math_oct_m Module Reference

This module is intended to contain "only mathematical" functions and procedures. More...

Detailed Description

This module is intended to contain "only mathematical" functions and procedures.

Data Types
interface	diagonal_matrix
interface	interpolate
	This is the common interface to a simple-minded polynomical interpolation procedure (simple use of the classical formula of Lagrange). More...
interface	is_close
interface	log2
interface	lower_triangular_to_hermitian
interface	pad
interface	upper_triangular_to_hermitian

Functions/Subroutines
elemental logical function	dis_close_scalar (x, y, rtol, atol)
	Are \(x\) and \(y\) equal within a tolerance. More...
elemental logical function	zis_close_scalar (x, y, rtol, atol)
	Same as dis_close_scalar for complex numbers. More...
pure integer function, dimension(dim, dim)	idiagonal_matrix (dim, diag)
	Currently only returns a matrix whose diagonal elements are all the same. Note that the real and complex versions are in math_inc.F90. More...
recursive real(real64) function, public	hermite (n, x)
recursive integer function, public	factorial (n)
subroutine, public	ylmr_cmplx (xx, li, mi, ylm)
	Computes spherical harmonics ylm at position (x, y, z) More...
subroutine, public	ylmr_real (xx, li, mi, ylm)
	This is a Numerical Recipes-based subroutine computes real spherical harmonics ylm at position (x, y, z): ylm = c * plm( cos(theta) ) * sin(m*phi) for m < 0 ylm = c * plm( cos(theta) ) * cos(m*phi) for m >= 0 with (theta,phi) the polar angles of r, c a positive normalization. More...
subroutine, public	weights (N, M, cc, side)
	Compute the weights for finite-difference calculations: More...
real(real64) pure function, public	ddelta (i, j)
subroutine, public	make_idx_set (n, out, length, in)
	Construct out(1:length) = (/1, ..., n/) if in is not present, out(1:length) = in otherwise. More...
logical function, public	member (n, a)
	Considers a(1:ubound(a, 1)) as an integer set and checks if n is a member of it. More...
subroutine, public	interpolation_coefficients (nn, xa, xx, cc)
logical pure function, public	even (n)
	Returns if n is even. More...
logical pure function, public	odd (n)
	Returns if n is odd. More...
subroutine, public	cartesian2hyperspherical (x, u)
	Performs a transformation of an n-dimensional vector from Cartesian coordinates to hyperspherical coordinates. More...
subroutine, public	hyperspherical2cartesian (u, x)
	Performs the inverse transformation of cartesian2hyperspherical. More...
subroutine, public	hypersphere_grad_matrix (grad_matrix, r, x)
	Gives the hyperspherical gradient matrix, which contains the derivatives of the Cartesian coordinates with respect to the hyperspherical angles. More...
integer(int64) pure function	pad88 (size, blk)
integer(int64) pure function	pad48 (size, blk)
integer(int64) pure function	pad8 (size, blk)
integer pure function	pad4 (size, blk)
integer pure function, public	pad_pow2 (size)
	create array size, which is padded to powers of 2 More...
real(real64) pure function	dlog2 (xx)
integer pure function	ilog2 (xx)
integer(int64) pure function	llog2 (xx)
complex(real64) pure function, public	exponential (z)
	Wrapper for exponential. More...
complex(real64) pure function, public	phi1 (z)
	Compute phi1(z) = (exp(z)-1)/z. More...
complex(real64) pure function, public	phi2 (z)
	Compute phi2(z) = (phi1(z)-1)/z = (exp(z) - z - 1)/z^2. More...
real(real64) pure function, public	square_root (x)
	Wrapper for sqrt. More...
logical function, public	is_prime (n)
subroutine, public	generate_rotation_matrix (R, ff, tt)
	Generates a rotation matrix R to rotate a vector f to t. More...
subroutine, public	numder_ridders (x, h, res, err, f)
	Numerical derivative (Ridder`s algorithm). More...
pure complex(real64) function, dimension(1:3), public	dzcross_product (a, b)
pure complex(real64) function, dimension(1:3), public	zdcross_product (a, b)
subroutine, public	generalized_laguerre_polynomial (np, nn, mm, xx, cx)
subroutine	dupper_triangular_to_hermitian (nn, aa)
subroutine	zupper_triangular_to_hermitian (nn, aa)
subroutine	dlower_triangular_to_hermitian (nn, aa)
subroutine	zlower_triangular_to_hermitian (nn, aa)
subroutine, public	dsymmetrize_matrix (nn, aa)
subroutine, public	dzero_small_elements_matrix (nn, aa, tol)
logical pure function, public	is_diagonal (dim, matrix, tol)
	Returns true is the matrix of size dim x dim is diagonal, given a relative tolerance. More...
pure complex(real64) function, dimension(dim, dim)	zdiagonal_matrix (dim, diag)
	Currently only returns a matrix whose diagonal elements are all the same. Note that the integer version is in math.F90. More...
subroutine	zinterpolate_2 (xa, ya, x, y)
subroutine	zinterpolate_1 (xa, ya, x, y)
subroutine	zinterpolate_0 (xa, ya, x, y)
pure complex(real64) function, dimension(1:3), public	zcross_product (a, b)
pure real(real64) function, dimension(dim, dim)	ddiagonal_matrix (dim, diag)
	Currently only returns a matrix whose diagonal elements are all the same. Note that the integer version is in math.F90. More...
subroutine	dinterpolate_2 (xa, ya, x, y)
subroutine	dinterpolate_1 (xa, ya, x, y)
subroutine	dinterpolate_0 (xa, ya, x, y)
pure real(real64) function, dimension(1:3), public	dcross_product (a, b)

