#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dtgsyl_(char *trans, integer *ijob, integer *m, integer * n, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *c__, integer *ldc, doublereal *d__, integer *ldd, doublereal *e, integer *lde, doublereal *f, integer *ldf, doublereal * scale, doublereal *dif, doublereal *work, integer *lwork, integer * iwork, integer *info) { /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 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. Decode and test input parameters Parameter adjustments */ /* Table of constant values */ static integer c__2 = 2; static integer c_n1 = -1; static integer c__5 = 5; static doublereal c_b14 = 0.; static integer c__1 = 1; static doublereal c_b51 = -1.; static doublereal c_b52 = 1.; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, c_dim1, c_offset, d_dim1, d_offset, e_dim1, e_offset, f_dim1, f_offset, i__1, i__2, i__3, i__4; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, j, k, p, q, ie, je, mb, nb, is, js, pq; static doublereal dsum; static integer ppqq; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *), dgemm_(char *, char *, integer *, integer *, integer * , doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *); extern logical lsame_(char *, char *); static integer ifunc, linfo, lwmin; static doublereal scale2; extern /* Subroutine */ int dtgsy2_(char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *); static doublereal dscale, scaloc; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), dlaset_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern /* Subroutine */ int xerbla_(char *, integer *); static integer iround; static logical notran; static integer isolve; static logical lquery; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; c_dim1 = *ldc; c_offset = 1 + c_dim1; c__ -= c_offset; d_dim1 = *ldd; d_offset = 1 + d_dim1; d__ -= d_offset; e_dim1 = *lde; e_offset = 1 + e_dim1; e -= e_offset; f_dim1 = *ldf; f_offset = 1 + f_dim1; f -= f_offset; --work; --iwork; /* Function Body */ *info = 0; notran = lsame_(trans, "N"); lquery = *lwork == -1; if (! notran && ! lsame_(trans, "T")) { *info = -1; } else if (notran) { if (*ijob < 0 || *ijob > 4) { *info = -2; } } if (*info == 0) { if (*m <= 0) { *info = -3; } else if (*n <= 0) { *info = -4; } else if (*lda < max(1,*m)) { *info = -6; } else if (*ldb < max(1,*n)) { *info = -8; } else if (*ldc < max(1,*m)) { *info = -10; } else if (*ldd < max(1,*m)) { *info = -12; } else if (*lde < max(1,*n)) { *info = -14; } else if (*ldf < max(1,*m)) { *info = -16; } } if (*info == 0) { if (notran) { if (*ijob == 1 || *ijob == 2) { /* Computing MAX */ i__1 = 1, i__2 = (*m << 1) * *n; lwmin = max(i__1,i__2); } else { lwmin = 1; } } else { lwmin = 1; } work[1] = (doublereal) lwmin; if (*lwork < lwmin && ! lquery) { *info = -20; } } if (*info != 0) { i__1 = -(*info); xerbla_("DTGSYL", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*m == 0 || *n == 0) { *scale = 1.; if (notran) { if (*ijob != 0) { *dif = 0.; } } return 0; } /* Determine optimal block sizes MB and NB */ mb = ilaenv_(&c__2, "DTGSYL", trans, m, n, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); nb = ilaenv_(&c__5, "DTGSYL", trans, m, n, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); isolve = 1; ifunc = 0; if (notran) { if (*ijob >= 3) { ifunc = *ijob - 