DAMASK_EICMD/src/quaternions.f90

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!---------------------------------------------------------------------------------------------------
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
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!> @author Philip Eisenlohr, Michigan State University
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!> @brief general quaternion math, not limited to unit quaternions
!> @details w is the real part, (x, y, z) are the imaginary parts.
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!> @details https://en.wikipedia.org/wiki/Quaternion
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!---------------------------------------------------------------------------------------------------
module quaternions
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use prec
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use IO
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implicit none
public
real(pReal), parameter, public :: P = -1.0_pReal !< parameter for orientation conversion.
type, public :: quaternion
real(pReal), private :: w = 0.0_pReal
real(pReal), private :: x = 0.0_pReal
real(pReal), private :: y = 0.0_pReal
real(pReal), private :: z = 0.0_pReal
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contains
procedure, private :: add__
procedure, private :: pos__
generic, public :: operator(+) => add__,pos__
procedure, private :: sub__
procedure, private :: neg__
generic, public :: operator(-) => sub__,neg__
procedure, private :: mul_quat__
procedure, private :: mul_scal__
generic, public :: operator(*) => mul_quat__, mul_scal__
procedure, private :: div_quat__
procedure, private :: div_scal__
generic, public :: operator(/) => div_quat__, div_scal__
procedure, private :: eq__
generic, public :: operator(==) => eq__
procedure, private :: neq__
generic, public :: operator(/=) => neq__
procedure, private :: pow_quat__
procedure, private :: pow_scal__
generic, public :: operator(**) => pow_quat__, pow_scal__
procedure, public :: abs => abs__
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procedure, public :: conjg => conjg__
procedure, public :: real => real__
procedure, public :: aimag => aimag__
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procedure, public :: homomorphed
procedure, public :: asArray
procedure, public :: inverse
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end type
interface assignment (=)
module procedure assign_quat__
module procedure assign_vec__
end interface assignment (=)
interface quaternion
module procedure init__
end interface quaternion
interface abs
procedure abs__
end interface abs
interface dot_product
procedure dot_product__
end interface dot_product
interface conjg
module procedure conjg__
end interface conjg
interface exp
module procedure exp__
end interface exp
interface log
module procedure log__
end interface log
interface real
module procedure real__
end interface real
interface aimag
module procedure aimag__
end interface aimag
private :: &
unitTest
contains
!--------------------------------------------------------------------------------------------------
!> @brief do self test
!--------------------------------------------------------------------------------------------------
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subroutine quaternions_init
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write(6,'(/,a)') ' <<<+- quaternions init -+>>>'; flush(6)
call unitTest
end subroutine quaternions_init
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!---------------------------------------------------------------------------------------------------
!> construct a quaternion from a 4-vector
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!---------------------------------------------------------------------------------------------------
type(quaternion) pure function init__(array)
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real(pReal), intent(in), dimension(4) :: array
init__%w = array(1)
init__%x = array(2)
init__%y = array(3)
init__%z = array(4)
end function init__
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!---------------------------------------------------------------------------------------------------
!> assign a quaternion
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!---------------------------------------------------------------------------------------------------
elemental pure subroutine assign_quat__(self,other)
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type(quaternion), intent(out) :: self
type(quaternion), intent(in) :: other
self = [other%w,other%x,other%y,other%z]
end subroutine assign_quat__
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!---------------------------------------------------------------------------------------------------
!> assign a 4-vector
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!---------------------------------------------------------------------------------------------------
pure subroutine assign_vec__(self,other)
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type(quaternion), intent(out) :: self
real(pReal), intent(in), dimension(4) :: other
self%w = other(1)
self%x = other(2)
self%y = other(3)
self%z = other(4)
end subroutine assign_vec__
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!---------------------------------------------------------------------------------------------------
!> add a quaternion
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function add__(self,other)
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class(quaternion), intent(in) :: self,other
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add__ = [ self%w, self%x, self%y ,self%z] &
+ [other%w, other%x, other%y,other%z]
end function add__
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!---------------------------------------------------------------------------------------------------
!> return (unary positive operator)
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function pos__(self)
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class(quaternion), intent(in) :: self
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pos__ = self * (+1.0_pReal)
end function pos__
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!---------------------------------------------------------------------------------------------------
!> subtract a quaternion
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function sub__(self,other)
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class(quaternion), intent(in) :: self,other
