clartg.f.html
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SUBROUTINE <a name="CLARTG.1"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( F, G, CS, SN, R )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> REAL CS
COMPLEX F, G, R, SN
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="CLARTG.15"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a> generates a plane rotation so that
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> [ CS SN ] [ F ] [ R ]
</span><span class="comment">*</span><span class="comment"> [ __ ] . [ ] = [ ] where CS**2 + |SN|**2 = 1.
</span><span class="comment">*</span><span class="comment"> [ -SN CS ] [ G ] [ 0 ]
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is a faster version of the BLAS1 routine CROTG, except for
</span><span class="comment">*</span><span class="comment"> the following differences:
</span><span class="comment">*</span><span class="comment"> F and G are unchanged on return.
</span><span class="comment">*</span><span class="comment"> If G=0, then CS=1 and SN=0.
</span><span class="comment">*</span><span class="comment"> If F=0, then CS=0 and SN is chosen so that R is real.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> F (input) COMPLEX
</span><span class="comment">*</span><span class="comment"> The first component of vector to be rotated.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> G (input) COMPLEX
</span><span class="comment">*</span><span class="comment"> The second component of vector to be rotated.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> CS (output) REAL
</span><span class="comment">*</span><span class="comment"> The cosine of the rotation.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SN (output) COMPLEX
</span><span class="comment">*</span><span class="comment"> The sine of the rotation.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> R (output) COMPLEX
</span><span class="comment">*</span><span class="comment"> The nonzero component of the rotated vector.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ======= =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 3-5-96 - Modified with a new algorithm by W. Kahan and J. Demmel
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This version has a few statements commented out for thread safety
</span><span class="comment">*</span><span class="comment"> (machine parameters are computed on each entry). 10 feb 03, SJH.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> REAL TWO, ONE, ZERO
PARAMETER ( TWO = 2.0E+0, ONE = 1.0E+0, ZERO = 0.0E+0 )
COMPLEX CZERO
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span><span class="comment">*</span><span class="comment"> LOGICAL FIRST
</span> INTEGER COUNT, I
REAL D, DI, DR, EPS, F2, F2S, G2, G2S, SAFMIN,
$ SAFMN2, SAFMX2, SCALE
COMPLEX FF, FS, GS
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> REAL <a name="SLAMCH.69"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>, <a name="SLAPY2.69"></a><a href="slapy2.f.html#SLAPY2.1">SLAPY2</a>
EXTERNAL <a name="SLAMCH.70"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>, <a name="SLAPY2.70"></a><a href="slapy2.f.html#SLAPY2.1">SLAPY2</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC ABS, AIMAG, CMPLX, CONJG, INT, LOG, MAX, REAL,
$ SQRT
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Statement Functions ..
</span> REAL ABS1, ABSSQ
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Save statement ..
</span><span class="comment">*</span><span class="comment"> SAVE FIRST, SAFMX2, SAFMIN, SAFMN2
</span><span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Data statements ..
</span><span class="comment">*</span><span class="comment"> DATA FIRST / .TRUE. /
</span><span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Statement Function definitions ..
</span> ABS1( FF ) = MAX( ABS( REAL( FF ) ), ABS( AIMAG( FF ) ) )
ABSSQ( FF ) = REAL( FF )**2 + AIMAG( FF )**2
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IF( FIRST ) THEN
</span> SAFMIN = <a name="SLAMCH.92"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>( <span class="string">'S'</span> )
EPS = <a name="SLAMCH.93"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>( <span class="string">'E'</span> )
SAFMN2 = <a name="SLAMCH.94"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>( <span class="string">'B'</span> )**INT( LOG( SAFMIN / EPS ) /
$ LOG( <a name="SLAMCH.95"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>( <span class="string">'B'</span> ) ) / TWO )
SAFMX2 = ONE / SAFMN2
<span class="comment">*</span><span class="comment"> FIRST = .FALSE.
</span><span class="comment">*</span><span class="comment"> END IF
</span> SCALE = MAX( ABS1( F ), ABS1( G ) )
FS = F
GS = G
COUNT = 0
IF( SCALE.GE.SAFMX2 ) THEN
10 CONTINUE
COUNT = COUNT + 1
FS = FS*SAFMN2
GS = GS*SAFMN2
SCALE = SCALE*SAFMN2
IF( SCALE.GE.SAFMX2 )
$ GO TO 10
ELSE IF( SCALE.LE.SAFMN2 ) THEN
IF( G.EQ.CZERO ) THEN
CS = ONE
SN = CZERO
R = F
RETURN
END IF
20 CONTINUE
COUNT = COUNT - 1
FS = FS*SAFMX2
GS = GS*SAFMX2
SCALE = SCALE*SAFMX2
IF( SCALE.LE.SAFMN2 )
$ GO TO 20
END IF
F2 = ABSSQ( FS )
G2 = ABSSQ( GS )
IF( F2.LE.MAX( G2, ONE )*SAFMIN ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is a rare case: F is very small.
