SUBROUTINE ZTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, \$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, \$ INFO ) * * -- LAPACK auxiliary routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER TRANS INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N DOUBLE PRECISION RDSCAL, RDSUM, SCALE * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ), \$ D( LDD, * ), E( LDE, * ), F( LDF, * ) * .. * * Purpose * ======= * * ZTGSY2 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. A, B, D and E are upper triangular * (i.e., (A,D) and (B,E) in generalized Schur form). * * 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 * Zx = 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. * * If TRANS = 'C', y in the conjugate transposed system Z'y = scale*b * is solved for, 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 ZLACON. * * ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL * of an upper bound on the separation between to matrix pairs. Then * the input (A, D), (B, E) are sub-pencils of two matrix pairs in * ZTGSYL. * * 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) COMPLEX*16 array, dimension (LDA, M) * On entry, A contains an upper triangular matrix. * * LDA (input) INTEGER * The leading dimension of the matrix A. LDA >= max(1, M). * * B (input) COMPLEX*16 array, dimension (LDB, N) * On entry, B contains an upper triangular matrix. * * LDB (input) INTEGER * The leading dimension of the matrix B. LDB >= max(1, N). * * C (input/output) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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 ZTGSYL, 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 ZTGSY2 is called by * ZTGSYL. * * 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 ZTGSY2 is called by * ZTGSYL. * * INFO (output) INTEGER * On exit, if INFO is set to * =0: Successful exit * <0: If INFO = -i, input argument number i is illegal. * >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. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE INTEGER LDZ PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, LDZ = 2 ) * .. * .. Local Scalars .. LOGICAL NOTRAN INTEGER I, IERR, J, K DOUBLE PRECISION SCALOC COMPLEX*16 ALPHA * .. * .. Local Arrays .. INTEGER IPIV( LDZ ), JPIV( LDZ ) COMPLEX*16 RHS( LDZ ), Z( LDZ, LDZ ) * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, ZAXPY, ZGESC2, ZGETC2, ZLATDF, ZSCAL * .. * .. Intrinsic Functions .. INTRINSIC DCMPLX, DCONJG, MAX * .. * .. Executable Statements .. * * Decode and test input parameters * INFO = 0 IERR = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'C' ) ) 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( 'ZTGSY2', -INFO ) RETURN END IF * IF( NOTRAN ) THEN * * Solve (I, J) - system * 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 = M, M - 1, ..., 1; J = 1, 2, ..., N * SCALE = ONE SCALOC = ONE DO 30 J = 1, N DO 20 I = M, 1, -1 * * Build 2 by 2 system * Z( 1, 1 ) = A( I, I ) Z( 2, 1 ) = D( I, I ) Z( 1, 2 ) = -B( J, J ) Z( 2, 2 ) = -E( J, J ) * * Set up right hand side(s) * RHS( 1 ) = C( I, J ) RHS( 2 ) = F( I, J ) * * Solve Z * x = RHS * CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) \$ INFO = IERR IF( IJOB.EQ.0 ) THEN CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 10 K = 1, N CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), \$ C( 1, K ), 1 ) CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), \$ F( 1, K ), 1 ) 10 CONTINUE SCALE = SCALE*SCALOC END IF ELSE CALL ZLATDF( IJOB, LDZ, Z, LDZ, RHS, RDSUM, RDSCAL, \$ IPIV, JPIV ) END IF * * Unpack solution vector(s) * C( I, J ) = RHS( 1 ) F( I, J ) = RHS( 2 ) * * Substitute R(I, J) and L(I, J) into remaining equation. * IF( I.GT.1 ) THEN ALPHA = -RHS( 1 ) CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, C( 1, J ), 1 ) CALL ZAXPY( I-1, ALPHA, D( 1, I ), 1, F( 1, J ), 1 ) END IF IF( J.LT.N ) THEN CALL ZAXPY( N-J, RHS( 2 ), B( J, J+1 ), LDB, \$ C( I, J+1 ), LDC ) CALL ZAXPY( N-J, RHS( 2 ), E( J, J+1 ), LDE, \$ F( I, J+1 ), LDF ) END IF * 20 CONTINUE 30 CONTINUE ELSE * * Solve transposed (I, J) - system: * 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, ..., M, J = N, N - 1, ..., 1 * SCALE = ONE SCALOC = ONE DO 80 I = 1, M DO 70 J = N, 1, -1 * * Build 2 by 2 system Z' * Z( 1, 1 ) = DCONJG( A( I, I ) ) Z( 2, 1 ) = -DCONJG( B( J, J ) ) Z( 1, 2 ) = DCONJG( D( I, I ) ) Z( 2, 2 ) = -DCONJG( E( J, J ) ) * * * Set up right hand side(s) * RHS( 1 ) = C( I, J ) RHS( 2 ) = F( I, J ) * * Solve Z' * x = RHS * CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) \$ INFO = IERR CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 40 K = 1, N CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), C( 1, K ), \$ 1 ) CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), F( 1, K ), \$ 1 ) 40 CONTINUE SCALE = SCALE*SCALOC END IF * * Unpack solution vector(s) * C( I, J ) = RHS( 1 ) F( I, J ) = RHS( 2 ) * * Substitute R(I, J) and L(I, J) into remaining equation. * DO 50 K = 1, J - 1 F( I, K ) = F( I, K ) + RHS( 1 )*DCONJG( B( K, J ) ) + \$ RHS( 2 )*DCONJG( E( K, J ) ) 50 CONTINUE DO 60 K = I + 1, M C( K, J ) = C( K, J ) - DCONJG( A( I, K ) )*RHS( 1 ) - \$ DCONJG( D( I, K ) )*RHS( 2 ) 60 CONTINUE * 70 CONTINUE 80 CONTINUE END IF RETURN * * End of ZTGSY2 * END