New algorithms for computing Givens rotations

BLAS/SRC/crotg.f90

@@ -122,7 +122,7 @@ subroutine CROTG( a, b, c, s )
  complex(wp) :: a, b, s
 ! ..
 ! .. Local Scalars ..
- real(wp) :: d, f1, f2, g1, g2, h2, p, u, uu, v, vv, w
+ real(wp) :: d, f1, f2, g1, g2, h2, u, v, w
  complex(wp) :: f, fs, g, gs, r, t
 ! ..
 ! .. Intrinsic Functions ..
@@ -149,19 +149,16 @@ subroutine CROTG( a, b, c, s )
 !
 ! Use unscaled algorithm
 !
- g2 = ABSSQ( g )
- d = sqrt( g2 )
+ d = abs( g )
  s = conjg( g ) / d
  r = d
  else
 !
 ! Use scaled algorithm
 !
  u = min( safmax, max( safmin, g1 ) )
- uu = one / u
- gs = g*uu
- g2 = ABSSQ( gs )
- d = sqrt( g2 )
+ gs = g / u
+ d = abs( gs )
  s = conjg( gs ) / d
  r = d*u
  end if
@@ -181,35 +178,32 @@ subroutine CROTG( a, b, c, s )
  else
  d = sqrt( f2 )*sqrt( h2 )
  end if
- p = 1 / d
- c = f2*p
- s = conjg( g )*( f*p )
- r = f*( h2*p )
+ c = f2 / d
+ s = conjg( g )*( f / d )
+ r = f*( h2 / d )
  else
 !
 ! Use scaled algorithm
 !
  u = min( safmax, max( safmin, f1, g1 ) )
- uu = one / u
- gs = g*uu
+ gs = g / u
  g2 = ABSSQ( gs )
- if( f1*uu < rtmin ) then
+ if( f1 / u < rtmin ) then
 !
 ! f is not well-scaled when scaled by g1.
 ! Use a different scaling for f.
 !
  v = min( safmax, max( safmin, f1 ) )
- vv = one / v
- w = v * uu
- fs = f*vv
+ w = v / u
+ fs = f / v
  f2 = ABSSQ( fs )
  h2 = f2*w**2 + g2
  else
 !
 ! Otherwise use the same scaling for f and g.
 !
  w = one
- fs = f*uu
+ fs = f / u
  f2 = ABSSQ( fs )
  h2 = f2 + g2
  end if
@@ -218,10 +212,9 @@ subroutine CROTG( a, b, c, s )
  else
  d = sqrt( f2 )*sqrt( h2 )
  end if
- p = 1 / d
- c = ( f2*p )*w
- s = conjg( gs )*( fs*p )
- r = ( fs*( h2*p ) )*u
+ c = ( f2 / d )*w
+ s = conjg( gs )*( fs / d )
+ r = ( fs*( h2 / d ) )*u
  end if
  end if
  a = r

BLAS/SRC/zrotg.f90

@@ -122,7 +122,7 @@ subroutine ZROTG( a, b, c, s )
  complex(wp) :: a, b, s
 ! ..
 ! .. Local Scalars ..
- real(wp) :: d, f1, f2, g1, g2, h2, p, u, uu, v, vv, w
+ real(wp) :: d, f1, f2, g1, g2, h2, u, v, w
  complex(wp) :: f, fs, g, gs, r, t
 ! ..
 ! .. Intrinsic Functions ..
@@ -149,19 +149,16 @@ subroutine ZROTG( a, b, c, s )
 !
 ! Use unscaled algorithm
 !
- g2 = ABSSQ( g )
- d = sqrt( g2 )
+ d = abs( g )
  s = conjg( g ) / d
  r = d
  else
 !
 ! Use scaled algorithm
 !
  u = min( safmax, max( safmin, g1 ) )
- uu = one / u
- gs = g*uu
- g2 = ABSSQ( gs )
- d = sqrt( g2 )
+ gs = g / u
+ d = abs( gs )
  s = conjg( gs ) / d
  r = d*u
  end if
@@ -181,35 +178,32 @@ subroutine ZROTG( a, b, c, s )
  else
  d = sqrt( f2 )*sqrt( h2 )
  end if
- p = 1 / d
- c = f2*p
- s = conjg( g )*( f*p )
- r = f*( h2*p )
+ c = f2 / d
+ s = conjg( g )*( f / d )
+ r = f*( h2 / d )
  else
 !
 ! Use scaled algorithm
 !
  u = min( safmax, max( safmin, f1, g1 ) )
- uu = one / u
- gs = g*uu
+ gs = g / u
  g2 = ABSSQ( gs )
- if( f1*uu < rtmin ) then
+ if( f1 / u < rtmin ) then
 !
 ! f is not well-scaled when scaled by g1.
 ! Use a different scaling for f.
 !
  v = min( safmax, max( safmin, f1 ) )
- vv = one / v
- w = v * uu
- fs = f*vv
+ w = v / u
+ fs = f / v
  f2 = ABSSQ( fs )
  h2 = f2*w**2 + g2
  else
 !
 ! Otherwise use the same scaling for f and g.
 !
  w = one
- fs = f*uu
+ fs = f / u
  f2 = ABSSQ( fs )
  h2 = f2 + g2
  end if
@@ -218,10 +212,9 @@ subroutine ZROTG( a, b, c, s )
  else
  d = sqrt( f2 )*sqrt( h2 )
  end if
- p = 1 / d
- c = ( f2*p )*w
- s = conjg( gs )*( fs*p )
- r = ( fs*( h2*p ) )*u
+ c = ( f2 / d )*w
+ s = conjg( gs )*( fs / d )
+ r = ( fs*( h2 / d ) )*u
  end if
  end if
  a = r

