path: root/SRC/slaln2.f

*> \brief \b SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
*                          LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            LTRANS
*       INTEGER            INFO, LDA, LDB, LDX, NA, NW
*       REAL               CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), B( LDB, * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLALN2 solves a system of the form  (ca A - w D ) X = s B
*> or (ca A**T - w D) X = s B   with possible scaling ("s") and
*> perturbation of A.  (A**T means A-transpose.)
*>
*> A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
*> real diagonal matrix, w is a real or complex value, and X and B are
*> NA x 1 matrices -- real if w is real, complex if w is complex.  NA
*> may be 1 or 2.
*>
*> If w is complex, X and B are represented as NA x 2 matrices,
*> the first column of each being the real part and the second
*> being the imaginary part.
*>
*> "s" is a scaling factor (.LE. 1), computed by SLALN2, which is
*> so chosen that X can be computed without overflow.  X is further
*> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
*> than overflow.
*>
*> If both singular values of (ca A - w D) are less than SMIN,
*> SMIN*identity will be used instead of (ca A - w D).  If only one
*> singular value is less than SMIN, one element of (ca A - w D) will be
*> perturbed enough to make the smallest singular value roughly SMIN.
*> If both singular values are at least SMIN, (ca A - w D) will not be
*> perturbed.  In any case, the perturbation will be at most some small
*> multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
*> are computed by infinity-norm approximations, and thus will only be
*> correct to a factor of 2 or so.
*>
*> Note: all input quantities are assumed to be smaller than overflow
*> by a reasonable factor.  (See BIGNUM.)
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] LTRANS
*> \verbatim
*>          LTRANS is LOGICAL
*>          =.TRUE.:  A-transpose will be used.
*>          =.FALSE.: A will be used (not transposed.)
*> \endverbatim
*>
*> \param[in] NA
*> \verbatim
*>          NA is INTEGER
*>          The size of the matrix A.  It may (only) be 1 or 2.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*>          NW is INTEGER
*>          1 if "w" is real, 2 if "w" is complex.  It may only be 1
*>          or 2.
*> \endverbatim
*>
*> \param[in] SMIN
*> \verbatim
*>          SMIN is REAL
*>          The desired lower bound on the singular values of A.  This
*>          should be a safe distance away from underflow or overflow,
*>          say, between (underflow/machine precision) and  (machine
*>          precision * overflow ).  (See BIGNUM and ULP.)
*> \endverbatim
*>
*> \param[in] CA
*> \verbatim
*>          CA is REAL
*>          The coefficient c, which A is multiplied by.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,NA)
*>          The NA x NA matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  It must be at least NA.
*> \endverbatim
*>
*> \param[in] D1
*> \verbatim
*>          D1 is REAL
*>          The 1,1 element in the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] D2
*> \verbatim
*>          D2 is REAL
*>          The 2,2 element in the diagonal matrix D.  Not used if NA=1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is REAL array, dimension (LDB,NW)
*>          The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
*>          complex), column 1 contains the real part of B and column 2
*>          contains the imaginary part.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  It must be at least NA.
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*>          WR is REAL
*>          The real part of the scalar "w".
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*>          WI is REAL
*>          The imaginary part of the scalar "w".  Not used if NW=1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is REAL array, dimension (LDX,NW)
*>          The NA x NW matrix X (unknowns), as computed by SLALN2.
*>          If NW=2 ("w" is complex), on exit, column 1 will contain
*>          the real part of X and column 2 will contain the imaginary
*>          part.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of X.  It must be at least NA.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is REAL
*>          The scale factor that B must be multiplied by to insure
*>          that overflow does not occur when computing X.  Thus,
*>          (ca A - w D) X  will be SCALE*B, not B (ignoring
*>          perturbations of A.)  It will be at most 1.
*> \endverbatim
*>
*> \param[out] XNORM
*> \verbatim
*>          XNORM is REAL
*>          The infinity-norm of X, when X is regarded as an NA x NW
*>          real matrix.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          An error flag.  It will be set to zero if no error occurs,
*>          a negative number if an argument is in error, or a positive
*>          number if  ca A - w D  had to be perturbed.
