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SUBROUTINE MB03TS( ISHAM, WANTU, N, A, LDA, G, LDG, U1, LDU1, U2,
$ LDU2, J1, N1, N2, DWORK, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To swap diagonal blocks A11 and A22 of order 1 or 2 in the upper
C quasi-triangular matrix A contained in a skew-Hamiltonian matrix
C
C [ A G ] T
C X = [ T ], G = -G,
C [ 0 A ]
C
C or in a Hamiltonian matrix
C
C [ A G ] T
C X = [ T ], G = G.
C [ 0 -A ]
C
C This routine is a modified version of the LAPACK subroutine
C DLAEX2.
C
C The matrix A must be in Schur canonical form (as returned by the
C LAPACK routine DHSEQR), that is, block upper triangular with
C 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
C its diagonal elements equal and its off-diagonal elements of
C opposite sign.
C
C ARGUMENTS
C
C Mode Parameters
C
C ISHAM LOGIGAL
C Specifies the type of X:
C = .TRUE.: X is a Hamiltonian matrix;
C = .FALSE.: X is a skew-Hamiltonian matrix.
C
C WANTU LOGIGAL
C = .TRUE.: update the matrices U1 and U2 containing the
C Schur vectors;
C = .FALSE.: do not update U1 and U2.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix A. N >= 0.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N part of this array must
C contain the upper quasi-triangular matrix A, in Schur
C canonical form.
C On exit, the leading N-by-N part of this array contains
C the reordered matrix A, again in Schur canonical form.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= MAX(1,N).
C
C G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
C On entry, the leading N-by-N part of this array must
C contain the upper triangular part of the symmetric
C matrix G, if ISHAM = .TRUE., or the strictly upper
C triangular part of the skew-symmetric matrix G, otherwise.
C The rest of this array is not referenced.
C On exit, the leading N-by-N part of this array contains
C the upper or strictly upper triangular part of the
C symmetric or skew-symmetric matrix G, respectively,
C updated by the orthogonal transformation which reorders A.
C
C LDG INTEGER
C The leading dimension of the array G. LDG >= MAX(1,N).
C
C U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
C On entry, if WANTU = .TRUE., the leading N-by-N part of
C this array must contain the matrix U1.
C On exit, if WANTU = .TRUE., the leading N-by-N part of
C this array contains U1, postmultiplied by the orthogonal
C transformation which reorders A. See the description in
C the SLICOT subroutine MB03TD for further details.
C If WANTU = .FALSE., this array is not referenced.
C
C LDU1 INTEGER
C The leading dimension of the array U1.
C LDU1 >= MAX(1,N), if WANTU = .TRUE.;
C LDU1 >= 1, otherwise.
C
C U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
C On entry, if WANTU = .TRUE., the leading N-by-N part of
C this array must contain the matrix U2.
C On exit, if WANTU = .TRUE., the leading N-by-N part of
C this array contains U2, postmultiplied by the orthogonal
C transformation which reorders A.
C If WANTU = .FALSE., this array is not referenced.
C
C LDU2 INTEGER
C The leading dimension of the array U2.
C LDU2 >= MAX(1,N), if WANTU = .TRUE.;
C LDU2 >= 1, otherwise.
C
C J1 (input) INTEGER
C The index of the first row of the first block A11.
C If J1+N1 < N, then A11 is swapped with the block starting
C at (J1+N1+1)-th diagonal element.
C If J1+N1 > N, then A11 is the last block in A and swapped
C with -A11', if ISHAM = .TRUE.,
C or A11', if ISHAM = .FALSE..
C
C N1 (input) INTEGER
C The order of the first block A11. N1 = 0, 1 or 2.
C
C N2 (input) INTEGER
C The order of the second block A22. N2 = 0, 1 or 2.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (N)
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C = 1: the transformed matrix A would be too far from Schur
C form; the blocks are not swapped and A, G, U1 and
C U2 are unchanged.
C
C REFERENCES
C
C [1] Bai, Z., and Demmel, J.W.
C On swapping diagonal blocks in real Schur form.
