#include "ESMF_LapackBlas.inc" !> \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 !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasy2.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasy2.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasy2.f"> !> [TXT]</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 ! DOUBLE PRECISION SCALE, XNORM ! .. ! .. Array Arguments .. ! DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ), ! $ X( LDX, * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> 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**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 DOUBLE PRECISION SCALE, XNORM ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ), & X( LDX, * ) ! .. ! ! ===================================================================== ! ! .. 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 ) 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