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diff --git a/src/fortran/lapack/dtgsyl.f b/src/fortran/lapack/dtgsyl.f new file mode 100644 index 0000000..0186671 --- /dev/null +++ b/src/fortran/lapack/dtgsyl.f @@ -0,0 +1,556 @@ + 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 |