Functions

Next: Integer (integer.h) Up: Natural (natural.h) Previous: Input/Output

Functions

Arithmetical Operations:

Algorithm:      c := sign(a)

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\texttt{bool}$ such that $\mbox{if }a = 0\mbox{ then }c = \texttt{false}\mbox{ else }c = \texttt{true}\quad\Box$

Algorithm:      div(a, b, q, r)

Input:      $a,b,q,r\in\textsf{Natural}\mbox{ where }b\not=0, q\not=r$ .

Output:      $q,r\in\textsf{Natural}$ such that $q = \lfloor\frac{a}{b}\rfloor, r = a-q\cdot b\quad\Box$

Algorithm:      div(a, b, q, r)

Input:      $a\in\textsf{Natural}, b\in\textsf{Digit}\mbox{ where }b\not=0$ .

Output:      $q\in\textsf{Natural}, r\in\textsf{Digit}$ such that $q = \lfloor\frac{a}{b}\rfloor, r = a-q\cdot b\quad\Box$

Algorithm:      c.split(n, a, b)

Input:      $a,b,c\in\textsf{Natural}, n\in\texttt{size\_t}\mbox{ where }a\not=b$ .

Output:      $a,b\in\textsf{Natural}$ such that $a = \lfloor\frac{c}{2^{\beta n}}\rfloor, b = c-a\cdot2^{\beta n}\quad\Box$

Algorithm:      c := abs(a, b)

Input:      $a,b\in\textsf{Natural}$ .

Output:      $c\in\textsf{Natural}$ such that $c = \vert a-b\vert\quad\Box$

Algorithm:      d := abs(a, b, c)

Input:      $a,b\in\textsf{Natural}$ .

Output:      $d\in\texttt{int}, c\in\textsf{Natural}$ such that $c = \vert a-b\vert, d\cdot c = a-b\quad\Box$

Algorithm:      b := log2(a)

Input:      $a\in\textsf{Natural}$ .

Output:      $b\in\textsf{Digit}$ such that $\mbox{if }a \gt 0\mbox{ then }b = \lfloor\log_2(a)\rfloor\mbox{ else }b = 0\quad\Box$

Algorithm:      sqrt(b, c, d)

Input:      $b\in\textsf{Natural}$ .

Output:      $c,d\in\textsf{Natural}$ such that $c = \lfloor\sqrt{b}\rfloor, d = b - c^2\quad\Box$

Algorithm:      b := sqrt(a)

Input:      $a\in\textsf{Natural}$ .

Output:      $b\in\textsf{Natural}$ such that $b = \lfloor\sqrt{a}\rfloor\quad\Box$

Algorithm:      b := root(a, n)

Input:      $a\in\textsf{Natural}, n\in\textsf{Digit}$ .

Output:      $b\in\textsf{Natural}$ such that $b = \lfloor\sqrt[n]{a}\rfloor\quad\Box$

Algorithm:      c := pow(a, b)

Input:      $a\in\textsf{Natural}, b\in\textsf{Digit}$ .

Output:      $c\in\textsf{Natural}$ such that $c = a^b\quad\Box$

Algorithm:      c := pow(a, b)

Input:      $a,b\in\textsf{Natural}$ .

Output:      $c\in\textsf{Natural}$ such that $c = a^b\quad\Box$
Basic Functions:

Algorithm:      c.setbit(a)

Input:      $a\in\texttt{size\_t}, c\in\textsf{Natural}$ .

Output:      $c\in\textsf{Natural}$ such that $c := c\vee 2^a\quad\Box$

Algorithm:      c.clearbit(a)

Input:      $a\in\texttt{size\_t}, c\in\textsf{Natural}$ .

Output:      $c\in\textsf{Natural}$ such that $c := c\wedge\neg 2^a\quad\Box$

Algorithm:      c := b.testbit(a)

Input:      $a\in\texttt{size\_t}, b\in\textsf{Natural}$ .

