string) : Write a string to an open pipe.
    execwriteln(string) : Write a string and a new line to an open pipe.
  
    execread() : Reads strings from an open pipe.
      The return value is a vector of strings, one for each
      line.
  
    message("This is a message") : Shows a dialog with this message. 
      Use "First line"+nl$+"Second Line"
      for line breaks.
  
  See: 
exec (Euler Core),
exec (Basic Utilities)
yacas, yacasclear, yacasmode
  Yacas is no more recommended for Euler. If you still want to use it,
  you need to enable it in the menu. Then you can enter a Yacas commands
  with >>...
  
    yacas(expr) : lets Yacas evaluate the expression. 
      It returns the result string or the Yacas error
      message.
  
    yacasclear() : starts a new Yacas session.
  
    yacasmode (on|off) : toggles or sets Yacas mode.
  
  In Yacas mode, it is possible to use Euler commands with euler ...
  
  >yacas ... : Sends a command to Yacas in Euler mode.
  >> ... : Sends a command to Yacas in Euler mode.
  
  >yacas("D(x) x^x")
   x^x*(Ln(x)+1)
  
  See: 
converty (Yacas),
y2matrix (Yacas),
y2vector (Yacas),
ynewton (Yacas),
mxm (Euler Core),
mxmeval (Maxima Functions for Euler)
eulerarg, script
    eulerarg(n) : Returns the n-th arg of the argument lines for scripts.
  
  Run euler with the "-script eulerfile.e" parameter and additional
  parameters to keep the GUI opening.
  

Python

Python code can be written and used from Euler. For this, Python 2.7 must be installed. It is not sufficient to download python27.dll. A full installation is required.

py, python, pyeval, pyget, pyset
  py: command 
  >>>> command
  
  Execute a Python command.
  
  For multi-line Python commands, multi-line Euler commands can be used.
  
  py: ...
  command ...
  command ...
  command
  
  Another method is the following, which uses a function body. Press F9
  to edit the function, or click into the function.
  
  function python ...
  Commands
  endfunction
  
  These two methods to run Python commands collect all output and print
  it after they have finished. They are not designed for interactive
  Python programs.
  
  function python f(x) ...
    Commands
    return value
  endfunction
  
  This defines the function f(x) in Python, which can called as f(x) in
  Euler. The function can have a help line ## ... like any other Euler
  function. In fact, an Euler function is defined, which is flagged to
  call Python.
  
    pyeval("function",argument1,argument2) : Evaluates a Python function.
    py$function(argument1,argument2) : Evaluates a Python function.
  
    pyset("varname",value) : Sets a variable in Python.
    pyget("varname") : Gets a variable from Python.
  
  Euler translates reals to float or integers, vectors to lists,
  matrices to lists of lists, strings to strings. The string
  "py$function" is translated to the Python function "function".
  
    python("Command") : Executes a Python command
    python(["Command1","Command2"]) : Executes a multi-line Python command
  
    loadpython(filename) : Loads a Python file.
      This will collect all output from the file
      like py: ...
  
  See: 

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