Version 2.1
Overview:
Class | Meaning |
ContFrac | Continued fractions |
Edge | Edges of a Graph |
Graph | Graphs |
Logic | propositional logic |
Matrix | Matrices |
Mod | Elements of the residue class rings Z[m] |
NumeralSystem | conversions between numeral systems |
Perm | Permutations (from dictionary or cycle representation) |
Poly | Polynomials over arbitrary rings |
Rational | Rational numbers |
Rsa | RSA Encryption |
Set | Polymorphic sets |
Vector | Vectors |
Function | Meaning |
allFactors (n) | List of all prime factors of n |
binomial (n, k) | Binomial coefficient |
ceil (x) | smallest integer n >= x |
chinese (mod1, *mod2) | Mod object from mod1, mod2, ... according to the Chinese remainder theorem |
closure (elements, operation) | The closure of the elements under the binary operation |
copy (x) | independent copy ("clone") of x |
divisors (n) | List of all divisors of n |
eulerPhi (n) | Euler's totient function |
expMod (a, x, m) | Calculates (a**x) % m efficiently. |
factor (n) | Prime factorization |
factorial (n) | n! = 1*2*3* ... *n (factorial of n) |
factors (n) | List of prime factors of n (without multiplicity) |
fibonacci (n) | n-th element of the Fibonacci series |
fit (data, values) | Interpolation using the Gaussian method of least squares |
floor (x) | highest integer n <= x |
fromTo (a, b, step=1) | Iterator over equidistant integers from a to b |
gcd (a,b) | greatest common divisor of a and b |
gcdExt (a,b) | Extended Euclidean algorithm. |
isInteger (n) | returns True if and only if n is of type int . |
isNatural (n) | |
isPrime (n,s=20) | Returns bool (is n a prime number?) |
join (l, filler="") | Joins the elements of l to a string |
lcm (a, b) | Least common multiple of a and b |
nextPrime (n) | next prime larger than or equal to n |
permutations (v, prefix=[]) | List of all permutations of v |
permute (v, fn, prefix=[]) | applies procedure fn to all permutations of liste v. |
plot (terms, x0, x1, **options) | |
plotFunctions (functions, x0, x1, **options) | |
prime (n) | n-th prime number |
printTable (height, width, cellFunction, **options) | displays a table of height rows and width columns, whose cells are calculated by cellFunction. |
printValueTable (values, variable="n", start=0, end=10, increment=1, vertical=True) | Displays a value table. |
product (obj, start=None, end=None, step=1) | The product of the elements of iterable obj or of all obj(i) for i in fromTo(start,end,step) |
rand (n) | Integer random number from the range 0 ... n-1 n must be a natural number (int). |
sum (obj, start=None, end=None, step=1) | The sum of the elements of iterable obj or of all obj(i) for i in fromTo(start,end,step) |
toNumber (s) | converts the string s into a rational or floating point number |
Usage: allFactors(n)
Description:
List of all prime factors of n
Example: allFactors(24)
returns [2, 2, 2, 3]
.
See also factor
,
factors
.
Usage: binomial(n, k)
Description:
Binomial coefficient
n must be a nonnegative integer and k in the range 0 .. n.
Usage: ceil(x)
Description:
smallest integer n >= x
Usage: chinese(mod1, *mod2)
Description:
Mod object from mod1, mod2, ... according to the Chinese remainder theorem
Requires that mod1, mod2, ... are Mod objects with pairwise coprime moduli.
chinese(Mod(a1, m1), Mod(a2, m2), ...)
returns Mod(x, m1*m2*...) with 0 <= x
< m1*m2*... and
x % m1 == a1
x % m2 == a2
, ...
Example:
chinese(Mod(1,5), Mod(2,7)) # -à Mod(16,35)
(16 mod 5 = 1, 16 mod 7 = 2).Mod
Usage: closure(elements, operation)
Description:
The closure of the elements under the binary operation
operation must be given as a function with two parameters.
Usage: copy(x)
Description:
independent copy ("clone") of x
Usage: divisors(n)
Description:
List of all divisors of n
Usage: eulerPhi(n)
Description:
Euler's totient function
Number of positive integers less than or equal to n that are coprime to n
Usage: expMod(a, x, m)
Description:
Calculates (a**x) % m efficiently.
Equivalent to the Python standard function pow
.
Usage: factor(n)
Description:
Prime factorization
Returns a list of pairs [factor, multiplicity]
Example:
factor(24)
returns [[2,3],[3,1]]
.
See also factors
,
allFactors
.
Usage: factorial(n)
Description:
n! = 1*2*3* ... *n (factorial of n)
Usage: factors(n)
Description:
List of prime factors of n (without multiplicity)
Example:
factors(24)
returns [2, 3]
.
See also factor
,
allFactors
.
