*> \brief \b DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
* INFO )
*
* .. Scalar Arguments ..
* LOGICAL WANTQ
* INTEGER INFO, J1, LDQ, LDT, N, N1, N2
* ..
* .. Array Arguments ..
* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
*> an upper quasi-triangular matrix T by an orthogonal similarity
*> transformation.
*>
*> T must be in Schur canonical form, that is, block upper triangular
*> with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
*> has its diagonal elements equal and its off-diagonal elements of
*> opposite sign.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTQ
*> \verbatim
*> WANTQ is LOGICAL
*> = .TRUE. : accumulate the transformation in the matrix Q;
*> = .FALSE.: do not accumulate the transformation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> On entry, the upper quasi-triangular matrix T, in Schur
*> canonical form.
*> On exit, the updated matrix T, again in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
*> On exit, if WANTQ is .TRUE., the updated matrix Q.
*> If WANTQ is .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q.
*> LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*> J1 is INTEGER
*> The index of the first row of the first block T11.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The order of the first block T11. N1 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[in] N2
*> \verbatim
*> N2 is INTEGER
*> The order of the second block T22. N2 = 0, 1 or 2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> = 1: the transformed matrix T would be too far from Schur
*> form; the blocks are not swapped and T and Q are
*> unchanged.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laexc
*
* =====================================================================
SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK,
$ INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL WANTQ
INTEGER INFO, J1, LDQ, LDT, N, N1, N2
* ..
* .. Array Arguments ..
DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION TEN
PARAMETER ( TEN = 1.0D+1 )
INTEGER LDD, LDX
PARAMETER ( LDD = 4, LDX = 2 )
* ..
* .. Local Scalars ..
INTEGER IERR, J2, J3, J4, K, ND
DOUBLE PRECISION CS, DNORM, EPS, SCALE, SMLNUM, SN, T11, T22,
$ T33, TAU, TAU1, TAU2, TEMP, THRESH, WI1, WI2,
$ WR1, WR2, XNORM
* ..
* .. Local Arrays ..
DOUBLE PRECISION D( LDD, 4 ), U( 3 ), U1( 3 ), U2( 3 ),
$ X( LDX, 2 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DLANV2, DLARFG, DLARFX, DLARTG,
$ DLASY2,
$ DROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 .OR. N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
IF( J1+N1.GT.N )
$ RETURN
*
J2 = J1 + 1
J3 = J1 + 2
J4 = J1 + 3
*
IF( N1.EQ.1 .AND. N2.EQ.1 ) THEN
*
* Swap two 1-by-1 blocks.
*
T11 = T( J1, J1 )
T22 = T( J2, J2 )
*
* Determine the transformation to perform the interchange.
*
CALL DLARTG( T( J1, J2 ), T22-T11, CS, SN, TEMP )
*
* Apply transformation to the matrix T.
*
IF( J3.LE.N )
$ CALL DROT( N-J1-1, T( J1, J3 ), LDT, T( J2, J3 ), LDT,
$ CS,
$ SN )
CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
*
T( J1, J1 ) = T22
T( J2, J2 ) = T11
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
END IF
*
ELSE
*
* Swapping involves at least one 2-by-2 block.
*
* Copy the diagonal block of order N1+N2 to the local array D
* and compute its norm.
*
ND = N1 + N2
CALL DLACPY( 'Full', ND, ND, T( J1, J1 ), LDT, D, LDD )
DNORM = DLANGE( 'Max', ND, ND, D, LDD, WORK )
*
* Compute machine-dependent threshold for test for accepting
* swap.
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
THRESH = MAX( TEN*EPS*DNORM, SMLNUM )
*
* Solve T11*X - X*T22 = scale*T12 for X.
*
CALL DLASY2( .FALSE., .FALSE., -1, N1, N2, D, LDD,
$ D( N1+1, N1+1 ), LDD, D( 1, N1+1 ), LDD, SCALE, X,
$ LDX, XNORM, IERR )
*
* Swap the adjacent diagonal blocks.
