Functions

Next: Rational (rational.h) Up: Integer (integer.h) Previous: Input/Output

Functions

Arithmetical Operations:

Algorithm:      div(a, b, c, d)

Input:      $a,b,c,d\in\textsf{Integer}\mbox{ where }b\not=0, c\not=d$ .

Output:      $c,d\in\textsf{Integer}$ such that $c = \mathrm{sign}(a\cdot b)\lfloor\vert\frac{a}{b}\vert\rfloor, d = a-c\cdot b\quad\Box$

Algorithm:      div(a, b, c, d)

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

Output:      $c,d\in\textsf{Integer}$ such that $c = \mathrm{sign}(a)\lfloor\frac{\vert a\vert}{b}\rfloor, d = a-c\cdot b\quad\Box$

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

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

Output:      $c,d\in\textsf{Integer}$ such that $c = \mathrm{sign}(a)\lfloor\frac{\vert a\vert}{2^{\beta b}}\rfloor, d = a-c\cdot2^{\beta b}\quad\Box$

Algorithm:      c := abs(a)

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

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

Algorithm:      c := sign(a)

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

Output:      $c\in\texttt{int}$ such that

     $\mbox{if }a = 0\mbox{ then }c = 0$

     else if $a \gt 0$ then c = 1

     else $c = -1\quad\Box$

Algorithm:      c := units(a)

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

Output:      $a\in\textsf{Integer}, c\in\texttt{int}$ such that $a := \vert a\vert, \mbox{if }a \geq 0\mbox{ then }c = 1\mbox{ else }c = -1\quad\Box$

Algorithm:      b := log2(a)

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

Output:      $b\in\textsf{Digit}$ such that $\mbox{if }\vert a\vert \gt 0\mbox{ then }b = \lfloor\log_2\vert a\vert\rfloor+1\mbox{ else }b = 0\quad\Box$

Algorithm:      sqrt(a, b, c)

Input:      $a\in\textsf{Integer}\mbox{ where }a\geq0$ .

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

Algorithm:      b := sqrt(a)

Input:      $a\in\textsf{Integer}\mbox{ where }a\geq0$ .

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

Algorithm:      c := root(a, b)

Input:      $a\in\textsf{Integer}, b\in\textsf{Digit}\mbox{ where }2\vert b\mbox{ or }a\geq0$ .

Output:      $c\in\textsf{Integer}$ such that $c = \mathrm{sign}(a)^b\lfloor \vert a\vert^\frac{1}{b}\rfloor\quad\Box$

Algorithm:      c := pow(a, b)

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

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

Algorithm:      c := pow(a, b)

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

Output:      $c\in\textsf{Integer}$ such that $c = \mathrm{sign}(a)^b\lfloor\vert a\vert^b\rfloor\quad\Box$
Basic Functions:

Algorithm:      c.setbit(a)

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

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

Algorithm:      c.clearbit(a)

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

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

Algorithm:      c := b.testbit(a)

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

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:      $a\in\textsf{Integer}$ .

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:      $a\in\textsf{Integer}$ .

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:      $a\in\textsf{Integer}$ .

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

Algorithm:      c := a.highest()

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

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

Algorithm:      c := a.lowest()

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

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

Algorithm:      swap(a, b)

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

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

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

Algorithm:      a.rand(n)

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

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

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

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

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

Algorithm:      c := atoI(a, b)

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

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

Algorithm:      gcd(a, b, x, y, z)

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

Output:      $x,y,z\in\textsf{Integer}$ such that $z = ax+by = \gcd(a, b)\quad\Box$

Algorithm:		`div(a, b, c, d)`
Input:		$a,b,c,d\in\textsf{Integer}\mbox{ where }b\not=0, c\not=d$ .
Output:		$c,d\in\textsf{Integer}$ such that $c = \mathrm{sign}(a\cdot b)\lfloor\vert\frac{a}{b}\vert\rfloor, d = a-c\cdot b\quad\Box$

Algorithm:		`div(a, b, c, d)`
Input:		$a\in\textsf{Integer}, b\in\textsf{Digit}\mbox{ where }b\not=0$ .
Output:		$c,d\in\textsf{Integer}$ such that $c = \mathrm{sign}(a)\lfloor\frac{\vert a\vert}{b}\rfloor, d = a-c\cdot b\quad\Box$

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

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

Algorithm:		`c := sign(a)`
Input:		$a\in\textsf{Integer}$ .
Output:		$c\in\texttt{int}$ such that
		$\mbox{if }a = 0\mbox{ then }c = 0$
		else if $a \gt 0$ then c = 1
		else $c = -1\quad\Box$

Algorithm:		`c := units(a)`
Input:		$a\in\textsf{Integer}$ .
Output:		$a\in\textsf{Integer}, c\in\texttt{int}$ such that $a := \vert a\vert, \mbox{if }a \geq 0\mbox{ then }c = 1\mbox{ else }c = -1\quad\Box$

Algorithm:		`b := log2(a)`
Input:		$a\in\textsf{Integer}$ .
Output:		$b\in\textsf{Digit}$ such that $\mbox{if }\vert a\vert \gt 0\mbox{ then }b = \lfloor\log_2\vert a\vert\rfloor+1\mbox{ else }b = 0\quad\Box$

Algorithm:		`sqrt(a, b, c)`
Input:		$a\in\textsf{Integer}\mbox{ where }a\geq0$ .
Output:		$b,c\in\textsf{Integer}$ such that $b = \lfloor\sqrt{a}\rfloor, c = a - b^2\quad\Box$

Algorithm:		`b := sqrt(a)`
Input:		$a\in\textsf{Integer}\mbox{ where }a\geq0$ .
Output:		$b\in\textsf{Integer}$ such that $b = \lfloor\sqrt{a}\rfloor\quad\Box$

Algorithm:		`c := root(a, b)`
Input:		$a\in\textsf{Integer}, b\in\textsf{Digit}\mbox{ where }2\vert b\mbox{ or }a\geq0$ .
Output:		$c\in\textsf{Integer}$ such that $c = \mathrm{sign}(a)^b\lfloor \vert a\vert^\frac{1}{b}\rfloor\quad\Box$

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

Algorithm:		`c := pow(a, b)`
Input:		$a,b\in\textsf{Integer}$ .
Output:		$c\in\textsf{Integer}$ such that $c = \mathrm{sign}(a)^b\lfloor\vert a\vert^b\rfloor\quad\Box$

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

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

Algorithm:		`c := b.testbit(a)`
Input:		$a\in\texttt{size\_t}, b\in\textsf{Integer}$ .
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:		$a\in\textsf{Integer}$ .
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:		$a\in\textsf{Integer}$ .
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:		$a\in\textsf{Integer}$ .
Output:		$c\in\texttt{size\_t}$ such that $c = \mathcal{L}(\vert a\vert)\quad\Box$

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

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

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

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

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

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

Algorithm:		`gcd(a, b, x, y, z)`
Input:		$a,b\in\textsf{Integer}$ .
Output:		$x,y,z\in\textsf{Integer}$ such that $z = ax+by = \gcd(a, b)\quad\Box$

Next: Rational (rational.h) Up: Integer (integer.h) Previous: Input/Output

Piologie Version 1.2
3/3/1998