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SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
$ IWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
$ LWORK, M, N
DOUBLE PRECISION DIF, SCALE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* DTGSYL solves the generalized Sylvester equation:
*
* A * R - L * B = scale * C (1)
* D * R - L * E = scale * F
*
* 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 (real) 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 (1) is equivalent to solve Zx = scale b, where
* Z is defined as
*
* Z = [ kron(In, A) -kron(B', Im) ] (2)
* [ kron(In, D) -kron(E', Im) ].
*
* Here 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.
*
* If TRANS = 'T', DTGSYL solves the transposed system Z'*y = scale*b,
* 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 (TRANS = 'T') is used to compute an one-norm-based estimate
* of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
* and (B,E), using DLACON.
*
* If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
* of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
* reciprocal of the smallest singular value of Z. See [1-2] for more
* information.
*
* This is a level 3 BLAS algorithm.
*
* 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: The functionality of 0 and 3.
* =2: The functionality of 0 and 4.
* =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
* (look ahead strategy IJOB = 1 is used).
* =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
* ( DGECON on sub-systems is used ).
* Not referenced if TRANS = 'T'.
*
* M (input) INTEGER
* The order of the matrices A and D, and the row dimension of
* the matrices C, F, R and L.
*
* N (input) INTEGER
* The order of the matrices B and E, and the column dimension
* of the matrices C, F, R and L.
*
* A (input) DOUBLE PRECISION array, dimension (LDA, M)
* The upper quasi triangular matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1, M).
*
* B (input) DOUBLE PRECISION array, dimension (LDB, N)
* The upper quasi triangular matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array 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) or (3).
* On exit, if IJOB = 0, 1 or 2, C has been overwritten by
* the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
* the solution achieved during the computation of the
* Dif-estimate.
*
* LDC (input) INTEGER
* The leading dimension of the array C. LDC >= max(1, M).
*
* D (input) DOUBLE PRECISION array, dimension (LDD, M)
* The upper triangular matrix D.
*
* LDD (input) INTEGER
* The leading dimension of the array D. LDD >= max(1, M).
*
* E (input) DOUBLE PRECISION array, dimension (LDE, N)
* The upper triangular matrix E.
*
* LDE (input) INTEGER
* The leading dimension of the array 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) or (3).
* On exit, if IJOB = 0, 1 or 2, F has been overwritten by
* the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
* the solution achieved during the computation of the
* Dif-estimate.
*
* LDF (input) INTEGER
* The leading dimension of the array F. LDF >= max(1, M).
*
* DIF (output) DOUBLE PRECISION
* On exit DIF is the reciprocal of a lower bound of the
* reciprocal of the Dif-function, i.e. DIF is an upper bound of
* Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
* IF IJOB = 0 or TRANS = 'T', DIF is not touched.
*
* SCALE (output) DOUBLE PRECISION
* On exit SCALE is the scaling factor in (1) or (3).
* If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
* to a slightly perturbed system but the input matrices A, B, D
* and E have not been changed. If SCALE = 0, C and F hold the
* solutions R and L, respectively, to the homogeneous system
* with C = F = 0. Normally, SCALE = 1.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK > = 1.
* If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* IWORK (workspace) INTEGER array, dimension (M+N+6)
*
* INFO (output) INTEGER
* =0: successful exit
* <0: If INFO = -i, the i-th argument had an illegal value.
* >0: (A, D) and (B, E) have common or close eigenvalues.
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
* for Solving the Generalized Sylvester Equation and Estimating the
* Separation between Regular Matrix Pairs, Report UMINF - 93.23,
* Department of Computing Science, Umea University, S-901 87 Umea,
* Sweden, December 1993, Revised April 1994, Also as LAPACK Working
* Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
* No 1, 1996.
*
* [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
* Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
* Appl., 15(4):1045-1060, 1994
*
* [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
* Condition Estimators for Solving the Generalized Sylvester
* Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
* July 1989, pp 745-751.
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET.
* Sven Hammarling, 1/5/02.
