Rødder af enhed
Givet et lille heltal n udskriver alle de n'th rødder af enhed op til 6 betydelige cifre. Vi har dybest set brug for at finde alle rødder af ligning x n - 1.
Eksempler:
Input : n = 1 Output : 1.000000 + i 0.000000 x - 1 = 0 has only one root i.e. 1 Input : 2 Output : 1.000000 + i 0.000000 -1.000000 + i 0.000000 x 2 - 1 = 0 has 2 distinct roots i.e. 1 and -1
Ethvert komplekst tal siges at være rod til enhed, hvis det giver 1, når det hæves til en vis magt.
Nth rod af enhed er ethvert komplekst tal, således at det giver 1, når den hæves til strømmen n.
Mathematically An nth root of unity where n is a positive integer (i.e. n = 1 2 3 …) is a number z satisfying the equation z^n = 1 or z^n - 1 = 0
Vi kan bruge De Moivres formel her
( Cos x + i Sin x )^k = Cos kx + i Sin kx Setting x = 2*pi/n we can obtain all the nth roots of unity using the fact that Nth roots are set of numbers given by Cos (2*pi*k/n) + i Sin(2*pi*k/n) Where 0 <= k < n
Ved hjælp af ovenstående kendsgerning kan vi nemt udskrive alle de niende enheder af enhed!
Nedenfor er programmet for det samme.
C++ // C++ program to print n'th roots of unity #include using namespace std ; // This function receives an integer n and prints // all the nth roots of unity void printRoots ( int n ) { // theta = 2*pi/n double theta = M_PI * 2 / n ; // print all nth roots with 6 significant digits for ( int k = 0 ; k < n ; k ++ ) { // calculate the real and imaginary part of root double real = cos ( k * theta ); double img = sin ( k * theta ); // Print real and imaginary parts printf ( '%.6f' real ); img >= 0 ? printf ( ' + i ' ) : printf ( ' - i ' ); printf ( '%.6f n ' abs ( img )); } } // Driver function to check the program int main () { printRoots ( 1 ); cout < < endl ; printRoots ( 2 ); cout < < endl ; printRoots ( 3 ); return 0 ; }
Java // Java program to print n'th roots of unity import java.io.* ; class GFG { // This function receives an integer n and prints // all the nth roots of unity static void printRoots ( int n ) { // theta = 2*pi/n double theta = 3.14 * 2 / n ; // print all nth roots with 6 significant digits for ( int k = 0 ; k < n ; k ++ ) { // calculate the real and imaginary part of root double real = Math . cos ( k * theta ); double img = Math . sin ( k * theta ); // Print real and imaginary parts System . out . println ( real ); if ( img >= 0 ) System . out . println ( ' + i ' ); else System . out . println ( ' - i ' ); System . out . println ( Math . abs ( img )); } } // Driver function to check the program public static void main ( String [] args ) { printRoots ( 1 ); //System.out.println(); printRoots ( 2 ); //System.out.println(); printRoots ( 3 ); } } // This code is contributed by Raj
Python3 # Python3 program to print n'th roots of unity import math # This function receives an integer n and prints # all the nth roots of unity def printRoots ( n ): # theta = 2*pi/n theta = math . pi * 2 / n # print all nth roots with 6 significant digits for k in range ( 0 n ): # calculate the real and imaginary part of root real = math . cos ( k * theta ) img = math . sin ( k * theta ) # Print real and imaginary parts print ( real end = ' ' ) if ( img >= 0 ): print ( ' + i ' end = ' ' ) else : print ( ' - i ' end = ' ' ) print ( abs ( img )) # Driver function to check the program if __name__ == '__main__' : printRoots ( 1 ) printRoots ( 2 ) printRoots ( 3 ) # This code is contributed by # Sanjit_Prasad
C# // C# program to print n'th roots of unity using System ; class GFG { // This function receives an integer n and prints // all the nth roots of unity static void printRoots ( int n ) { // theta = 2*pi/n double theta = 3.14 * 2 / n ; // print all nth roots with 6 significant digits for ( int k = 0 ; k < n ; k ++ ) { // calculate the real and imaginary part of root double real = Math . Cos ( k * theta ); double img = Math . Sin ( k * theta ); // Print real and imaginary parts Console . Write ( real ); if ( img >= 0 ) Console . Write ( ' + i ' ); else Console . Write ( ' - i ' ); Console . WriteLine ( Math . Abs ( img )); } } // Driver function to check the program static void Main () { printRoots ( 1 ); printRoots ( 2 ); printRoots ( 3 ); } } // This code is contributed by mits
PHP // PHP program to print n'th roots of unity // This function receives an integer n // and prints all the nth roots of unity function printRoots ( $n ) { // theta = 2*pi/n $theta = pi () * 2 / $n ; // print all nth roots with 6 // significant digits for ( $k = 0 ; $k < $n ; $k ++ ) { // calculate the real and imaginary // part of root $real = cos ( $k * $theta ); $img = sin ( $k * $theta ); // Print real and imaginary parts print ( round ( $real 6 )); $img >= 0 ? print ( ' + i ' ) : print ( ' - i ' ); printf ( round ( abs ( $img ) 6 ) . ' n ' ); } } // Driver Code printRoots ( 1 ); printRoots ( 2 ); printRoots ( 3 ); // This code is contributed by mits ?>
JavaScript < script > // javascript program to print n'th roots of unity // This function receives an integer n and prints // all the nth roots of unity function printRoots ( n ) { // theta = 2*pi/n var theta = ( 3.14 * 2 / n ); // print all nth roots with 6 significant digits for ( k = 0 ; k < n ; k ++ ) { // calculate the real and imaginary part of root var real = Math . cos ( k * theta ); var img = Math . sin ( k * theta ); // Print real and imaginary parts document . write ( real . toFixed ( 6 )); if ( img >= 0 ) document . write ( ' + i ' ); else document . write ( ' - i ' ); document . write ( Math . abs ( img ). toFixed ( 6 ) + '
' ); } } // Driver function to check the program printRoots ( 1 ); //document.write('
'); printRoots ( 2 ); //document.write('
'); printRoots ( 3 ); // This code is contributed by shikhasingrajput < /script>
Produktion:
1.000000 + i 0.000000 1.000000 + i 0.000000 -1.000000 + i 0.000000 1.000000 + i 0.000000 -0.500000 + i 0.866025 -0.500000 - i 0.866025
Referencer: Wikipedia
