SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
$ IWORK, PQ, INFO )
*
* -- LAPACK auxiliary routine (version 3.1.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* January 2007
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
$ PQ
DOUBLE PRECISION RDSCAL, RDSUM, SCALE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
* ..
*
* Purpose
* =======
*
* DTGSY2 solves the generalized Sylvester equation:
*
* A * R - L * B = scale * C (1)
* D * R - L * E = scale * F,
*
* using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
* (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
* N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
* must be in generalized Schur canonical form, i.e. A, B are upper
* quasi triangular and D, E are upper triangular. The solution (R, L)
* overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
* chosen to avoid overflow.
*
* In matrix notation solving equation (1) corresponds to solve
* Z*x = scale*b, where Z is defined as
*
* Z = [ kron(In, A) -kron(B', Im) ] (2)
* [ kron(In, D) -kron(E', Im) ],
*
* Ik is the identity matrix of size k and X' is the transpose of X.
* kron(X, Y) is the Kronecker product between the matrices X and Y.
* In the process of solving (1), we solve a number of such systems
* where Dim(In), Dim(In) = 1 or 2.
*
* If TRANS = 'T', solve the transposed system Z'*y = scale*b for y,
* which is equivalent to solve for R and L in
*
* A' * R + D' * L = scale * C (3)
* R * B' + L * E' = scale * -F
*
* This case is used to compute an estimate of Dif[(A, D), (B, E)] =
* sigma_min(Z) using reverse communicaton with DLACON.
*
* DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
* of an upper bound on the separation between to matrix pairs. Then
* the input (A, D), (B, E) are sub-pencils of the matrix pair in
* DTGSYL. See DTGSYL for details.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* = 'N', solve the generalized Sylvester equation (1).
* = 'T': solve the 'transposed' system (3).
*
* IJOB (input) INTEGER
* Specifies what kind of functionality to be performed.
* = 0: solve (1) only.
* = 1: A contribution from this subsystem to a Frobenius
* norm-based estimate of the separation between two matrix
* pairs is computed. (look ahead strategy is used).
* = 2: A contribution from this subsystem to a Frobenius
* norm-based estimate of the separation between two matrix
* pairs is computed. (DGECON on sub-systems is used.)
* Not referenced if TRANS = 'T'.
*
* M (input) INTEGER
* On entry, M specifies the order of A and D, and the row
* dimension of C, F, R and L.
*
* N (input) INTEGER
* On entry, N specifies the order of B and E, and the column
* dimension of C, F, R and L.
*
* A (input) DOUBLE PRECISION array, dimension (LDA, M)
* On entry, A contains an upper quasi triangular matrix.
*
* LDA (input) INTEGER
* The leading dimension of the matrix A. LDA >= max(1, M).
*
* B (input) DOUBLE PRECISION array, dimension (LDB, N)
* On entry, B contains an upper quasi triangular matrix.
*
* LDB (input) INTEGER
* The leading dimension of the matrix B. LDB >= max(1, N).
*
* C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
* On entry, C contains the right-hand-side of the first matrix
* equation in (1).
* On exit, if IJOB = 0, C has been overwritten by the
* solution R.
*
* LDC (input) INTEGER
* The leading dimension of the matrix C. LDC >= max(1, M).
*
* D (input) DOUBLE PRECISION array, dimension (LDD, M)
* On entry, D contains an upper triangular matrix.
*
* LDD (input) INTEGER
* The leading dimension of the matrix D. LDD >= max(1, M).
*
* E (input) DOUBLE PRECISION array, dimension (LDE, N)
* On entry, E contains an upper triangular matrix.
*
* LDE (input) INTEGER
* The leading dimension of the matrix E. LDE >= max(1, N).
*
* F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
* On entry, F contains the right-hand-side of the second matrix
* equation in (1).
* On exit, if IJOB = 0, F has been overwritten by the
* solution L.
*
* LDF (input) INTEGER
* The leading dimension of the matrix F. LDF >= max(1, M).