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sub__ = [ self%w, self%x, self%y ,self%z] &
- [other%w, other%x, other%y,other%z]
end function sub__
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!---------------------------------------------------------------------------------------------------
!> negate (unary negative operator)
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function neg__(self)
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class(quaternion), intent(in) :: self
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neg__ = self * (-1.0_pReal)
end function neg__
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!---------------------------------------------------------------------------------------------------
!> multiply with a quaternion
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function mul_quat__(self,other)
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class(quaternion), intent(in) :: self, other
mul_quat__%w = self%w*other%w - self%x*other%x - self%y*other%y - self%z*other%z
mul_quat__%x = self%w*other%x + self%x*other%w + P * (self%y*other%z - self%z*other%y)
mul_quat__%y = self%w*other%y + self%y*other%w + P * (self%z*other%x - self%x*other%z)
mul_quat__%z = self%w*other%z + self%z*other%w + P * (self%x*other%y - self%y*other%x)
end function mul_quat__
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!---------------------------------------------------------------------------------------------------
!> multiply with a scalar
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function mul_scal__(self,scal)
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class(quaternion), intent(in) :: self
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real(pReal), intent(in) :: scal
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mul_scal__ = [self%w,self%x,self%y,self%z]*scal
end function mul_scal__
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!---------------------------------------------------------------------------------------------------
!> divide by a quaternion
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function div_quat__(self,other)
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class(quaternion), intent(in) :: self, other
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div_quat__ = self * (conjg(other)/(abs(other)**2.0_pReal))
end function div_quat__
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!---------------------------------------------------------------------------------------------------
!> divide by a scalar
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function div_scal__(self,scal)
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class(quaternion), intent(in) :: self
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real(pReal), intent(in) :: scal
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div_scal__ = [self%w,self%x,self%y,self%z]/scal
end function div_scal__
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!---------------------------------------------------------------------------------------------------
!> test equality
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!---------------------------------------------------------------------------------------------------
logical elemental pure function eq__(self,other)
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class(quaternion), intent(in) :: self,other
eq__ = all(dEq([ self%w, self%x, self%y, self%z], &
[other%w,other%x,other%y,other%z]))
end function eq__
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!---------------------------------------------------------------------------------------------------
!> test inequality
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!---------------------------------------------------------------------------------------------------
logical elemental pure function neq__(self,other)
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class(quaternion), intent(in) :: self,other
neq__ = .not. self%eq__(other)
end function neq__
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!---------------------------------------------------------------------------------------------------
!> raise to the power of a quaternion
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function pow_quat__(self,expon)
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class(quaternion), intent(in) :: self
type(quaternion), intent(in) :: expon
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pow_quat__ = exp(log(self)*expon)
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end function pow_quat__
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!---------------------------------------------------------------------------------------------------
!> raise to the power of a scalar
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function pow_scal__(self,expon)
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class(quaternion), intent(in) :: self
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real(pReal), intent(in) :: expon
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pow_scal__ = exp(log(self)*expon)
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end function pow_scal__
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!---------------------------------------------------------------------------------------------------
!> take exponential
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function exp__(a)
class(quaternion), intent(in) :: a
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real(pReal) :: absImag
absImag = norm2(aimag(a))
exp__ = merge(exp(a%w) * [ cos(absImag), &
a%x/absImag * sin(absImag), &
a%y/absImag * sin(absImag), &
a%z/absImag * sin(absImag)], &
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IEEE_value(1.0_pReal,IEEE_SIGNALING_NAN), &
dNeq0(absImag))
end function exp__
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!---------------------------------------------------------------------------------------------------
!> take logarithm
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function log__(a)
class(quaternion), intent(in) :: a
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real(pReal) :: absImag
absImag = norm2(aimag(a))
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log__ = merge([log(abs(a)), &
a%x/absImag * acos(a%w/abs(a)), &
a%y/absImag * acos(a%w/abs(a)), &
a%z/absImag * acos(a%w/abs(a))], &
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IEEE_value(1.0_pReal,IEEE_SIGNALING_NAN), &
dNeq0(absImag))
end function log__
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!---------------------------------------------------------------------------------------------------