Function/Subroutine Documentation

dis_close_scalar()

elemental logical function math_oct_m::dis_close_scalar ( real(real64), intent(in)  x, real(real64), intent(in)  y, real(real64), intent(in), optional  rtol, real(real64), intent(in), optional  atol  )

private

Are \(x\) and \(y\) equal within a tolerance.

The function evaluates the expression:

\[ |x - y| \leq (atol * rtol) * |y| \]

The tolerance values are positive, typically very small numbers. The relative difference \((rtol * |y|)\) and the absolute difference \( atol \) are added together to compare against the absolute difference between \(x\) and \(y\). Default tolerances are based on numpy''s [implementation](https:

Parameters

[in]	x	Scalar.
[in]	y	Scalar.
[in]	rtol	Optional, relative tolerance.
[in]	atol	Optional, absolute tolerance.

Returns

is_close .true. if \(x\) and \(y\) are close.

Definition at line 221 of file math.F90.

zis_close_scalar()

elemental logical function math_oct_m::zis_close_scalar ( complex(real64), intent(in)  x, complex(real64), intent(in)  y, real(real64), intent(in), optional  rtol, real(real64), intent(in), optional  atol  )

private

Same as dis_close_scalar for complex numbers.

Definition at line 234 of file math.F90.

idiagonal_matrix()

pure integer function, dimension(dim, dim) math_oct_m::idiagonal_matrix ( integer, intent(in)  dim, integer, intent(in)  diag  )

private

Currently only returns a matrix whose diagonal elements are all the same. Note that the real and complex versions are in math_inc.F90.

Definition at line 250 of file math.F90.

hermite()

recursive real(real64) function, public math_oct_m::hermite	(	integer, intent(in)	n,
		real(real64), intent(in)	x
	)

Definition at line 265 of file math.F90.

factorial()

recursive integer function, public math_oct_m::factorial

(

integer, intent(in)

n

)

Definition at line 285 of file math.F90.

ylmr_cmplx()

subroutine, public math_oct_m::ylmr_cmplx	(	real(real64), dimension(3), intent(in)	xx,
		integer, intent(in)	li,
		integer, intent(in)	mi,
		complex(real64), intent(out)	ylm
	)

Computes spherical harmonics ylm at position (x, y, z)

Definition at line 301 of file math.F90.

ylmr_real()

subroutine, public math_oct_m::ylmr_real	(	real(real64), dimension(3), intent(in)	xx,
		integer, intent(in)	li,
		integer, intent(in)	mi,
		real(real64), intent(out)	ylm
	)

This is a Numerical Recipes-based subroutine computes real spherical harmonics ylm at position (x, y, z): ylm = c * plm( cos(theta) ) * sin(m*phi) for m < 0 ylm = c * plm( cos(theta) ) * cos(m*phi) for m >= 0 with (theta,phi) the polar angles of r, c a positive normalization.