C Linear Algebra Appl., 186, pp. 73-95, 1993.
C
C [2] Benner, P., Kressner, D., and Mehrmann, V.
C Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory,
C Algorithms and Applications. Techn. Report, TU Berlin, 2003.
C
C CONTRIBUTORS
C
C D. Kressner, Technical Univ. Berlin, Germany, and
C P. Benner, Technical Univ. Chemnitz, Germany, December 2003.
C
C REVISIONS
C
C V. Sima, May 2008 (SLICOT version of the HAPACK routine DHAEX2).
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE, TWO, THIRTY, FORTY
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
$ TWO = 2.0D+0, THIRTY = 3.0D+1,
$ FORTY = 4.0D+1 )
INTEGER LDD, LDX
PARAMETER ( LDD = 4, LDX = 2 )
C .. Scalar Arguments ..
LOGICAL ISHAM, WANTU
INTEGER INFO, J1, LDA, LDG, LDU1, LDU2, N, N1, N2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), G(LDG,*), U1(LDU1,*),
$ U2(LDU2,*)
C .. Local Scalars ..
LOGICAL LBLK
INTEGER IERR, J2, J3, J4, K, ND
DOUBLE PRECISION A11, A22, A33, CS, DNORM, EPS, SCALE, SMLNUM,
$ SN, TAU, TAU1, TAU2, TEMP, THRESH, WI1, WI2,
$ WR1, WR2, XNORM
C .. Local Arrays ..
DOUBLE PRECISION D(LDD,4), V(3), V1(3), V2(3), X(LDX,2)
C .. External Functions ..
DOUBLE PRECISION DDOT, DLAMCH, DLANGE
EXTERNAL DDOT, DLAMCH, DLANGE
C .. External Subroutines ..
EXTERNAL DAXPY, DLACPY, DLANV2, DLARFG, DLARFX, DLARTG,
$ DLASET, DLASY2, DROT, DSCAL, DSWAP, DSYMV,
$ DSYR2, MB01MD, MB01ND
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX
C
C .. Executable Statements ..
C
INFO = 0
C
C Quick return if possible.
C
IF ( N.EQ.0 .OR. N1.EQ.0 .OR. N2.EQ.0 )
$ RETURN
LBLK = ( J1+N1.GT.N )
C
J2 = J1 + 1
J3 = J1 + 2
J4 = J1 + 3
C
IF ( LBLK .AND. N1.EQ.1 ) THEN
C
IF ( ISHAM ) THEN
A11 = A(N,N)
CALL DLARTG( G(N,N), -TWO*A11, CS, SN, TEMP )
CALL DROT( N-1, A(1,N), 1, G(1,N), 1, CS, SN )
A(N,N) = -A11
IF ( WANTU )
$ CALL DROT( N, U1(1,N), 1, U2(1,N), 1, CS, SN )
ELSE
CALL DSWAP( N-1, A(1,N), 1, G(1,N), 1 )
CALL DSCAL( N-1, -ONE, A(1,N), 1 )
IF ( WANTU ) THEN
CALL DSWAP( N, U1(1,N), 1, U2(1,N), 1 )
CALL DSCAL( N, -ONE, U1(1,N), 1 )
END IF
END IF
C
ELSE IF ( LBLK .AND. N1.EQ.2 ) THEN
C
IF ( ISHAM ) THEN
C
C Reorder Hamiltonian matrix:
C
C [ A11 G11 ]
C [ T ].
C [ 0 -A11 ]
C
ND = 4
CALL DLACPY( 'Full', 2, 2, A(N-1,N-1), LDA, D, LDD )
CALL DLASET( 'All', 2, 2, ZERO, ZERO, D(3,1), LDD )
CALL DLACPY( 'Upper', 2, 2, G(N-1,N-1), LDG, D(1,3), LDD )
D(2,3) = D(1,4)
D(3,3) = -D(1,1)
D(4,3) = -D(1,2)
D(3,4) = -D(2,1)
D(4,4) = -D(2,2)
DNORM = DLANGE( 'Max', ND, ND, D, LDD, DWORK )
C
C Compute machine-dependent threshold for test for accepting
C swap.