Usage: fibonacci(n)
Description:
n-th element of the Fibonacci series
fibonacci(0) = 0
fibonacci(1) = 1
fibonacci(n) = fibonacci(n-1) + fibonacci(n-2) für n >= 2
Usage: fit(data, values)
Description:
Interpolation using the Gaussian method of least squares
fit(data, values)
calculates (using the method of least squares)
a linear combination of the functions mapping x
to the terms in values
.
(corresponds to the Mathematica function Fit
).
Example:
fit([(-1,2), (1,1), (2,1), (3,0), (5,3)], ["1", "x", "x**2"])calculates
[c1, c2, c3]
such that the functionf(x) = c1*1 + c2*x + c3 * x**2
Usage: floor(x)
Description:
highest integer n <= x
Usage: fromTo(a, b, step=1)
Description:
Iterator over equidistant integers from a to b
If no step is given, step is set to 1.
Examples:
list(fromTo(3, 6)) # -à [3,4,5,6] list(fromTo(3, 6, 2)) # -à [3,5] list(fromTo(6, 3)) # -à [] list(fromTo(6, 3, -2)) # -à [6,4] [2*i for i in fromTo(6, 3, -2)] # -à [12,8] sum(fromTo(1, 10)) # -à 55
Usage: gcd(a,b)
Description:
greatest common divisor of a and b
Calculated using the Euclidean algorithm
See also lcm
Usage: gcdExt(a,b)
Description:
Extended Euclidean algorithm.
Returns (d, x, y) with d == gcd(a,b) and x*a + y*b == d
Usage: isInteger(n)
Description:
returns True
if and only if n is of type int
.
Usage: isNatural(n)
Usage: isPrime(n,s=20)
Description:
Returns bool (is n a prime number?)
Implementation using Miller Rabin test
(see. Cormen, Leiserson, Rivest: Algorithms, p. 841)
See also prime
,
nextPrime
.
Usage: join(l, filler="")
Description:
Joins the elements of l
to a string
Between each two adjacent elements filler
is inserted.
This is a replacement for the string method join()
.
In addition to the string method the list elements are converted to strings.
Example:
join(fromTo(1,5), "-")
returns "1-2-3-4-5"
.
Usage: lcm(a, b)
Description:
Least common multiple of a and b
See also gcd
Usage: nextPrime(n)
Description:
next prime larger than or equal to n
See also isPrime
,
prime
.
Usage: permutations(v, prefix=[])
Description:
List of all permutations of v
If v is sorted, the permutations are also generated in sorted order.
Usage: permute(v, fn, prefix=[])
Description:
applies procedure fn
to all permutations of liste v.
If v is sorted, the permutations are also generated in sorted order.
Usage: plot(terms, x0, x1, **options)
Usage: plotFunctions(functions, x0, x1, **options)
Usage: prime(n)
Description:
n-th prime number
See also isPrime
,
nextPrime
.
Usage: printTable(height, width, cellFunction, **options)
Description:
displays a table of height rows and width columns, whose cells are calculated by cellFunction.
cellFunction must be a function with two parameters (row, column).
The following optional named parameters are supported:
rowHeadFn: if given, row header cells are added and calculated using this function
colHeadFn: if given, column header cells are added and calculated using this function
headAlign: "left" (default), "right" or "center"
cellAlign: "left", "right" (default) or "center"
title: if given, a title line is added, using this text
corner: if given (and rowHeadFn and colHeadFn are given), this text is displayed in the top left table corner.
This function is used by the menu command Insert -- Table.
Example:
printTable(7, 7, lambda i,k: i*k % 7, rowHeadFn = lambda i: i, colHeadFn = lambda k: k, title = "Multiplication mod 7", corner = "*")prints a multiplication table modulo 7.
Usage: printValueTable(values, variable="n", start=0, end=10, increment=1, vertical=True)
Description:
Displays a value table.
This function is used by the menu command Insert -- Value Table.
Example:
printValueTable([(lambda n:n^2, "n²"), (lambda n:2^n, "2n")], "n", 0, 10, True)displays the values of n² and 2n for n from 0 to 10.
Usage: product(obj, start=None, end=None, step=1)
Description:
The product of the elements of iterable obj or of all obj(i) for i in fromTo(start,end,step)
The element type must define the operator *.
Examples:
product([i^2 for i n fromTo(1,10)])
product(fibonacci, 0, 10)
product(lambda i:i^2, 2, 10, 2)
Usage: rand(n)
Description:
Integer random number from the range 0 ... n-1
n must be a natural number (int).
Usage: sum(obj, start=None, end=None, step=1)
Description:
The sum of the elements of iterable obj or of all obj(i) for i in fromTo(start,end,step)
The element type must define the operator + (e.g. numbers or strings).
Examples:
sum([i^2 for i n fromTo(1,10)])
sum(fibonacci, 0, 10)
sum(lambda i:i^2, 2, 10, 2)
Usage: toNumber(s)
Description:
converts the string s into a rational or floating point number
Examples:
toNumber("3.1")
toNumber("7/3")