Definition at line 372 of file math.F90.

weights()

subroutine, public math_oct_m::weights	(	integer, intent(in)	N,
		integer, intent(in)	M,
		real(real64), dimension(0:,0:,0:), intent(out)	cc,
		integer, intent(in), optional	side
	)

Compute the weights for finite-difference calculations:

N -> highest order of the derivative to be approximated M -> number of grid points to be used in the approximation.

c(j,k,i) -> ith order derivative at kth-order approximation j=0,k: the coefficients acting of each point

side -> -1 left-sided, +1 right-sided, 0 centered (default)

Parameters

[out]

cc

(0:M, 0:M, 0:N)

Definition at line 522 of file math.F90.

ddelta()

real(real64) pure function, public math_oct_m::ddelta	(	integer, intent(in)	i,
		integer, intent(in)	j
	)

Definition at line 598 of file math.F90.

make_idx_set()

subroutine, public math_oct_m::make_idx_set	(	integer, intent(in)	n,
		integer, dimension(:), intent(out), allocatable	out,
		integer, intent(out)	length,
		integer, dimension(:), intent(in), optional	in
	)

Construct out(1:length) = (/1, ..., n/) if in is not present, out(1:length) = in otherwise.

Definition at line 616 of file math.F90.

member()

logical function, public math_oct_m::member	(	integer, intent(in)	n,
		integer, dimension(:), intent(in)	a
	)

Considers a(1:ubound(a, 1)) as an integer set and checks if n is a member of it.

Definition at line 645 of file math.F90.

interpolation_coefficients()

subroutine, public math_oct_m::interpolation_coefficients	(	integer, intent(in)	nn,
		real(real64), dimension(:), intent(in)	xa,
		real(real64), intent(in)	xx,
		real(real64), dimension(:), intent(out)	cc
	)

Parameters

[in]	nn	the number of points and coefficients
[in]	xa	the nn points where we know the function
[in]	xx	the point where we want the function
[out]	cc	the coefficients

Definition at line 666 of file math.F90.

even()

logical pure function, public math_oct_m::even

(

integer, intent(in)

n

)

Returns if n is even.

Definition at line 689 of file math.F90.

odd()

logical pure function, public math_oct_m::odd

(

integer, intent(in)

n

)

Returns if n is odd.

Definition at line 699 of file math.F90.

cartesian2hyperspherical()

subroutine, public math_oct_m::cartesian2hyperspherical	(	real(real64), dimension(:), intent(in)	x,
		real(real64), dimension(:), intent(out)	u
	)

Performs a transformation of an n-dimensional vector from Cartesian coordinates to hyperspherical coordinates.

Definition at line 709 of file math.F90.

hyperspherical2cartesian()

subroutine, public math_oct_m::hyperspherical2cartesian	(	real(real64), dimension(:), intent(in)	u,
		real(real64), dimension(:), intent(out)	x
	)

Performs the inverse transformation of cartesian2hyperspherical.

Definition at line 756 of file math.F90.

hypersphere_grad_matrix()

subroutine, public math_oct_m::hypersphere_grad_matrix	(	real(real64), dimension(:,:), intent(out)	grad_matrix,
		real(real64), intent(in)	r,
		real(real64), dimension(:), intent(in)	x
	)

Gives the hyperspherical gradient matrix, which contains the derivatives of the Cartesian coordinates with respect to the hyperspherical angles.

Parameters

[in]	r	radius of hypersphere
[in]	x	array of hyperspherical angles

Definition at line 802 of file math.F90.

pad88()

integer(int64) pure function math_oct_m::pad88 ( integer(int64), intent(in)  size, integer(int64), intent(in)  blk  )

private

Definition at line 842 of file math.F90.

pad48()

integer(int64) pure function math_oct_m::pad48 ( integer, intent(in)  size, integer(int64), intent(in)  blk  )

private

Definition at line 857 of file math.F90.

pad8()

integer(int64) pure function math_oct_m::pad8 ( integer(int64), intent(in)  size, integer, intent(in)  blk  )

private

Definition at line 865 of file math.F90.

pad4()

integer pure function math_oct_m::pad4 ( integer, intent(in)  size, integer, intent(in)  blk  )

private

Definition at line 873 of file math.F90.

pad_pow2()

integer pure function, public math_oct_m::pad_pow2

(

integer, intent(in)

size

)

create array size, which is padded to powers of 2

This helps alignment of data, and is necessary for some device hosted arrays

Definition at line 886 of file math.F90.

dlog2()

real(real64) pure function math_oct_m::dlog2 ( real(real64), intent(in)  xx )

private

Definition at line 915 of file math.F90.

ilog2()

integer pure function math_oct_m::ilog2 ( integer, intent(in)  xx )

private

Definition at line 923 of file math.F90.

llog2()

integer(int64) pure function math_oct_m::llog2 ( integer(int64), intent(in)  xx )

private

Definition at line 931 of file math.F90.

exponential()

complex(real64) pure function, public math_oct_m::exponential

(

complex(real64), intent(in)

z

)

Wrapper for exponential.

