/* 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; /* Local variables */ integer i__, k, n1, n2, kk, ks, mn2, ijb; doublereal eps; integer kase; logical pair; integer ierr; doublereal dsum; logical swap; integer isave[3]; logical wantd; integer lwmin; logical wantp; logical wantd1, wantd2; doublereal dscale, rdscal; integer liwmin; doublereal smlnum; logical lquery; /* -- LAPACK routine (version 3.2) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* January 2007 */ /* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */ /* 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 */ /* 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) */ /* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i */ /* 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. */ /* ===================================================================== */ /* 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); } } } } 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); } 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; } } } } 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]; } } } alphar[k] = a[k + k * a_dim1]; alphai[k] = 0.; beta[k] = b[k + k * b_dim1]; } } } work[1] = (doublereal) lwmin; iwork[1] = liwmin; return 0; /* End of DTGSEN */ } /* dtgsen_ */
/* Subroutine */ int dtgsna_(char *job, char *howmny, logical *select, integer *n, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr, doublereal *s, doublereal *dif, integer *mm, integer *m, doublereal * work, integer *lwork, integer *iwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1, i__2; doublereal d__1, d__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ integer i__, k; doublereal c1, c2; integer n1, n2, ks, iz; doublereal eps, beta, cond; extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, integer *); logical pair; integer ierr; doublereal uhav, uhbv; integer ifst; doublereal lnrm; integer ilst; doublereal rnrm; extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); extern doublereal dnrm2_(integer *, doublereal *, integer *); doublereal root1, root2, scale; extern logical lsame_(char *, char *); extern /* Subroutine */ int dgemv_(char *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *); doublereal uhavi, uhbvi, tmpii; integer lwmin; logical wants; doublereal tmpir, tmpri, dummy[1], tmprr; extern doublereal dlapy2_(doublereal *, doublereal *); doublereal dummy1[1]; extern doublereal dlamch_(char *); doublereal alphai, alphar; 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 *); logical wantbh, wantdf, somcon; doublereal alprqt; 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 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* DTGSNA estimates reciprocal condition numbers for specified */ /* eigenvalues and/or eigenvectors of a matrix pair (A, B) in */ /* generalized real Schur canonical form (or of any matrix pair */ /* (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where */ /* Z' denotes the transpose of Z. */ /* (A, B) must be in generalized real Schur 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. */ /* Arguments */ /* ========= */ /* JOB (input) CHARACTER*1 */ /* Specifies whether condition numbers are required for */ /* eigenvalues (S) or eigenvectors (DIF): */ /* = 'E': for eigenvalues only (S); */ /* = 'V': for eigenvectors only (DIF); */ /* = 'B': for both eigenvalues and eigenvectors (S and DIF). */ /* HOWMNY (input) CHARACTER*1 */ /* = 'A': compute condition numbers for all eigenpairs; */ /* = 'S': compute condition numbers for selected eigenpairs */ /* specified by the array SELECT. */ /* SELECT (input) LOGICAL array, dimension (N) */ /* If HOWMNY = 'S', SELECT specifies the eigenpairs for which */ /* condition numbers are required. To select condition numbers */ /* for the eigenpair corresponding to a real eigenvalue w(j), */ /* SELECT(j) must be set to .TRUE.. To select condition numbers */ /* corresponding to a complex conjugate pair of eigenvalues w(j) */ /* and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be */ /* set to .TRUE.. */ /* If HOWMNY = 'A', SELECT is not referenced. */ /* N (input) INTEGER */ /* The order of the square matrix pair (A, B). N >= 0. */ /* A (input) DOUBLE PRECISION array, dimension (LDA,N) */ /* The upper quasi-triangular matrix A in the pair (A,B). */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,N). */ /* B (input) DOUBLE PRECISION array, dimension (LDB,N) */ /* The upper triangular matrix B in the pair (A,B). */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,N). */ /* VL (input) DOUBLE PRECISION array, dimension (LDVL,M) */ /* If JOB = 'E' or 'B', VL must contain left eigenvectors of */ /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ /* and SELECT. The eigenvectors must be stored in consecutive */ /* columns of VL, as returned by DTGEVC. */ /* If JOB = 'V', VL is not referenced. */ /* LDVL (input) INTEGER */ /* The leading dimension of the array VL. LDVL >= 1. */ /* If JOB = 'E' or 'B', LDVL >= N. */ /* VR (input) DOUBLE PRECISION array, dimension (LDVR,M) */ /* If JOB = 'E' or 'B', VR must contain right eigenvectors of */ /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ /* and SELECT. The eigenvectors must be stored in consecutive */ /* columns ov