*
* SCALE (output) DOUBLE PRECISION
* On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
* R and L (C and F on entry) will hold the solutions to a
* slightly perturbed system but the input matrices A, B, D and
* E have not been changed. If SCALE = 0, R and L will hold the
* solutions to the homogeneous system with C = F = 0. Normally,
* SCALE = 1.
*
* RDSUM (input/output) DOUBLE PRECISION
* On entry, the sum of squares of computed contributions to
* the Dif-estimate under computation by DTGSYL, where the
* scaling factor RDSCAL (see below) has been factored out.
* On exit, the corresponding sum of squares updated with the
* contributions from the current sub-system.
* If TRANS = 'T' RDSUM is not touched.
* NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.
*
* RDSCAL (input/output) DOUBLE PRECISION
* On entry, scaling factor used to prevent overflow in RDSUM.
* On exit, RDSCAL is updated w.r.t. the current contributions
* in RDSUM.
* If TRANS = 'T', RDSCAL is not touched.
* NOTE: RDSCAL only makes sense when DTGSY2 is called by
* DTGSYL.
*
* IWORK (workspace) INTEGER array, dimension (M+N+2)
*
* PQ (output) INTEGER
* On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
* 8-by-8) solved by this routine.
*
* INFO (output) INTEGER
* On exit, if INFO is set to
* =0: Successful exit
* <0: If INFO = -i, the i-th argument had an illegal value.
* >0: The matrix pairs (A, D) and (B, E) have common or very
* close eigenvalues.
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET.
* Sven Hammarling, 27/5/02.
*
* .. Parameters ..
INTEGER LDZ
PARAMETER ( LDZ = 8 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER I, IE, IERR, II, IS, ISP1, J, JE, JJ, JS, JSP1,
$ K, MB, NB, P, Q, ZDIM
DOUBLE PRECISION ALPHA, SCALOC
* ..
* .. Local Arrays ..
INTEGER IPIV( LDZ ), JPIV( LDZ )
DOUBLE PRECISION RHS( LDZ ), Z( LDZ, LDZ )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DGER, DGESC2,
$ DGETC2, DLASET, DLATDF, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode and test input parameters
*
INFO = 0
IERR = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -1
ELSE IF( NOTRAN ) THEN
IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
INFO = -2
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( M.LE.0 ) THEN
INFO = -3
ELSE IF( N.LE.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
INFO = -12
ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
INFO = -16
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSY2', -INFO )
RETURN
END IF
*
* Determine block structure of A
*
PQ = 0
P = 0
I = 1
10 CONTINUE
IF( I.GT.M )
$ GO TO 20
P = P + 1
IWORK( P ) = I
IF( I.EQ.M )
$ GO TO 20
IF( A( I+1, I ).NE.ZERO ) THEN
I = I + 2
ELSE
I = I + 1
END IF
GO TO 10
20 CONTINUE
IWORK( P+1 ) = M + 1
*
* Determine block structure of B
*
Q = P + 1
J = 1
30 CONTINUE
IF( J.GT.N )
$ GO TO 40
Q = Q + 1
IWORK( Q ) = J
IF( J.EQ.N )
$ GO TO 40
IF( B( J+1, J ).NE.ZERO ) THEN
J = J + 2
ELSE
J = J + 1
END IF
GO TO 30
40 CONTINUE
IWORK( Q+1 ) = N + 1
PQ = P*( Q-P-1 )
*
IF( NOTRAN ) THEN
*
* Solve (I, J) - subsystem
* A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
* D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
* for I = P, P - 1, ..., 1; J = 1, 2, ..., Q
*
SCALE = ONE
SCALOC = ONE
DO 120 J = P + 2, Q
JS = IWORK( J )
JSP1 = JS + 1
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
DO 110 I = P, 1, -1
*
IS = IWORK( I )
ISP1 = IS + 1
IE = IWORK( I+1 ) - 1
MB = IE - IS + 1
ZDIM = MB*NB*2
*
IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 2-by-2 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = D( IS, IS )
