Defines a polynomial, and associated routines for performing polynomial operations. More...

Public Member Functions
generic, public	initialize (this, order, err)
	Initializes the polynomial instance.

generic, public	initialize (this, c, err)
	Initializes the polynomial instance.

procedure, public	order (this)
	Returns the order of the polynomial object.

procedure, public	fit (this, x, y, order, err)
	Fits a polynomial of the specified order to a data set.

procedure, public	fit_thru_zero (this, x, y, order, err)
	Fits a polynomial of the specified order that passes through zero to a data set.

generic, public	evaluate evaluate_real
	Evaluates a polynomial at the specified points.

generic, public	evaluate evaluate_complex
	Evaluates a polynomial at the specified points.

procedure, public	companion_mtx (this)
	Returns the companion matrix for the polynomial.

procedure, public	roots (this, err)
	Computes all the roots of a polynomial.

procedure, public	get (this, ind, err)
	Gets the requested polynomial coefficient.

procedure, public	get_all (this)
	Gets an array containing all the coefficients of the polynomial.

procedure, public	set (this, ind, c, err)
	Sets the requested polynomial coefficient by index.

procedure	evaluate_real (this, x)
	Evaluates a polynomial at the specified points.

procedure	evaluate_complex (this, x)
	Evaluates a polynomial at the specified points.

procedure	init_poly (this, order, err)
	Initializes the polynomial instance, and sets all coefficients to zero.

procedure	init_poly_coeffs (this, c, err)
	Initializes the polynomial instance.

Private Attributes
real(real64), dimension(:), allocatable	m_coeffs
	An array that contains the polynomial coefficients in ascending order.

Detailed Description

Defines a polynomial, and associated routines for performing polynomial operations.

Definition at line 60 of file nonlin_polynomials.f90.

Member Function/Subroutine Documentation

◆ companion_mtx()

procedure, public nonlin_polynomials::polynomial::companion_mtx ( class(polynomial), intent(in) this )

Returns the companion matrix for the polynomial.

Parameters

[in] this The polynomial object.

Returns: The companion matrix.

See Also

Definition at line 77 of file nonlin_polynomials.f90.

◆ evaluate() [1/2]

generic, public nonlin_polynomials::polynomial::evaluate

Evaluates a polynomial at the specified points.

Definition at line 75 of file nonlin_polynomials.f90.

◆ evaluate() [2/2]

generic, public nonlin_polynomials::polynomial::evaluate

Evaluates a polynomial at the specified points.

Definition at line 75 of file nonlin_polynomials.f90.

◆ evaluate_complex()

procedure nonlin_polynomials::polynomial::evaluate_complex	(	class(polynomial), intent(in)	this,
		complex(real64), intent(in)	x )

Evaluates a polynomial at the specified points.

Parameters

[in]	this	The polynomial object.
[in]	x	The value(s) at which to evaluate the polynomial.

Returns: The value(s) of the polynomial at x.

Definition at line 88 of file nonlin_polynomials.f90.

◆ evaluate_real()

procedure nonlin_polynomials::polynomial::evaluate_real	(	class(polynomial), intent(in)	this,
		real(real64), intent(in)	x )

Evaluates a polynomial at the specified points.

Parameters

[in]	this	The polynomial object.
[in]	x	The value(s) at which to evaluate the polynomial.

Returns: The value(s) of the polynomial at x.

Definition at line 87 of file nonlin_polynomials.f90.

◆ fit()

procedure, public nonlin_polynomials::polynomial::fit	(	class(polynomial), intent(inout)	this,
		real(real64), dimension(:), intent(in)	x,
		real(real64), dimension(:), intent(inout)	y,
		integer(int32), intent(in)	order,
		class(errors), intent(inout), optional, target	err )

Fits a polynomial of the specified order to a data set.

Parameters

[in,out]	this	The polynomial object.
[in]	x	An N-element array containing the independent variable data points. Notice, must be N > `order`.
[in,out]	y	On input, an N-element array containing the dependent variable data points. On output, the contents are overwritten.
[in]	order	The order of the polynomial (must be >= 1).
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if a zero or negative polynomial order was specified, or if order is too large for the data set. NL_OUT_OF_MEMORY_ERROR: Occurs if insufficient memory is available. NL_ARRAY_SIZE_ERROR: Occurs if `x` and `y` are different sizes.

