polynomial Module Reference

polynomials provides a framework for simple (and quite naive) work with polynomials Further description in Submodule interpolate More...

Data Types
interface	polyval
	Computes the value of the polynomial when applied to a number or list of numbers. More...

Functions/Subroutines
real(dp) function, dimension(:), allocatable, public	polyder (p, m)
	polyder Computes the derivative of a polynomial. Returns an array with the coefficients
real(dp) function, dimension(:), allocatable, public	polyint (p, m, k)
	polyint Computes m-esima antiderivative
subroutine, public	bisect_pol (x0, dx, p, toler, x)
	bisect_pol Classical bisection method for root finding on polynomials

Detailed Description

polynomials provides a framework for simple (and quite naive) work with polynomials Further description in Submodule interpolate

It allows to easily evaluate, derivate, and integrate a polynomial

Examples:

program ex_polynomial
  USE numfor, only: dp, zero, m_pi, center, arange, str
  USE numfor, only: polyval, polyder, polyint
  implicit none
 
  real(dp), dimension(5) :: p1
 
  p1 = arange(5, 1, -1)
 
  print "(A)", center(' Polynomials ', 40, '*')
  print "(A)", 'Coefficients:'//str(p1)
  print "(A)", center('Evaluations', 40, '-')
  
  p1 = arange(5, 1, -1)
  print "(A)", "P(-1)= "//str(polyval(p1, -1._dp)) ! P(-1)= 3
  print "(A)", "P(0) = "//str(polyval(p1, 0._dp))  ! P(0) = 1
  print "(A)", "P(1) = "//str(polyval(p1, 1._dp))  ! P(1) = 15
  ! Polynomial evaluated in an array:
  print "(A)", "P([-1,0,1]) = "//str(polyval(p1, [-1._dp, 0._dp, 1._dp]))
  ! P([-1,0,1]) = [ 3, 1, 15]
  
  print "(A)", center('Derivatives', 40, '-')
  
  print "(A)", "Coefficients of P^(1)(x) = "//str(polyder(p1, 1))
  ! Coefficients of P^(1)(x) = [ 20, 12, 6, 2]
  print "(A)", "Coefficients of P^(2)(x) = "//str(polyder(p1, 2))
  ! Coefficients of P^(2)(x) = [ 60, 24, 6]
  print "(A)", "Coefficients of P^(3)(x) = "//str(polyder(p1, 3))
  ! Coefficients of P^(3)(x) = [120, 24]
  print "(A)", "P'([-1,0,1]) = "//str(polyval(polyder(p1, 1), [-1._dp, 0._dp, 1._dp]))
  ! P'([-1,0,1]) = [-12, 2, 40]
  
  print "(A)", center('Integration', 40, '-')
  
  print "(A)", "Coefficients of first antiderivative  (cte=0) ="//str(polyint(p1, 1))
  ! Coefficients of first antiderivative  (cte=0) =[ 1, 1, 1, 1, 1, 0]
  print "(A)", "Coefficients of first antiderivative  (cte=3) ="//str(polyint(p1, 1, 3._dp))
  ! Coefficients of first antiderivative  (cte=3) =[ 1, 1, 1, 1, 1, 3]
  print "(A)", "Coefficients of double antiderivative (ctes=0)="//str(polyint(p1, 2))
  ! Coefficients of double antiderivative (ctes=0)=[ 0.1666666666667, 0.2, 0.25, 0.3333333333333, 0.5, 0, 0]
  print "(A)", "(\int P(x) dx)([-1,0,1]) = "//str(polyval(polyint(p1, 1), [-1._dp, 0._dp, 1._dp]))
  ! (\int P(x) dx)([-1,0,1]) = [ -1, 0, 5]
  
end program ex_polynomial
 

Function/Subroutine Documentation

bisect_pol()

subroutine, public bisect_pol	(	real(dp), intent(in)	x0,
		real(dp), intent(inout)	dx,
		real(dp), dimension(:), intent(in)	p,
		real(dp), intent(in)	toler,
		real(dp), intent(out)	x )

bisect_pol Classical bisection method for root finding on polynomials

Parameters

[in]	x0	Initial value
[in,out]	dx	range. It will probe in the range (x0-dx, x0+dx). On return it will have an estimation of error
[in]	p	Array with coefficients of polynomial
[in]	toler	Tolerance in the root determination
[out]	x	Value of the root

polyder()

real(dp) function, dimension(:), allocatable, public polyder	(	real(dp), dimension(:), intent(in)	p,
		integer, intent(in), optional	m )

polyder Computes the derivative of a polynomial. Returns an array with the coefficients

Parameters

[in]	p	Array of coefficients, from highest degree to constant term
[in]	m	Order of derivation

Returns

Derivative of polynomial

Examples:

  print "(A)", "Coefficients of P^(1)(x) = "//str(polyder(p1, 1))
  ! Coefficients of P^(1)(x) = [ 20, 12, 6, 2]
  print "(A)", "Coefficients of P^(2)(x) = "//str(polyder(p1, 2))
  ! Coefficients of P^(2)(x) = [ 60, 24, 6]
  print "(A)", "Coefficients of P^(3)(x) = "//str(polyder(p1, 3))
  ! Coefficients of P^(3)(x) = [120, 24]
  print "(A)", "P'([-1,0,1]) = "//str(polyval(polyder(p1, 1), [-1._dp, 0._dp, 1._dp]))
  ! P'([-1,0,1]) = [-12, 2, 40]

References basic::print_msg(), and basic::zero.

Referenced by csplevder::cspl_interpdev(), csplevder::cspl_interpdev_tab(), and csplines::csplder().

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polyint()

real(dp) function, dimension(:), allocatable, public polyint	(	real(dp), dimension(:), intent(in)	p,
		integer, intent(in), optional	m,
		real(dp), intent(in), optional	k )

polyint Computes m-esima antiderivative

Parameters

[in]	p	Array of coefficients, from highest degree to constant term
[in]	m	Number of times that p must be integrated
[in]	k	Additive Constant

Returns

Antiderivative polynomial

Examples:

  print "(A)", "Coefficients of first antiderivative  (cte=0) ="//str(polyint(p1, 1))
  ! Coefficients of first antiderivative  (cte=0) =[ 1, 1, 1, 1, 1, 0]
  print "(A)", "Coefficients of first antiderivative  (cte=3) ="//str(polyint(p1, 1, 3._dp))
  ! Coefficients of first antiderivative  (cte=3) =[ 1, 1, 1, 1, 1, 3]
  print "(A)", "Coefficients of double antiderivative (ctes=0)="//str(polyint(p1, 2))
  ! Coefficients of double antiderivative (ctes=0)=[ 0.1666666666667, 0.2, 0.25, 0.3333333333333, 0.5, 0, 0]
  print "(A)", "(\int P(x) dx)([-1,0,1]) = "//str(polyval(polyint(p1, 1), [-1._dp, 0._dp, 1._dp]))
  ! (\int P(x) dx)([-1,0,1]) = [ -1, 0, 5]

References basic::print_msg(), and basic::zero.

Referenced by csplines::csplantider().

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