Collection of simple mathematical functions and subroutines. More...

Data Types
interface	det_3x3

interface	invert_matrix

interface	set_diag

interface	unit_matrix

Functions/Subroutines
real(kind=dp) function, public	pswitch (x, a, b, order)
	Polynomial (5th degree) switching function f(a) = 1 .... f(b) = 0 with f'(a) = f"(a) = f'(b) = f"(b) = 0.

logical function, public	abnormal_value (a)
	determines if a value is not normal (e.g. for Inf and Nan) based on IO to work also under optimization.

pure real(kind=dp) function, public	angle (a, b)
	Calculation of the angle between the vectors a and b. The angle is returned in radians.

elemental real(kind=dp) function, public	binomial (n, k)
	The binomial coefficient n over k for 0 <= k <= n is calculated, otherwise zero is returned.

elemental real(kind=dp) function, public	binomial_gen (z, k)
	The generalized binomial coefficient z over k for 0 <= k <= n is calculated. (z) z(z-1)...(z-k+2)(z-k+1) ( ) = ------------------------— (k) k!

pure real(kind=dp) function, public	multinomial (n, k)
	Calculates the multinomial coefficients.

pure subroutine, public	build_rotmat (phi, a, rotmat)
	The rotation matrix rotmat which rotates a vector about a rotation axis defined by the vector a is build up. The rotation angle is phi (radians).

subroutine, public	diamat_all (a, eigval, dac)
	Diagonalize the symmetric n by n matrix a using the LAPACK library. Only the upper triangle of matrix a is used. Externals (LAPACK 3.0)

pure real(kind=dp) function, public	dihedral_angle (ab, bc, cd)
	Returns the dihedral angle, i.e. the angle between the planes defined by the vectors (-ab,bc) and (cd,-bc). The dihedral angle is returned in radians.

pure real(kind=dp) function, dimension(min(size(a, 1), size(a, 2))), public	get_diag (a)
	Return the diagonal elements of matrix a as a vector.

pure real(kind=dp) function, dimension(3, 3), public	inv_3x3 (a)
	Returns the inverse of the 3 x 3 matrix a.

subroutine, public	invmat (a, info)
	returns inverse of matrix using the lapack routines DGETRF and DGETRI

subroutine, public	invmat_symm (a, potrf, uplo)
	returns inverse of real symmetric, positive definite matrix

subroutine, public	get_pseudo_inverse_svd (a, a_pinverse, rskip, determinant, sval)
	returns the pseudoinverse of a real, square matrix using singular value decomposition

subroutine, public	get_pseudo_inverse_diag (a, a_pinverse, rskip)
	returns the pseudoinverse of a real, symmetric and positive definite matrix using diagonalization.

pure real(kind=dp) function, dimension(3), public	reflect_vector (a, b)
	Reflection of the vector a through a mirror plane defined by the normal vector b. The reflected vector a is stored in a_mirror.

pure real(kind=dp) function, dimension(3), public	rotate_vector (a, phi, b)
	Rotation of the vector a about an rotation axis defined by the vector b. The rotation angle is phi (radians). The rotated vector a is stored in a_rot.

subroutine, public	symmetrize_matrix (a, option)
	Symmetrize the matrix a.

pure real(kind=dp) function, dimension(3), public	vector_product (a, b)
	Calculation of the vector product c = a x b.

elemental integer function, public	gcd (a, b)
	computes the greatest common divisor of two number

elemental integer function, public	lcm (a, b)
	computes the least common multiplier of two numbers

elemental impure real(dp) function, public	expint (n, x)
	computes the exponential integral En(x) = Int(exp(-x*t)/t^n,t=1..infinity) x>0, n=0,1,.. Note: Ei(-x) = -E1(x)

subroutine, public	jacobi (a, d, v)
	Jacobi matrix diagonalization. The eigenvalues are returned in vector d and the eigenvectors are returned in matrix v in ascending order.

subroutine, public	diag (n, a, d, v)
	Diagonalize matrix a. The eigenvalues are returned in vector d and the eigenvectors are returned in matrix v.

subroutine, public	erfc_cutoff (eps, omg, r_cutoff)
	compute a truncation radius for the shortrange operator

subroutine, public	diag_complex (matrix, eigenvectors, eigenvalues)
	Diagonalizes a local complex Hermitian matrix using LAPACK. Based on cp_cfm_heevd.

subroutine, public	diag_antisym (matrix, evecs, evals)
	Helper routine for diagonalizing anti symmetric matrices.

Detailed Description

Collection of simple mathematical functions and subroutines.