Usage: The following code provides an example of how to fit a polynomial to a set of data.
program example

use linalg_constants, only : dp, i32

use nonlin_polynomials

! Local Variables

real(real64), dimension(21) :: xp, yp, yf, yc, err

real(real64) :: res

type(polynomial) :: p

! Data to fit

xp = [0.0d0, 0.1d0, 0.2d0, 0.3d0, 0.4d0, 0.5d0, 0.6d0, 0.7d0, 0.8d0, &

0.9d0, 1.0d0, 1.1d0, 1.2d0, 1.3d0, 1.4d0, 1.5d0, 1.6d0, 1.7d0, &

1.8d0, 1.9d0, 2.0d0]

yp = [1.216737514d0, 1.250032542d0, 1.305579195d0, 1.040182335d0, &

1.751867738d0, 1.109716707d0, 2.018141531d0, 1.992418729d0, &

1.807916923d0, 2.078806005d0, 2.698801324d0, 2.644662712d0, &

3.412756702d0, 4.406137221d0, 4.567156645d0, 4.999550779d0, &

5.652854194d0, 6.784320119d0, 8.307936836d0, 8.395126494d0, &

10.30252404d0]

! Create a copy of yp as it will be overwritten in the fit command

yc = yp

! Fit the polynomial

call p%fit(xp, yp, 3)

! Evaluate the polynomial at xp, and then determine the residual

yf = p%evaluate(xp)

err = abs(yf - yc)

res = maxval(err)

! Print out the coefficients

print '(A)', "Polynomial Coefficients (c0 + c1*x + c2*x**2 + c3*x**3):"

do i = 1, 4

print '(AI0AF12.9)', "c", i - 1, " = ", p%get(i)

end do

print '(AE9.4)', "Residual: ", res

end program

nonlin_polynomials
polynomials
Definition nonlin_polynomials.f90:13

The above program returns the following results.
Polynomial Coefficients (c0 + c1*x + c2*x**2 + c3*x**3):

c0 = 1.186614186

c1 = 0.446613631

c2 = -0.122320499

c3 = 1.064762822

Residual: .5064E+00

Definition at line 70 of file nonlin_polynomials.f90.

◆ fit_thru_zero()

procedure, public nonlin_polynomials::polynomial::fit_thru_zero	(	class(polynomial), intent(inout)	this,
		real(real64), dimension(:), intent(in)	x,
		real(real64), dimension(:), intent(inout)	y,
		integer(int32), intent(in)	order,
		class(errors), intent(inout), optional, target	err )

Fits a polynomial of the specified order that passes through zero to a data set.

Parameters

[in,out]	this	The polynomial object.
[in]	x	An N-element array containing the independent variable data points. Notice, must be N > `order`.
[in,out]	y	On input, an N-element array containing the dependent variable data points. On output, the contents are overwritten.
[in]	order	The order of the polynomial (must be >= 1).
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if a zero or negative polynomial order was specified, or if order is too large for the data set. NL_OUT_OF_MEMORY_ERROR: Occurs if insufficient memory is available. NL_ARRAY_SIZE_ERROR: Occurs if `x` and `y` are different sizes.

Definition at line 73 of file nonlin_polynomials.f90.

◆ get()

procedure, public nonlin_polynomials::polynomial::get	(	class(polynomial), intent(in)	this,
		integer(int32), intent(in)	ind,
		class(errors), intent(inout), optional, target	err )

Gets the requested polynomial coefficient.

Parameters

[in]	this	The polynomial.
[in]	ind	The polynomial coefficient index (0 < ind <= order + 1).
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if the requested index is less than or equal to zero, or if the requested index exceeds the number of polynomial coefficients.

Returns: The requested coefficient.

Definition at line 81 of file nonlin_polynomials.f90.

◆ get_all()

procedure, public nonlin_polynomials::polynomial::get_all ( class(polynomial), intent(in) this )

Gets an array containing all the coefficients of the polynomial.

Parameters

[in] this The polynomial object.

Returns: The array of coefficients.

Definition at line 83 of file nonlin_polynomials.f90.

◆ init_poly()

procedure nonlin_polynomials::polynomial::init_poly	(	class(polynomial), intent(inout)	this,
		integer(int32), intent(in)	order,
		class(errors), intent(inout), optional, target	err )

Initializes the polynomial instance, and sets all coefficients to zero.

Parameters

[in,out]	this	The polynomial object.
[in]	order	The order of the polynomial (must be >= 0).
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if a zero or negative polynomial order was specified. NL_OUT_OF_MEMORY_ERROR: Occurs if insufficient memory is available.

Definition at line 89 of file nonlin_polynomials.f90.

◆ init_poly_coeffs()

procedure nonlin_polynomials::polynomial::init_poly_coeffs	(	class(polynomial), intent(inout)	this,
		real(real64), dimension(:), intent(in)	c,
		class(errors), intent(inout), optional, target	err )

Initializes the polynomial instance.

Parameters

[in,out]	this	The polynomial object.
[in]	c	The array of polynomial coefficients. The coefficients are established as follows: c(1) + c(2) * x + c(3) * x*2 + ... c(n) x**n-1.
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if a zero or negative polynomial order was specified. NL_OUT_OF_MEMORY_ERROR: Occurs if insufficient memory is available.

Definition at line 90 of file nonlin_polynomials.f90.

◆ initialize() [1/2]

generic, public nonlin_polynomials::polynomial::initialize	(	class(polynomial), intent(inout)	this,
		real(real64), dimension(:), intent(in)	c,
		class(errors), intent(inout), optional, target	err )

Initializes the polynomial instance.

