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dlaln2.f
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*> \brief \b DLALN2 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/
*
*> \htmlonly
*> Download DLALN2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaln2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaln2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaln2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLALN2( 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
* DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLALN2 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 (<= 1), computed by DLALN2, 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 DOUBLE PRECISION
*> 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 DOUBLE PRECISION
*> The coefficient c, which A is multiplied by.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION 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 DOUBLE PRECISION
*> The 1,1 element in the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] D2
*> \verbatim
*> D2 is DOUBLE PRECISION
*> The 2,2 element in the diagonal matrix D. Not used if NA=1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION 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 DOUBLE PRECISION
*> The real part of the scalar "w".
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is DOUBLE PRECISION
*> The imaginary part of the scalar "w". Not used if NW=1.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NW)
*> The NA x NW matrix X (unknowns), as computed by DLALN2.
*> 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 DOUBLE PRECISION
*> 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 DOUBLE PRECISION
*> 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 doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLALN2( 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
DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
INTEGER ICMAX, J
DOUBLE PRECISION 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 RSWAP( 4 ), ZSWAP( 4 )
INTEGER IPIVOT( 4, 4 )
DOUBLE PRECISION CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLADIV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Equivalences ..
EQUIVALENCE ( CI( 1, 1 ), CIV( 1 ) ),
$ ( CR( 1, 1 ), CRV( 1 ) )
* ..
* .. Data statements ..
DATA ZSWAP / .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*DLAMCH( '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 DLADIV( 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( ZSWAP( 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 DLADIV( 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( ZSWAP( 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 DLALN2
*
END