Z( 1, 2 ) = -B( JS, JS )
Z( 2, 2 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = F( IS, JS )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 50 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
50 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
F( IS, JS ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
ALPHA = -RHS( 1 )
CALL DAXPY( IS-1, ALPHA, A( 1, IS ), 1, C( 1, JS ),
$ 1 )
CALL DAXPY( IS-1, ALPHA, D( 1, IS ), 1, F( 1, JS ),
$ 1 )
END IF
IF( J.LT.Q ) THEN
CALL DAXPY( N-JE, RHS( 2 ), B( JS, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 2 ), E( JS, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build a 4-by-4 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = ZERO
Z( 3, 1 ) = D( IS, IS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = ZERO
Z( 2, 2 ) = A( IS, IS )
Z( 3, 2 ) = ZERO
Z( 4, 2 ) = D( IS, IS )
*
Z( 1, 3 ) = -B( JS, JS )
Z( 2, 3 ) = -B( JS, JSP1 )
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = -E( JS, JSP1 )
*
Z( 1, 4 ) = -B( JSP1, JS )
Z( 2, 4 ) = -B( JSP1, JSP1 )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( IS, JSP1 )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( IS, JSP1 )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 60 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
60 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( IS, JSP1 ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( IS, JSP1 ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGER( IS-1, NB, -ONE, A( 1, IS ), 1, RHS( 1 ),
$ 1, C( 1, JS ), LDC )
CALL DGER( IS-1, NB, -ONE, D( 1, IS ), 1, RHS( 1 ),
$ 1, F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
CALL DAXPY( N-JE, RHS( 3 ), B( JS, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 3 ), E( JS, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
CALL DAXPY( N-JE, RHS( 4 ), B( JSP1, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 4 ), E( JSP1, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 4-by-4 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( ISP1, IS )
Z( 3, 1 ) = D( IS, IS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = A( IS, ISP1 )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 3, 2 ) = D( IS, ISP1 )
Z( 4, 2 ) = D( ISP1, ISP1 )
*
Z( 1, 3 ) = -B( JS, JS )
Z( 2, 3 ) = ZERO
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = -B( JS, JS )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( ISP1, JS )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( ISP1, JS )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 70 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
70 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( ISP1, JS ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( ISP1, JS ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGEMV( 'N', IS-1, MB, -ONE, A( 1, IS ), LDA,
$ RHS( 1 ), 1, ONE, C( 1, JS ), 1 )
CALL DGEMV( 'N', IS-1, MB, -ONE, D( 1, IS ), LDD,
$ RHS( 1 ), 1, ONE, F( 1, JS ), 1 )
END IF
IF( J.LT.Q ) THEN
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
$ B( JS, JE+1 ), LDB, C( IS, JE+1 ), LDC )
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
$ E( JS, JE+1 ), LDE, F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build an 8-by-8 system Z * x = RHS
*
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( ISP1, IS )
Z( 5, 1 ) = D( IS, IS )
*
Z( 1, 2 ) = A( IS, ISP1 )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 5, 2 ) = D( IS, ISP1 )
Z( 6, 2 ) = D( ISP1, ISP1 )
*
Z( 3, 3 ) = A( IS, IS )
Z( 4, 3 ) = A( ISP1, IS )
Z( 7, 3 ) = D( IS, IS )
*
Z( 3, 4 ) = A( IS, ISP1 )
Z( 4, 4 ) = A( ISP1, ISP1 )
