Polynomial roots

Tools to compute the complex roots of quadratic, cubic, and quartic polynomials. More...

Functions/Subroutines
pure subroutine, public	mod_polynomial_roots::solve_quadratic (a, b, x)
	Solve quadratic equation in monic form.
pure subroutine, public	mod_polynomial_roots::solve_cubic (a, b, c, x)
	Solve cubic equation in monic form.
pure subroutine, public	mod_polynomial_roots::solve_quartic (a, b, c, d, x)
	Solve quartic equation in monic form.
pure subroutine	mod_polynomial_roots::solve_complex_quadratic (a, b, x)
	Solve quadratic equation in monic form with complex coefficients.
pure subroutine	mod_polynomial_roots::fitcubic (a, b, c, x)
	Solve cubic equation using a fitting algorithm.
pure subroutine	mod_polynomial_roots::sort_roots (x)
	Sort complex root using the Insertion Sort algorithm from larger to lower magnitude.
pure subroutine	mod_polynomial_roots::quartic_closed_form (a, b, c, d, x)
	Ferrari's algorithm to solve quartic equations in monic form (closed form root finding)

Detailed Description

Tools to compute the complex roots of quadratic, cubic, and quartic polynomials.

The equations must be represented in the monic form, that is the leading coefficient of the polynomial is equal to 1:

\[ x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0 \]

The coefficients must be real numbers.

Function/Subroutine Documentation

fitcubic()

pure subroutine mod_polynomial_roots::fitcubic ( double precision, intent(in) a, double precision, intent(in) b, double precision, intent(in) c, complex(kind=8), dimension(3), intent(out) x )

private

Solve cubic equation using a fitting algorithm.

References:

• Strobach, P. (2011). Solving cubics by polynomial fitting. Journal of computational and applied mathematics, 235(9), 3033-3052. doi:10.1016/j.cam.2010.12.025

Parameters

[in]	a,b,c	Coefficients of the cubic polynomial in monic form.
[out]	x	Roots of the equation.

quartic_closed_form()

pure subroutine mod_polynomial_roots::quartic_closed_form ( double precision, intent(in) a, double precision, intent(in) b, double precision, intent(in) c, double precision, intent(in) d, complex(kind=8), dimension(4), intent(out) x )

private

Ferrari's algorithm to solve quartic equations in monic form (closed form root finding)

\[ x^4 + ax^3 + bx^2 + cx + d = 0 \]

Parameters

[in]	a,b,c,d	Coefficients of the quartic polynomial in monic form.
[out]	x	Roots of the quartic equation.

solve_complex_quadratic()

pure subroutine mod_polynomial_roots::solve_complex_quadratic ( complex(kind=8), intent(in) a, complex(kind=8), intent(in) b, complex(kind=8), dimension(2), intent(out) x )

private

Solve quadratic equation in monic form with complex coefficients.

\[ x^2 + ax + b = 0 \]

Parameters

[in]	a,b	Complex cefficients of the second order polynomial in monic form.
[out]	x	The two complex roots.

solve_cubic()

pure subroutine, public mod_polynomial_roots::solve_cubic	(	double precision, intent(in)	a,
		double precision, intent(in)	b,
		double precision, intent(in)	c,
		complex(kind=8), dimension(3), intent(out)	x )

Solve cubic equation in monic form.

The cubic equation to solve must be in the form:

\[ x^3 + ax^2 + bx + c = 0 \]

References:

• Strobach, P. (2011). Solving cubics by polynomial fitting. Journal of computational and applied mathematics, 235(9), 3033-3052. doi:10.1016/j.cam.2010.12.025

Parameters

[in]	a,b,c	Coefficients of the third order polynomial in monic form.
[out]	x	The three complex roots.

solve_quadratic()

pure subroutine, public mod_polynomial_roots::solve_quadratic	(	double precision, intent(in)	a,
		double precision, intent(in)	b,
		complex(kind=8), dimension(2), intent(out)	x )

Solve quadratic equation in monic form.

The quadratic equation to solve must be in the form:

\[ x^2 + ax + b = 0 \]

Reference:

• Vetterling, W. T., Teukolsky, S. A., Press, W. H., & Flannery, B. P. (1992). Numerical recipes in Fortran 77: the art of scientific computing. Cambridge University Press, London. ISBN:978-0521430647

Parameters

[in]	a,b	Coefficients of the second order polynomial in monic form.
[out]	x	The two complex roots.

solve_quartic()

pure subroutine, public mod_polynomial_roots::solve_quartic	(	double precision, intent(in)	a,
		double precision, intent(in)	b,
		double precision, intent(in)	c,
		double precision, intent(in)	d,
		complex(kind=8), dimension(4), intent(out)	x )

Solve quartic equation in monic form.

The quartic equation to solve must be in the form:

\[ x^4 + ax^3 + bx^2 + cx + d = 0 \]

References:

• Strobach, P. (2010). The fast quartic solver. Journal of computational and applied mathematics, 234(10), 3007-3024. doi:10.1016/j.cam.2010.04.015

Parameters

[in]	a,b,c,d	Coefficients of the fourth order polynomial in monic form.
[out]	x	The four complex roots.

sort_roots()

pure subroutine mod_polynomial_roots::sort_roots ( complex(kind=8), dimension(:), intent(inout) x )

private

Sort complex root using the Insertion Sort algorithm from larger to lower magnitude.

Parameters

[in,out]

x

Array of complex roots