#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int ztgsyl_(char *trans, integer *ijob, integer *m, integer * n, doublecomplex *a, integer *lda, doublecomplex *b, integer *ldb, doublecomplex *c__, integer *ldc, doublecomplex *d__, integer *ldd, doublecomplex *e, integer *lde, doublecomplex *f, integer *ldf, doublereal *scale, doublereal *dif, doublecomplex *work, integer * lwork, integer *iwork, integer *info) { /* -- LAPACK routine (version 3.1.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. January 2007 Purpose ======= ZTGSYL 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 complex entries. 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 (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 Ix is the identity matrix of size x and X' is the conjugate 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 (TRANS = 'C') 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 ZLACON. If IJOB >= 1, ZTGSYL 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. This is a level-3 BLAS algorithm. Arguments ========= TRANS (input) CHARACTER*1 = 'N': solve the generalized sylvester equation (1). = 'C': solve the "conjugate 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 is used). =4: Only an estimate of Dif[(A,D), (B,E)] is computed. (ZGECON on sub-systems is used). Not referenced if TRANS = 'C'. 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) COMPLEX*16 array, dimension (LDA, M) The upper triangular matrix A. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1, M). B (input) COMPLEX*16 array, dimension (LDB, N) The upper triangular matrix B. LDB (input) INTEGER The leading dimension of the array 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) 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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 = 'C', DIF is not referenced. 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, R and L will hold the solutions to the homogenious system with C = F = 0. WORK (workspace/output) COMPLEX*16 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+2) 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 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. [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 CCOPY by calls to CLASET. Sven Hammarling, 1/5/02. Decode and test input parameters Parameter adjustments */ /* Table of constant values */ static doublecomplex c_b1 = {0.,0.}; static integer c__2 = 2; static integer c_n1 = -1; static integer c__5 = 5; static integer c__1 = 1; static doublecomplex c_b44 = {-1.,0.}; static doublecomplex c_b45 = {1.,0.}; /* 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; doublecomplex z__1; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, j, k, p, q, ie, je, mb, nb, is, js, pq; static doublereal dsum; extern logical lsame_(char *, char *); static integer ifunc, linfo; extern /* Subroutine */ int zscal_(integer *, doublecomplex *, doublecomplex *, integer *), zgemm_(char *, char *, integer *, integer *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *); static integer lwmin; static doublereal scale2, dscale; extern /* Subroutine */ int ztgsy2_(char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublereal *, integer *); static doublereal scaloc; extern /* Subroutine */ int xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static integer iround; static logical notran; static integer isolve; extern /* Subroutine */ int zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *), zlaset_(char *, integer *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *); 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, "C")) { *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].r = (doublereal) lwmin, work[1].i = 0.; if (*lwork < lwmin && ! lquery) { *info = -20; } } if (*info != 0) { i__1 = -(*info); xerbla_("ZTGSYL", &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, "ZTGSYL", trans, m, n, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); nb = ilaenv_(&c__5, "ZTGSYL", trans, m, n, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); isolve = 1; ifunc = 0; if (notran) { if (*ijob >= 3) { ifunc = *ijob - 2; zlaset_("F", m, n, &c_b1, &c_b1, &c__[c_offset], ldc); zlaset_("F", m, n, &c_b1, &c_b1, &f[f_offset], ldf); } else if (*ijob >= 1 && notran) { isolve = 2; } } if (mb <= 1 && nb <= 1 || mb >= *m && nb >= *n) { /* Use unblocked Level 2 solver */ i__1 = isolve; for (iround = 1; iround <= i__1; ++iround) { *scale = 1.; dscale = 0.; dsum = 1.; pq = *m * *n; ztgsy2_(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, 