C
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
THRESH = MAX( FORTY*EPS*DNORM, SMLNUM )
C
C Solve A11*X + X*A11' = scale*G11 for X.
C
CALL DLASY2( .FALSE., .FALSE., -1, 2, 2, D, LDD, D(3,3),
$ LDD, D(1,3), LDD, SCALE, X, LDX, XNORM, IERR )
C
C Compute symplectic QR decomposition of
C
C ( -X11 -X12 )
C ( -X21 -X22 ).
C ( scale 0 )
C ( 0 scale )
C
TEMP = -X(1,1)
CALL DLARTG( TEMP, SCALE, V1(1), V2(1), X(1,1) )
CALL DLARTG( X(1,1), -X(2,1), V1(2), V2(2), TEMP )
X(1,2) = -X(1,2)
X(2,2) = -X(2,2)
X(1,1) = ZERO
X(2,1) = SCALE
CALL DROT( 1, X(1,2), 1, X(1,1), 1, V1(1), V2(1) )
CALL DROT( 1, X(1,2), 1, X(2,2), 1, V1(2), V2(2) )
CALL DROT( 1, X(1,1), 1, X(2,1), 1, V1(2), V2(2) )
CALL DLARTG( X(2,2), X(2,1), V1(3), V2(3), TEMP )
C
C Perform swap provisionally on D.
C
CALL DROT( 4, D(1,1), LDD, D(3,1), LDD, V1(1), V2(1) )
CALL DROT( 4, D(1,1), LDD, D(2,1), LDD, V1(2), V2(2) )
CALL DROT( 4, D(3,1), LDD, D(4,1), LDD, V1(2), V2(2) )
CALL DROT( 4, D(2,1), LDD, D(4,1), LDD, V1(3), V2(3) )
CALL DROT( 4, D(1,1), 1, D(1,3), 1, V1(1), V2(1) )
CALL DROT( 4, D(1,1), 1, D(1,2), 1, V1(2), V2(2) )
CALL DROT( 4, D(1,3), 1, D(1,4), 1, V1(2), V2(2) )
CALL DROT( 4, D(1,2), 1, D(1,4), 1, V1(3), V2(3) )
C
C Test whether to reject swap.
C
IF ( MAX( ABS( D(3,1) ), ABS( D(3,2) ), ABS( D(4,1) ),
$ ABS( D(4,2) ) ).GT.THRESH ) GO TO 50
C
CALL DLACPY( 'All', 2, 2, D(1,1), LDD, A(N-1,N-1), LDA )
CALL DLACPY( 'Upper', 2, 2, D(1,3), LDD, G(N-1,N-1), LDG )
C
IF ( N.GT.2 ) THEN
CALL DROT( N-2, A(1,N-1), 1, G(1,N-1), 1, V1(1), V2(1) )
CALL DROT( N-2, A(1,N-1), 1, A(1,N), 1, V1(2), V2(2) )
CALL DROT( N-2, G(1,N-1), 1, G(1,N), 1, V1(2), V2(2) )
CALL DROT( N-2, A(1,N), 1, G(1,N), 1, V1(3), V2(3) )
END IF
C
IF ( WANTU ) THEN
CALL DROT( N, U1(1,N-1), 1, U2(1,N-1), 1, V1(1), V2(1) )
CALL DROT( N, U1(1,N-1), 1, U1(1,N), 1, V1(2), V2(2) )
CALL DROT( N, U2(1,N-1), 1, U2(1,N), 1, V1(2), V2(2) )
CALL DROT( N, U1(1,N), 1, U2(1,N), 1, V1(3), V2(3) )
END IF
C
ELSE
C
IF ( ABS( A(N-1,N) ).GT.ABS( A(N,N-1) ) ) THEN
TEMP = G(N-1,N)
CALL DLARTG( TEMP, A(N-1,N), CS, SN, G(N-1,N) )
SN = -SN
CALL DROT(N-2, A(1,N), 1, G(1,N), 1, CS, SN )
C
A(N-1,N) = -SN*A(N,N-1)
TEMP = -CS*A(N,N-1)
A(N,N-1) = G(N-1,N)
