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!> \brief \b DZNRM2
!
! =========== DOCUMENTATION ===========
!
! Online html documentation available at
! http://www.netlib.org/lapack/explore-html/
!
! Definition:
! ===========
!
! DOUBLE PRECISION FUNCTION DZNRM2(N,X,INCX)
!
! .. Scalar Arguments ..
! INTEGER INCX,N
! ..
! .. Array Arguments ..
! DOUBLE COMPLEX X(*)
! ..
!
!
!> \par Purpose:
! =============
!>
!> \verbatim
!>
!> DZNRM2 returns the euclidean norm of a vector via the function
!> name, so that
!>
!> DZNRM2 := sqrt( x**H*x )
!> \endverbatim
!
! Arguments:
! ==========
!
!> \param[in] N
!> \verbatim
!> N is INTEGER
!> number of elements in input vector(s)
!> \endverbatim
!>
!> \param[in] X
!> \verbatim
!> X is COMPLEX*16 array, dimension (N)
!> complex vector with N elements
!> \endverbatim
!>
!> \param[in] INCX
!> \verbatim
!> INCX is INTEGER, storage spacing between elements of X
!> If INCX > 0, X(1+(i-1)*INCX) = x(i) for 1 <= i <= n
!> If INCX < 0, X(1-(n-i)*INCX) = x(i) for 1 <= i <= n
!> If INCX = 0, x isn't a vector so there is no need to call
!> this subroutine. If you call it anyway, it will count x(1)
!> in the vector norm N times.
!> \endverbatim
!
! Authors:
! ========
!
!> \author Edward Anderson, Lockheed Martin
!
!> \date August 2016
!
!> \ingroup nrm2
!
!> \par Contributors:
! ==================
!>
!> Weslley Pereira, University of Colorado Denver, USA
!
!> \par Further Details:
! =====================
!>
!> \verbatim
!>
!> Anderson E. (2017)
!> Algorithm 978: Safe Scaling in the Level 1 BLAS
!> ACM Trans Math Softw 44:1--28
!> https://doi.org/10.1145/3061665
!>
!> Blue, James L. (1978)
!> A Portable Fortran Program to Find the Euclidean Norm of a Vector
!> ACM Trans Math Softw 4:15--23
!> https://doi.org/10.1145/355769.355771
!>
!> \endverbatim
!>
! =====================================================================
function DZNRM2( n, x, incx )
integer, parameter :: wp = kind(1.d0)
real(wp) :: DZNRM2
!
! -- Reference BLAS level1 routine (version 3.9.1) --
! -- Reference BLAS is a software package provided by Univ. of Tennessee, --
! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! March 2021
!
! .. Constants ..
real(wp), parameter :: zero = 0.0_wp
real(wp), parameter :: one = 1.0_wp
real(wp), parameter :: maxN = huge(0.0_wp)
! ..
! .. Blue's scaling constants ..
real(wp), parameter :: tsml = real(radix(0._wp), wp)**ceiling( &
(minexponent(0._wp) - 1) * 0.5_wp)
real(wp), parameter :: tbig = real(radix(0._wp), wp)**floor( &
(maxexponent(0._wp) - digits(0._wp) + 1) * 0.5_wp)
real(wp), parameter :: ssml = real(radix(0._wp), wp)**( - floor( &
(minexponent(0._wp) - digits(0._wp)) * 0.5_wp))
real(wp), parameter :: sbig = real(radix(0._wp), wp)**( - ceiling( &
(maxexponent(0._wp) + digits(0._wp) - 1) * 0.5_wp))
! ..
! .. Scalar Arguments ..
integer :: incx, n
! ..
! .. Array Arguments ..
complex(wp) :: x(*)
! ..
! .. Local Scalars ..
integer :: i, ix
logical :: notbig
real(wp) :: abig, amed, asml, ax, scl, sumsq, ymax, ymin
!
! Quick return if possible
!
DZNRM2 = zero
if( n <= 0 ) return
!
scl = one
sumsq = zero
!
! Compute the sum of squares in 3 accumulators:
! abig -- sums of squares scaled down to avoid overflow
! asml -- sums of squares scaled up to avoid underflow
! amed -- sums of squares that do not require scaling
! The thresholds and multipliers are
! tbig -- values bigger than this are scaled down by sbig
! tsml -- values smaller than this are scaled up by ssml
!
notbig = .true.
asml = zero
amed = zero
abig = zero
ix = 1
if( incx < 0 ) ix = 1 - (n-1)*incx
do i = 1, n
ax = abs(real(x(ix)))
if (ax > tbig) then
abig = abig + (ax*sbig)**2
notbig = .false.
else if (ax < tsml) then
if (notbig) asml = asml + (ax*ssml)**2
else
amed = amed + ax**2
end if
ax = abs(aimag(x(ix)))
if (ax > tbig) then
abig = abig + (ax*sbig)**2
notbig = .false.
else if (ax < tsml) then
if (notbig) asml = asml + (ax*ssml)**2
else
amed = amed + ax**2
end if
ix = ix + incx
end do
!
! Combine abig and amed or amed and asml if more than one
! accumulator was used.
!
if (abig > zero) then
!
! Combine abig and amed if abig > 0.
!
if ( (amed > zero) .or. (amed > maxN) .or. (amed /= amed) ) then
abig = abig + (amed*sbig)*sbig
end if
scl = one / sbig
sumsq = abig
else if (asml > zero) then
!
! Combine amed and asml if asml > 0.
!
if ( (amed > zero) .or. (amed > maxN) .or. (amed /= amed) ) then
amed = sqrt(amed)
asml = sqrt(asml) / ssml
if (asml > amed) then
ymin = amed
ymax = asml
else
ymin = asml
ymax = amed
end if
scl = one
sumsq = ymax**2*( one + (ymin/ymax)**2 )
else
scl = one / ssml
sumsq = asml
end if
else
!
! Otherwise all values are mid-range
!
scl = one
sumsq = amed
end if
DZNRM2 = scl*sqrt( sumsq )
return
end function
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