/* dtgsen.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__1 = 1; static integer c__2 = 2; static doublereal c_b28 = 1.; /* Subroutine */ int dtgsen_(integer *ijob, logical *wantq, logical *wantz, logical *select, integer *n, doublereal *a, integer *lda, doublereal * b, integer *ldb, doublereal *alphar, doublereal *alphai, doublereal * beta, doublereal *q, integer *ldq, doublereal *z__, integer *ldz, integer *m, doublereal *pl, doublereal *pr, doublereal *dif, doublereal *work, integer *lwork, integer *iwork, integer *liwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1, z_offset, i__1, i__2; doublereal d__1; /* Builtin functions */ double sqrt(doublereal), d_sign(doublereal *, doublereal *); /* Local variables */ integer i__, k, n1, n2, kk, ks, mn2, ijb; doublereal eps; integer kase; logical pair; integer ierr; doublereal dsum; logical swap; extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); integer isave[3]; logical wantd; integer lwmin; logical wantp; extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); logical wantd1, wantd2; extern doublereal dlamch_(char *); doublereal dscale, rdscal; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *), dtgexc_(logical *, logical *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *, integer *), dlassq_(integer *, doublereal *, integer *, doublereal *, doublereal *); integer liwmin; extern /* Subroutine */ int dtgsyl_(char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *); doublereal smlnum; logical lquery; /* -- LAPACK routine (version 3.2) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* January 2007 */ /* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* DTGSEN reorders the generalized real Schur decomposition of a real */ /* matrix pair (A, B) (in terms of an orthonormal equivalence trans- */ /* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */ /* appears in the leading diagonal blocks of the upper quasi-triangular */ /* matrix A and the upper triangular B. The leading columns of Q and */ /* Z form orthonormal bases of the corresponding left and right eigen- */ /* spaces (deflating subspaces). (A, B) must be in generalized real */ /* Schur canonical form (as returned by DGGES), i.e. A is block upper */ /* triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper */ /* triangular. */ /* DTGSEN also computes the generalized eigenvalues */ /* w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j) */ /* of the reordered matrix pair (A, B). */ /* Optionally, DTGSEN computes the estimates of reciprocal condition */ /* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */ /* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */ /* between the matrix pairs (A11, B11) and (A22,B22) that correspond to */ /* the selected cluster and the eigenvalues outside the cluster, resp., */ /* and norms of "projections" onto left and right eigenspaces w.r.t. */ /* the selected cluster in the (1,1)-block. */ /* Arguments */ /* ========= */ /* IJOB (input) INTEGER */ /* Specifies whether condition numbers are required for the */ /* cluster of eigenvalues (PL and PR) or the deflating subspaces */ /* (Difu and Difl): */ /* =0: Only reorder w.r.t. SELECT. No extras. */ /* =1: Reciprocal of norms of "projections" onto left and right */ /* eigenspaces w.r.t. the selected cluster (PL and PR). */ /* =2: Upper bounds on Difu and Difl. F-norm-based estimate */ /* (DIF(1:2)). */ /* =3: Estimate of Difu and Difl. 1-norm-based estimate */ /* (DIF(1:2)). */ /* About 5 times as expensive as IJOB = 2. */ /* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */ /* version to get it all. */ /* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */ /* WANTQ (input) LOGICAL */ /* .TRUE. : update the left transformation matrix Q; */ /* .FALSE.: do not update Q. */ /* WANTZ (input) LOGICAL */ /* .TRUE. : update the right transformation matrix Z; */ /* .FALSE.: do not update Z. */ /* SELECT (input) LOGICAL array, dimension (N) */ /* SELECT specifies the eigenvalues in the selected cluster. */ /* To select a real eigenvalue w(j), SELECT(j) must be set to */ /* .TRUE.. To select a complex