Output:      $c\in\texttt{bool}$ such that $\mbox{if }b\wedge 2^a\mbox{ then }c = \texttt{true}\mbox{ else }c = \texttt{false}\quad\Box$

Algorithm:      c := a.odd()

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\texttt{bool}$ such that $\mbox{if }2\vert a\mbox{ then }c = \texttt{false}\mbox{ else }c = \texttt{true}\quad\Box$

Algorithm:      c := a.even()

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\texttt{bool}$ such that $\mbox{if }2\vert a\mbox{ then }c = \texttt{true}\mbox{ else }c = \texttt{false}\quad\Box$

Algorithm:      c := a.length()

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\texttt{size\_t}$ such that $c = \mathcal{L}(a)\quad\Box$

Algorithm:      c := a.highest()

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\textsf{Digit}$ such that $c = \lfloor\frac{a}{2^{\beta(\mathcal{L}(a)-1)}}\rfloor\quad\Box$

Algorithm:      c := a.lowest()

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\textsf{Digit}$ such that $c = a\wedge\gamma\quad\Box$

Algorithm:      swap(a, b)

Input:      $a,b\in\textsf{Natural}$ .

Output:      $a,b\in\textsf{Natural}$ such that t := a, a := b, b := t

     where $t\in\textsf{Natural}\quad\Box$

Algorithm:      a.rand(n)

Input:      $n\in\texttt{size\_t}$ .

Output:      $a\in\textsf{Natural}$ such that $a < 2^n\mbox{ (random number)}\quad\Box$
Conversions:

Algorithm:      c := Ntoa(a, c, b)

Input:      $a\in\textsf{Natural}, b\in\textsf{Digit}, c\in\texttt{char*}\mbox{ where }2\leq b\leq 36, \mathtt{sizeof}(c) \gt \beta\frac{\mathcal{L}(a)}{\log_2(b)}$ .

Output:      $c\in\texttt{char*}$ such that $c = a\quad\Box$

Algorithm:      c := atoN(a, b)

Input:      $a\in\texttt{char*}, b\in\textsf{Digit}\mbox{ where }2\leq b\leq 36$ .

Output:      $c\in\textsf{Natural}$ such that $c = a\quad\Box$
Functions of Number Theory:

Algorithm:      b := factorial(a)

Input:      $a\in\textsf{Digit}$ .

Output:      $b\in\textsf{Natural}$ such that $b = a!\quad\Box$

Algorithm:      c := bin_coefficient(a, b)

Input:      $a,b\in\textsf{Digit}$ .

Output:      $c\in\textsf{Natural}$ such that $c = {a\choose {b}}\quad\Box$

Algorithm:      c := fibonacci(n)

Input:      $n\in\textsf{Digit}$ .

Output:      $c\in\textsf{Natural}$ such that c = F_n

     where $F_0 = 0, F_1 = 1, F_k = F_{k-1}+F_{k-2}\mbox{ for }k\geq2\quad\Box$

Algorithm:      c := gcd(a, b)

Input:      $a,b\in\textsf{Natural}$ .

Output:      $c\in\textsf{Natural}$ such that $c = \max\{x\in\textsf{Natural}\mid x\vert a, x\vert b\}\quad\Box$

Algorithm:      c := lcm(a, b)

Input:      $a,b\in\textsf{Natural}$ .

Output:      $c\in\textsf{Natural}$ such that $c = \min\{x\in\textsf{Natural}\mid a\vert x, b\vert x\}\quad\Box$
Functions of Number Theory which uses STL (primes.h):

Algorithm:      c := a.isprime()

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\texttt{bool}$ such that $\mbox{if }a\mbox{ is a prime element}\mbox{ then }c = \texttt{true}\mbox{ else }c = \texttt{false}\quad\Box$

Note:      Error possible:``the primality could not be determined".

Algorithm:      factoring(a, c)

Input:      $a\in\textsf{Natural}$ .

Output:      c is a container over Naturals with an output-iterator $\begin{displaymath} such that \end{displaymath}$

     $\mbox{if }a\geq1\mbox{ then }a = \prod_{\mathtt{c.begin()\leq i<c.end()}}\mathtt{*i}\mbox{ else }\mathtt{c.begin() = c.end()},$

     for all $i\in\mathtt{[c.begin(), c.end()[, j\in [i, c.end()[\; : *i \leq *j}\quad\Box$

Algorithm:      c := euler(a)

Input:      $a\in\textsf{Natural}$ .