G(N-1,N) = TEMP
IF ( WANTU )
$ CALL DROT( N, U1(1,N), 1, U2(1,N), 1, CS, SN )
CALL DSWAP( N-2, A(1,N-1), 1, G(1,N-1), 1 )
CALL DSCAL( N-2, -ONE, A(1,N-1), 1 )
IF ( WANTU ) THEN
CALL DSWAP( N, U1(1,N-1), 1, U2(1,N-1), 1 )
CALL DSCAL( N, -ONE, U1(1,N-1), 1 )
END IF
ELSE
TEMP = G(N-1,N)
CALL DLARTG( TEMP, A(N,N-1), CS, SN, G(N-1,N) )
CALL DROT( N-2, A(1,N-1), 1, G(1,N-1), 1, CS, SN )
A(N,N-1) = -SN*A(N-1,N)
A(N-1,N) = CS*A(N-1,N)
IF ( WANTU )
$ CALL DROT( N, U1(1,N-1), 1, U2(1,N-1), 1, CS, SN )
CALL DSWAP( N-1, A(1,N), 1, G(1,N), 1 )
CALL DSCAL( N-1, -ONE, A(1,N), 1 )
IF ( WANTU ) THEN
CALL DSWAP( N, U1(1,N), 1, U2(1,N), 1 )
CALL DSCAL( N, -ONE, U1(1,N), 1 )
END IF
END IF
END IF
C
C Standardize new 2-by-2 block.
C
CALL DLANV2( A(N-1,N-1), A(N-1,N), A(N,N-1),
$ A(N,N), WR1, WI1, WR2, WI2, CS, SN )
CALL DROT( N-2, A(1,N-1), 1, A(1,N), 1, CS, SN )
IF ( ISHAM ) THEN
TEMP = G(N-1,N)
CALL DROT( N-1, G(1,N-1), 1, G(1,N), 1, CS, SN )
TAU = CS*TEMP + SN*G(N,N)
G(N,N) = CS*G(N,N) - SN*TEMP
G(N-1,N-1) = CS*G(N-1,N-1) + SN*TAU
CALL DROT( 1, G(N-1,N), LDG, G(N,N), LDG, CS, SN )
ELSE
CALL DROT( N-2, G(1,N-1), 1, G(1,N), 1, CS, SN )
END IF
IF ( WANTU ) THEN
CALL DROT( N, U1(1,N-1), 1, U1(1,N), 1, CS, SN )
CALL DROT( N, U2(1,N-1), 1, U2(1,N), 1, CS, SN )
END IF
C
ELSE IF ( N1.EQ.1 .AND. N2.EQ.1 ) THEN
C
C Swap two 1-by-1 blocks.
C
A11 = A(J1,J1)
A22 = A(J2,J2)
C
C Determine the transformation to perform the interchange.
C
CALL DLARTG( A(J1,J2), A22-A11, CS, SN, TEMP )
C
C Apply transformation to the matrix A.
C
IF ( J3.LE.N )
$ CALL DROT( N-J1-1, A(J1,J3), LDA, A(J2,J3), LDA, CS, SN )
CALL DROT( J1-1, A(1,J1), 1, A(1,J2), 1, CS, SN )
C
A(J1,J1) = A22
A(J2,J2) = A11
C
C Apply transformation to the matrix G.
C
IF ( ISHAM ) THEN
TEMP = G(J1,J2)
CALL DROT( J1, G(1,J1), 1, G(1,J2), 1, CS, SN )
TAU = CS*TEMP + SN*G(J2,J2)
G(J2,J2) = CS*G(J2,J2) - SN*TEMP
G(J1,J1) = CS*G(J1,J1) + SN*TAU
CALL DROT( N-J1, G(J1,J2), LDG, G(J2,J2), LDG, CS, SN )
ELSE
IF ( N.GT.J1+1 )
$ CALL DROT( N-J1-1, G(J1,J1+2), LDG, G(J2,J1+2), LDG, CS,
$ SN )
CALL DROT( J1-1, G(1,J1), 1, G(1,J2), 1, CS, SN )
END IF
IF ( WANTU ) THEN
C
C Accumulate transformation in the matrices U1 and U2.