conjugate pair of eigenvalues */ /* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, */ /* either SELECT(j) or SELECT(j+1) or both must be set to */ /* .TRUE.; a complex conjugate pair of eigenvalues must be */ /* either both included in the cluster or both excluded. */ /* N (input) INTEGER */ /* The order of the matrices A and B. N >= 0. */ /* A (input/output) DOUBLE PRECISION array, dimension(LDA,N) */ /* On entry, the upper quasi-triangular matrix A, with (A, B) in */ /* generalized real Schur canonical form. */ /* On exit, A is overwritten by the reordered matrix A. */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,N). */ /* B (input/output) DOUBLE PRECISION array, dimension(LDB,N) */ /* On entry, the upper triangular matrix B, with (A, B) in */ /* generalized real Schur canonical form. */ /* On exit, B is overwritten by the reordered matrix B. */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,N). */ /* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */ /* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */ /* BETA (output) DOUBLE PRECISION array, dimension (N) */ /* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will */ /* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i */ /* and BETA(j),j=1,...,N are the diagonals of the complex Schur */ /* form (S,T) that would result if the 2-by-2 diagonal blocks of */ /* the real generalized Schur form of (A,B) were further reduced */ /* to triangular form using complex unitary transformations. */ /* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if */ /* positive, then the j-th and (j+1)-st eigenvalues are a */ /* complex conjugate pair, with ALPHAI(j+1) negative. */ /* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) */ /* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */ /* On exit, Q has been postmultiplied by the left orthogonal */ /* transformation matrix which reorder (A, B); The leading M */ /* columns of Q form orthonormal bases for the specified pair of */ /* left eigenspaces (deflating subspaces). */ /* If WANTQ = .FALSE., Q is not referenced. */ /* LDQ (input) INTEGER */ /* The leading dimension of the array Q. LDQ >= 1; */ /* and if WANTQ = .TRUE., LDQ >= N. */ /* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) */ /* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */ /* On exit, Z has been postmultiplied by the left orthogonal */ /* transformation matrix which reorder (A, B); The leading M */ /* columns of Z form orthonormal bases for the specified pair of */ /* left eigenspaces (deflating subspaces). */ /* If WANTZ = .FALSE., Z is not referenced. */ /* LDZ (input) INTEGER */ /* The leading dimension of the array Z. LDZ >= 1; */ /* If WANTZ = .TRUE., LDZ >= N. */ /* M (output) INTEGER */ /* The dimension of the specified pair of left and right eigen- */ /* spaces (deflating subspaces). 0 <= M <= N. */ /* PL (output) DOUBLE PRECISION */ /* PR (output) DOUBLE PRECISION */ /* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */ /* reciprocal of the norm of "projections" onto left and right */ /* eigenspaces with respect to the selected cluster. */ /* 0 < PL, PR <= 1. */ /* If M = 0 or M = N, PL = PR = 1. */ /* If IJOB = 0, 2 or 3, PL and PR are not referenced. */ /* DIF (output) DOUBLE PRECISION array, dimension (2). */ /* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */ /* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */ /* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */ /* estimates of Difu and Difl. */ /* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */ /* If IJOB = 0 or 1, DIF is not referenced. */ /* 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 >= 4*N+16. */ /* If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). */ /* If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)). */ /* 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/output) INTEGER array, dimension (MAX(1,LIWORK)) */ /* IF IJOB = 0, IWORK is not referenced. Otherwise, */ /* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */ /* LIWORK (input) INTEGER */ /* The dimension of the array IWORK. LIWORK >= 1. */ /* If IJOB = 1, 2 or 4, LIWORK >= N+6. */ /* If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). */ /* If LIWORK = -1, then a workspace query is assumed; the */ /* routine only calculates the optimal size of the IWORK array, */ /* returns this value as the first entry of the IWORK array, and */ /* no error message related to LIWORK is issued by XERBLA. */ /* INFO (output) INTEGER */ /* =0: Successful exit. */ /* <0: If INFO = -i, the i-th argument had an illegal value. */ /* =1: Reordering of (A, B) failed because the transformed */ /* matrix pair (A, B) would be too far from generalized */ /* Schur form; the problem is very ill-conditioned. */ /* (A, B) may have been partially reordered. */ /* If requested, 0 is returned in DIF(*), PL and PR. */ /* Further Details */ /* =============== */ /* DTGSEN first collects the selected eigenvalues by computing */ /* orthogonal U and W that move them to the top left corner of (A, B). */ /* In other words, the selected eigenvalues are the eigenvalues of */ /* (A11, B11) in: */ /* U'*(A, B)*W = (A11 A12) (B11 B12) n1 */ /* ( 0 A22),( 0 B22) n2 */ /* n1 n2 n1 n2 */ /* where N = n1+n2 and U' means the transpose of U. The first n1 columns */ /* of U and W span the specified pair of left and right eigenspaces */ /* (deflating subspaces) of (A, B). */ /* If (A, B) has been obtained from the generalized real Schur */ /* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */ /* reordered generalized real Schur form of (C, D) is given by */ /* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', */ /* and the first n1 columns of Q*U and Z*W span the corresponding */ /* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */ /* Note that if the selected eigenvalue is sufficiently ill-conditioned, */ /* then its value may differ significantly from its value before */ /* reordering. */ /* The reciprocal condition numbers of the left and right eigenspaces */ /* spanned by the first n1 columns of U and W (or Q*U and Z*W) may */ /* be returned in DIF(1:2), corresponding to Difu and Difl, resp. */ /* The Difu and Difl are defined as: */ /* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) */ /* and */ /* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */ /* where sigma-min(Zu) is the smallest singular value of the */ /* (2*n1*n2)-by-(2*n1*n2) matrix */ /* Zu = [ kron(In2, A11) -kron(A22', In1) ] */ /* [ kron(In2, B11) -kron(B22', In1) ]. */ /* Here, Inx is the identity matrix of size nx and A22' is the */ /* transpose of A22. kron(X, Y) is the Kronecker product between */ /* the matrices X and Y. */ /* When DIF(2) is small, small changes in (A, B) can cause large changes */ /* in the deflating subspace. An approximate (asymptotic) bound on the */ /* maximum angular error in the computed deflating subspaces is */ /* EPS * norm((A, B)) / DIF(2), */ /* where EPS is the machine precision. */ /* The reciprocal norm of the projectors on the left and right */ /* eigenspaces associated with (A11, B11) may be returned in PL and PR. */ /* They are computed as follows. First we compute L and R so that */ /* P*(A, B)*Q is block diagonal, where */ /* P = ( I -L ) n1 Q = ( I R ) n1 */ /* ( 0 I ) n2 and ( 0 I ) n2 */ /* n1 n2 n1 n2 */ /* and (L, R) is the solution to the generalized Sylvester equation */ /* A11*R - L*A22 = -A12 */ /* B11*R - L*B22 = -B12 */ /* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */ /* An approximate (asymptotic) bound on the average absolute error of */ /* the selected eigenvalues is */ /* EPS * norm((A, B)) / PL. */ /* There are also global error bounds which valid for perturbations up */ /* to a certain restriction: A lower bound (x) on the smallest */ /* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */ /* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */ /* (i.e. (A + E, B + F), is */ /* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). */ /* An approximate bound on x can be computed from DIF(1:2), PL and PR. */ /* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */ /* (L', R') and unperturbed (L, R) left and right deflating subspaces */ /* associated with