VR, as returned by DTGEVC. */ /* If JOB = 'V', VR is not referenced. */ /* LDVR (input) INTEGER */ /* The leading dimension of the array VR. LDVR >= 1. */ /* If JOB = 'E' or 'B', LDVR >= N. */ /* S (output) DOUBLE PRECISION array, dimension (MM) */ /* If JOB = 'E' or 'B', the reciprocal condition numbers of the */ /* selected eigenvalues, stored in consecutive elements of the */ /* array. For a complex conjugate pair of eigenvalues two */ /* consecutive elements of S are set to the same value. Thus */ /* S(j), DIF(j), and the j-th columns of VL and VR all */ /* correspond to the same eigenpair (but not in general the */ /* j-th eigenpair, unless all eigenpairs are selected). */ /* If JOB = 'V', S is not referenced. */ /* DIF (output) DOUBLE PRECISION array, dimension (MM) */ /* If JOB = 'V' or 'B', the estimated reciprocal condition */ /* numbers of the selected eigenvectors, stored in consecutive */ /* elements of the array. For a complex eigenvector two */ /* consecutive elements of DIF are set to the same value. If */ /* the eigenvalues cannot be reordered to compute DIF(j), DIF(j) */ /* is set to 0; this can only occur when the true value would be */ /* very small anyway. */ /* If JOB = 'E', DIF is not referenced. */ /* MM (input) INTEGER */ /* The number of elements in the arrays S and DIF. MM >= M. */ /* M (output) INTEGER */ /* The number of elements of the arrays S and DIF used to store */ /* the specified condition numbers; for each selected real */ /* eigenvalue one element is used, and for each selected complex */ /* conjugate pair of eigenvalues, two elements are used. */ /* If HOWMNY = 'A', M is set to N. */ /* 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 >= max(1,N). */ /* If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. */ /* 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 (N + 6) */ /* If JOB = 'E', IWORK is not referenced. */ /* INFO (output) INTEGER */ /* =0: Successful exit */ /* <0: If INFO = -i, the i-th argument had an illegal value */ /* Further Details */ /* =============== */ /* The reciprocal of the condition number of a generalized eigenvalue */ /* w = (a, b) is defined as */ /* S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v)) */ /* where u and v are the left and right eigenvectors of (A, B) */ /* corresponding to w; |z| denotes the absolute value of the complex */ /* number, and norm(u) denotes the 2-norm of the vector u. */ /* The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv) */ /* of the matrix pair (A, B). If both a and b equal zero, then (A B) is */ /* singular and S(I) = -1 is returned. */ /* An approximate error bound on the chordal distance between the i-th */ /* computed generalized eigenvalue w and the corresponding exact */ /* eigenvalue lambda is */ /* chord(w, lambda) <= EPS * norm(A, B) / S(I) */ /* where EPS is the machine precision. */ /* The reciprocal of the condition number DIF(i) of right eigenvector u */ /* and left eigenvector v corresponding to the generalized eigenvalue w */ /* is defined as follows: */ /* a) If the i-th eigenvalue w = (a,b) is real */ /* Suppose U and V are orthogonal transformations such that */ /* U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 */ /* ( 0 S22 ),( 0 T22 ) n-1 */ /* 1 n-1 1 n-1 */ /* Then the reciprocal condition number DIF(i) is */ /* Difl((a, b), (S22, T22)) = sigma-min( Zl ), */ /* where sigma-min(Zl) denotes the smallest singular value of the */ /* 2(n-1)-by-2(n-1) matrix */ /* Zl = [ kron(a, In-1) -kron(1, S22) ] */ /* [ kron(b, In-1) -kron(1, T22) ] . */ /* Here In-1 is the identity matrix of size n-1. kron(X, Y) is the */ /* Kronecker product between the matrices X and Y. */ /* Note that if the default method for computing DIF(i) is wanted */ /* (see DLATDF), then the parameter DIFDRI (see below) should be */ /* changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). */ /* See DTGSYL for more details. */ /* b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair, */ /* Suppose U and V are orthogonal transformations such that */ /* U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 */ /* ( 0 S22 ),( 0 T22) n-2 */ /* 2 n-2 2 n-2 */ /* and (S11, T11) corresponds to the complex conjugate eigenvalue */ /* pair (w, conjg(w)). There exist unitary matrices U1 and V1 such */ /* that */ /* U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 ) */ /* ( 0 s22 ) ( 0 t22 ) */ /* where the generalized eigenvalues w = s11/t11 and */ /* conjg(w) = s22/t22. */ /* Then the reciprocal condition number DIF(i) is bounded by */ /* min( d1, max( 1, |real(s11)/real(s22)| )*d2 ) */ /* where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where */ /* Z1 is the complex 2-by-2 matrix */ /* Z1 = [ s11 -s22 ] */ /* [ t11 -t22 ], */ /* This is done by computing (using real arithmetic) the */ /* roots of the characteristical polynomial det(Z1' * Z1 - lambda I), */ /* where Z1' denotes the conjugate transpose of Z1 and det(X) denotes */ /* the determinant of X. */ /* and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an */ /* upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) */ /* Z2 = [ kron(S11', In-2) -kron(I2, S22) ] */ /* [ kron(T11', In-2) -kron(I2, T22) ] */ /* Note that if the default method for computing DIF is wanted (see */ /* DLATDF), then the parameter DIFDRI (see below) should be changed */ /* from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL */ /* for more details. */ /* For each eigenvalue/vector specified by SELECT, DIF stores a */ /* Frobenius norm-based estimate of Difl. */ /* An approximate error bound for the i-th computed eigenvector VL(i) or */ /* VR(i) is given by */ /* EPS * norm(A, B) / DIF(i). */ /* See ref. [2-3] for more details and further references. */ /* 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 Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. 