dlasy2.f

          SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
         $                   LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          LOGICAL            LTRANL, LTRANR
          INTEGER            INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
          DOUBLE PRECISION   SCALE, XNORM
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
         $                   X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLASY2 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', where T' denotes the transpose of T.
    *
    *  Arguments
    *  =========
    *
    *  LTRANL  (input) LOGICAL
    *          On entry, LTRANL specifies the op(TL):
    *             = .FALSE., op(TL) = TL,
    *             = .TRUE., op(TL) = TL'.
    *
    *  LTRANR  (input) LOGICAL
    *          On entry, LTRANR specifies the op(TR):
    *            = .FALSE., op(TR) = TR,
    *            = .TRUE., op(TR) = TR'.
    *
    *  ISGN    (input) INTEGER
    *          On entry, ISGN specifies the sign of the equation
    *          as described before. ISGN may only be 1 or -1.
    *
    *  N1      (input) INTEGER
    *          On entry, N1 specifies the order of matrix TL.
    *          N1 may only be 0, 1 or 2.
    *
    *  N2      (input) INTEGER
    *          On entry, N2 specifies the order of matrix TR.
    *          N2 may only be 0, 1 or 2.
    *
    *  TL      (input) DOUBLE PRECISION array, dimension (LDTL,2)
    *          On entry, TL contains an N1 by N1 matrix.
    *
    *  LDTL    (input) INTEGER
    *          The leading dimension of the matrix TL. LDTL >= max(1,N1).
    *
    *  TR      (input) DOUBLE PRECISION array, dimension (LDTR,2)
    *          On entry, TR contains an N2 by N2 matrix.
    *
    *  LDTR    (input) INTEGER
    *          The leading dimension of the matrix TR. LDTR >= max(1,N2).
    *
    *  B       (input) DOUBLE PRECISION array, dimension (LDB,2)
    *          On entry, the N1 by N2 matrix B contains the right-hand
    *          side of the equation.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the matrix B. LDB >= max(1,N1).
    *
    *  SCALE   (output) DOUBLE PRECISION
    *          On exit, SCALE contains the scale factor. SCALE is chosen
    *          less than or equal to 1 to prevent the solution overflowing.
    *
    *  X       (output) DOUBLE PRECISION array, dimension (LDX,2)
    *          On exit, X contains the N1 by N2 solution.
    *
    *  LDX     (input) INTEGER
    *          The leading dimension of the matrix X. LDX >= max(1,N1).
    *
    *  XNORM   (output) DOUBLE PRECISION
    *          On exit, XNORM is the infinity-norm of the solution.
    *
    *  INFO    (output) 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.
    *
    * =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
          DOUBLE PRECISION   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
          DOUBLE PRECISION   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 )
          DOUBLE PRECISION   BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
    *     ..
    *     .. External Functions ..
          INTEGER            IDAMAX
          DOUBLE PRECISION   DLAMCH
          EXTERNAL           IDAMAX, DLAMCH
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DSWAP
    *     ..
    *     .. 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 = 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 )
         $   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 DLASY2
    *
          END