*> \brief \b SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLASY2 + dependencies
*>
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*
* Definition:
* ===========
*
* SUBROUTINE SLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
* LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
* .. Scalar Arguments ..
* LOGICAL LTRANL, LTRANR
* INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
* REAL SCALE, XNORM
* ..
* .. Array Arguments ..
* REAL B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
*>
*> op(TL)*X + ISGN*X*op(TR) = SCALE*B,
*>
*> where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
*> -1. op(T) = T or T**T, where T**T denotes the transpose of T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] LTRANL
*> \verbatim
*> LTRANL is LOGICAL
*> On entry, LTRANL specifies the op(TL):
*> = .FALSE., op(TL) = TL,
*> = .TRUE., op(TL) = TL**T.
*> \endverbatim
*>
*> \param[in] LTRANR
*> \verbatim
*> LTRANR is LOGICAL
*> On entry, LTRANR specifies the op(TR):
*> = .FALSE., op(TR) = TR,
*> = .TRUE., op(TR) = TR**T.
*> \endverbatim
*>
*> \param[in] ISGN
*> \verbatim
*> ISGN is INTEGER
*> On entry, ISGN specifies the sign of the equation
*> as described before. ISGN may only be 1 or -1.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> On entry, N1 specifies the order of matrix TL.
*> N1 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> On entry, N2 specifies the order of matrix TR.
*> N2 may only be 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] TL
*> \verbatim
*> TL is REAL array, dimension (LDTL,2)
*> On entry, TL contains an N1 by N1 matrix.
*> \endverbatim
*>
*> \param[in] LDTL
*> \verbatim
*> LDTL is INTEGER
*> The leading dimension of the matrix TL. LDTL >= max(1,N1).
*> \endverbatim
*>
*> \param[in] TR
*> \verbatim
*> TR is REAL array, dimension (LDTR,2)
*> On entry, TR contains an N2 by N2 matrix.
*> \endverbatim
*>
*> \param[in] LDTR
*> \verbatim
*> LDTR is INTEGER
*> The leading dimension of the matrix TR. LDTR >= max(1,N2).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is REAL array, dimension (LDB,2)
*> On entry, the N1 by N2 matrix B contains the right-hand
*> side of the equation.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the matrix B. LDB >= max(1,N1).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is REAL
*> On exit, SCALE contains the scale factor. SCALE is chosen
*> less than or equal to 1 to prevent the solution overflowing.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is REAL array, dimension (LDX,2)
*> On exit, X contains the N1 by N2 solution.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the matrix X. LDX >= max(1,N1).
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*> XNORM is REAL
*> On exit, XNORM is the infinity-norm of the solution.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> On exit, INFO is set to
*> 0: successful exit.
*> 1: TL and TR have too close eigenvalues, so TL or
*> TR is perturbed to get a nonsingular equation.
*> NOTE: In the interests of speed, this routine does not
*> check the inputs for errors.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup realSYauxiliary
*
* =====================================================================
SUBROUTINE SLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
LOGICAL LTRANL, LTRANR
INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
REAL SCALE, XNORM
* ..
* .. Array Arguments ..
REAL B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
REAL TWO, HALF, EIGHT
PARAMETER ( TWO = 2.0E+0, HALF = 0.5E+0, EIGHT = 8.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL BSWAP, XSWAP
INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
REAL BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
$ TEMP, U11, U12, U22, XMAX
* ..
* .. Local Arrays ..
LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
$ LOCU22( 4 )
REAL BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SLAMCH
EXTERNAL ISAMAX, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Data statements ..
DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
$ LOCU22 / 4, 3, 2, 1 /
DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
* ..
* .. Executable Statements ..