*
K = N1 + N1 + N2 - 3
GO TO ( 10, 20, 30 )K
*
10 CONTINUE
*
* N1 = 1, N2 = 2: generate elementary reflector H so that:
*
* ( scale, X11, X12 ) H = ( 0, 0, * )
*
U( 1 ) = SCALE
U( 2 ) = X( 1, 1 )
U( 3 ) = X( 1, 2 )
CALL DLARFG( 3, U( 3 ), U, 1, TAU )
U( 3 ) = ONE
T11 = T( J1, J1 )
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 3,
$ 3 )-T11 ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'L', 3, N-J1+1, U, TAU, T( J1, J1 ), LDT,
$ WORK )
CALL DLARFX( 'R', J2, 3, U, TAU, T( 1, J1 ), LDT, WORK )
*
T( J3, J1 ) = ZERO
T( J3, J2 ) = ZERO
T( J3, J3 ) = T11
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
END IF
GO TO 40
*
20 CONTINUE
*
* N1 = 2, N2 = 1: generate elementary reflector H so that:
*
* H ( -X11 ) = ( * )
* ( -X21 ) = ( 0 )
* ( scale ) = ( 0 )
*
U( 1 ) = -X( 1, 1 )
U( 2 ) = -X( 2, 1 )
U( 3 ) = SCALE
CALL DLARFG( 3, U( 1 ), U( 2 ), 1, TAU )
U( 1 ) = ONE
T33 = T( J3, J3 )
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK )
CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 2, 1 ) ), ABS( D( 3, 1 ) ), ABS( D( 1,
$ 1 )-T33 ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'R', J3, 3, U, TAU, T( 1, J1 ), LDT, WORK )
CALL DLARFX( 'L', 3, N-J1, U, TAU, T( J1, J2 ), LDT, WORK )
*
T( J1, J1 ) = T33
T( J2, J1 ) = ZERO
T( J3, J1 ) = ZERO
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK )
END IF
GO TO 40
*
30 CONTINUE
*
* N1 = 2, N2 = 2: generate elementary reflectors H(1) and H(2) so
* that:
*
* H(2) H(1) ( -X11 -X12 ) = ( * * )
* ( -X21 -X22 ) ( 0 * )
* ( scale 0 ) ( 0 0 )
* ( 0 scale ) ( 0 0 )
*
U1( 1 ) = -X( 1, 1 )
U1( 2 ) = -X( 2, 1 )
U1( 3 ) = SCALE
CALL DLARFG( 3, U1( 1 ), U1( 2 ), 1, TAU1 )
U1( 1 ) = ONE
*
TEMP = -TAU1*( X( 1, 2 )+U1( 2 )*X( 2, 2 ) )
U2( 1 ) = -TEMP*U1( 2 ) - X( 2, 2 )
U2( 2 ) = -TEMP*U1( 3 )
U2( 3 ) = SCALE
CALL DLARFG( 3, U2( 1 ), U2( 2 ), 1, TAU2 )
U2( 1 ) = ONE
*
* Perform swap provisionally on diagonal block in D.
*
CALL DLARFX( 'L', 3, 4, U1, TAU1, D, LDD, WORK )
CALL DLARFX( 'R', 4, 3, U1, TAU1, D, LDD, WORK )
CALL DLARFX( 'L', 3, 4, U2, TAU2, D( 2, 1 ), LDD, WORK )
CALL DLARFX( 'R', 4, 3, U2, TAU2, D( 1, 2 ), LDD, WORK )
*
* Test whether to reject swap.
*
IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 4, 1 ) ),
$ ABS( D( 4, 2 ) ) ).GT.THRESH )GO TO 50
*
* Accept swap: apply transformation to the entire matrix T.
*
CALL DLARFX( 'L', 3, N-J1+1, U1, TAU1, T( J1, J1 ), LDT,
$ WORK )
CALL DLARFX( 'R', J4, 3, U1, TAU1, T( 1, J1 ), LDT, WORK )
CALL DLARFX( 'L', 3, N-J1+1, U2, TAU2, T( J2, J1 ), LDT,
$ WORK )
CALL DLARFX( 'R', J4, 3, U2, TAU2, T( 1, J2 ), LDT, WORK )
*
T( J3, J1 ) = ZERO
T( J3, J2 ) = ZERO
T( J4, J1 ) = ZERO
T( J4, J2 ) = ZERO
*
IF( WANTQ ) THEN
*
* Accumulate transformation in the matrix Q.
*
CALL DLARFX( 'R', N, 3, U1, TAU1, Q( 1, J1 ), LDQ, WORK )
CALL DLARFX( 'R', N, 3, U2, TAU2, Q( 1, J2 ), LDQ, WORK )
END IF
*
40 CONTINUE
*
IF( N2.EQ.2 ) THEN
*
* Standardize new 2-by-2 block T11
*
CALL DLANV2( T( J1, J1 ), T( J1, J2 ), T( J2, J1 ),
$ T( J2, J2 ), WR1, WI1, WR2, WI2, CS, SN )
CALL DROT( N-J1-1, T( J1, J1+2 ), LDT, T( J2, J1+2 ),
$ LDT,
$ CS, SN )
CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN )
IF( WANTQ )
$ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN )
END IF
*
IF( N1.EQ.2 ) THEN
*
* Standardize new 2-by-2 block T22
*
J3 = J1 + N2
J4 = J3 + 1
CALL DLANV2( T( J3, J3 ), T( J3, J4 ), T( J4, J3 ),
$ T( J4, J4 ), WR1, WI1, WR2, WI2, CS, SN )
IF( J3+2.LE.N )
$ CALL DROT( N-J3-1, T( J3, J3+2 ), LDT, T( J4, J3+2 ),
$ LDT, CS, SN )
CALL DROT( J3-1, T( 1, J3 ), 1, T( 1, J4 ), 1, CS, SN )
IF( WANTQ )
$ CALL DROT( N, Q( 1, J3 ), 1, Q( 1, J4 ), 1, CS, SN )
END IF
*
END IF
RETURN
*
* Exit with INFO = 1 if swap was rejected.
*
50 CONTINUE
INFO = 1
RETURN
*
* End of DLAEXC
*
END