Z( 7, 4 ) = D( IS, ISP1 )
Z( 8, 4 ) = D( ISP1, ISP1 )
*
Z( 1, 5 ) = -B( JS, JS )
Z( 3, 5 ) = -B( JS, JSP1 )
Z( 5, 5 ) = -E( JS, JS )
Z( 7, 5 ) = -E( JS, JSP1 )
*
Z( 2, 6 ) = -B( JS, JS )
Z( 4, 6 ) = -B( JS, JSP1 )
Z( 6, 6 ) = -E( JS, JS )
Z( 8, 6 ) = -E( JS, JSP1 )
*
Z( 1, 7 ) = -B( JSP1, JS )
Z( 3, 7 ) = -B( JSP1, JSP1 )
Z( 7, 7 ) = -E( JSP1, JSP1 )
*
Z( 2, 8 ) = -B( JSP1, JS )
Z( 4, 8 ) = -B( JSP1, JSP1 )
Z( 8, 8 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
K = 1
II = MB*NB + 1
DO 80 JJ = 0, NB - 1
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 )
K = K + MB
II = II + MB
80 CONTINUE
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 90 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
90 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
K = 1
II = MB*NB + 1
DO 100 JJ = 0, NB - 1
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 )
K = K + MB
II = II + MB
100 CONTINUE
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ A( 1, IS ), LDA, RHS( 1 ), MB, ONE,
$ C( 1, JS ), LDC )
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ D( 1, IS ), LDD, RHS( 1 ), MB, ONE,
$ F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
K = MB*NB + 1
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ),
$ MB, B( JS, JE+1 ), LDB, ONE,
$ C( IS, JE+1 ), LDC )
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ),
$ MB, E( JS, JE+1 ), LDE, ONE,
$ F( IS, JE+1 ), LDF )
END IF
*
END IF
*
110 CONTINUE
120 CONTINUE
ELSE
*
* Solve (I, J) - subsystem
* A(I, I)' * R(I, J) + D(I, I)' * L(J, J) = C(I, J)
* R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
* for I = 1, 2, ..., P, J = Q, Q - 1, ..., 1
*
SCALE = ONE
SCALOC = ONE
DO 200 I = 1, P
*
IS = IWORK( I )
ISP1 = IS + 1
IE = ( I+1 ) - 1
MB = IE - IS + 1
DO 190 J = Q, P + 2, -1
*
JS = IWORK( J )
JSP1 = JS + 1
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
ZDIM = MB*NB*2
IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 2-by-2 system Z' * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = -B( JS, JS )
Z( 1, 2 ) = D( IS, IS )
Z( 2, 2 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = F( IS, JS )
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 130 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
130 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
F( IS, JS ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
ALPHA = RHS( 1 )
CALL DAXPY( JS-1, ALPHA, B( 1, JS ), 1, F( IS, 1 ),
$ LDF )
ALPHA = RHS( 2 )
CALL DAXPY( JS-1, ALPHA, E( 1, JS ), 1, F( IS, 1 ),
$ LDF )
END IF
IF( I.LT.P ) THEN
ALPHA = -RHS( 1 )
CALL DAXPY( M-IE, ALPHA, A( IS, IE+1 ), LDA,
$ C( IE+1, JS ), 1 )
ALPHA = -RHS( 2 )
CALL DAXPY( M-IE, ALPHA, D( IS, IE+1 ), LDD,
$ C( IE+1, JS ), 1 )
END IF
*
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build a 4-by-4 system Z' * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = ZERO
Z( 3, 1 ) = -B( JS, JS )
Z( 4, 1 ) = -B( JSP1, JS )
*
Z( 1, 2 ) = ZERO
Z( 2, 2 ) = A( IS, IS )
Z( 3, 2 ) = -B( JS, JSP1 )
Z( 4, 2 ) = -B( JSP1, JSP1 )
*
Z( 1, 3 ) = D( IS, IS )
Z( 2, 3 ) = ZERO
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = D( IS, IS )
Z( 3, 4 ) = -E( JS, JSP1 )
Z( 4, 4 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( IS, JSP1 )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( IS, JSP1 )
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 140 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
