#include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__2 = 2; static integer c_n1 = -1; static integer c__5 = 5; static real c_b14 = 0.f; static integer c__1 = 1; static real c_b51 = -1.f; static real c_b52 = 1.f; /* Subroutine */ int stgsyl_(char *trans, integer *ijob, integer *m, integer * n, real *a, integer *lda, real *b, integer *ldb, real *c__, integer * ldc, real *d__, integer *ldd, real *e, integer *lde, real *f, integer *ldf, real *scale, real *dif, real *work, integer *lwork, integer * iwork, integer *info) { /* 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 */ integer i__, j, k, p, q, ie, je, mb, nb, is, js, pq; real dsum; integer ppqq; extern logical lsame_(char *, char *); integer ifunc; extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); integer linfo; extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); integer lwmin; real scale2, dscale; extern /* Subroutine */ int stgsy2_(char *, integer *, integer *, integer *, real *, integer *, real *, integer *, real *, integer *, real * , integer *, real *, integer *, real *, integer *, real *, real *, real *, integer *, integer *, integer *); real scaloc; extern /* Subroutine */ int xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *); extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *), slaset_(char *, integer *, integer *, real *, real *, real *, integer *); integer iround; logical notran; integer isolve; logical lquery; /* -- LAPACK routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* STGSYL 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', STGSYL 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 SLACON. */ /* If IJOB >= 1, STGSYL 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. */ /* ( SGECON 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL */ /* 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) REAL */ /* 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) REAL 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 SCOPY by calls to SLASET. */ /* Sven Hammarling, 1/5/02. */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode and test input parameters */ /* Parameter adjustments */ 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] = (real) lwmin; if (*lwork < lwmin && ! lquery) { *info = -20; } } if (*info != 0) { i__1 = -(*info); xerbla_("STGSYL", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*m == 0 || *n == 0) { *scale = 1.f; if (notran) { if (*ijob != 0) { *dif = 0.f; } } return 0; } /* Determine optimal block sizes MB and NB */ mb = ilaenv_(&c__2, "STGSYL", trans, m, n, &c_n1, &c_n1); nb = ilaenv_(&c__5, "STGSYL", trans, m, n, &c_n1, &c_n1); isolve = 1; ifunc = 0; if (notran) { if (*ijob >= 3) { ifunc = *ijob - 2; slaset_("F", m, n, &c_b14, &c_b14, &c__[c_offset], ldc) ; slaset_("F", m, n, &c_b14, &c_b14, &f[f_offset], ldf); } else if (*ijob >= 1 && notran) { 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.f; dsum = 1.f; pq = 0; stgsy2_(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.f) { if (*ijob == 1 || *ijob == 3) { *dif = sqrt((real) ((*m << 1) * *n)) / (dscale * sqrt( dsum)); } else { *dif = sqrt((real) pq) / (dscale * sqrt(dsum)); } } if (isolve == 2 && iround == 1) { if (notran) { ifunc = *ijob; } scale2 = *scale; slacpy_("F", m, n, &c__[c_offset], ldc, &work[1], m); slacpy_("F", m, n, &f[f_offset], ldf, &work[*m * *n + 1], m); slaset_("F", m, n, &c_b14, &c_b14, &c__[c_offset], ldc); slaset_("F", m, n, &c_b14, &c_b14, &f[f_offset], ldf); } else if (isolve == 2 && iround == 2) { slacpy_("F", m, n, &work[1], m, &c__[c_offset], ldc); slacpy_("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.f) { ++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.f) { ++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.f; dsum = 1.f; pq = 0; *scale = 1.f; 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; stgsy2_(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.f) { i__3 = js - 1; for (k = 1; k <= i__3; ++k) { sscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); sscal_(m, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L80: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = is - 1; sscal_(&i__4, &scaloc, &c__[k * c_dim1 + 1], & c__1); i__4 = is - 1; sscal_(&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; sscal_(&i__4, &scaloc, &c__[ie + 1 + k * c_dim1], &c__1); i__4 = *m - ie; sscal_(&i__4, &scaloc, &f[ie + 1 + k * f_dim1], & c__1); /* L100: */ } i__3 = *n; for (k = je + 1; k <= i__3; ++k) { sscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); sscal_(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; sgemm_("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; sgemm_("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; sgemm_("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; sgemm_("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.f) { if (*ijob == 1 || *ijob == 3) { *dif = sqrt((real) ((*m << 1) * *n)) / (dscale * sqrt( dsum)); } else { *dif = sqrt((real) pq) / (dscale * sqrt(dsum)); } } if (isolve == 2 && iround == 1) { if (notran) { ifunc = *ijob; } scale2 = *scale; slacpy_("F", m, n, &c__[c_offset], ldc, &work[1], m); slacpy_("F", m, n, &f[f_offset], ldf, &work[*m * *n + 1], m); slaset_("F", m, n, &c_b14, &c_b14, &c__[c_offset], ldc); slaset_("F", m, n, &c_b14, &c_b14, &f[f_offset], ldf); } else if (isolve == 2 && iround == 2) { slacpy_("F", m, n, &work[1], m, &c__[c_offset], ldc); slacpy_("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.f; 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; stgsy2_(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.f) { i__3 = js - 1; for (k = 1; k <= i__3; ++k) { sscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); sscal_(m, &scaloc, &f[k * f_dim1 + 1], &c__1); /* L160: */ } i__3 = je; for (k = js; k <= i__3; ++k) { i__4 = is - 1; sscal_(&i__4, &scaloc, &c__[k * c_dim1 + 1], &c__1); i__4 = is - 1; sscal_(&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; sscal_(&i__4, &scaloc, &c__[ie + 1 + k * c_dim1], & c__1); i__4 = *m - ie; sscal_(&i__4, &scaloc, &f[ie + 1 + k * f_dim1], &c__1) ; /* L180: */ } i__3 = *n; for (k = je + 1; k <= i__3; ++k) { sscal_(m, &scaloc, &c__[k * c_dim1 + 1], &c__1); sscal_(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; sgemm_("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; sgemm_("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; sgemm_("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; sgemm_("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] = (real) lwmin; return 0; /* End of STGSYL */ } /* stgsyl_ */