C
CALL DROT( N, U1(1,J1), 1, U1(1,J2), 1, CS, SN )
CALL DROT( N, U2(1,J1), 1, U2(1,J2), 1, CS, SN )
END IF
C
ELSE
C
C Swapping involves at least one 2-by-2 block.
C
C Copy the diagonal block of order N1+N2 to the local array D
C and compute its norm.
C
ND = N1 + N2
CALL DLACPY( 'Full', ND, ND, A(J1,J1), LDA, D, LDD )
DNORM = DLANGE( 'Max', ND, ND, D, LDD, DWORK )
C
C Compute machine-dependent threshold for test for accepting
C swap.
C
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
THRESH = MAX( THIRTY*EPS*DNORM, SMLNUM )
C
C Solve A11*X - X*A22 = scale*A12 for X.
C
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 )
C
C Swap the adjacent diagonal blocks.
C
K = N1 + N1 + N2 - 3
GO TO ( 10, 20, 30 )K
C
10 CONTINUE
C
C N1 = 1, N2 = 2: generate elementary reflector H so that:
C
C ( scale, X11, X12 ) H = ( 0, 0, * ).
C
V(1) = SCALE
V(2) = X(1,1)
V(3) = X(1,2)
CALL DLARFG( 3, V(3), V, 1, TAU )
V(3) = ONE
A11 = A(J1,J1)
C
C Perform swap provisionally on diagonal block in D.
C
CALL DLARFX( 'Left', 3, 3, V, TAU, D, LDD, DWORK )
CALL DLARFX( 'Right', 3, 3, V, TAU, D, LDD, DWORK )
C
C Test whether to reject swap.
C
IF ( MAX( ABS( D(3,1) ), ABS( D(3,2) ), ABS( D(3,3)-A11 ) )
$ .GT.THRESH ) GO TO 50
C
C Accept swap: apply transformation to the entire matrix A.
C
CALL DLARFX( 'Left', 3, N-J1+1, V, TAU, A(J1,J1), LDA, DWORK )
CALL DLARFX( 'Right', J2, 3, V, TAU, A(1,J1), LDA, DWORK )
C
A(J3,J1) = ZERO
A(J3,J2) = ZERO
A(J3,J3) = A11
C
C Apply transformation to G.
C
IF ( ISHAM ) THEN
CALL DLARFX( 'Right', J1-1, 3, V, TAU, G(1,J1), LDG, DWORK )
CALL DSYMV( 'Upper', 3, TAU, G(J1,J1), LDG, V, 1, ZERO,
$ DWORK, 1 )
TEMP = -HALF*TAU*DDOT( 3, DWORK, 1, V, 1 )
CALL DAXPY( 3, TEMP, V, 1, DWORK, 1 )
CALL DSYR2( 'Upper', 3, -ONE, V, 1, DWORK, 1,
$ G(J1,J1), LDG )
IF ( N.GT.J1+2 )
$ CALL DLARFX( 'Left', 3, N-J1-2, V, TAU, G(J1,J1+3), LDG,
$ DWORK )
ELSE
CALL DLARFX( 'Right', J1-1, 3, V, TAU, G(1,J1), LDG, DWORK )
CALL MB01MD( 'Upper', 3, TAU, G(J1,J1), LDG, V, 1, ZERO,
$ DWORK, 1 )
CALL MB01ND( 'Upper', 3, ONE, V, 1, DWORK, 1, G(J1,J1),
$ LDG )
IF ( N.GT.J1+2 )
$ CALL DLARFX( 'Left', 3, N-J1-2, V, TAU, G(J1,J1+3), LDG,
$ DWORK )
END IF
C
IF ( WANTU ) THEN
C
C Accumulate transformation in the matrices U1 and U2.