140 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( IS, JSP1 ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( IS, JSP1 ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DAXPY( JS-1, RHS( 1 ), B( 1, JS ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 2 ), B( 1, JSP1 ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 3 ), E( 1, JS ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 4 ), E( 1, JSP1 ), 1,
$ F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGER( M-IE, NB, -ONE, A( IS, IE+1 ), LDA,
$ RHS( 1 ), 1, C( IE+1, JS ), LDC )
CALL DGER( M-IE, NB, -ONE, D( IS, IE+1 ), LDD,
$ RHS( 3 ), 1, C( IE+1, JS ), LDC )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 4-by-4 system Z' * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( IS, ISP1 )
Z( 3, 1 ) = -B( JS, JS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = A( ISP1, IS )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 3, 2 ) = ZERO
Z( 4, 2 ) = -B( JS, JS )
*
Z( 1, 3 ) = D( IS, IS )
Z( 2, 3 ) = D( IS, ISP1 )
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = D( ISP1, ISP1 )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( ISP1, JS )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( ISP1, JS )
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 150 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
150 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( ISP1, JS ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( ISP1, JS ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DGER( MB, JS-1, ONE, RHS( 1 ), 1, B( 1, JS ),
$ 1, F( IS, 1 ), LDF )
CALL DGER( MB, JS-1, ONE, RHS( 3 ), 1, E( 1, JS ),
$ 1, F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGEMV( 'T', MB, M-IE, -ONE, A( IS, IE+1 ),
$ LDA, RHS( 1 ), 1, ONE, C( IE+1, JS ),
$ 1 )
CALL DGEMV( 'T', MB, M-IE, -ONE, D( IS, IE+1 ),
$ LDD, RHS( 3 ), 1, ONE, C( IE+1, JS ),
$ 1 )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build an 8-by-8 system Z' * x = RHS
*
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( IS, ISP1 )
Z( 5, 1 ) = -B( JS, JS )
Z( 7, 1 ) = -B( JSP1, JS )
*
Z( 1, 2 ) = A( ISP1, IS )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 6, 2 ) = -B( JS, JS )
Z( 8, 2 ) = -B( JSP1, JS )
*
Z( 3, 3 ) = A( IS, IS )
Z( 4, 3 ) = A( IS, ISP1 )
Z( 5, 3 ) = -B( JS, JSP1 )
Z( 7, 3 ) = -B( JSP1, JSP1 )
*
Z( 3, 4 ) = A( ISP1, IS )
Z( 4, 4 ) = A( ISP1, ISP1 )
Z( 6, 4 ) = -B( JS, JSP1 )
Z( 8, 4 ) = -B( JSP1, JSP1 )
*
Z( 1, 5 ) = D( IS, IS )
Z( 2, 5 ) = D( IS, ISP1 )
Z( 5, 5 ) = -E( JS, JS )
*
Z( 2, 6 ) = D( ISP1, ISP1 )
Z( 6, 6 ) = -E( JS, JS )
*
Z( 3, 7 ) = D( IS, IS )
Z( 4, 7 ) = D( IS, ISP1 )
Z( 5, 7 ) = -E( JS, JSP1 )
Z( 7, 7 ) = -E( JSP1, JSP1 )
*
Z( 4, 8 ) = D( ISP1, ISP1 )
Z( 6, 8 ) = -E( JS, JSP1 )
Z( 8, 8 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
K = 1
II = MB*NB + 1
DO 160 JJ = 0, NB - 1
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 )
K = K + MB
II = II + MB
160 CONTINUE
*
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 170 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
170 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
K = 1
II = MB*NB + 1
DO 180 JJ = 0, NB - 1
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 )
K = K + MB
II = II + MB
180 CONTINUE
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
$ C( IS, JS ), LDC, B( 1, JS ), LDB, ONE,
$ F( IS, 1 ), LDF )
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
$ F( IS, JS ), LDF, E( 1, JS ), LDE, ONE,
$ F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ A( IS, IE+1 ), LDA, C( IS, JS ), LDC,
$ ONE, C( IE+1, JS ), LDC )
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ D( IS, IE+1 ), LDD, F( IS, JS ), LDF,
$ ONE, C( IE+1, JS ), LDC )
END IF
*
END IF
*
190 CONTINUE
200 CONTINUE
*
END IF
RETURN
*
* End of DTGSY2
*
END