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; zlacpy_("F", m, n, &c__[c_offset], ldc, &work[1], m); zlacpy_("F", m, n, &f[f_offset], ldf, &work[*m * *n + 1], m); zlaset_("F", m, n, &c_b1, &c_b1, &c__[c_offset], ldc); zlaset_("F", m, n, &c_b1, &c_b1, &f[f_offset], ldf) ; } else if (isolve == 2 && iround == 2) { zlacpy_("F", m, n, &work[1], m, &c__[c_offset], ldc); zlacpy_("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; } 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; } 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 */ pq = 0; *scale = 1.; dscale = 0.; dsum = 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; ztgsy2_(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, &linfo); if (linfo > 0) { *info = linfo; } pq += mb * nb; if (scaloc != 1.) { i__3 = js - 1; for (k = 1; k <= i__3; ++k) { z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &c__[k * c_dim1 + 1], &c__1); z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &f[k * f_dim1 + 1], &c__1); /* L80: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = is - 1; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &c__[k * c_dim1 + 1], &c__1); i__4 = is - 1; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &f[k * f_dim1 + 1], &c__1); /* L90: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = *m - ie; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &c__[ie + 1 + k * c_dim1], & c__1); i__4 = *m - ie; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &f[ie + 1 + k * f_dim1], & c__1); /* L100: */ } i__3 = *n; for (k = je + 1; k <= i__3; ++k) { z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &c__[k * c_dim1 + 1], &c__1); z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &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; zgemm_("N", "N", &i__3, &nb, &mb, &c_b44, &a[is * a_dim1 + 1], lda, &c__[is + js * c_dim1], ldc, &c_b45, &c__[js * c_dim1 + 1], ldc); i__3 = is - 1; zgemm_("N", "N", &i__3, &nb, &mb, &c_b44, &d__[is * d_dim1 + 1], ldd, &c__[is + js * c_dim1], ldc, &c_b45, &f[js * f_dim1 + 1], ldf); } if (j < q) { i__3 = *n - je; zgemm_("N", "N", &mb, &i__3, &nb, &c_b45, &f[is + js * f_dim1], ldf, &b[js + (je + 1) * b_dim1], ldb, &c_b45, &c__[is + (je + 1) * c_dim1], ldc); i__3 = *n - je; zgemm_("N", "N", &mb, &i__3, &nb, &c_b45, &f[is + js * f_dim1], ldf, &e[js + (je + 1) * e_dim1], lde, &c_b45, &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; zlacpy_("F", m, n, &c__[c_offset], ldc, &work[1], m); zlacpy_("F", m, n, &f[f_offset], ldf, &work[*m * *n + 1], m); zlaset_("F", m, n, &c_b1, &c_b1, &c__[c_offset], ldc); zlaset_("F", m, n, &c_b1, &c_b1, &f[f_offset], ldf) ; } else if (isolve == 2 && iround == 2) { zlacpy_("F", m, n, &work[1], m, &c__[c_offset], ldc); zlacpy_("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; ztgsy2_(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, &linfo); if (linfo > 0) { *info = linfo; } if (scaloc != 1.) { i__3 = js - 1; for (k = 1; k <= i__3; ++k) { z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &c__[k * c_dim1 + 1], &c__1); z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &f[k * f_dim1 + 1], &c__1); /* L160: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = is - 1; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &c__[k * c_dim1 + 1], &c__1); i__4 = is - 1; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &f[k * f_dim1 + 1], &c__1); /* L170: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = *m - ie; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &c__[ie + 1 + k * c_dim1], &c__1) ; i__4 = *m - ie; z__1.r = scaloc, z__1.i = 0.; zscal_(&i__4, &z__1, &f[ie + 1 + k * f_dim1], &c__1); /* L180: */ } i__3 = *n; for (k = je + 1; k <= i__3; ++k) { z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &c__[k * c_dim1 + 1], &c__1); z__1.r = scaloc, z__1.i = 0.; zscal_(m, &z__1, &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; zgemm_("N", "C", &mb, &i__3, &nb, &c_b45, &c__[is + js * c_dim1], ldc, &b[js * b_dim1 + 1], ldb, &c_b45, & f[is + f_dim1], ldf); i__3 = js - 1; zgemm_("N", "C", &mb, &i__3, &nb, &c_b45, &f[is + js * f_dim1], ldf, &e[js * e_dim1 + 1], lde, &c_b45, & f[is + f_dim1], ldf); } if (i__ < p) { i__3 = *m - ie; zgemm_("C", "N", &i__3, &nb, &mb, &c_b44, &a[is + (ie + 1) * a_dim1], lda, &c__[is + js * c_dim1], ldc, & c_b45, &c__[ie + 1 + js * c_dim1], ldc); i__3 = *m - ie; zgemm_("C", "N", &i__3, &nb, &mb, &c_b44, &d__[is + (ie + 1) * d_dim1], ldd, &f[is + js * f_dim1], ldf, & c_b45, &c__[ie + 1 + js * c_dim1], ldc); } /* L200: */ } /* L210: */ } } work[1].r = (doublereal) lwmin, work[1].i = 0.; return 0; /* End of ZTGSYL */ } /* ztgsyl_ */