Parameters

[in,out]	this	The polynomial object.
[in]	c	The array of polynomial coefficients. The coefficients are established as follows: c(1) + c(2) * x + c(3) * x*2 + ... c(n) x**n-1.
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if a zero or negative polynomial order was specified. NL_OUT_OF_MEMORY_ERROR: Occurs if insufficient memory is available.

Definition at line 66 of file nonlin_polynomials.f90.

◆ initialize() [2/2]

generic, public nonlin_polynomials::polynomial::initialize	(	class(polynomial), intent(inout)	this,
		integer(int32), intent(in)	order,
		class(errors), intent(inout), optional, target	err )

Initializes the polynomial instance.

Parameters

[in,out]	this	The polynomial object.
[in]	order	The order of the polynomial (must be >= 0).
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if a zero or negative polynomial order was specified. NL_OUT_OF_MEMORY_ERROR: Occurs if insufficient memory is available.

Definition at line 66 of file nonlin_polynomials.f90.

◆ order()

procedure, public nonlin_polynomials::polynomial::order ( class(polynomial), intent(in) this )

Returns the order of the polynomial object.

Parameters

[in] this The polynomial object.

Returns: The order of the polynomial. Returns -1 in the event no polynomial coefficients have been defined.

Definition at line 68 of file nonlin_polynomials.f90.

◆ roots()

procedure, public nonlin_polynomials::polynomial::roots	(	class(polynomial), intent(in)	this,
		class(errors), intent(inout), optional, target	err )

Computes all the roots of a polynomial.

Parameters

[in]	this	The polynomial object.
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_OUT_OF_MEMORY_ERROR: Occurs if local memory must be allocated, and there is insufficient memory available. NL_CONVERGENCE_ERROR: Occurs if the algorithm failed to converge.

Usage: The following code provides an example of how to compute the roots of a polynomial. This examples uses a tenth order polynomial; however, this process is applicable to any order.
program example

use linalg_constants, only : dp, i32

use nonlin_polynomials

! Parameters

integer(int32), parameter :: order = 10

! Local Variables

integer(int32) :: i

type(polynomial) :: p

real(real64), dimension(order+1) :: coeff

complex(real64), allocatable, dimension(:) :: rts, sol

! Define the polynomial

call random_number(coeff)

call p%initialize(order)

do i = 1, size(coeff)

call p%set(i, coeff(i))

end do

! Compute the roots via the polynomial routine

rts = p%roots()

! Compute the value of the polynomial at each root and ensure it

! is sufficiently close to zero.

sol = p%evaluate(rts)

do i = 1, size(sol)

print '(AE9.3AE9.3A)', "(", real(sol(i)), ", ", aimag(sol(i)), ")"

end do

end program

The above program returns the following results.
(-.466E-14, -.161E-14)

(-.466E-14, 0.161E-14)

(-.999E-15, 0.211E-14)

(-.999E-15, -.211E-14)

(0.444E-15, 0.108E-14)

(0.444E-15, -.108E-14)

(-.144E-14, -.433E-14)

(-.144E-14, 0.433E-14)

(0.644E-14, -.100E-13)

(0.644E-14, 0.100E-13)

Definition at line 79 of file nonlin_polynomials.f90.

◆ set()

procedure, public nonlin_polynomials::polynomial::set	(	class(polynomial), intent(inout)	this,
		integer(int32), intent(in)	ind,
		real(real64), intent(in)	c,
		class(errors), intent(inout), optional, target	err )

Sets the requested polynomial coefficient by index.

Parameters

[in,out]	this	The polynomial.
[in]	ind	The polynomial coefficient index (0 < ind <= order + 1).
[in]	c	The polynomial coefficient.
[out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. NL_INVALID_INPUT_ERROR: Occurs if the requested index is less than or equal to zero, or if the requested index exceeds the number of polynomial coefficients.

Definition at line 85 of file nonlin_polynomials.f90.

Member Data Documentation

◆ m_coeffs

real(real64), dimension(:), allocatable nonlin_polynomials::polynomial::m_coeffs

private

An array that contains the polynomial coefficients in ascending order.

Definition at line 63 of file nonlin_polynomials.f90.

The documentation for this type was generated from the following file:

D:/Code/nonlin/src/nonlin_polynomials.f90

Public Member Functions

Private Attributes

Detailed Description

Member Function/Subroutine Documentation

◆ companion_mtx()

◆ evaluate() [1/2]

◆ evaluate() [2/2]

◆ evaluate_complex()

◆ evaluate_real()

◆ fit()

◆ fit_thru_zero()

◆ get()

◆ get_all()

◆ init_poly()

◆ init_poly_coeffs()

◆ initialize() [1/2]

◆ initialize() [2/2]

◆ order()

◆ roots()

◆ set()

Member Data Documentation

◆ m_coeffs