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, NOTRAN
INTEGER I, IE, IFUNC, IROUND, IS, ISOLVE, J, JE, JS, K,
$ LINFO, LWMIN, MB, NB, P, PPQQ, PQ, Q
DOUBLE PRECISION DSCALE, DSUM, SCALE2, SCALOC
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, DSCAL, DTGSY2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test input parameters
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -1
ELSE IF( NOTRAN ) THEN
IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.4 ) ) 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 = -6
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.EQ.0 ) THEN
IF( NOTRAN ) THEN
IF( IJOB.EQ.1 .OR. IJOB.EQ.2 ) THEN
LWMIN = MAX( 1, 2*M*N )
ELSE
LWMIN = 1
END IF
ELSE
LWMIN = 1
END IF
WORK( 1 ) = LWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -20
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSYL', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
SCALE = 1
IF( NOTRAN ) THEN
IF( IJOB.NE.0 ) THEN
DIF = 0
END IF
END IF
RETURN
END IF
*
* Determine optimal block sizes MB and NB
*
MB = ILAENV( 2, 'DTGSYL', TRANS, M, N, -1, -1 )
NB = ILAENV( 5, 'DTGSYL', TRANS, M, N, -1, -1 )
*
ISOLVE = 1
IFUNC = 0
IF( NOTRAN ) THEN
IF( IJOB.GE.3 ) THEN
IFUNC = IJOB - 2
CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
ELSE IF( IJOB.GE.1 ) THEN
ISOLVE = 2
END IF
END IF
*
IF( ( MB.LE.1 .AND. NB.LE.1 ) .OR. ( MB.GE.M .AND. NB.GE.N ) )
$ THEN
*
DO 30 IROUND = 1, ISOLVE
*
* Use unblocked Level 2 solver
*
DSCALE = ZERO
DSUM = ONE
PQ = 0
CALL DTGSY2( TRANS, IFUNC, M, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, E, LDE, F, LDF, SCALE, DSUM, DSCALE,
$ IWORK, PQ, INFO )
IF( DSCALE.NE.ZERO ) THEN
IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
ELSE
DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
END IF
END IF
*
IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
IF( NOTRAN ) THEN
IFUNC = IJOB
END IF
SCALE2 = SCALE
CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
SCALE = SCALE2
END IF
30 CONTINUE
*
RETURN
END IF
*
* Determine block structure of A
*
P = 0
I = 1
40 CONTINUE
IF( I.GT.M )
$ GO TO 50
P = P + 1
IWORK( P ) = I
I = I + MB
IF( I.GE.M )
$ GO TO 50
IF( A( I, I-1 ).NE.ZERO )
$ I = I + 1
GO TO 40
50 CONTINUE
*
IWORK( P+1 ) = M + 1
IF( IWORK( P ).EQ.IWORK( P+1 ) )
$ P = P - 1
*
* Determine block structure of B
*
Q = P + 1
J = 1
60 CONTINUE
IF( J.GT.N )
$ GO TO 70
Q = Q + 1
IWORK( Q ) = J
J = J + NB
IF( J.GE.N )
$ GO TO 70
IF( B( J, J-1 ).NE.ZERO )
$ J = J + 1
GO TO 60
70 CONTINUE
*
IWORK( Q+1 ) = N + 1
IF( IWORK( Q ).EQ.IWORK( Q+1 ) )
$ Q = Q - 1
*
IF( NOTRAN ) THEN
*
DO 150 IROUND = 1, ISOLVE
*
* 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
*
DSCALE = ZERO
DSUM = ONE
PQ = 0
SCALE = ONE
DO 130 J = P + 2, Q
JS = IWORK( J )
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
DO 120 I = P, 1, -1
IS = IWORK( I )
IE = IWORK( I+1 ) - 1
MB = IE - IS + 1
PPQQ = 0
CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
$ B( JS, JS ), LDB, C( IS, JS ), LDC,
$ D( IS, IS ), LDD, E( JS, JS ), LDE,
$ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
$ IWORK( Q+2 ), PPQQ, LINFO )
IF( LINFO.GT.0 )
$ INFO = LINFO
*
PQ = PQ + PPQQ
IF( SCALOC.NE.ONE ) THEN
DO 80 K = 1, JS - 1
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
80 CONTINUE
DO 90 K = JS, JE
CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
90 CONTINUE
DO 100 K = JS, JE
CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
100 CONTINUE
DO 110 K = JE + 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
110 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* 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, C( IS, JS ), LDC, ONE,
$ C( 1, JS ), LDC )
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ D( 1, IS ), LDD, C( IS, JS ), LDC, ONE,
$ F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
$ F( IS, JS ), LDF, B( JS, JE+1 ), LDB,
$ ONE, C( IS, JE+1 ), LDC )
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
$ F( IS, JS ), LDF, E( JS, JE+1 ), LDE,
$ ONE, F( IS, JE+1 ), LDF )
END IF
120 CONTINUE
130 CONTINUE
IF( DSCALE.NE.ZERO ) THEN
IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
ELSE
DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
END IF
END IF
IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
IF( NOTRAN ) THEN
IFUNC = IJOB
END IF
SCALE2 = SCALE
CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
SCALE = SCALE2
END IF
150 CONTINUE
*
ELSE
*
* Solve transposed (I, J)-subsystem
* A(I, I)' * R(I, J) + D(I, I)' * L(I, J) = C(I, J)
* R(I, J) * B(J, J)' + L(I, J) * E(J, J)' = -F(I, J)
* for I = 1,2,..., P; J = Q, Q-1,..., 1
*
SCALE = ONE
DO 210 I = 1, P
IS = IWORK( I )
IE = IWORK( I+1 ) - 1
MB = IE - IS + 1
DO 200 J = Q, P + 2, -1
JS = IWORK( J )
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
$ B( JS, JS ), LDB, C( IS, JS ), LDC,
$ D( IS, IS ), LDD, E( JS, JS ), LDE,
$ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
$ IWORK( Q+2 ), PPQQ, LINFO )
IF( LINFO.GT.0 )
$ INFO = LINFO
IF( SCALOC.NE.ONE ) THEN
DO 160 K = 1, JS - 1
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
160 CONTINUE
DO 170 K = JS, JE
CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
170 CONTINUE
DO 180 K = JS, JE
CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
180 CONTINUE
DO 190 K = JE + 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
190 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* 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
200 CONTINUE
210 CONTINUE
*
END IF
*
WORK( 1 ) = LWMIN
*
RETURN
*
* End of DTGSYL
*
END
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