Algorithm precise and with no bias in the error

Author: weslleyspereira

SRC/clartg.f90

@@ -117,7 +117,7 @@
 subroutine CLARTG( f, g, c, s, r )
    use LA_CONSTANTS, &
    only: wp=>sp, zero=>szero, one=>sone, two=>stwo, czero, &
-         rtmin=>srtmin, rtmax=>srtmax, safmin=>ssafmin, safmax=>ssafmax
+         safmin=>ssafmin, safmax=>ssafmax
 !
 !  -- LAPACK auxiliary routine (version 3.10.0) --
 !  -- LAPACK is a software package provided by Univ. of Tennessee,    --
@@ -129,7 +129,7 @@ subroutine CLARTG( f, g, c, s, r )
    complex(wp)        f, g, r, s
 !  ..
 !  .. Local Scalars ..
-   real(wp) :: d, f1, f2, g1, g2, h2, u, v, w
+   real(wp) :: d, f1, f2, g1, g2, h2, u, v, w, rtmin, rtmax
    complex(wp) :: fs, gs, t
 !  ..
 !  .. Intrinsic Functions ..
@@ -141,6 +141,9 @@ subroutine CLARTG( f, g, c, s, r )
 !  .. Statement Function definitions ..
    ABSSQ( t ) = real( t )**2 + aimag( t )**2
 !  ..
+!  .. Constants ..
+   rtmin = sqrt( safmin )
+!  ..
 !  .. Executable Statements ..
 !
    if( g == czero ) then
@@ -150,6 +153,7 @@ subroutine CLARTG( f, g, c, s, r )
    else if( f == czero ) then
       c = zero
       g1 = max( abs(real(g)), abs(aimag(g)) )
+      rtmax = sqrt( safmax/2 )
       if( g1 > rtmin .and. g1 < rtmax ) then
 !
 !        Use unscaled algorithm
@@ -170,6 +174,7 @@ subroutine CLARTG( f, g, c, s, r )
    else
       f1 = max( abs(real(f)), abs(aimag(f)) )
       g1 = max( abs(real(g)), abs(aimag(g)) )
+      rtmax = sqrt( safmax/4 )
       if( f1 > rtmin .and. f1 < rtmax .and. &
           g1 > rtmin .and. g1 < rtmax ) then
 !
@@ -178,14 +183,36 @@ subroutine CLARTG( f, g, c, s, r )
          f2 = ABSSQ( f )
          g2 = ABSSQ( g )
          h2 = f2 + g2
-         if( f2 > rtmin .and. h2 < rtmax ) then
-            d = sqrt( f2*h2 )
+         ! safmin <= f2 <= h2 <= safmax 
+         if( f2 >= h2 * safmin ) then
+            ! safmin <= f2/h2 <= 1, and h2/f2 is finite
+            c = sqrt( f2 / h2 )
+            r = f / c
+            rtmax = rtmax * 2
+            if( f2 > rtmin .and. h2 < rtmax ) then
+               ! safmin <= sqrt( f2*h2 ) <= safmax
+               s = conjg( g ) * ( f / sqrt( f2*h2 ) )
+            else
+               s = conjg( g ) * ( r / h2 )
+            end if
          else
-            d = sqrt( f2 )*sqrt( h2 )
+            ! f2/h2 <= safmin may be subnormal, and h2/f2 may overflow.
+            ! Moreover,
+            !  safmin <= f2*f2 * safmax < f2 * h2 < h2*h2 * safmin <= safmax,
+            !  sqrt(safmin) <= sqrt(f2 * h2) <= sqrt(safmax).
+            ! Also,
+            !  g2 >> f2, which means that h2 = g2.
+            d = sqrt( f2 * h2 )
+            c = f2 / d
+            if( c >= safmin ) then
+               r = f / c
+            else
+               ! f2 / sqrt(f2 * h2) < safmin, then
+               !  h2 / sqrt(f2 * h2) <= h2 * (safmin / f2) <= h2 <= safmax
+               r = f * ( h2 / d )
+            end if
+            s = conjg( g ) * ( f / d )
          end if
-         c = f2 / d
-         s = conjg( g )*( f / d )
-         r = f*( h2 / d )
       else
 !
 !        Use scaled algorithm
@@ -212,14 +239,39 @@ subroutine CLARTG( f, g, c, s, r )
             f2 = ABSSQ( fs )
             h2 = f2 + g2
          end if
-         if( f2 > rtmin .and. h2 < rtmax ) then
-            d = sqrt( f2*h2 )
+         ! safmin <= f2 <= h2 <= safmax 
+         if( f2 >= h2 * safmin ) then
+            ! safmin <= f2/h2 <= 1, and h2/f2 is finite
+            c = sqrt( f2 / h2 )
+            r = fs / c
+            rtmax = rtmax * 2
+            if( f2 > rtmin .and. h2 < rtmax ) then
+               ! safmin <= sqrt( f2*h2 ) <= safmax
+               s = conjg( gs ) * ( fs / sqrt( f2*h2 ) )
+            else
+               s = conjg( gs ) * ( r / h2 )
+            end if
          else
-            d = sqrt( f2 )*sqrt( h2 )
+            ! f2/h2 <= safmin may be subnormal, and h2/f2 may overflow.
+            ! Moreover,
+            !  safmin <= f2*f2 * safmax < f2 * h2 < h2*h2 * safmin <= safmax,
+            !  sqrt(safmin) <= sqrt(f2 * h2) <= sqrt(safmax).
+            ! Also,
+            !  g2 >> f2, which means that h2 = g2.
+            d = sqrt( f2 * h2 )
+            c = f2 / d
+            if( c >= safmin ) then
+               r = fs / c
+            else
+               ! f2 / sqrt(f2 * h2) < safmin, then
+               !  h2 / sqrt(f2 * h2) <= h2 * (safmin / f2) <= h2 <= safmax
+               r = fs * ( h2 / d )
+            end if
+            s = conjg( gs ) * ( fs / d )
          end if
-         c = ( f2 / d )*w
-         s = conjg( gs )*( fs / d )
-         r = ( fs*( h2 / d ) )*u
+         ! Rescale c and r
+         c = c * w
+         r = r * u
       end if
    end if
    return