*
* Do not check the input parameters for errors
*
INFO = 0
*
* Quick return if possible
*
IF( N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
*
* Set constants to control overflow
*
EPS = SLAMCH( 'P' )
SMLNUM = SLAMCH( 'S' ) / EPS
SGN = ISGN
*
K = N1 + N1 + N2 - 2
GO TO ( 10, 20, 30, 50 )K
*
* 1 by 1: TL11*X + SGN*X*TR11 = B11
*
10 CONTINUE
TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
BET = ABS( TAU1 )
IF( BET.LE.SMLNUM ) THEN
TAU1 = SMLNUM
BET = SMLNUM
INFO = 1
END IF
*
SCALE = ONE
GAM = ABS( B( 1, 1 ) )
IF( SMLNUM*GAM.GT.BET )
$ SCALE = ONE / GAM
*
X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
XNORM = ABS( X( 1, 1 ) )
RETURN
*
* 1 by 2:
* TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
* [TR21 TR22]
*
20 CONTINUE
*
SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
$ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
IF( LTRANR ) THEN
TMP( 2 ) = SGN*TR( 2, 1 )
TMP( 3 ) = SGN*TR( 1, 2 )
ELSE
TMP( 2 ) = SGN*TR( 1, 2 )
TMP( 3 ) = SGN*TR( 2, 1 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 1, 2 )
GO TO 40
*
* 2 by 1:
* op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
* [TL21 TL22] [X21] [X21] [B21]
*
30 CONTINUE
SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
$ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
$ SMLNUM )
TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
IF( LTRANL ) THEN
TMP( 2 ) = TL( 1, 2 )
TMP( 3 ) = TL( 2, 1 )
ELSE
TMP( 2 ) = TL( 2, 1 )
TMP( 3 ) = TL( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
40 CONTINUE
*
* Solve 2 by 2 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
IPIV = ISAMAX( 4, TMP, 1 )
U11 = TMP( IPIV )
IF( ABS( U11 ).LE.SMIN ) THEN
INFO = 1
U11 = SMIN
END IF
U12 = TMP( LOCU12( IPIV ) )
L21 = TMP( LOCL21( IPIV ) ) / U11
U22 = TMP( LOCU22( IPIV ) ) - U12*L21
XSWAP = XSWPIV( IPIV )
BSWAP = BSWPIV( IPIV )
IF( ABS( U22 ).LE.SMIN ) THEN
INFO = 1
U22 = SMIN
END IF
IF( BSWAP ) THEN
TEMP = BTMP( 2 )
BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
BTMP( 1 ) = TEMP
ELSE
BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
END IF
SCALE = ONE
IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
$ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
END IF
X2( 2 ) = BTMP( 2 ) / U22
X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
IF( XSWAP ) THEN
TEMP = X2( 2 )
X2( 2 ) = X2( 1 )
X2( 1 ) = TEMP
END IF
X( 1, 1 ) = X2( 1 )
IF( N1.EQ.1 ) THEN
X( 1, 2 ) = X2( 2 )
XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
ELSE
X( 2, 1 ) = X2( 2 )
XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
END IF
RETURN
*
* 2 by 2:
* op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
* [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
*
* Solve equivalent 4 by 4 system using complete pivoting.
* Set pivots less than SMIN to SMIN.
*
50 CONTINUE
SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
$ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
$ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
SMIN = MAX( EPS*SMIN, SMLNUM )
BTMP( 1 ) = ZERO
CALL SCOPY( 16, BTMP, 0, T16, 1 )
T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
IF( LTRANL ) THEN
T16( 1, 2 ) = TL( 2, 1 )
T16( 2, 1 ) = TL( 1, 2 )
T16( 3, 4 ) = TL( 2, 1 )
T16( 4, 3 ) = TL( 1, 2 )
ELSE
T16( 1, 2 ) = TL( 1, 2 )
T16( 2, 1 ) = TL( 2, 1 )
T16( 3, 4 ) = TL( 1, 2 )
T16( 4, 3 ) = TL( 2, 1 )
END IF
IF( LTRANR ) THEN
T16( 1, 3 ) = SGN*TR( 1, 2 )
T16( 2, 4 ) = SGN*TR( 1, 2 )
T16( 3, 1 ) = SGN*TR( 2, 1 )
T16( 4, 2 ) = SGN*TR( 2, 1 )
ELSE
T16( 1, 3 ) = SGN*TR( 2, 1 )
T16( 2, 4 ) = SGN*TR( 2, 1 )
T16( 3, 1 ) = SGN*TR( 1, 2 )
T16( 4, 2 ) = SGN*TR( 1, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
BTMP( 2 ) = B( 2, 1 )
BTMP( 3 ) = B( 1, 2 )
BTMP( 4 ) = B( 2, 2 )
*
* Perform elimination
*
DO 100 I = 1, 3
XMAX = ZERO
DO 70 IP = I, 4
DO 60 JP = I, 4
IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
XMAX = ABS( T16( IP, JP ) )
IPSV = IP
JPSV = JP
END IF
60 CONTINUE
70 CONTINUE
IF( IPSV.NE.I ) THEN
CALL SSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
TEMP = BTMP( I )
BTMP( I ) = BTMP( IPSV )
BTMP( IPSV ) = TEMP
END IF
IF( JPSV.NE.I )
$ CALL SSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
JPIV( I ) = JPSV
IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
INFO = 1
T16( I, I ) = SMIN
END IF
DO 90 J = I + 1, 4
T16( J, I ) = T16( J, I ) / T16( I, I )
BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
DO 80 K = I + 1, 4
T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
80 CONTINUE
90 CONTINUE
100 CONTINUE
IF( ABS( T16( 4, 4 ) ).LT.SMIN )
$ T16( 4, 4 ) = SMIN
SCALE = ONE
IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
$ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
BTMP( 3 ) = BTMP( 3 )*SCALE
BTMP( 4 ) = BTMP( 4 )*SCALE
END IF
DO 120 I = 1, 4
K = 5 - I
TEMP = ONE / T16( K, K )
TMP( K ) = BTMP( K )*TEMP
DO 110 J = K + 1, 4
TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
110 CONTINUE
120 CONTINUE
DO 130 I = 1, 3
IF( JPIV( 4-I ).NE.4-I ) THEN
TEMP = TMP( 4-I )
TMP( 4-I ) = TMP( JPIV( 4-I ) )
TMP( JPIV( 4-I ) ) = TEMP
END IF
130 CONTINUE
X( 1, 1 ) = TMP( 1 )
X( 2, 1 ) = TMP( 2 )
X( 1, 2 ) = TMP( 3 )
X( 2, 2 ) = TMP( 4 )
XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
$ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
RETURN
*
* End of SLASY2
*
END