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C DLASY2    SOURCE    BP208322  18/07/10    21:15:21     9872           *> \brief \b DLASY2 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 DLASY2 + dependencies*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasy2.f">*> [TGZ]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasy2.f">*> [ZIP]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasy2.f">*> [TXT]&lt;/a>*> \endhtmlonly**  Definition:*  ===========**       SUBROUTINE DLASY2( 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*8   SCALE, XNORM*       ..*       .. Array Arguments ..*       REAL*8   B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),*      $X( LDX, * )* ..***> \par Purpose:* =============*>*> \verbatim*>*> DLASY2 solves for the N1 by N2 matrix X, 1 &lt;= N1,N2 &lt;= 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION*> 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 DOUBLE PRECISION 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 DOUBLE PRECISION*> 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 June 2016**> \ingroup doubleSYauxiliary** ===================================================================== SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,$                   LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )**  -- LAPACK auxiliary routine (version 3.7.0) --*  -- LAPACK is a software package provided by Univ. of Tennessee,    --*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--*     June 2016**     .. Scalar Arguments ..      LOGICAL            LTRANL, LTRANR      INTEGER            INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2      REAL*8   SCALE, XNORM*     ..*     .. Array Arguments ..      REAL*8   B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),     $X( LDX, * )* ..** =====================================================================** .. Parameters .. REAL*8 ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) REAL*8 TWO, HALF, EIGHT PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )* ..* .. Local Scalars .. LOGICAL BSWAP, XSWAP INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K REAL*8 BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,$                   TEMP, U11, U12, U22, XMAX*     ..*     .. Local Arrays ..      LOGICAL            BSWPIV( 4 ), XSWPIV( 4 )      INTEGER            JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),     $LOCU22( 4 ) REAL*8 BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )* ..* .. External Functions .. INTEGER IDAMAX REAL*8 DLAMCH EXTERNAL IDAMAX, DLAMCH* ..* .. External Subroutines .. EXTERNAL DCOPY, DSWAP* ..** .. Intrinsic Functions ..* INTRINSIC ABS, MAX** ..** .. Data statements .. DATA LOCU12 / 3, 4, 1, 2 / DATA LOCL21 / 2, 1, 4, 3 / DATA LOCU22 / 4, 3, 2, 1 / DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. / DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /** ..** .. Executable Statements ..** Do not check the input parameters for errors* INFO = 0** Quick return if possible* IF( N1.EQ.0 .OR. N2.EQ.0 )$   RETURN**     Set constants to control overflow*      EPS = DLAMCH( 'P' )      SMLNUM = DLAMCH( '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 ) ) ),     $SMLNUM ) TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) IF( LTRANL ) THEN TMP( 2 ) = TL( 1, 2 ) TMP( 3 ) = TL( 2, 1 ) ELSE TMP( 2 ) = TL( 2, 1 ) TMP( 3 ) = TL( 1, 2 ) END IF BTMP( 1 ) = B( 1, 1 ) BTMP( 2 ) = B( 2, 1 ) 40 CONTINUE** Solve 2 by 2 system using complete pivoting.* Set pivots less than SMIN to SMIN.* IPIV = IDAMAX( 4, TMP, 1 ) U11 = TMP( IPIV ) IF( ABS( U11 ).LE.SMIN ) THEN INFO = 1 U11 = SMIN END IF U12 = TMP( LOCU12( IPIV ) ) L21 = TMP( LOCL21( IPIV ) ) / U11 U22 = TMP( LOCU22( IPIV ) ) - U12*L21 XSWAP = XSWPIV( IPIV ) BSWAP = BSWPIV( IPIV ) IF( ABS( U22 ).LE.SMIN ) THEN INFO = 1 U22 = SMIN END IF IF( BSWAP ) THEN TEMP = BTMP( 2 ) BTMP( 2 ) = BTMP( 1 ) - L21*TEMP BTMP( 1 ) = TEMP ELSE BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 ) END IF SCALE = ONE IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.$    ( 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 DCOPY( 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 DSWAP( 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 )     $CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 ) JPIV( I ) = JPSV IF( ABS( T16( I, I ) ).LT.SMIN ) THEN INFO = 1 T16( I, I ) = SMIN END IF DO 90 J = I + 1, 4 T16( J, I ) = T16( J, I ) / T16( I, I ) BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I ) DO 80 K = I + 1, 4 T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K ) 80 CONTINUE 90 CONTINUE 100 CONTINUE IF( ABS( T16( 4, 4 ) ).LT.SMIN ) THEN INFO = 1 T16( 4, 4 ) = SMIN END IF SCALE = ONE IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.$    ( 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 DLASY2*      END

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