C
CALL DLARFX( 'R', N, 3, V, TAU, U1(1,J1), LDU1, DWORK )
CALL DLARFX( 'R', N, 3, V, TAU, U2(1,J1), LDU2, DWORK )
END IF
GO TO 40
C
20 CONTINUE
C
C N1 = 2, N2 = 1: generate elementary reflector H so that:
C
C H ( -X11 ) = ( * )
C ( -X21 ) = ( 0 ).
C ( scale ) = ( 0 )
C
V(1) = -X(1,1)
V(2) = -X(2,1)
V(3) = SCALE
CALL DLARFG( 3, V(1), V(2), 1, TAU )
V(1) = ONE
A33 = A(J3,J3)
C
C Perform swap provisionally on diagonal block in D.
C
CALL DLARFX( 'L', 3, 3, V, TAU, D, LDD, DWORK )
CALL DLARFX( 'R', 3, 3, V, TAU, D, LDD, DWORK )
C
C Test whether to reject swap.
C
IF ( MAX( ABS( D(2,1) ), ABS( D(3,1) ), ABS( D(1,1)-A33 ) )
$ .GT. THRESH ) GO TO 50
C
C Accept swap: apply transformation to the entire matrix A.
C
CALL DLARFX( 'Right', J3, 3, V, TAU, A(1,J1), LDA, DWORK )
CALL DLARFX( 'Left', 3, N-J1, V, TAU, A(J1,J2), LDA, DWORK )
C
A(J1,J1) = A33
A(J2,J1) = ZERO
A(J3,J1) = ZERO
C
C Apply transformation to G.
C
IF ( ISHAM ) THEN
CALL DLARFX( 'Right', J1-1, 3, V, TAU, G(1,J1), LDG, DWORK )
CALL DSYMV( 'Upper', 3, TAU, G(J1,J1), LDG, V, 1, ZERO,
$ DWORK, 1 )
TEMP = -HALF*TAU*DDOT( 3, DWORK, 1, V, 1 )
CALL DAXPY( 3, TEMP, V, 1, DWORK, 1 )
CALL DSYR2( 'Upper', 3, -ONE, V, 1, DWORK, 1, G(J1,J1),
$ LDG )
IF ( N.GT.J1+2 )
$ CALL DLARFX( 'Left', 3, N-J1-2, V, TAU, G(J1,J1+3), LDG,
$ DWORK )
ELSE
CALL DLARFX( 'Right', J1-1, 3, V, TAU, G(1,J1), LDG, DWORK )
CALL MB01MD( 'Upper', 3, TAU, G(J1,J1), LDG, V, 1, ZERO,
$ DWORK, 1 )
CALL MB01ND( 'Upper', 3, ONE, V, 1, DWORK, 1, G(J1,J1),
$ LDG )
IF ( N.GT.J1+2 )
$ CALL DLARFX( 'Left', 3, N-J1-2, V, TAU, G(J1,J1+3), LDG,
$ DWORK )
END IF
C
IF ( WANTU ) THEN
C
C Accumulate transformation in the matrices U1 and U2.
C
CALL DLARFX( 'R', N, 3, V, TAU, U1(1,J1), LDU1, DWORK )
CALL DLARFX( 'R', N, 3, V, TAU, U2(1,J1), LDU2, DWORK )
END IF
GO TO 40
C
30 CONTINUE
C
C N1 = 2, N2 = 2: generate elementary reflectors H(1) and H(2) so
C that:
C
C H(2) H(1) ( -X11 -X12 ) = ( * * )
C ( -X21 -X22 ) ( 0 * ).
C ( scale 0 ) ( 0 0 )
C ( 0 scale ) ( 0 0 )
C
V1(1) = -X(1,1)
V1(2) = -X(2,1)
V1(3) = SCALE
CALL DLARFG( 3, V1(1), V1(2), 1, TAU1 )
V1(1) = ONE
C
TEMP = -TAU1*( X(1,2)+V1(2)*X(2,2) )
V2(1) = -TEMP*V1(2) - X(2,2)
V2(2) = -TEMP*V1(3)
V2(3) = SCALE
CALL DLARFG( 3, V2(1), V2(2), 1, TAU2 )
V2(1) = ONE
C
C Perform swap provisionally on diagonal block in D.