the selected cluster in the (1,1)-blocks can be */ /* bounded as */ /* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */ /* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */ /* See LAPACK User's Guide section 4.11 or the following references */ /* for more information. */ /* Note that if the default method for computing the Frobenius-norm- */ /* based estimate DIF is not wanted (see DLATDF), then the parameter */ /* IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF */ /* (IJOB = 2 will be used)). See DTGSYL for more details. */ /* Based on contributions by */ /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ /* Umea University, S-901 87 Umea, Sweden. */ /* References */ /* ========== */ /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ /* Estimation: Theory, Algorithms and Software, */ /* Report UMINF - 94.04, Department of Computing Science, Umea */ /* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working */ /* Note 87. To appear in Numerical Algorithms, 1996. */ /* [3] 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. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode and test the input parameters */ /* Parameter adjustments */ --select; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --alphar; --alphai; --beta; q_dim1 = *ldq; q_offset = 1 + q_dim1; q -= q_offset; z_dim1 = *ldz; z_offset = 1 + z_dim1; z__ -= z_offset; --dif; --work; --iwork; /* Function Body */ *info = 0; lquery = *lwork == -1 || *liwork == -1; if (*ijob < 0 || *ijob > 5) { *info = -1; } else if (*n < 0) { *info = -5; } else if (*lda < max(1,*n)) { *info = -7; } else if (*ldb < max(1,*n)) { *info = -9; } else if (*ldq < 1 || *wantq && *ldq < *n) { *info = -14; } else if (*ldz < 1 || *wantz && *ldz < *n) { *info = -16; } if (*info != 0) { i__1 = -(*info); xerbla_("DTGSEN", &i__1); return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S") / eps; ierr = 0; wantp = *ijob == 1 || *ijob >= 4; wantd1 = *ijob == 2 || *ijob == 4; wantd2 = *ijob == 3 || *ijob == 5; wantd = wantd1 || wantd2; /* Set M to the dimension of the specified pair of deflating */ /* subspaces. */ *m = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { if (k < *n) { if (a[k + 1 + k * a_dim1] == 0.) { if (select[k]) { ++(*m); } } else { pair = TRUE_; if (select[k] || select[k + 1]) { *m += 2; } } } else { if (select[*n]) { ++(*m); } } } /* L10: */ } if (*ijob == 1 || *ijob == 2 || *ijob == 4) { /* Computing MAX */ i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), i__2 = (*m << 1) * (*n - *m); lwmin = max(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = *n + 6; liwmin = max(i__1,i__2); } else if (*ijob == 3 || *ijob == 5) { /* Computing MAX */ i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), i__2 = (*m << 2) * (*n - *m); lwmin = max(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = max(i__1,i__2), i__2 = *n + 6; liwmin = max(i__1,i__2); } else { /* Computing MAX */ i__1 = 1, i__2 = (*n << 2) + 16; lwmin = max(i__1,i__2); liwmin = 1; } work[1] = (doublereal) lwmin; iwork[1] = liwmin; if (*lwork < lwmin && ! lquery) { *info = -22; } else if (*liwork < liwmin && ! lquery) { *info = -24; } if (*info != 0) { i__1 = -(*info); xerbla_("DTGSEN", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible. */ if (*m == *n || *m == 0) { if (wantp) { *pl = 1.; *pr = 1.; } if (wantd) { dscale = 0.; dsum = 1.; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { dlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum); dlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum); /* L20: */ } dif[1] = dscale * sqrt(dsum); dif[2] = dif[1]; } goto L60; } /* Collect the selected blocks at the top-left corner of (A, B). */ ks = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { swap = select[k]; if (k < *n) { if (a[k + 1 + k * a_dim1] != 0.) { pair = TRUE_; swap = swap || select[k + 1]; } } if (swap) { ++ks; /* Swap the K-th block to position KS. */ /* Perform the reordering of diagonal blocks in (A, B) */ /* by orthogonal transformation matrices and update */ /* Q and Z accordingly (if requested): */ kk = k; if (k != ks) { dtgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb, &q[q_offset], ldq, &z__[z_offset], ldz, &kk, &ks, &work[1], lwork, &ierr); } if (ierr > 0) { /* Swap is rejected: exit. */ *info = 1; if (wantp) { *pl = 0.; *pr = 0.; } if (wantd) { dif[1] = 0.; dif[2] = 0.; } goto L60; } if (pair) { ++ks; } } } /* L30: */ } if (wantp) { /* Solve generalized Sylvester equation for R and L */ /* and compute PL and PR. */ n1 = *m; n2 = *n - *m; i__ = n1 + 1; ijb = 0; dlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1); dlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 + 1], &n1); i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1] , lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], & work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr); /* Estimate the reciprocal of norms of "projections" onto left */ /* and right eigenspaces. */ rdscal = 0.; dsum = 1.; i__1 = n1 * n2; dlassq_(&i__1, &work[1], &c__1, &rdscal, &dsum); *pl = rdscal * sqrt(dsum); if (*pl == 0.) { *pl = 1.; } else { *pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl)); } rdscal = 0.; dsum = 1.; i__1 = n1 * n2; dlassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum); *pr = rdscal * sqrt(dsum); if (*pr == 0.) { *pr = 1.; } else { *pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr)); } } if (wantd) { /* Compute estimates of Difu and Difl. */ if (wantd1) { n1 = *m; n2 = *n - *m; i__ = n1 + 1; ijb = 3; /* Frobenius norm-based Difu-estimate. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, & dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & ierr); /* Frobenius norm-based Difl-estimate. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[ a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & ierr); } else { /* Compute 1-norm-based estimates of Difu and Difl using */ /* reversed communication with DLACN2. In each step a */ /* generalized Sylvester equation or a transposed variant */ /* is solved. */ kase = 0; n1 = *m; n2 = *n - *m; i__ = n1 + 1; ijb = 0; mn2 = (n1 << 1) * n2; /* 1-norm-based estimate of Difu. */ L40: dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[1], &kase, isave); if (kase != 0) { if (kase == 1) { /* Solve generalized Sylvester equation. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } else { /* Solve the transposed variant. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("T", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } goto L40; } dif[1] = dscale / dif[1]; /* 1-norm-based estimate of Difl. */ L50: dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[2], &kase, isave); if (kase != 0) { if (kase == 1) { /* Solve generalized Sylvester equation. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } else { /* Solve the transposed variant. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("T", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } goto L50; } dif[2] = dscale / dif[2]; } } L60: /* Compute generalized eigenvalues of reordered pair (A, B) and */ /* normalize the generalized Schur form. */ pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { if (k < *n) { if (a[k + 1 + k * a_dim1] != 0.) { pair = TRUE_; } } if (pair) { /* Compute the eigenvalue(s) at position K. */ work[1] = a[k + k * a_dim1]; work[2] = a[k + 1 + k * a_dim1]; work[3] = a[k + (k + 1) * a_dim1]; work[4] = a[k + 1 + (k + 1) * a_dim1]; work[5] = b[k + k * b_dim1]; work[6] = b[k + 1 + k * b_dim1]; work[7] = b[k + (k + 1) * b_dim1]; work[8] = b[k + 1 + (k + 1) * b_dim1]; d__1 = smlnum * eps; dlag2_(&work[1], &c__2, &work[5], &c__2, &d__1, &beta[k], & beta[k + 1], &alphar[k], &alphar[k + 1], &alphai[k]); alphai[k + 1] = -alphai[k]; } else { if (d_sign(&c_b28, &b[k + k * b_dim1]) < 0.) { /* If B(K,K) is negative, make it positive */ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { a[k + i__ * a_dim1] = -a[k + i__ * a_dim1]; b[k + i__ * b_dim1] = -b[k + i__ * b_dim1]; if (*wantq) { q[i__ + k * q_dim1] = -q[i__ + k * q_dim1]; } /* L70: */ } } alphar[k] = a[k + k * a_dim1]; alphai[k] = 0.; beta[k] = b[k + k * b_dim1]; } } /* L80: */ } work[1] = (doublereal) lwmin; iwork[1] = liwmin; return 0; /* End of DTGSEN */ } /* dtgsen_ */