!> return norm
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!---------------------------------------------------------------------------------------------------
real(pReal) elemental pure function abs__(self)
class(quaternion), intent(in) :: self
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abs__ = norm2([self%w,self%x,self%y,self%z])
end function abs__
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!---------------------------------------------------------------------------------------------------
!> calculate dot product
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!---------------------------------------------------------------------------------------------------
real(pReal) elemental pure function dot_product__(a,b)
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class(quaternion), intent(in) :: a,b
dot_product__ = a%w*b%w + a%x*b%x + a%y*b%y + a%z*b%z
end function dot_product__
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!---------------------------------------------------------------------------------------------------
!> take conjugate complex
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!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function conjg__(self)
class(quaternion), intent(in) :: self
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conjg__ = [self%w,-self%x,-self%y,-self%z]
end function conjg__
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!---------------------------------------------------------------------------------------------------
!> homomorph
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!---------------------------------------------------------------------------------------------------
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type(quaternion) elemental pure function homomorphed(self)
class(quaternion), intent(in) :: self
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homomorphed = - self
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end function homomorphed
!---------------------------------------------------------------------------------------------------
!> return as plain array
!---------------------------------------------------------------------------------------------------
pure function asArray(self)
real(pReal), dimension(4) :: asArray
class(quaternion), intent(in) :: self
asArray = [self%w,self%x,self%y,self%z]
end function asArray
!---------------------------------------------------------------------------------------------------
!> real part (scalar)
!---------------------------------------------------------------------------------------------------
pure function real__(self)
real(pReal) :: real__
class(quaternion), intent(in) :: self
real__ = self%w
end function real__
!---------------------------------------------------------------------------------------------------
!> imaginary part (3-vector)
!---------------------------------------------------------------------------------------------------
pure function aimag__(self)
real(pReal), dimension(3) :: aimag__
class(quaternion), intent(in) :: self
aimag__ = [self%x,self%y,self%z]
end function aimag__
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!---------------------------------------------------------------------------------------------------
!> inverse
!---------------------------------------------------------------------------------------------------
type(quaternion) elemental pure function inverse(self)
class(quaternion), intent(in) :: self
inverse = conjg(self)/abs(self)**2.0_pReal
end function inverse
!--------------------------------------------------------------------------------------------------
!> @brief check correctness of (some) quaternions functions
!--------------------------------------------------------------------------------------------------
subroutine unitTest
real(pReal), dimension(4) :: qu
type(quaternion) :: q, q_2
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call random_number(qu)
qu = (qu-0.5_pReal) * 2.0_pReal
q = quaternion(qu)
q_2= qu
if(any(dNeq(q%asArray(),q_2%asArray()))) call IO_error(0,ext_msg='assign_vec__')
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q_2 = q + q
if(any(dNeq(q_2%asArray(),2.0_pReal*qu))) call IO_error(0,ext_msg='add__')
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q_2 = q - q
if(any(dNeq0(q_2%asArray()))) call IO_error(0,ext_msg='sub__')
q_2 = q * 5.0_pReal
if(any(dNeq(q_2%asArray(),5.0_pReal*qu))) call IO_error(0,ext_msg='mul__')
q_2 = q / 0.5_pReal
if(any(dNeq(q_2%asArray(),2.0_pReal*qu))) call IO_error(0,ext_msg='div__')
q_2 = q * 0.3_pReal
if(dNeq0(abs(q)) .and. q_2 == q) call IO_error(0,ext_msg='eq__')
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q_2 = q
if(q_2 /= q) call IO_error(0,ext_msg='neq__')
if(dNeq(abs(q),norm2(qu))) call IO_error(0,ext_msg='abs__')
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if(dNeq(abs(q)**2.0_pReal, real(q*q%conjg()),1.0e-14_pReal)) &
call IO_error(0,ext_msg='abs__/*conjg')
if(any(dNeq(q%asArray(),qu))) call IO_error(0,ext_msg='eq__')
if(dNeq(q%real(), qu(1))) call IO_error(0,ext_msg='real()')
if(any(dNeq(q%aimag(), qu(2:4)))) call IO_error(0,ext_msg='aimag()')
q_2 = q%homomorphed()
if(q /= q_2* (-1.0_pReal)) call IO_error(0,ext_msg='homomorphed')
if(dNeq(q_2%real(), qu(1)* (-1.0_pReal))) call IO_error(0,ext_msg='homomorphed/real')
if(any(dNeq(q_2%aimag(),qu(2:4)*(-1.0_pReal)))) call IO_error(0,ext_msg='homomorphed/aimag')
q_2 = conjg(q)
if(dNeq(abs(q),abs(q_2))) call IO_error(0,ext_msg='conjg/abs')
if(q /= conjg(q_2)) call IO_error(0,ext_msg='conjg/involution')
if(dNeq(q_2%real(), q%real())) call IO_error(0,ext_msg='conjg/real')
if(any(dNeq(q_2%aimag(),q%aimag()*(-1.0_pReal)))) call IO_error(0,ext_msg='conjg/aimag')
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if(abs(q) > 0.0_pReal) then
q_2 = q * q%inverse()
if( dNeq(real(q_2), 1.0_pReal,1.0e-15_pReal)) call IO_error(0,ext_msg='inverse/real')
if(any(dNeq0(aimag(q_2), 1.0e-15_pReal))) call IO_error(0,ext_msg='inverse/aimag')
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q_2 = q/abs(q)
q_2 = conjg(q_2) - inverse(q_2)
if(any(dNeq0(q_2%asArray(),1.0e-15_pReal))) call IO_error(0,ext_msg='inverse/conjg')
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endif
#if !(defined(__GFORTRAN__) && __GNUC__ < 9)
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if (norm2(aimag(q)) > 0.0_pReal) then
if (dNeq0(abs(q-exp(log(q))),1.0e-13_pReal)) call IO_error(0,ext_msg='exp/log')
if (dNeq0(abs(q-log(exp(q))),1.0e-13_pReal)) call IO_error(0,ext_msg='log/exp')
endif
#endif
end subroutine unitTest
end module quaternions