C
CALL DLARFX( 'L', 3, 4, V1, TAU1, D, LDD, DWORK )
CALL DLARFX( 'R', 4, 3, V1, TAU1, D, LDD, DWORK )
CALL DLARFX( 'L', 3, 4, V2, TAU2, D(2,1), LDD, DWORK )
CALL DLARFX( 'R', 4, 3, V2, TAU2, D(1,2), LDD, DWORK )
C
C Test whether to reject swap.
C
IF ( MAX( ABS( D(3,1) ), ABS( D(3,2) ), ABS( D(4,1) ),
$ ABS( D(4,2) ) ).GT.THRESH ) GO TO 50
C
C Accept swap: apply transformation to the entire matrix A.
C
CALL DLARFX( 'L', 3, N-J1+1, V1, TAU1, A(J1,J1), LDA, DWORK )
CALL DLARFX( 'R', J4, 3, V1, TAU1, A(1,J1), LDA, DWORK )
CALL DLARFX( 'L', 3, N-J1+1, V2, TAU2, A(J2,J1), LDA, DWORK )
CALL DLARFX( 'R', J4, 3, V2, TAU2, A(1,J2), LDA, DWORK )
C
A(J3,J1) = ZERO
A(J3,J2) = ZERO
A(J4,J1) = ZERO
A(J4,J2) = ZERO
C
C Apply transformation to G.
C
IF ( ISHAM ) THEN
CALL DLARFX( 'Right', J1-1, 3, V1, TAU1, G(1,J1), LDG,
$ DWORK )
CALL DSYMV( 'Upper', 3, TAU1, G(J1,J1), LDG, V1, 1, ZERO,
$ DWORK, 1 )
TEMP = -HALF*TAU1*DDOT( 3, DWORK, 1, V1, 1 )
CALL DAXPY( 3, TEMP, V1, 1, DWORK, 1 )
CALL DSYR2( 'Upper', 3, -ONE, V1, 1, DWORK, 1,
$ G(J1,J1), LDG )
IF ( N.GT.J1+2 )
$ CALL DLARFX( 'Left', 3, N-J1-2, V1, TAU1, G(J1,J1+3),
$ LDG, DWORK )
C
CALL DLARFX( 'Right', J2-1, 3, V2, TAU2, G(1,J2), LDG,
$ DWORK )
CALL DSYMV( 'Upper', 3, TAU2, G(J2,J2), LDG, V2, 1, ZERO,
$ DWORK, 1 )
TEMP = -HALF*TAU2*DDOT( 3, DWORK, 1, V2, 1 )
CALL DAXPY( 3, TEMP, V2, 1, DWORK, 1 )
CALL DSYR2( 'Upper', 3, -ONE, V2, 1, DWORK, 1, G(J2,J2),
$ LDG )
IF ( N.GT.J2+2 )
$ CALL DLARFX( 'Left', 3, N-J2-2, V2, TAU2, G(J2,J2+3),
$ LDG, DWORK )
ELSE
CALL DLARFX( 'Right', J1-1, 3, V1, TAU1, G(1,J1), LDG,
$ DWORK )
CALL MB01MD( 'Upper', 3, TAU1, G(J1,J1), LDG, V1, 1, ZERO,
$ DWORK, 1 )
CALL MB01ND( 'Upper', 3, ONE, V1, 1, DWORK, 1, G(J1,J1),
$ LDG )
IF ( N.GT.J1+2 )
$ CALL DLARFX( 'Left', 3, N-J1-2, V1, TAU1, G(J1,J1+3),
$ LDG, DWORK )
CALL DLARFX( 'Right', J2-1, 3, V2, TAU2, G(1,J2), LDG,
$ DWORK )
CALL MB01MD( 'Upper', 3, TAU2, G(J2,J2), LDG, V2, 1, ZERO,
$ DWORK, 1 )
CALL MB01ND( 'Upper', 3, ONE, V2, 1, DWORK, 1, G(J2,J2),
$ LDG )
IF ( N.GT.J2+2 )
$ CALL DLARFX( 'Left', 3, N-J2-2, V2, TAU2, G(J2,J2+3),
$ LDG, DWORK )
END IF
C
IF ( WANTU ) THEN
C
C Accumulate transformation in the matrices U1 and U2.
