#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int ctgsy2_(char *trans, integer *ijob, integer *m, integer * n, complex *a, integer *lda, complex *b, integer *ldb, complex *c__, integer *ldc, complex *d__, integer *ldd, complex *e, integer *lde, complex *f, integer *ldf, real *scale, real *rdsum, real *rdscal, integer *info) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= CTGSY2 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 CLACON. CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL 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 CTGSYL. 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. (SGECON 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 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 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 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 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 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 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) REAL 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) REAL On entry, the sum of squares of computed contributions to the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by CTGSYL. RDSCAL (input/output) REAL 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 CTGSY2 is called by CTGSYL. 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. ===================================================================== Decode and test input parameters Parameter adjustments */ /* Table of constant values */ static integer c__2 = 2; static integer c__1 = 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; complex q__1, q__2, q__3, q__4, q__5, q__6; /* Builtin functions */ void r_cnjg(complex *, complex *); /* Local variables */ static integer i__, j, k; static complex z__[4] /* was [2][2] */, rhs[2]; static integer ierr, ipiv[2], jpiv[2]; static complex alpha; extern /* Subroutine */ int cscal_(integer *, complex *, complex *, integer *); extern logical lsame_(char *, char *); extern /* Subroutine */ int caxpy_(integer *, complex *, complex *, integer *, complex *, integer *), cgesc2_(integer *, complex *, integer *, complex *, integer *, integer *, real *), cgetc2_( integer *, complex *, integer *, integer *, integer *, integer *), clatdf_(integer *, integer *, complex *, integer *, complex *, real *, real *, integer *, integer *); static real scaloc; extern /* Subroutine */ int xerbla_(char *, integer *); static logical notran; 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; /* Function Body */ *info = 0; ierr = 0; notran = lsame_(trans, "N"); if (! notran && ! lsame_(trans, "C")) { *info = -1; } else if (notran) { if (*ijob < 0 || *ijob > 2) { *info = -2; } } if (*info == 0) { if (*m <= 0) { *info = -3; } else if (*n <= 0) { *info = -4; } else if (*lda < max(1,*m)) { *info = -5; } 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) { i__1 = -(*info); xerbla_("CTGSY2", &i__1); return 0; } if (notran) { /* 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 = 1.f; scaloc = 1.f; i__1 = *n; for (j = 1; j <= i__1; ++j) { for (i__ = *m; i__ >= 1; --i__) { /* Build 2 by 2 system */ i__2 = i__ + i__ * a_dim1; z__[0].r = a[i__2].r, z__[0].i = a[i__2].i; i__2 = i__ + i__ * d_dim1; z__[1].r = d__[i__2].r, z__[1].i = d__[i__2].i; i__2 = j + j * b_dim1; q__1.r = -b[i__2].r, q__1.i = -b[i__2].i; z__[2].r = q__1.r, z__[2].i = q__1.i; i__2 = j + j * e_dim1; q__1.r = -e[i__2].r, q__1.i = -e[i__2].i; z__[3].r = q__1.r, z__[3].i = q__1.i; /* Set up right hand side(s) */ i__2 = i__ + j * c_dim1; rhs[0].r = c__[i__2].r, rhs[0].i = c__[i__2].i; i__2 = i__ + j * f_dim1; rhs[1].r = f[i__2].r, rhs[1].i = f[i__2].i; /* Solve Z * x = RHS */ cgetc2_(&c__2, z__, &c__2, ipiv, jpiv, &ierr); if (ierr > 0) { *info = ierr; } if (*ijob == 0) { cgesc2_(&c__2, z__, &c__2, rhs, ipiv, jpiv, &scaloc); if (scaloc != 1.f) { i__2 = *n; for (k = 1; k <= i__2; ++k) { q__1.r = scaloc, q__1.i = 0.f; cscal_(m, &q__1, &c__[k * c_dim1 + 1], &c__1); q__1.r = scaloc, q__1.i = 0.f; cscal_(m, &q__1, &f[k * f_dim1 + 1], &c__1); /* L10: */ } *scale *= scaloc; } } else { clatdf_(ijob, &c__2, z__, &c__2, rhs, rdsum, rdscal, ipiv, jpiv); } /* Unpack solution vector(s) */ i__2 = i__ + j * c_dim1; c__[i__2].r = rhs[0].r, c__[i__2].i = rhs[0].i; i__2 = i__ + j * f_dim1; f[i__2].r = rhs[1].r, f[i__2].i = rhs[1].i; /* Substitute R(I, J) and L(I, J) into remaining equation. */ if (i__ > 1) { q__1.r = -rhs[0].r, q__1.i = -rhs[0].i; alpha.r = q__1.r, alpha.i = q__1.i; i__2 = i__ - 1; caxpy_(&i__2, &alpha, &a[i__ * a_dim1 + 1], &c__1, &c__[j * c_dim1 + 1], &c__1); i__2 = i__ - 1; caxpy_(&i__2, &alpha, &d__[i__ * d_dim1 + 1], &c__1, &f[j * f_dim1 + 1], &c__1); } if (j < *n) { i__2 = *n - j; caxpy_(&i__2, &rhs[1], &b[j + (j + 1) * b_dim1], ldb, & c__[i__ + (j + 1) * c_dim1], ldc); i__2 = *n - j; caxpy_(&i__2, &rhs[1], &e[j + (j + 1) * e_dim1], lde, &f[ i__ + (j + 1) * f_dim1], ldf); } /* L20: */ } /* L30: */ } } 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 = 1.f; scaloc = 1.f; i__1 = *m; for (i__ = 1; i__ <= i__1; ++i__) { for (j = *n; j >= 1; --j) { /* Build 2 by 2 system Z' */ r_cnjg(&q__1, &a[i__ + i__ * a_dim1]); z__[0].r = q__1.r, z__[0].i = q__1.i; r_cnjg(&q__2, &b[j + j * b_dim1]); q__1.r = -q__2.r, q__1.i = -q__2.i; z__[1].r = q__1.r, z__[1].i = q__1.i; r_cnjg(&q__1, &d__[i__ + i__ * d_dim1]); z__[2].r = q__1.r, z__[2].i = q__1.i; r_cnjg(&q__2, &e[j + j * e_dim1]); q__1.r = -q__2.r, q__1.i = -q__2.i; z__[3].r = q__1.r, z__[3].i = q__1.i; /* Set up right hand side(s) */ i__2 = i__ + j * c_dim1; rhs[0].r = c__[i__2].r, rhs[0].i = c__[i__2].i; i__2 = i__ + j * f_dim1; rhs[1].r = f[i__2].r, rhs[1].i = f[i__2].i; /* Solve Z' * x = RHS */ cgetc2_(&c__2, z__, &c__2, ipiv, jpiv, &ierr); if (ierr > 0) { *info = ierr; } cgesc2_(&c__2, z__, &c__2, rhs, ipiv, jpiv, &scaloc); if (scaloc != 1.f) { i__2 = *n; for (k = 1; k <= i__2; ++k) { q__1.r = scaloc, q__1.i = 0.f; cscal_(m, &q__1, &c__[k * c_dim1 + 1], &c__1); q__1.r = scaloc, q__1.i = 0.f; cscal_(m, &q__1, &f[k * f_dim1 + 1], &c__1); /* L40: */ } *scale *= scaloc; } /* Unpack solution vector(s) */ i__2 = i__ + j * c_dim1; c__[i__2].r = rhs[0].r, c__[i__2].i = rhs[0].i; i__2 = i__ + j * f_dim1; f[i__2].r = rhs[1].r, f[i__2].i = rhs[1].i; /* Substitute R(I, J) and L(I, J) into remaining equation. */ i__2 = j - 1; for (k = 1; k <= i__2; ++k) { i__3 = i__ + k * f_dim1; i__4 = i__ + k * f_dim1; r_cnjg(&q__4, &b[k + j * b_dim1]); q__3.r = rhs[0].r * q__4.r - rhs[0].i * q__4.i, q__3.i = rhs[0].r * q__4.i + rhs[0].i * q__4.r; q__2.r = f[i__4].r + q__3.r, q__2.i = f[i__4].i + q__3.i; r_cnjg(&q__6, &e[k + j * e_dim1]); q__5.r = rhs[1].r * q__6.r - rhs[1].i * q__6.i, q__5.i = rhs[1].r * q__6.i + rhs[1].i * q__6.r; q__1.r = q__2.r + q__5.r, q__1.i = q__2.i + q__5.i; f[i__3].r = q__1.r, f[i__3].i = q__1.i; /* L50: */ } i__2 = *m; for (k = i__ + 1; k <= i__2; ++k) { i__3 = k + j * c_dim1; i__4 = k + j * c_dim1; r_cnjg(&q__4, &a[i__ + k * a_dim1]); q__3.r = q__4.r * rhs[0].r - q__4.i * rhs[0].i, q__3.i = q__4.r * rhs[0].i + q__4.i * rhs[0].r; q__2.r = c__[i__4].r - q__3.r, q__2.i = c__[i__4].i - q__3.i; r_cnjg(&q__6, &d__[i__ + k * d_dim1]); q__5.r = q__6.r * rhs[1].r - q__6.i * rhs[1].i, q__5.i = q__6.r * rhs[1].i + q__6.i * rhs[1].r; q__1.r = q__2.r - q__5.r, q__1.i = q__2.i - q__5.i; c__[i__3].r = q__1.r, c__[i__3].i = q__1.i; /* L60: */ } /* L70: */ } /* L80: */ } } return 0; /* End of CTGSY2 */ } /* ctgsy2_ */