*>          The possible values are:
*>          = 0: No error occurred, and (ca A - w D) did not have to be
*>                 perturbed.
*>          = 1: (ca A - w D) had to be perturbed to make its smallest
*>               (or only) singular value greater than SMIN.
*>          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 December 2016
*
*> \ingroup realOTHERauxiliary
*
*  =====================================================================
      SUBROUTINE SLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
     $                   LDB, WR, WI, X, LDX, SCALE, 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..--
*     December 2016
*
*     .. Scalar Arguments ..
      LOGICAL            LTRANS
      INTEGER            INFO, LDA, LDB, LDX, NA, NW
      REAL               CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * ), X( LDX, * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
      REAL               TWO
      PARAMETER          ( TWO = 2.0E0 )
*     ..
*     .. Local Scalars ..
      INTEGER            ICMAX, J
      REAL               BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21,
     $                   CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21,
     $                   LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R,
     $                   UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S,
     $                   UR22, XI1, XI2, XR1, XR2
*     ..
*     .. Local Arrays ..
      LOGICAL            CSWAP( 4 ), RSWAP( 4 )
      INTEGER            IPIVOT( 4, 4 )
      REAL               CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 )
*     ..
*     .. External Functions ..
      REAL               SLAMCH
      EXTERNAL           SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           SLADIV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Equivalences ..
      EQUIVALENCE        ( CI( 1, 1 ), CIV( 1 ) ),
     $                   ( CR( 1, 1 ), CRV( 1 ) )
*     ..
*     .. Data statements ..
      DATA               CSWAP / .FALSE., .FALSE., .TRUE., .TRUE. /
      DATA               RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. /
      DATA               IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4,
     $                   3, 2, 1 /
*     ..
*     .. Executable Statements ..
*
*     Compute BIGNUM
*
      SMLNUM = TWO*SLAMCH( 'Safe minimum' )
      BIGNUM = ONE / SMLNUM
      SMINI = MAX( SMIN, SMLNUM )
*
*     Don't check for input errors
*
      INFO = 0
*
*     Standard Initializations
*
      SCALE = ONE
*
      IF( NA.EQ.1 ) THEN
*
*        1 x 1  (i.e., scalar) system   C X = B
*
         IF( NW.EQ.1 ) THEN
*
*           Real 1x1 system.
*
*           C = ca A - w D
*
            CSR = CA*A( 1, 1 ) - WR*D1
            CNORM = ABS( CSR )
*
*           If | C | < SMINI, use C = SMINI
*
            IF( CNORM.LT.SMINI ) THEN
               CSR = SMINI
               CNORM = SMINI
               INFO = 1
            END IF
*
*           Check scaling for  X = B / C
*
            BNORM = ABS( B( 1, 1 ) )
            IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
               IF( BNORM.GT.BIGNUM*CNORM )
     $            SCALE = ONE / BNORM
            END IF
*
*           Compute X
*
            X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR
            XNORM = ABS( X( 1, 1 ) )
         ELSE
*
*           Complex 1x1 system (w is complex)
*
*           C = ca A - w D
*
            CSR = CA*A( 1, 1 ) - WR*D1
            CSI = -WI*D1
            CNORM = ABS( CSR ) + ABS( CSI )
*
*           If | C | < SMINI, use C = SMINI
*
            IF( CNORM.LT.SMINI ) THEN
               CSR = SMINI
               CSI = ZERO
               CNORM = SMINI
               INFO = 1
            END IF
*
*           Check scaling for  X = B / C
*
            BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) )
            IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
               IF( BNORM.GT.BIGNUM*CNORM )
     $            SCALE = ONE / BNORM
            END IF
*
*           Compute X
*
            CALL SLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI,
     $                   X( 1, 1 ), X( 1, 2 ) )
            XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
         END IF
*
      ELSE
*
*        2x2 System
*
*        Compute the real part of  C = ca A - w D  (or  ca A**T - w D )
*
         CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1
         CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2
         IF( LTRANS ) THEN
            CR( 1, 2 ) = CA*A( 2, 1 )
            CR( 2, 1 ) = CA*A( 1, 2 )
         ELSE
            CR( 2, 1 ) = CA*A( 2, 1 )
            CR( 1, 2 ) = CA*A( 1, 2 )
         END IF
*
         IF( NW.EQ.1 ) THEN
*
*           Real 2x2 system  (w is real)
*
*           Find the largest element in C
*
            CMAX = ZERO
            ICMAX = 0
*
            DO 10 J = 1, 4
               IF( ABS( CRV( J ) ).GT.CMAX ) THEN
                  CMAX = ABS( CRV( J ) )
                  ICMAX = J
               END IF
   10       CONTINUE
*
*           If norm(C) < SMINI, use SMINI*identity.