Definition at line 939 of file math.F90.

phi1()

complex(real64) pure function, public math_oct_m::phi1

(

complex(real64), intent(in)

z

)

Compute phi1(z) = (exp(z)-1)/z.

Use the power series \( \sum^{4}_{k=0} z^k/(k+1)! \) for small arguments to avoid round-off errors. The expression and the cut value are similar to those from [GSL](https:

Definition at line 951 of file math.F90.

phi2()

complex(real64) pure function, public math_oct_m::phi2

(

complex(real64), intent(in)

z

)

Compute phi2(z) = (phi1(z)-1)/z = (exp(z) - z - 1)/z^2.

Use the power series \( \sum^{4}_{k=0} z^k/(k+2)! \) for small arguments to avoid round-off errors. The expression and the cut value are similar to those from [GSL](https: (there is a difference by a factor of 2 between the two definitions).

Definition at line 970 of file math.F90.

square_root()

real(real64) pure function, public math_oct_m::square_root

(

real(real64), intent(in)

x

)

Wrapper for sqrt.

Definition at line 984 of file math.F90.

is_prime()

logical function, public math_oct_m::is_prime

(

integer, intent(in)

n

)

Definition at line 992 of file math.F90.

generate_rotation_matrix()

subroutine, public math_oct_m::generate_rotation_matrix	(	real(real64), dimension(:,:), intent(out)	R,
		real(real64), dimension(:), intent(in)	ff,
		real(real64), dimension(:), intent(in)	tt
	)

Generates a rotation matrix R to rotate a vector f to t.

T. M\U000000f6ller and J. F. Hughes, Journal of Graphics Tools 4, 1 (1999)

Definition at line 1027 of file math.F90.

numder_ridders()

subroutine, public math_oct_m::numder_ridders	(	real(real64), intent(in)	x,
		real(real64), intent(in)	h,
		real(real64), intent(out)	res,
		real(real64), intent(out)	err,
			f
	)

Numerical derivative (Ridder`s algorithm).

This is an alternative to "loct_numerical_derivative" (which is just an interface to the GSL numerical derivative). This version is an implementation of Ridders algorithm [C. J. F. Ridders, Adv. Eng. Software 4, 75 (1982); also described in Numerical Recipes]. It is more precise, but also typically more expensive, than the simpler 4-point algorithm implemented in the GSL library.

Definition at line 1171 of file math.F90.

dzcross_product()

pure complex(real64) function, dimension(1:3), public math_oct_m::dzcross_product	(	real(real64), dimension(:), intent(in)	a,
		complex(real64), dimension(:), intent(in)	b
	)

Parameters

[in]	a	(3)
[in]	b	(3)

Definition at line 1226 of file math.F90.

zdcross_product()

pure complex(real64) function, dimension(1:3), public math_oct_m::zdcross_product	(	complex(real64), dimension(:), intent(in)	a,
		real(real64), dimension(:), intent(in)	b
	)

Parameters

[in]	a	(3)
[in]	b	(3)

Definition at line 1239 of file math.F90.

generalized_laguerre_polynomial()

subroutine, public math_oct_m::generalized_laguerre_polynomial	(	integer, intent(in)	np,
		integer, intent(in)	nn,
		integer, intent(in)	mm,
		real(real64), dimension(np), intent(in)	xx,
		real(real64), dimension(np), intent(out)	cx
	)

Definition at line 1352 of file math.F90.

dupper_triangular_to_hermitian()

subroutine math_oct_m::dupper_triangular_to_hermitian ( integer, intent(in)  nn, real(real64), dimension(:, :), intent(inout)  aa  )

private

Definition at line 1390 of file math.F90.

zupper_triangular_to_hermitian()

subroutine math_oct_m::zupper_triangular_to_hermitian ( integer, intent(in)  nn, complex(real64), dimension(:, :), intent(inout)  aa  )

private

Definition at line 1403 of file math.F90.

dlower_triangular_to_hermitian()

subroutine math_oct_m::dlower_triangular_to_hermitian ( integer, intent(in)  nn, real(real64), dimension(:, :), intent(inout)  aa  )

private

Definition at line 1417 of file math.F90.

