I have forgotten
my Password

Or login with:

  • Facebookhttp://facebook.com/
  • Googlehttps://www.google.com/accounts/o8/id
  • Yahoohttps://me.yahoo.com
get GPL
COST (GBP)
this unit 2.00
sub units 0.00
+
0

Euler

viewed 2998 times and licensed 41 times
Calculates Euler numbers by means of recurrent relation
Controller: CodeCogs

get GPL add to cart

Interface

C++

Overview

Euler numbers, which are also called secant numbers or 'zig' numbers are defined for |z| <\pi/2 by

The Euler number give the number of odd alternating permutations and are related to Genocchi numbers.

A slightly different convention, which is also used here, defines

Thus, the Euler numbers E_n and polynomials E_n(x) are defined through equations

The relation between Euler polynomials and numbers:

References:

  • Higher Transcendental Functions, vol.1, (1.14) by H.Bateman and A.Erdelyi (Bateman Manuscript Project), 1953
  • M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, 1964 chapt.23

Authors

Will Bateman (September 2005)

EulerA

 
voidEulerAintiMax
double*daEuler )
Calculates Euler numbers using the recurrent relation

Example 1

#include <stdio.h>
#include <codecogs/maths/discrete/number_theory/euler.h>
 
#define MAX_INDEX 16
int main()
{
  double dEulerA[MAX_INDEX+1];
  double dEulerB[MAX_INDEX+1];
 
  // Table headings
  printf( "%10s %20s %20s\n", "2n", "E_A(2n) (recurrent)", "E_B(2n) (series)" );
  printf( "%8s", " " );
  for( int i = 0; i < 52; i++ )
    printf( "%c", '-' );
  printf( "\n" );
 
  // array calculation
  Maths::NumberTheory::EulerA( MAX_INDEX, dEulerA );
  Maths::NumberTheory::EulerB( MAX_INDEX, dEulerB );
 
  // results output
  for( int i = 0; i <= MAX_INDEX; i+=2 )
    printf( "%10d %20.6f %20.6f\n", i, dEulerA[i], dEulerB[i]);
  return 0;
}
Output:
2n  E_A(2n) (recurrent)     E_B(2n) (series)
----------------------------------------------------
   0             1.000000             1.000000
   2            -1.000000            -1.000000
   4             5.000000             5.000000
   6           -61.000000           -61.000000
   8          1385.000000          1385.000000
  10        -50521.000000        -50521.000000
  12       2702765.000000       2702765.000000
  14    -199360981.000000    -199360981.000000
  16   19391512145.000000   19391512145.000027

Parameters

iMaxinput maximal index requested. Must be a power of 2.
daEulerarray into which to place the result, with iMax parameters.

Returns

pointer on the resulting array of type double

Authors

Anatoly Prognimack (Mar 19, 2005)
Developed & tested with: Borland C++ 3.1 for DOS Microsoft Visual C++ 5.0, 6.0
Updated by Will Bateman (March 2005)
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.


EulerB

 
voidEulerBintiMax
double*daEuler )
Calculates Euler numbers using the recurrent series expansion

Parameters

iMaxinput maximal index requested. Must be a power of 2.
daEulerarray into which to place the result, with iMax parameters.

Returns

pointer on the resulting array of type double

Authors

Anatoly Prognimack (Mar 19, 2005)
Developed & tested with: Borland C++ 3.1 for DOS Microsoft Visual C++ 5.0, 6.0
Will Bateman (March 2005)
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.