2; dlaset_("F", m, n, &c_b14, &c_b14, &c__[c_offset], ldc) ; dlaset_("F", m, n, &c_b14, &c_b14, &f[f_offset], ldf); } else if (*ijob >= 1) { isolve = 2; } } if (mb <= 1 && nb <= 1 || mb >= *m && nb >= *n) { i__1 = isolve; for (iround = 1; iround <= i__1; ++iround) { /* Use unblocked Level 2 solver */ dscale = 0.; dsum = 1.; pq = 0; dtgsy2_(trans, &ifunc, m, n, &a[a_offset], lda, &b[b_offset], ldb, &c__[c_offset], ldc, &d__[d_offset], ldd, &e[e_offset], lde, &f[f_offset], ldf, scale, &dsum, &dscale, &iwork[1], &pq, info); if (dscale != 0.) { if (*ijob == 1 || *ijob == 3) { *dif = sqrt((doublereal) ((*m << 1) * *n)) / (dscale * sqrt(dsum)); } else { *dif = sqrt((doublereal) pq) / (dscale * sqrt(dsum)); } } if (isolve == 2 && iround == 1) { if (notran) { ifunc = *ijob; } scale2 = *scale; dlacpy_("F", m, n, &c__[c_offset], ldc, &work[1], m); dlacpy_("F", m, n, &f[f_offset], ldf, &work[*m * *n + 1], m); dlaset_("F", m, n, &c_b14, &c_b14, &c__[c_offset], ldc); dlaset_("F", m, n, &c_b14, &c_b14, &f[f_offset], ldf); } else if (isolve == 2 && iround == 2) { dlacpy_("F", m, n, &work[1], m, &c__[c_offset], ldc); dlacpy_("F", m, n, &work[*m * *n + 1], m, &f[f_offset], ldf); *scale = scale2; } /* L30: */ } return 0; } /* Determine block structure of A */ p = 0; i__ = 1; L40: if (i__ > *m) { goto L50; } ++p; iwork[p] = i__; i__ += mb; if (i__ >= *m) { goto L50; } if (a[i__ + (i__ - 1) * a_dim1] != 0.) { ++i__; } goto L40; L50: iwork[p + 1] = *m + 1; if (iwork[p] == iwork[p + 1]) { --p; } /* Determine block structure of B */ q = p + 1; j = 1; L60: if (j > *n) { goto L70; } ++q; iwork[q] = j; j += nb; if (j >= *n) { goto L70; } if (b[j + (j - 1) * b_dim1] != 0.) { ++j; } goto L60; L70: iwork[q + 1] = *n + 1; if (iwork[q] == iwork[q + 1]) { --q; } if (notran) { i__1 = isolve; for (iround = 1; iround <= i__1; ++iround) { /* 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 = 0.; dsum = 1.; pq = 0; *scale = 1.; i__2 = q; for (j = p + 2; j <= i__2; ++j) { js = iwork[j]; je = iwork[j + 1] - 1; nb = je - js + 1; for (i__ = p; i__ >= 1; --i__) { is = iwork[i__]; ie = iwork[i__ + 1] - 1; mb = ie - is + 1; ppqq = 0; dtgsy2_(trans, &ifunc, &mb, &nb, &a[is + is * a_dim1], lda, &b[js + js * b_dim1], ldb, &c__[is + js * c_dim1], ldc, &d__[is + is * d_dim1], ldd, &e[js + js * e_dim1], lde, &f[is + js * f_dim1], ldf, & scaloc, &dsum, &dscale, &iwork[q + 2], &ppqq, & linfo); if (linfo > 0) { *info = linfo; } pq += ppqq; if (scaloc != 1.) { i__3 = js - 1; for (k = 1; k <= i__3; ++k) { dscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); dscal_(m, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L80: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = is - 1; dscal_(&i__4, &scaloc, &c__[k * c_dim1 + 1], & c__1); i__4 = is - 1; dscal_(&i__4, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L90: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = *m - ie; dscal_(&i__4, &scaloc, &c__[ie + 1 + k * c_dim1], &c__1); i__4 = *m - ie; dscal_(&i__4, &scaloc, &f[ie + 1 + k * f_dim1], & c__1); /* L100: */ } i__3 = *n; for (k = je + 1; k <= i__3; ++k) { dscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); dscal_(m, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L110: */ } *scale *= scaloc; } /* Substitute R(I, J) and L(I, J) into remaining equation. */ if (i__ > 1) { i__3 = is - 1; dgemm_("N", "N", &i__3, &nb, &mb, &c_b51, &a[is * a_dim1 + 1], lda, &c__[is + js * c_dim1], ldc, &c_b52, &c__[js * c_dim1 + 1], ldc); i__3 = is - 1; dgemm_("N", "N", &i__3, &nb, &mb, &c_b51, &d__[is * d_dim1 + 1], ldd, &c__[is + js * c_dim1], ldc, &c_b52, &f[js * f_dim1 + 1], ldf); } if (j < q) { i__3 = *n - je; dgemm_("N", "N", &mb, &i__3, &nb, &c_b52, &f[is + js * f_dim1], ldf, &b[js + (je + 1) * b_dim1], ldb, &c_b52, &c__[is + (je + 1) * c_dim1], ldc); i__3 = *n - je; dgemm_("N", "N", &mb, &i__3, &nb, &c_b52, &f[is + js * f_dim1], ldf, &e[js + (je + 1) * e_dim1], lde, &c_b52, &f[is + (je + 1) * f_dim1], ldf); } /* L120: */ } /* L130: */ } if (dscale != 0.) { if (*ijob == 1 || *ijob == 3) { *dif = sqrt((doublereal) ((*m << 1) * *n)) / (dscale * sqrt(dsum)); } else { *dif = sqrt((doublereal) pq) / (dscale * sqrt(dsum)); } } if (isolve == 2 && iround == 1) { if (notran) { ifunc = *ijob; } scale2 = *scale; dlacpy_("F", m, n, &c__[c_offset], ldc, &work[1], m); dlacpy_("F", m, n, &f[f_offset], ldf, &work[*m * *n + 1], m); dlaset_("F", m, n, &c_b14, &c_b14, &c__[c_offset], ldc); dlaset_("F", m, n, &c_b14, &c_b14, &f[f_offset], ldf); } else if (isolve == 2 && iround == 2) { dlacpy_("F", m, n, &work[1], m, &c__[c_offset], ldc); dlacpy_("F", m, n, &work[*m * *n + 1], m, &f[f_offset], ldf); *scale = scale2; } /* L150: */ } } 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 = 1.; i__1 = p; for (i__ = 1; i__ <= i__1; ++i__) { is = iwork[i__]; ie = iwork[i__ + 1] - 1; mb = ie - is + 1; i__2 = p + 2; for (j = q; j >= i__2; --j) { js = iwork[j]; je = iwork[j + 1] - 1; nb = je - js + 1; dtgsy2_(trans, &ifunc, &mb, &nb, &a[is + is * a_dim1], lda, & b[js + js * b_dim1], ldb, &c__[is + js * c_dim1], ldc, &d__[is + is * d_dim1], ldd, &e[js + js * e_dim1], lde, &f[is + js * f_dim1], ldf, &scaloc, &dsum, & dscale, &iwork[q + 2], &ppqq, &linfo); if (linfo > 0) { *info = linfo; } if (scaloc != 1.) { i__3 = js - 1; for (k = 1; k <= i__3; ++k) { dscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); dscal_(m, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L160: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = is - 1; dscal_(&i__4, &scaloc, &c__[k * c_dim1 + 1], &c__1); i__4 = is - 1; dscal_(&i__4, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L170: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = *m - ie; dscal_(&i__4, &scaloc, &c__[ie + 1 + k * c_dim1], & c__1); i__4 = *m - ie; dscal_(&i__4, &scaloc, &f[ie + 1 + k * f_dim1], &c__1) ; /* L180: */ } i__3 = *n; for (k = je + 1; k <= i__3; ++k) { dscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); dscal_(m, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L190: */ } *scale *= scaloc; } /* Substitute R(I, J) and L(I, J) into remaining equation. */ if (j > p + 2) { i__3 = js - 1; dgemm_("N", "T", &mb, &i__3, &nb, &c_b52, &c__[is + js * c_dim1], ldc, &b[js * b_dim1 + 1], ldb, &c_b52, & f[is + f_dim1], ldf); i__3 = js - 1; dgemm_("N", "T", &mb, &i__3, &nb, &c_b52, &f[is + js * f_dim1], ldf, &e[js * e_dim1 + 1], lde, &c_b52, & f[is + f_dim1], ldf); } if (i__ < p) { i__3 = *m - ie; dgemm_("T", "N", &i__3, &nb, &mb, &c_b51, &a[is + (ie + 1) * a_dim1], lda, &c__[is + js * c_dim1], ldc, & c_b52, &c__[ie + 1 + js * c_dim1], ldc); i__3 = *m - ie; dgemm_("T", "N", &i__3, &nb, &mb, &c_b51, &d__[is + (ie + 1) * d_dim1], ldd, &f[is + js * f_dim1], ldf, & c_b52, &c__[ie + 1 + js * c_dim1], ldc); } /* L200: */ } /* L210: */ } } work[1] = (doublereal) lwmin; return 0; /* End of DTGSYL */ } /* dtgsyl_ */