History: FUNCTION angle updated and FUNCTION dihedral angle added; cleaned (13.03.2004,MK)

Author: MK (15.11.1998)

Function/Subroutine Documentation

◆ pswitch()

real(kind=dp) function, public mathlib::pswitch	(	real(kind=dp)	x,
		real(kind=dp)	a,
		real(kind=dp)	b,
		integer	order
	)

Polynomial (5th degree) switching function f(a) = 1 .... f(b) = 0 with f'(a) = f"(a) = f'(b) = f"(b) = 0.

Parameters

x	...
a	...
b	...
order	...

Returns: =0 : f(x); =1 : f'(x); =2 : f"(x)

Definition at line 99 of file mathlib.F.

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◆ abnormal_value()

logical function, public mathlib::abnormal_value ( real(kind=dp) a )

determines if a value is not normal (e.g. for Inf and Nan) based on IO to work also under optimization.

Parameters

a	input value

Returns: TRUE for NaN and Inf

Definition at line 150 of file mathlib.F.

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◆ angle()

pure real(kind=dp) function, public mathlib::angle	(	real(kind=dp), dimension(:), intent(in)	a,
		real(kind=dp), dimension(:), intent(in)	b
	)

Calculation of the angle between the vectors a and b. The angle is returned in radians.

Parameters

a	...
b	...

Returns: ...

Date: 14.10.1998

Author: MK

Version: 1.0

Definition at line 175 of file mathlib.F.

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◆ binomial()

elemental real(kind=dp) function, public mathlib::binomial	(	integer, intent(in)	n,
		integer, intent(in)	k
	)

The binomial coefficient n over k for 0 <= k <= n is calculated, otherwise zero is returned.

Parameters

n	...
k	...

Returns: ...

Date: 08.03.1999

Author: MK

Version: 1.0

Definition at line 205 of file mathlib.F.

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◆ binomial_gen()

elemental real(kind=dp) function, public mathlib::binomial_gen	(	real(kind=dp), intent(in)	z,
		integer, intent(in)	k
	)

The generalized binomial coefficient z over k for 0 <= k <= n is calculated. (z) z*(z-1)*...*(z-k+2)*(z-k+1) ( ) = ------------------------— (k) k!

Parameters

z	...
k	...

Returns: ...

Date: 11.11.2019

Author: FS

Version: 1.0

Definition at line 229 of file mathlib.F.

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◆ multinomial()

pure real(kind=dp) function, public mathlib::multinomial	(	integer, intent(in)	n,
		integer, dimension(:), intent(in)	k
	)

Calculates the multinomial coefficients.

Parameters

n	...
k	...

Returns: ...

Author: Ole Schuett

Definition at line 254 of file mathlib.F.

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◆ build_rotmat()

pure subroutine, public mathlib::build_rotmat	(	real(kind=dp), intent(in)	phi,
		real(kind=dp), dimension(3), intent(in)	a,
		real(kind=dp), dimension(3, 3), intent(out)	rotmat
	)

The rotation matrix rotmat which rotates a vector about a rotation axis defined by the vector a is build up. The rotation angle is phi (radians).

Parameters

phi	...
a	...
rotmat	...

Date: 16.10.1998

Author: MK

Version: 1.0

Definition at line 285 of file mathlib.F.

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◆ diamat_all()

subroutine, public mathlib::diamat_all	(	real(kind=dp), dimension(:, :), intent(inout)	a,
		real(kind=dp), dimension(:), intent(out)	eigval,
		logical, intent(in), optional	dac
	)

Diagonalize the symmetric n by n matrix a using the LAPACK library. Only the upper triangle of matrix a is used. Externals (LAPACK 3.0)

Parameters

a	...
eigval	...
dac	...

Date: 29.03.1999

Variables

a : Symmetric matrix to be diagonalized (input; upper triangle) ->
eigenvectors of the matrix a (output).
dac : If true, then the divide-and-conquer algorithm is applied.
eigval : Eigenvalues of the matrix a (output).

Author: MK

Version: 1.0

Definition at line 372 of file mathlib.F.

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◆ dihedral_angle()

pure real(kind=dp) function, public mathlib::dihedral_angle	(	real(kind=dp), dimension(3), intent(in)	ab,
		real(kind=dp), dimension(3), intent(in)	bc,
		real(kind=dp), dimension(3), intent(in)	cd
	)

Returns the dihedral angle, i.e. the angle between the planes defined by the vectors (-ab,bc) and (cd,-bc). The dihedral angle is returned in radians.

Parameters

ab	...
bc	...
cd	...

Returns: ...

Date: 13.03.2004

Author: MK

Version: 1.0

Definition at line 467 of file mathlib.F.

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◆ get_diag()

pure real(kind=dp) function, dimension(min(size(a, 1), size(a, 2))), public mathlib::get_diag ( real(kind=dp), dimension(:, :), intent(in) a )

Return the diagonal elements of matrix a as a vector.