C
CALL DLARFX( 'R', N, 3, V1, TAU1, U1(1,J1), LDU1, DWORK )
CALL DLARFX( 'R', N, 3, V2, TAU2, U1(1,J2), LDU1, DWORK )
CALL DLARFX( 'R', N, 3, V1, TAU1, U2(1,J1), LDU2, DWORK )
CALL DLARFX( 'R', N, 3, V2, TAU2, U2(1,J2), LDU2, DWORK )
END IF
C
40 CONTINUE
C
IF ( N2.EQ.2 ) THEN
C
C Standardize new 2-by-2 block A11.
C
CALL DLANV2( A(J1,J1), A(J1,J2), A(J2,J1), A(J2,J2), WR1,
$ WI1, WR2, WI2, CS, SN )
CALL DROT( N-J1-1, A(J1,J1+2), LDA, A(J2,J1+2), LDA, CS,
$ SN )
CALL DROT( J1-1, A(1,J1), 1, A(1,J2), 1, CS, SN )
IF ( ISHAM ) THEN
TEMP = G(J1,J2)
CALL DROT( J1, G(1,J1), 1, G(1,J2), 1, CS, SN )
TAU = CS*TEMP + SN*G(J2,J2)
G(J2,J2) = CS*G(J2,J2) - SN*TEMP
G(J1,J1) = CS*G(J1,J1) + SN*TAU
CALL DROT( N-J1, G(J1,J2), LDG, G(J2,J2), LDG, CS, SN )
ELSE
IF ( N.GT.J1+1 )
$ CALL DROT( N-J1-1, G(J1,J1+2), LDG, G(J2,J1+2), LDG,
$ CS, SN )
CALL DROT( J1-1, G(1,J1), 1, G(1,J2), 1, CS, SN )
END IF
IF ( WANTU ) THEN
CALL DROT( N, U1(1,J1), 1, U1(1,J2), 1, CS, SN )
CALL DROT( N, U2(1,J1), 1, U2(1,J2), 1, CS, SN )
END IF
END IF
C
IF ( N1.EQ.2 ) THEN
C
C Standardize new 2-by-2 block A22.
C
J3 = J1 + N2
J4 = J3 + 1
CALL DLANV2( A(J3,J3), A(J3,J4), A(J4,J3), A(J4,J4), WR1,
$ WI1, WR2, WI2, CS, SN )
IF ( J3+2.LE.N )
$ CALL DROT( N-J3-1, A(J3,J3+2), LDA, A(J4,J3+2), LDA, CS,
$ SN )
CALL DROT( J3-1, A(1,J3), 1, A(1,J4), 1, CS, SN )
IF ( ISHAM ) THEN
TEMP = G(J3,J4)
CALL DROT( J3, G(1,J3), 1, G(1,J4), 1, CS, SN )
TAU = CS*TEMP + SN*G(J4,J4)
G(J4,J4) = CS*G(J4,J4) - SN*TEMP
G(J3,J3) = CS*G(J3,J3) + SN*TAU
CALL DROT( N-J3, G(J3,J4), LDG, G(J4,J4), LDG, CS, SN )
ELSE
IF ( N.GT.J3+1 )
$ CALL DROT( N-J3-1, G(J3,J3+2), LDG, G(J4,J3+2), LDG,
$ CS, SN )
CALL DROT( J3-1, G(1,J3), 1, G(1,J4), 1, CS, SN )
END IF
IF ( WANTU ) THEN
CALL DROT( N, U1(1,J3), 1, U1(1,J4), 1, CS, SN )
CALL DROT( N, U2(1,J3), 1, U2(1,J4), 1, CS, SN )
END IF
END IF
C
END IF
RETURN
C
C Exit with INFO = 1 if swap was rejected.
C
50 CONTINUE
INFO = 1
RETURN
C *** Last line of MB03TS ***
END
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