zlower_triangular_to_hermitian()

subroutine math_oct_m::zlower_triangular_to_hermitian ( integer, intent(in)  nn, complex(real64), dimension(:, :), intent(inout)  aa  )

private

Definition at line 1430 of file math.F90.

dsymmetrize_matrix()

subroutine, public math_oct_m::dsymmetrize_matrix	(	integer, intent(in)	nn,
		real(real64), dimension(:, :), intent(inout)	aa
	)

Definition at line 1445 of file math.F90.

dzero_small_elements_matrix()

subroutine, public math_oct_m::dzero_small_elements_matrix	(	integer, intent(in)	nn,
		real(real64), dimension(:, :), intent(inout)	aa,
		real(real64)	tol
	)

Definition at line 1459 of file math.F90.

is_diagonal()

logical pure function, public math_oct_m::is_diagonal	(	integer, intent(in)	dim,
		real(real64), dimension(:, :), intent(in)	matrix,
		real(real64), intent(in)	tol
	)

Returns true is the matrix of size dim x dim is diagonal, given a relative tolerance.

Definition at line 1471 of file math.F90.

zdiagonal_matrix()

pure complex(real64) function, dimension(dim, dim) math_oct_m::zdiagonal_matrix ( integer, intent(in)  dim, complex(real64), intent(in)  diag  )

private

Currently only returns a matrix whose diagonal elements are all the same. Note that the integer version is in math.F90.

Definition at line 1564 of file math.F90.

zinterpolate_2()

subroutine math_oct_m::zinterpolate_2 ( real(real64), dimension(:), intent(in)  xa, complex(real64), dimension(:, :, :), intent(in), contiguous  ya, real(real64), intent(in)  x, complex(real64), dimension(:, :), intent(out), contiguous  y  )

private

Definition at line 1579 of file math.F90.

zinterpolate_1()

subroutine math_oct_m::zinterpolate_1 ( real(real64), dimension(:), intent(in)  xa, complex(real64), dimension(:, :), intent(in)  ya, real(real64), intent(in)  x, complex(real64), dimension(:), intent(out)  y  )

private

Definition at line 1614 of file math.F90.

zinterpolate_0()

subroutine math_oct_m::zinterpolate_0 ( real(real64), dimension(:), intent(in)  xa, complex(real64), dimension(:), intent(in)  ya, real(real64), intent(in)  x, complex(real64), intent(out)  y  )

private

Definition at line 1642 of file math.F90.

zcross_product()

pure complex(real64) function, dimension(1:3), public math_oct_m::zcross_product	(	complex(real64), dimension(:), intent(in)	a,
		complex(real64), dimension(:), intent(in)	b
	)

Parameters

[in]	a	(3)
[in]	b	(3)

Definition at line 1667 of file math.F90.

ddiagonal_matrix()

pure real(real64) function, dimension(dim, dim) math_oct_m::ddiagonal_matrix ( integer, intent(in)  dim, real(real64), intent(in)  diag  )

private

Currently only returns a matrix whose diagonal elements are all the same. Note that the integer version is in math.F90.

Definition at line 1755 of file math.F90.

dinterpolate_2()

subroutine math_oct_m::dinterpolate_2 ( real(real64), dimension(:), intent(in)  xa, real(real64), dimension(:, :, :), intent(in), contiguous  ya, real(real64), intent(in)  x, real(real64), dimension(:, :), intent(out), contiguous  y  )

private

Definition at line 1770 of file math.F90.

dinterpolate_1()

subroutine math_oct_m::dinterpolate_1 ( real(real64), dimension(:), intent(in)  xa, real(real64), dimension(:, :), intent(in)  ya, real(real64), intent(in)  x, real(real64), dimension(:), intent(out)  y  )

private

Definition at line 1805 of file math.F90.

dinterpolate_0()

subroutine math_oct_m::dinterpolate_0 ( real(real64), dimension(:), intent(in)  xa, real(real64), dimension(:), intent(in)  ya, real(real64), intent(in)  x, real(real64), intent(out)  y  )

private

Definition at line 1833 of file math.F90.

dcross_product()

pure real(real64) function, dimension(1:3), public math_oct_m::dcross_product	(	real(real64), dimension(:), intent(in)	a,
		real(real64), dimension(:), intent(in)	b
	)

Parameters

[in]	a	(3)
[in]	b	(3)

Definition at line 1858 of file math.F90.