rootfinding Module Reference

RootFinding Module. More...

Data Types
interface	func
	Interface of the function whose root needs to be computed. More...

Functions/Subroutines
subroutine	polyroots (c, r, error, BalanceOn)
	Computes the complex roots of the given polynomial. More...
pure subroutine	root (a, b, ya, yb, tol, f, x, fx, niter, error)
	Subroutine for computing the root using Brent algorithm. More...

Variables
integer, parameter	max_iterations = 50
	Max iterations for all the root-finding. More...

Detailed Description

RootFinding Module.

This module provides functions for root-finding of polynomials by solving associated eigenvalue problems and nonlinear equations using algorithms like Brent's Algorithm.

Author

Bharat Mahajan

Date

Created: 07/31/2019

Function/Subroutine Documentation

polyroots()

subroutine rootfinding::polyroots	(	complex(wp), dimension(1:), intent(in)	c,
		complex(wp), dimension(:), intent(out), allocatable	r,
		integer, intent(out)	error,
		logical, intent(in), optional	BalanceOn
	)

Computes the complex roots of the given polynomial.

PolyRoots finds the roots of the polynomial defined by complex coefficients given in array "c" (c(1) is the highest degree coeff.). It forms the companion matrix and then solves its eigenvalues using LAPACk routines from Intel MKL library. The algorithm is same as that used by MATLAB's "roots" command. References:

1. https://blogs.mathworks.com/cleve/2016/06/27/19-dubious-ways-to-compute-the-zeros-of-a-polynomial/
2. Math77: http://www.netlib.org/math/docpdf/ch07-01.pdf

   Parameters

   [in]	c	Complex coefficients of the polynomial defined by p(x) = c(1)*x^n + ... + c(n)*x + c(n+1)
   [out]	r	Complex roots array
   [out]	error	Error status returned to the user. "0" : Success, valid results in "r" "-1": Error, input coefficients array has length less than two "-2": Error, all the coefficients are zero "-3": Error, companion matrix balancing failed "-4": Error, LAPACK routine failed to compute eigenvalues
   [in]	balanceon	If True, then the companion matrix is balanced first before finding its eigenvalues, otherwise not. Default is True. It may produce inaccurate results in case of very poorly-scaled matrices.

Remarks

return if input coefficient array is empty

Return if all coefficients are zero

Leading zeros are thrown away while the trailing zero coefficients are remembered by introducing roots at zero.

Definition at line 66 of file rootfinding.f90.

References fmutilbase::inf.

root()

pure subroutine rootfinding::root	(	real(wp), intent(in)	a,
		real(wp), intent(in)	b,
		real(wp), intent(in)	ya,
		real(wp), intent(in)	yb,
		real(wp), intent(in)	tol,
		procedure(func)	f,
		real(wp), intent(out)	x,
		real(wp), intent(out)	fx,
		integer, intent(out)	niter,
		integer, intent(out)	error
	)

Subroutine for computing the root using Brent algorithm.

For details on the algorithm, see Numerical receipes in C book at https://www2.units.it/ipl/students_area/imm2/files/Numerical_Recipes.pdf

Parameters

[in]	a	Lower bound on the root
[in]	b	Upper bound on the root
[in]	ya	f(a)
[in]	yb	f(b)
[in]	tol	tolerance
	f	function
[out]	x	root
[out]	fx	value at root
[out]	niter	no. of iterations
[out]	error	Error status returned to the user. "0" means root has been found. "1" means provided bounds do not contain the root, i.e., root is not bracketed. "2" means maximum number of interations reached.

Definition at line 208 of file rootfinding.f90.

References max_iterations.

Variable Documentation

max_iterations

integer, parameter rootfinding::max_iterations = 50

Max iterations for all the root-finding.

Definition at line 42 of file rootfinding.f90.

Referenced by root().