</span><span class="comment">*</span><span class="comment">
</span> IF( F.EQ.CZERO ) THEN
CS = ZERO
R = <a name="SLAPY2.134"></a><a href="slapy2.f.html#SLAPY2.1">SLAPY2</a>( REAL( G ), AIMAG( G ) )
<span class="comment">*</span><span class="comment"> Do complex/real division explicitly with two real divisions
</span> D = <a name="SLAPY2.136"></a><a href="slapy2.f.html#SLAPY2.1">SLAPY2</a>( REAL( GS ), AIMAG( GS ) )
SN = CMPLX( REAL( GS ) / D, -AIMAG( GS ) / D )
RETURN
END IF
F2S = <a name="SLAPY2.140"></a><a href="slapy2.f.html#SLAPY2.1">SLAPY2</a>( REAL( FS ), AIMAG( FS ) )
<span class="comment">*</span><span class="comment"> G2 and G2S are accurate
</span><span class="comment">*</span><span class="comment"> G2 is at least SAFMIN, and G2S is at least SAFMN2
</span> G2S = SQRT( G2 )
<span class="comment">*</span><span class="comment"> Error in CS from underflow in F2S is at most
</span><span class="comment">*</span><span class="comment"> UNFL / SAFMN2 .lt. sqrt(UNFL*EPS) .lt. EPS
</span><span class="comment">*</span><span class="comment"> If MAX(G2,ONE)=G2, then F2 .lt. G2*SAFMIN,
</span><span class="comment">*</span><span class="comment"> and so CS .lt. sqrt(SAFMIN)
</span><span class="comment">*</span><span class="comment"> If MAX(G2,ONE)=ONE, then F2 .lt. SAFMIN
</span><span class="comment">*</span><span class="comment"> and so CS .lt. sqrt(SAFMIN)/SAFMN2 = sqrt(EPS)
</span><span class="comment">*</span><span class="comment"> Therefore, CS = F2S/G2S / sqrt( 1 + (F2S/G2S)**2 ) = F2S/G2S
</span> CS = F2S / G2S
<span class="comment">*</span><span class="comment"> Make sure abs(FF) = 1
</span><span class="comment">*</span><span class="comment"> Do complex/real division explicitly with 2 real divisions
</span> IF( ABS1( F ).GT.ONE ) THEN
D = <a name="SLAPY2.155"></a><a href="slapy2.f.html#SLAPY2.1">SLAPY2</a>( REAL( F ), AIMAG( F ) )
FF = CMPLX( REAL( F ) / D, AIMAG( F ) / D )
ELSE
DR = SAFMX2*REAL( F )
DI = SAFMX2*AIMAG( F )
D = <a name="SLAPY2.160"></a><a href="slapy2.f.html#SLAPY2.1">SLAPY2</a>( DR, DI )
FF = CMPLX( DR / D, DI / D )
END IF
SN = FF*CMPLX( REAL( GS ) / G2S, -AIMAG( GS ) / G2S )
R = CS*F + SN*G
ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is the most common case.
</span><span class="comment">*</span><span class="comment"> Neither F2 nor F2/G2 are less than SAFMIN
</span><span class="comment">*</span><span class="comment"> F2S cannot overflow, and it is accurate
</span><span class="comment">*</span><span class="comment">
</span> F2S = SQRT( ONE+G2 / F2 )
<span class="comment">*</span><span class="comment"> Do the F2S(real)*FS(complex) multiply with two real multiplies
</span> R = CMPLX( F2S*REAL( FS ), F2S*AIMAG( FS ) )
CS = ONE / F2S
D = F2 + G2
<span class="comment">*</span><span class="comment"> Do complex/real division explicitly with two real divisions
</span> SN = CMPLX( REAL( R ) / D, AIMAG( R ) / D )
SN = SN*CONJG( GS )
IF( COUNT.NE.0 ) THEN
IF( COUNT.GT.0 ) THEN
DO 30 I = 1, COUNT
R = R*SAFMX2
30 CONTINUE
ELSE
DO 40 I = 1, -COUNT
R = R*SAFMN2
40 CONTINUE
END IF
END IF
END IF
RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> End of <a name="CLARTG.193"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>
</span><span class="comment">*</span><span class="comment">
</span> END
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