1!> \brief \b DZNRM2
2!
3!  =========== DOCUMENTATION ===========
4!
5! Online html documentation available at
6!            http://www.netlib.org/lapack/explore-html/
7!
8!  Definition:
9!  ===========
10!
11!       DOUBLE PRECISION FUNCTION DZNRM2(N,X,INCX)
12!
13!       .. Scalar Arguments ..
14!       INTEGER INCX,N
15!       ..
16!       .. Array Arguments ..
17!       DOUBLE COMPLEX X(*)
18!       ..
19!
20!
21!> \par Purpose:
22!  =============
23!>
24!> \verbatim
25!>
26!> DZNRM2 returns the euclidean norm of a vector via the function
27!> name, so that
28!>
29!>    DZNRM2 := sqrt( x**H*x )
30!> \endverbatim
31!
32!  Arguments:
33!  ==========
34!
35!> \param[in] N
36!> \verbatim
37!>          N is INTEGER
38!>         number of elements in input vector(s)
39!> \endverbatim
40!>
41!> \param[in] X
42!> \verbatim
43!>          X is COMPLEX*16 array, dimension (N)
44!>         complex vector with N elements
45!> \endverbatim
46!>
47!> \param[in] INCX
48!> \verbatim
49!>          INCX is INTEGER, storage spacing between elements of X
50!>          If INCX > 0, X(1+(i-1)*INCX) = x(i) for 1 <= i <= n
51!>          If INCX < 0, X(1-(n-i)*INCX) = x(i) for 1 <= i <= n
52!>          If INCX = 0, x isn't a vector so there is no need to call
53!>          this subroutine.  If you call it anyway, it will count x(1)
54!>          in the vector norm N times.
55!> \endverbatim
56!
57!  Authors:
58!  ========
59!
60!> \author Edward Anderson, Lockheed Martin
61!
62!> \date August 2016
63!
64!> \ingroup single_blas_level1
65!
66!> \par Contributors:
67!  ==================
68!>
69!> Weslley Pereira, University of Colorado Denver, USA
70!
71!> \par Further Details:
72!  =====================
73!>
74!> \verbatim
75!>
76!>  Anderson E. (2017)
77!>  Algorithm 978: Safe Scaling in the Level 1 BLAS
78!>  ACM Trans Math Softw 44:1--28
79!>  https://doi.org/10.1145/3061665
80!>
81!>  Blue, James L. (1978)
82!>  A Portable Fortran Program to Find the Euclidean Norm of a Vector
83!>  ACM Trans Math Softw 4:15--23
84!>  https://doi.org/10.1145/355769.355771
85!>
86!> \endverbatim
87!>
88!  =====================================================================
89function DZNRM2( n, x, incx )
90   integer, parameter :: wp = kind(1.d0)
91   real(wp) :: DZNRM2
92!
93!  -- Reference BLAS level1 routine (version 3.9.1) --
94!  -- Reference BLAS is a software package provided by Univ. of Tennessee,    --
95!  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
96!     March 2021
97!
98!  .. Constants ..
99   real(wp), parameter :: zero = 0.0_wp
100   real(wp), parameter :: one  = 1.0_wp
101   real(wp), parameter :: maxN = huge(0.0_wp)
102!  ..
103!  .. Blue's ccaling constants ..
104   real(wp), parameter :: tsml = real(radix(0._wp), wp)**ceiling( &
105       (minexponent(0._wp) - 1) * 0.5_wp)
106   real(wp), parameter :: tbig = real(radix(0._wp), wp)**floor( &
107       (maxexponent(0._wp) - digits(0._wp) + 1) * 0.5_wp)
108   real(wp), parameter :: ssml = real(radix(0._wp), wp)**( - floor( &
109       (minexponent(0._wp) - 1) * 0.5_wp))
110   real(wp), parameter :: sbig = real(radix(0._wp), wp)**( - ceiling( &
111       (maxexponent(0._wp) - digits(0._wp) + 1) * 0.5_wp))
112!  ..
113!  .. Scalar Arguments ..
114   integer :: incx, n
115!  ..
116!  .. Array Arguments ..
117   complex(wp) :: x(*)
118!  ..
119!  .. Local Scalars ..
120   integer :: i, ix
121   logical :: notbig
122   real(wp) :: abig, amed, asml, ax, scl, sumsq, ymax, ymin
123!
124!  Quick return if possible
125!
126   DZNRM2 = zero
127   if( n <= 0 ) return
128!
129   scl = one
130   sumsq = zero
131!
132!  Compute the sum of squares in 3 accumulators:
133!     abig -- sums of squares scaled down to avoid overflow
134!     asml -- sums of squares scaled up to avoid underflow
135!     amed -- sums of squares that do not require scaling
136!  The thresholds and multipliers are
137!     tbig -- values bigger than this are scaled down by sbig
138!     tsml -- values smaller than this are scaled up by ssml
139!
140   notbig = .true.
141   asml = zero
142   amed = zero
143   abig = zero
144   ix = 1
145   if( incx < 0 ) ix = 1 - (n-1)*incx
146   do i = 1, n
147      ax = abs(real(x(ix)))
148      if (ax > tbig) then
149         abig = abig + (ax*sbig)**2
150         notbig = .false.
151      else if (ax < tsml) then
152         if (notbig) asml = asml + (ax*ssml)**2
153      else
154         amed = amed + ax**2
155      end if
156      ax = abs(aimag(x(ix)))
157      if (ax > tbig) then
158         abig = abig + (ax*sbig)**2
159         notbig = .false.
160      else if (ax < tsml) then
161         if (notbig) asml = asml + (ax*ssml)**2
162      else
163         amed = amed + ax**2
164      end if
165      ix = ix + incx
166   end do
167!
168!  Combine abig and amed or amed and asml if more than one
169!  accumulator was used.
170!
171   if (abig > zero) then
172!
173!     Combine abig and amed if abig > 0.
174!
175      if ( (amed > zero) .or. (amed > maxN) .or. (amed /= amed) ) then
176         abig = abig + (amed*sbig)*sbig
177      end if
178      scl = one / sbig
179      sumsq = abig
180   else if (asml > zero) then
181!
182!     Combine amed and asml if asml > 0.
183!
184      if ( (amed > zero) .or. (amed > maxN) .or. (amed /= amed) ) then
185         amed = sqrt(amed)
186         asml = sqrt(asml) / ssml
187         if (asml > amed) then
188            ymin = amed
189            ymax = asml
190         else
191            ymin = asml
192            ymax = amed
193         end if
194         scl = one
195         sumsq = ymax**2*( one + (ymin/ymax)**2 )
196      else
197         scl = one / ssml
198         sumsq = asml
199      end if
200   else
201!
202!     Otherwise all values are mid-range
203!
204      scl = one
205      sumsq = amed
206   end if
207   DZNRM2 = scl*sqrt( sumsq )
208   return
209end function
210