Parameters

a ...

Returns: ...

Date: 20.11.1998

Author: MK

Version: 1.0

Definition at line 492 of file mathlib.F.

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◆ inv_3x3()

pure real(kind=dp) function, dimension(3, 3), public mathlib::inv_3x3 ( real(kind=dp), dimension(3, 3), intent(in) a )

Returns the inverse of the 3 x 3 matrix a.

Parameters

a ...

Returns: ...

Date: 13.03.2004

Author: MK

Version: 1.0

Definition at line 515 of file mathlib.F.

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◆ invmat()

subroutine, public mathlib::invmat	(	real(kind=dp), dimension(:, :), intent(inout)	a,
		integer, intent(out)	info
	)

returns inverse of matrix using the lapack routines DGETRF and DGETRI

Parameters

a	...
info	...

Definition at line 542 of file mathlib.F.

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◆ invmat_symm()

subroutine, public mathlib::invmat_symm	(	real(kind=dp), dimension(:, :), intent(inout)	a,
		logical, intent(in), optional	potrf,
		character(len=1), intent(in), optional	uplo
	)

returns inverse of real symmetric, positive definite matrix

Parameters

a	matrix
potrf	if cholesky decomposition of a was already done using dpotrf. If not given, cholesky decomposition of a will be done before inversion.
uplo	indicating if the upper or lower triangle of a is stored.

Author: Dorothea Golze [02.2015]

Definition at line 579 of file mathlib.F.

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◆ get_pseudo_inverse_svd()

subroutine, public mathlib::get_pseudo_inverse_svd	(	real(kind=dp), dimension(:, :)	a,
		real(kind=dp), dimension(:, :)	a_pinverse,
		real(kind=dp), intent(in)	rskip,
		real(kind=dp), intent(out), optional	determinant,
		real(kind=dp), dimension(:), intent(inout), optional, pointer	sval
	)

returns the pseudoinverse of a real, square matrix using singular value decomposition

Parameters

a	matrix a
a_pinverse	pseudoinverse of matrix a
rskip	parameter for setting small singular values to zero
determinant	determinant of matrix a (optional output)
sval	array holding singular values of matrix a (optional output)

Author: Dorothea Golze [02.2015]

Definition at line 937 of file mathlib.F.

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◆ get_pseudo_inverse_diag()

subroutine, public mathlib::get_pseudo_inverse_diag	(	real(kind=dp), dimension(:, :)	a,
		real(kind=dp), dimension(:, :)	a_pinverse,
		real(kind=dp), intent(in)	rskip
	)

returns the pseudoinverse of a real, symmetric and positive definite matrix using diagonalization.

Parameters

a	matrix a
a_pinverse	pseudoinverse of matrix a
rskip	parameter for setting small eigenvalues to zero

Author: Dorothea Golze [02.2015]

Definition at line 1018 of file mathlib.F.

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◆ reflect_vector()

pure real(kind=dp) function, dimension(3), public mathlib::reflect_vector	(	real(kind=dp), dimension(3), intent(in)	a,
		real(kind=dp), dimension(3), intent(in)	b
	)

Reflection of the vector a through a mirror plane defined by the normal vector b. The reflected vector a is stored in a_mirror.

Parameters

a	...
b	...

Returns: ...

Date: 16.10.1998

Author: MK

Version: 1.0

Definition at line 1085 of file mathlib.F.

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◆ rotate_vector()

pure real(kind=dp) function, dimension(3), public mathlib::rotate_vector	(	real(kind=dp), dimension(3), intent(in)	a,
		real(kind=dp), intent(in)	phi,
		real(kind=dp), dimension(3), intent(in)	b
	)

Rotation of the vector a about an rotation axis defined by the vector b. The rotation angle is phi (radians). The rotated vector a is stored in a_rot.

Parameters

a	...
phi	...
b	...

Returns: ...

Date: 16.10.1998

Author: MK

Version: 1.0

Definition at line 1123 of file mathlib.F.

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◆ symmetrize_matrix()

subroutine, public mathlib::symmetrize_matrix	(	real(kind=dp), dimension(:, :), intent(inout)	a,
		character(len=*), intent(in)	option
	)

Symmetrize the matrix a.

Parameters

a	...
option	...

Date: 16.10.1998

Author: MK

Version: 1.0

Definition at line 1196 of file mathlib.F.

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◆ vector_product()

pure real(kind=dp) function, dimension(3), public mathlib::vector_product	(	real(kind=dp), dimension(3), intent(in)	a,
		real(kind=dp), dimension(3), intent(in)	b
	)

Calculation of the vector product c = a x b.

Parameters

a	...
b	...