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, p, u, uu, v, vv, 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
@@ -150,29 +153,28 @@ 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
 !
- g2 = ABSSQ( g )
- d = sqrt( g2 )
+ d = abs( g )
  s = conjg( g ) / d
  r = d
  else
 !
 ! Use scaled algorithm
 !
  u = min( safmax, max( safmin, g1 ) )
- uu = one / u
- gs = g*uu
- g2 = ABSSQ( gs )
- d = sqrt( g2 )
+ gs = g / u
+ d = abs( gs )
  s = conjg( gs ) / d
  r = d*u
  end if
  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
 !
@@ -181,52 +183,95 @@ 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
- p = 1 / d
- c = f2*p
- s = conjg( g )*( f*p )
- r = f*( h2*p )
  else
 !
 ! Use scaled algorithm
 !
  u = min( safmax, max( safmin, f1, g1 ) )
- uu = one / u
- gs = g*uu
+ gs = g / u
  g2 = ABSSQ( gs )
- if( f1*uu < rtmin ) then
+ if( f1 / u < rtmin ) then
 !
 ! f is not well-scaled when scaled by g1.
 ! Use a different scaling for f.
 !
  v = min( safmax, max( safmin, f1 ) )
- vv = one / v
- w = v * uu
- fs = f*vv
+ w = v / u
+ fs = f / v
  f2 = ABSSQ( fs )
  h2 = f2*w**2 + g2
  else
 !
 ! Otherwise use the same scaling for f and g.
 !
  w = one
- fs = f*uu
+ fs = f / u
  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
- p = 1 / d
- c = ( f2*p )*w
- s = conjg( gs )*( fs*p )
- r = ( fs*( h2*p ) )*u
+ ! Rescale c and r
+ c = c * w
+ r = r * u
  end if
  end if
  return

SRC/dlartg.f90

@@ -123,7 +123,7 @@ subroutine DLARTG( f, g, c, s, r )
  real(wp) :: c, f, g, r, s
 ! ..
 ! .. Local Scalars ..
- real(wp) :: d, f1, fs, g1, gs, p, u, uu
+ real(wp) :: d, f1, fs, g1, gs, u
 ! ..
 ! .. Intrinsic Functions ..
  intrinsic :: abs, sign, sqrt
@@ -143,20 +143,18 @@ subroutine DLARTG( f, g, c, s, r )
  else if( f1 > rtmin .and. f1 < rtmax .and. &
  g1 > rtmin .and. g1 < rtmax ) then
  d = sqrt( f*f + g*g )
- p = one / d
- c = f1*p
- s = g*sign( p, f )
+ c = f1 / d
  r = sign( d, f )
+ s = g / r
  else
  u = min( safmax, max( safmin, f1, g1 ) )
- uu = one / u
- fs = f*uu
- gs = g*uu
+ fs = f / u
+ gs = g / u
  d = sqrt( fs*fs + gs*gs )
- p = one / d
- c = abs( fs )*p
- s = gs*sign( p, f )
- r = sign( d, f )*u
+ c = abs( fs ) / d
+ r = sign( d, f )
+ s = gs / r
+ r = r * u
  end if
  return
 end subroutine