*
            IF( CMAX.LT.SMINI ) THEN
               BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) )
               IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
                  IF( BNORM.GT.BIGNUM*SMINI )
     $               SCALE = ONE / BNORM
               END IF
               TEMP = SCALE / SMINI
               X( 1, 1 ) = TEMP*B( 1, 1 )
               X( 2, 1 ) = TEMP*B( 2, 1 )
               XNORM = TEMP*BNORM
               INFO = 1
               RETURN
            END IF
*
*           Gaussian elimination with complete pivoting.
*
            UR11 = CRV( ICMAX )
            CR21 = CRV( IPIVOT( 2, ICMAX ) )
            UR12 = CRV( IPIVOT( 3, ICMAX ) )
            CR22 = CRV( IPIVOT( 4, ICMAX ) )
            UR11R = ONE / UR11
            LR21 = UR11R*CR21
            UR22 = CR22 - UR12*LR21
*
*           If smaller pivot < SMINI, use SMINI
*
            IF( ABS( UR22 ).LT.SMINI ) THEN
               UR22 = SMINI
               INFO = 1
            END IF
            IF( RSWAP( ICMAX ) ) THEN
               BR1 = B( 2, 1 )
               BR2 = B( 1, 1 )
            ELSE
               BR1 = B( 1, 1 )
               BR2 = B( 2, 1 )
            END IF
            BR2 = BR2 - LR21*BR1
            BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) )
            IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN
               IF( BBND.GE.BIGNUM*ABS( UR22 ) )
     $            SCALE = ONE / BBND
            END IF
*
            XR2 = ( BR2*SCALE ) / UR22
            XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 )
            IF( CSWAP( ICMAX ) ) THEN
               X( 1, 1 ) = XR2
               X( 2, 1 ) = XR1
            ELSE
               X( 1, 1 ) = XR1
               X( 2, 1 ) = XR2
            END IF
            XNORM = MAX( ABS( XR1 ), ABS( XR2 ) )
*
*           Further scaling if  norm(A) norm(X) > overflow
*
            IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
               IF( XNORM.GT.BIGNUM / CMAX ) THEN
                  TEMP = CMAX / BIGNUM
                  X( 1, 1 ) = TEMP*X( 1, 1 )
                  X( 2, 1 ) = TEMP*X( 2, 1 )
                  XNORM = TEMP*XNORM
                  SCALE = TEMP*SCALE
               END IF
            END IF
         ELSE
*
*           Complex 2x2 system  (w is complex)
*
*           Find the largest element in C
*
            CI( 1, 1 ) = -WI*D1
            CI( 2, 1 ) = ZERO
            CI( 1, 2 ) = ZERO
            CI( 2, 2 ) = -WI*D2
            CMAX = ZERO
            ICMAX = 0
*
            DO 20 J = 1, 4
               IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN
                  CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) )
                  ICMAX = J
               END IF
   20       CONTINUE
*
*           If norm(C) < SMINI, use SMINI*identity.
*
            IF( CMAX.LT.SMINI ) THEN
               BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
     $                 ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
               IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
                  IF( BNORM.GT.BIGNUM*SMINI )
     $               SCALE = ONE / BNORM
               END IF
               TEMP = SCALE / SMINI
               X( 1, 1 ) = TEMP*B( 1, 1 )
               X( 2, 1 ) = TEMP*B( 2, 1 )
               X( 1, 2 ) = TEMP*B( 1, 2 )
               X( 2, 2 ) = TEMP*B( 2, 2 )
               XNORM = TEMP*BNORM
               INFO = 1
               RETURN
            END IF
*
*           Gaussian elimination with complete pivoting.
