*> \brief \b DTGSY2 solves the generalized Sylvester equation (unblocked algorithm). * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DTGSY2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, * LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, * IWORK, PQ, INFO ) * * .. 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, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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**T, Im) ] (2) *> [ kron(In, D) -kron(E**T, Im) ], *> *> Ik is the identity matrix of size k and X**T 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**T*y = scale*b for y, *> which is equivalent to solve for R and L in *> *> A**T * R + D**T * L = scale * C (3) *> R * B**T + L * E**T = 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] TRANS *> \verbatim *> TRANS is CHARACTER*1 *> = 'N', solve the generalized Sylvester equation (1). *> = 'T': solve the 'transposed' system (3). *> \endverbatim *> *> \param[in] IJOB *> \verbatim *> IJOB is 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'. *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> On entry, M specifies the order of A and D, and the row *> dimension of C, F, R and L. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> On entry, N specifies the order of B and E, and the column *> dimension of C, F, R and L. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is DOUBLE PRECISION array, dimension (LDA, M) *> On entry, A contains an upper quasi triangular matrix. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the matrix A. LDA >= max(1, M). *> \endverbatim *> *> \param[in] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB, N) *> On entry, B contains an upper quasi triangular matrix. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the matrix B. LDB >= max(1, N). *> \endverbatim *> *> \param[in,out] C *> \verbatim *> C is 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. *> \endverbatim *> *> \param[in] LDC *> \verbatim *> LDC is INTEGER *> The leading dimension of the matrix C. LDC >= max(1, M). *> \endverbatim *> *> \param[in] D *> \verbatim *> D is DOUBLE PRECISION array, dimension (LDD, M) *> On entry, D contains an upper triangular matrix. *> \endverbatim *> *> \param[in] LDD *> \verbatim *> LDD is INTEGER *> The leading dimension of the matrix D. LDD >= max(1, M). *> \endverbatim *> *> \param[in] E *> \verbatim *> E is DOUBLE PRECISION array, dimension (LDE, N) *> On entry, E contains an upper triangular matrix. *> \endverbatim *> *> \param[in] LDE *> \verbatim *> LDE is INTEGER *> The leading dimension of the matrix E. LDE >= max(1, N). *> \endverbatim *> *> \param[in,out] F *> \verbatim *> F is 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. *> \endverbatim *> *> \param[in] LDF *> \verbatim *> LDF is INTEGER *> The leading dimension of the matrix F. LDF >= max(1, M). *> \endverbatim *> *> \param[out] SCALE *> \verbatim *> SCALE is 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. *> \endverbatim *> *> \param[in,out] RDSUM *> \verbatim *> RDSUM is 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. *> \endverbatim *> *> \param[in,out] RDSCAL *> \verbatim *> RDSCAL is 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. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (M+N+2) *> \endverbatim *> *> \param[out] PQ *> \verbatim *> PQ is INTEGER *> On exit, the number of subsystems (of size 2-by-2, 4-by-4 and *> 8-by-8) solved by this routine. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup doubleSYauxiliary * *> \par Contributors: * ================== *> *> Bo Kagstrom and Peter Poromaa, Department of Computing Science, *> Umea University, S-901 87 Umea, Sweden. * * ===================================================================== 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 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. 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, * ) * .. * * ===================================================================== * 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 = -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.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)**T * R(I, J) + D(I, I)**T * 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 = IWORK ( 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**T * 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**T * 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**T * 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**T * 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**T * 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**T * 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**T * 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**T * 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