Returns: ...

Date: 16.10.1998

Author: MK

Version: 1.0

Definition at line 1262 of file mathlib.F.

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◆ gcd()

elemental integer function, public mathlib::gcd	(	integer, intent(in)	a,
		integer, intent(in)	b
	)

computes the greatest common divisor of two number

Parameters

a	...
b	...

Returns: ...

Author: Joost VandeVondele

Definition at line 1279 of file mathlib.F.

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◆ lcm()

elemental integer function, public mathlib::lcm	(	integer, intent(in)	a,
		integer, intent(in)	b
	)

computes the least common multiplier of two numbers

Parameters

a	...
b	...

Returns: ...

Author: Joost VandeVondele

Definition at line 1314 of file mathlib.F.

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◆ expint()

elemental impure real(dp) function, public mathlib::expint	(	integer, intent(in)	n,
		real(dp), intent(in)	x
	)

computes the exponential integral En(x) = Int(exp(-x*t)/t^n,t=1..infinity) x>0, n=0,1,.. Note: Ei(-x) = -E1(x)

Parameters

n	...
x	...

Returns: ...

History: 05.2007 Created

Author: Manuel Guidon (adapted from Numerical recipies)

Definition at line 1392 of file mathlib.F.

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◆ jacobi()

subroutine, public mathlib::jacobi	(	real(kind=dp), dimension(:, :), intent(inout)	a,
		real(kind=dp), dimension(:), intent(out)	d,
		real(kind=dp), dimension(:, :), intent(out)	v
	)

Jacobi matrix diagonalization. The eigenvalues are returned in vector d and the eigenvectors are returned in matrix v in ascending order.

Parameters

a	...
d	...
v	...

History

Creation (20.11.98, Matthias Krack)

Definition at line 1467 of file mathlib.F.

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◆ diag()

subroutine, public mathlib::diag	(	integer, intent(in)	n,
		real(kind=dp), dimension(:, :), intent(inout)	a,
		real(kind=dp), dimension(:), intent(out)	d,
		real(kind=dp), dimension(:, :), intent(out)	v
	)

Diagonalize matrix a. The eigenvalues are returned in vector d and the eigenvectors are returned in matrix v.

Parameters

n	matrix/vector extent (problem size)
a	matrix to be diagonalised
d	vector of eigenvalues
v	matrix of eigenvectors

History

Creation (20.11.98, Matthias Krack)

Definition at line 1495 of file mathlib.F.

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◆ erfc_cutoff()

subroutine, public mathlib::erfc_cutoff	(	real(dp), intent(in)	eps,
		real(dp), intent(in)	omg,
		real(dp), intent(out)	r_cutoff
	)

compute a truncation radius for the shortrange operator

Parameters

eps	target accuracy!>
omg	screening parameter
omg	...
r_cutoff	cutoff radius

History: 10.2012 created [Hossein Banihashemian] 05.2019 moved here from hfx_types (A. Bussy)

Author: Hossein Banihashemian

Definition at line 1685 of file mathlib.F.

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◆ diag_complex()

subroutine, public mathlib::diag_complex	(	complex(kind=dp), dimension(:, :), intent(in)	matrix,
		complex(kind=dp), dimension(:, :), intent(out)	eigenvectors,
		real(kind=dp), dimension(:), intent(out)	eigenvalues
	)

Diagonalizes a local complex Hermitian matrix using LAPACK. Based on cp_cfm_heevd.

Parameters

matrix	Hermitian matrix is preserved
eigenvectors	...
eigenvalues	...

Author: A. Bussy

Definition at line 1742 of file mathlib.F.

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◆ diag_antisym()

subroutine, public mathlib::diag_antisym	(	real(dp), dimension(:, :)	matrix,
		complex(dp), dimension(:, :)	evecs,
		complex(dp), dimension(:)	evals
	)

Helper routine for diagonalizing anti symmetric matrices.

Parameters

matrix	...
evecs	...
evals	...

Definition at line 1794 of file mathlib.F.

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Data Types

Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ pswitch()

◆ abnormal_value()

◆ angle()

◆ binomial()

◆ binomial_gen()

◆ multinomial()

◆ build_rotmat()

◆ diamat_all()

◆ dihedral_angle()

◆ get_diag()

◆ inv_3x3()

◆ invmat()

◆ invmat_symm()

◆ get_pseudo_inverse_svd()

◆ get_pseudo_inverse_diag()

◆ reflect_vector()

◆ rotate_vector()

◆ symmetrize_matrix()

◆ vector_product()

◆ gcd()

◆ lcm()

◆ expint()

◆ jacobi()

◆ diag()

◆ erfc_cutoff()

◆ diag_complex()

◆ diag_antisym()