*
            UR11 = CRV( ICMAX )
            UI11 = CIV( ICMAX )
            CR21 = CRV( IPIVOT( 2, ICMAX ) )
            CI21 = CIV( IPIVOT( 2, ICMAX ) )
            UR12 = CRV( IPIVOT( 3, ICMAX ) )
            UI12 = CIV( IPIVOT( 3, ICMAX ) )
            CR22 = CRV( IPIVOT( 4, ICMAX ) )
            CI22 = CIV( IPIVOT( 4, ICMAX ) )
            IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN
*
*              Code when off-diagonals of pivoted C are real
*
               IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN
                  TEMP = UI11 / UR11
                  UR11R = ONE / ( UR11*( ONE+TEMP**2 ) )
                  UI11R = -TEMP*UR11R
               ELSE
                  TEMP = UR11 / UI11
                  UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) )
                  UR11R = -TEMP*UI11R
               END IF
               LR21 = CR21*UR11R
               LI21 = CR21*UI11R
               UR12S = UR12*UR11R
               UI12S = UR12*UI11R
               UR22 = CR22 - UR12*LR21
               UI22 = CI22 - UR12*LI21
            ELSE
*
*              Code when diagonals of pivoted C are real
*
               UR11R = ONE / UR11
               UI11R = ZERO
               LR21 = CR21*UR11R
               LI21 = CI21*UR11R
               UR12S = UR12*UR11R
               UI12S = UI12*UR11R
               UR22 = CR22 - UR12*LR21 + UI12*LI21
               UI22 = -UR12*LI21 - UI12*LR21
            END IF
            U22ABS = ABS( UR22 ) + ABS( UI22 )
*
*           If smaller pivot < SMINI, use SMINI
*
            IF( U22ABS.LT.SMINI ) THEN
               UR22 = SMINI
               UI22 = ZERO
               INFO = 1
            END IF
            IF( RSWAP( ICMAX ) ) THEN
               BR2 = B( 1, 1 )
               BR1 = B( 2, 1 )
               BI2 = B( 1, 2 )
               BI1 = B( 2, 2 )
            ELSE
               BR1 = B( 1, 1 )
               BR2 = B( 2, 1 )
               BI1 = B( 1, 2 )
               BI2 = B( 2, 2 )
            END IF
            BR2 = BR2 - LR21*BR1 + LI21*BI1
            BI2 = BI2 - LI21*BR1 - LR21*BI1
            BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )*
     $             ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ),
     $             ABS( BR2 )+ABS( BI2 ) )
            IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN
               IF( BBND.GE.BIGNUM*U22ABS ) THEN
                  SCALE = ONE / BBND
                  BR1 = SCALE*BR1
                  BI1 = SCALE*BI1
                  BR2 = SCALE*BR2
                  BI2 = SCALE*BI2
               END IF
            END IF
*
            CALL SLADIV( BR2, BI2, UR22, UI22, XR2, XI2 )
            XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2
            XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2
            IF( CSWAP( ICMAX ) ) THEN
               X( 1, 1 ) = XR2
               X( 2, 1 ) = XR1
               X( 1, 2 ) = XI2
               X( 2, 2 ) = XI1
            ELSE
               X( 1, 1 ) = XR1
               X( 2, 1 ) = XR2
               X( 1, 2 ) = XI1
               X( 2, 2 ) = XI2
            END IF
            XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) )
*
*           Further scaling if  norm(A) norm(X) > overflow
*
            IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
               IF( XNORM.GT.BIGNUM / CMAX ) THEN
                  TEMP = CMAX / BIGNUM
                  X( 1, 1 ) = TEMP*X( 1, 1 )
                  X( 2, 1 ) = TEMP*X( 2, 1 )
                  X( 1, 2 ) = TEMP*X( 1, 2 )
                  X( 2, 2 ) = TEMP*X( 2, 2 )
                  XNORM = TEMP*XNORM
                  SCALE = TEMP*SCALE
               END IF
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of SLALN2
*
      END