Output:      $c\in\textsf{Natural}$ such that $c = \varphi(a) := a\prod_{i=0}^t(1-\frac{1}{p_i})$

     where $a = \prod_{i = 0}^t p_i^{n_i}\qquad\mbox{for $t\in\mathbb{N}, n_i\in\mathbb{N}_+, p_i\in\mathbb{P}, 0\leq i\leq t$}\quad\Box$

Algorithm:		`c := sign(a)`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\texttt{bool}$ such that $\mbox{if }a = 0\mbox{ then }c = \texttt{false}\mbox{ else }c = \texttt{true}\quad\Box$

Algorithm:		`div(a, b, q, r)`
Input:		$a,b,q,r\in\textsf{Natural}\mbox{ where }b\not=0, q\not=r$ .
Output:		$q,r\in\textsf{Natural}$ such that $q = \lfloor\frac{a}{b}\rfloor, r = a-q\cdot b\quad\Box$

Algorithm:		`div(a, b, q, r)`
Input:		$a\in\textsf{Natural}, b\in\textsf{Digit}\mbox{ where }b\not=0$ .
Output:		$q\in\textsf{Natural}, r\in\textsf{Digit}$ such that $q = \lfloor\frac{a}{b}\rfloor, r = a-q\cdot b\quad\Box$

Algorithm:		`c.split(n, a, b)`
Input:		$a,b,c\in\textsf{Natural}, n\in\texttt{size\_t}\mbox{ where }a\not=b$ .
Output:		$a,b\in\textsf{Natural}$ such that $a = \lfloor\frac{c}{2^{\beta n}}\rfloor, b = c-a\cdot2^{\beta n}\quad\Box$

Algorithm:		`c := abs(a, b)`
Input:		$a,b\in\textsf{Natural}$ .
Output:		$c\in\textsf{Natural}$ such that $c = \vert a-b\vert\quad\Box$

Algorithm:		`d := abs(a, b, c)`
Input:		$a,b\in\textsf{Natural}$ .
Output:		$d\in\texttt{int}, c\in\textsf{Natural}$ such that $c = \vert a-b\vert, d\cdot c = a-b\quad\Box$

Algorithm:		`b := log2(a)`
Input:		$a\in\textsf{Natural}$ .
Output:		$b\in\textsf{Digit}$ such that $\mbox{if }a \gt 0\mbox{ then }b = \lfloor\log_2(a)\rfloor\mbox{ else }b = 0\quad\Box$

Algorithm:		`sqrt(b, c, d)`
Input:		$b\in\textsf{Natural}$ .
Output:		$c,d\in\textsf{Natural}$ such that $c = \lfloor\sqrt{b}\rfloor, d = b - c^2\quad\Box$

Algorithm:		`b := sqrt(a)`
Input:		$a\in\textsf{Natural}$ .
Output:		$b\in\textsf{Natural}$ such that $b = \lfloor\sqrt{a}\rfloor\quad\Box$

Algorithm:		`b := root(a, n)`
Input:		$a\in\textsf{Natural}, n\in\textsf{Digit}$ .
Output:		$b\in\textsf{Natural}$ such that $b = \lfloor\sqrt[n]{a}\rfloor\quad\Box$

Algorithm:		`c := pow(a, b)`
Input:		$a\in\textsf{Natural}, b\in\textsf{Digit}$ .
Output:		$c\in\textsf{Natural}$ such that $c = a^b\quad\Box$

Algorithm:		`c := pow(a, b)`
Input:		$a,b\in\textsf{Natural}$ .
Output:		$c\in\textsf{Natural}$ such that $c = a^b\quad\Box$

Algorithm:		`c.setbit(a)`
Input:		$a\in\texttt{size\_t}, c\in\textsf{Natural}$ .
Output:		$c\in\textsf{Natural}$ such that $c := c\vee 2^a\quad\Box$

Algorithm:		`c.clearbit(a)`
Input:		$a\in\texttt{size\_t}, c\in\textsf{Natural}$ .
Output:		$c\in\textsf{Natural}$ such that $c := c\wedge\neg 2^a\quad\Box$

Algorithm:		`c := b.testbit(a)`
Input:		$a\in\texttt{size\_t}, b\in\textsf{Natural}$ .
Output:		$c\in\texttt{bool}$ such that $\mbox{if }b\wedge 2^a\mbox{ then }c = \texttt{true}\mbox{ else }c = \texttt{false}\quad\Box$

Algorithm:		`c := a.odd()`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\texttt{bool}$ such that $\mbox{if }2\vert a\mbox{ then }c = \texttt{false}\mbox{ else }c = \texttt{true}\quad\Box$

Algorithm:		`c := a.even()`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\texttt{bool}$ such that $\mbox{if }2\vert a\mbox{ then }c = \texttt{true}\mbox{ else }c = \texttt{false}\quad\Box$

