ZLARTG generates a plane rotation so that


    [  C         S  ] . [ F ]  =  [ R ]


    [ -conjg(S)  C  ]   [ G ]     [ 0 ]


 where C is real and C**2 + |S|**2 = 1.


 The mathematical formulas used for C and S are


    sgn(x) = {  x / |x|,   x != 0


             {  1,         x  = 0


    R = sgn(F) * sqrt(|F|**2 + |G|**2)


    C = |F| / sqrt(|F|**2 + |G|**2)


    S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2)


 Special conditions:


    If G=0, then C=1 and S=0.


    If F=0, then C=0 and S is chosen so that R is real.


 When F and G are real, the formulas simplify to C = F/R and


 S = G/R, and the returned values of C, S, and R should be


 identical to those returned by DLARTG.


 The algorithm used to compute these quantities incorporates scaling


 to avoid overflow or underflow in computing the square root of the


 sum of squares.


 This is the same routine ZROTG fom BLAS1, except that


 F and G are unchanged on return.


 Below, wp=>dp stands for double precision from LA_CONSTANTS module.

F

          F is COMPLEX(wp)


          The first component of vector to be rotated.

G

          G is COMPLEX(wp)


          The second component of vector to be rotated.

C

          C is REAL(wp)


          The cosine of the rotation.

S

          S is COMPLEX(wp)


          The sine of the rotation.

R

          R is COMPLEX(wp)


          The nonzero component of the rotated vector.

Weslley Pereira, University of Colorado Denver, USA

December 2021

 Based on the algorithm from


  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