Algorithm:		`c := a.length()`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\texttt{size\_t}$ such that $c = \mathcal{L}(a)\quad\Box$

Algorithm:		`c := a.highest()`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\textsf{Digit}$ such that $c = \lfloor\frac{a}{2^{\beta(\mathcal{L}(a)-1)}}\rfloor\quad\Box$

Algorithm:		`c := a.lowest()`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\textsf{Digit}$ such that $c = a\wedge\gamma\quad\Box$

Algorithm:		`swap(a, b)`
Input:		$a,b\in\textsf{Natural}$ .
Output:		$a,b\in\textsf{Natural}$ such that t := a, a := b, b := t
		where $t\in\textsf{Natural}\quad\Box$

Algorithm:		`a.rand(n)`
Input:		$n\in\texttt{size\_t}$ .
Output:		$a\in\textsf{Natural}$ such that $a < 2^n\mbox{ (random number)}\quad\Box$

Algorithm:		`c := Ntoa(a, c, b)`
Input:		$a\in\textsf{Natural}, b\in\textsf{Digit}, c\in\texttt{char*}\mbox{ where }2\leq b\leq 36, \mathtt{sizeof}(c) \gt \beta\frac{\mathcal{L}(a)}{\log_2(b)}$ .
Output:		$c\in\texttt{char*}$ such that $c = a\quad\Box$

Algorithm:		`c := atoN(a, b)`
Input:		$a\in\texttt{char*}, b\in\textsf{Digit}\mbox{ where }2\leq b\leq 36$ .
Output:		$c\in\textsf{Natural}$ such that $c = a\quad\Box$

Algorithm:		`b := factorial(a)`
Input:		$a\in\textsf{Digit}$ .
Output:		$b\in\textsf{Natural}$ such that $b = a!\quad\Box$

Algorithm:		`c := bin_coefficient(a, b)`
Input:		$a,b\in\textsf{Digit}$ .
Output:		$c\in\textsf{Natural}$ such that $c = {a\choose {b}}\quad\Box$

Algorithm:		`c := fibonacci(n)`
Input:		$n\in\textsf{Digit}$ .
Output:		$c\in\textsf{Natural}$ such that c = F_n
		where $F_0 = 0, F_1 = 1, F_k = F_{k-1}+F_{k-2}\mbox{ for }k\geq2\quad\Box$

Algorithm:		`c := gcd(a, b)`
Input:		$a,b\in\textsf{Natural}$ .
Output:		$c\in\textsf{Natural}$ such that $c = \max\{x\in\textsf{Natural}\mid x\vert a, x\vert b\}\quad\Box$

Algorithm:		`c := lcm(a, b)`
Input:		$a,b\in\textsf{Natural}$ .
Output:		$c\in\textsf{Natural}$ such that $c = \min\{x\in\textsf{Natural}\mid a\vert x, b\vert x\}\quad\Box$

Algorithm:		`c := a.isprime()`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\texttt{bool}$ such that $\mbox{if }a\mbox{ is a prime element}\mbox{ then }c = \texttt{true}\mbox{ else }c = \texttt{false}\quad\Box$
Note:		Error possible:``the primality could not be determined".

Algorithm:		`factoring(a, c)`
Input:		$a\in\textsf{Natural}$ .
Output:		c is a container over Naturals with an output-iterator $\begin{displaymath} such that \end{displaymath}$
		$\mbox{if }a\geq1\mbox{ then }a = \prod_{\mathtt{c.begin()\leq i<c.end()}}\mathtt{*i}\mbox{ else }\mathtt{c.begin() = c.end()},$
		for all $i\in\mathtt{[c.begin(), c.end()[, j\in [i, c.end()[\; : i \leq j}\quad\Box$

Algorithm:		`c := euler(a)`
Input:		$a\in\textsf{Natural}$ .
Output:		$c\in\textsf{Natural}$ such that $c = \varphi(a) := a\prod_{i=0}^t(1-\frac{1}{p_i})$
		where $a = \prod_{i = 0}^t p_i^{n_i}\qquad\mbox{for $t\in\mathbb{N}, n_i\in\mathbb{N}_+, p_i\in\mathbb{P}, 0\leq i\leq t$}\quad\Box$

Next: Integer (integer.h) Up: Natural (natural.h) Previous: Input/Output

Piologie Version 1.2
3/3/1998