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; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1; vr -= vr_offset; --s; --dif; --work; --iwork; /* Function Body */ wantbh = lsame_(job, "B"); wants = lsame_(job, "E") || wantbh; wantdf = lsame_(job, "V") || wantbh; somcon = lsame_(howmny, "S"); *info = 0; lquery = *lwork == -1; if (! wants && ! wantdf) { *info = -1; } else if (! lsame_(howmny, "A") && ! somcon) { *info = -2; } else if (*n < 0) { *info = -4; } else if (*lda < max(1,*n)) { *info = -6; } else if (*ldb < max(1,*n)) { *info = -8; } else if (wants && *ldvl < *n) { *info = -10; } else if (wants && *ldvr < *n) { *info = -12; } else { /* Set M to the number of eigenpairs for which condition numbers */ /* are required, and test MM. */ if (somcon) { *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: */ } } else { *m = *n; } if (*n == 0) { lwmin = 1; } else if (lsame_(job, "V") || lsame_(job, "B")) { lwmin = (*n << 1) * (*n + 2) + 16; } else { lwmin = *n; } work[1] = (doublereal) lwmin; if (*mm < *m) { *info = -15; } else if (*lwork < lwmin && ! lquery) { *info = -18; } } if (*info != 0) { i__1 = -(*info); xerbla_("DTGSNA", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S") / eps; ks = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { /* Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block. */ if (pair) { pair = FALSE_; goto L20; } else { if (k < *n) { pair = a[k + 1 + k * a_dim1] != 0.; } } /* Determine whether condition numbers are required for the k-th */ /* eigenpair. */ if (somcon) { if (pair) { if (! select[k] && ! select[k + 1]) { goto L20; } } else { if (! select[k]) { goto L20; } } } ++ks; if (wants) { /* Compute the reciprocal condition number of the k-th */ /* eigenvalue. */ if (pair) { /* Complex eigenvalue pair. */ d__1 = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); d__2 = dnrm2_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1); rnrm = dlapy2_(&d__1, &d__2); d__1 = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); d__2 = dnrm2_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1); lnrm = dlapy2_(&d__1, &d__2); dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmprr = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); tmpri = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[(ks + 1) * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmpii = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); tmpir = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); uhav = tmprr + tmpii; uhavi = tmpir - tmpri; dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmprr = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); tmpri = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[(ks + 1) * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmpii = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); tmpir = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); uhbv = tmprr + tmpii; uhbvi = tmpir - tmpri; uhav = dlapy2_(&uhav, &uhavi); uhbv = dlapy2_(&uhbv, &uhbvi); cond = dlapy2_(&uhav, &uhbv); s[ks] = cond / (rnrm * lnrm); s[ks + 1] = s[ks]; } else { /* Real eigenvalue. */ rnrm = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); lnrm = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); uhav = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1) ; dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); uhbv = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1) ; cond = dlapy2_(&uhav, &uhbv); if (cond == 0.) { s[ks] = -1.; } else { s[ks] = cond / (rnrm * lnrm); } } } if (wantdf) { if (*n == 1) { dif[ks] = dlapy2_(&a[a_dim1 + 1], &b[b_dim1 + 1]); goto L20; } /* Estimate the reciprocal condition number of the k-th */ /* eigenvectors. */ if (pair) { /* Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)). */ /* 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, dummy1, &alphar, dummy, &alphai); alprqt = 1.; c1 = (alphar * alphar + alphai * alphai + beta * beta) * 2.; c2 = beta * 4. * beta * alphai * alphai; root1 = c1 + sqrt(c1 * c1 - c2 * 4.); root2 = c2 / root1; root1 /= 2.; /* Computing MIN */ d__1 = sqrt(root1), d__2 = sqrt(root2); cond = min(d__1,d__2); } /* Copy the matrix (A, B) to the array WORK and swap the */ /* diagonal block beginning at A(k,k) to the (1,1) position. */ dlacpy_("Full", n, n, &a[a_offset], lda, &work[1], n); dlacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1], n); ifst = k; ilst = 1; i__2 = *lwork - (*n << 1) * *n; dtgexc_(&c_false, &c_false, n, &work[1], n, &work[*n * *n + 1], n, dummy, &c__1, dummy1, &c__1, &ifst, &ilst, &work[(*n * * n << 1) + 1], &i__2, &ierr); if (ierr > 0) { /* Ill-conditioned problem - swap rejected. */ dif[ks] = 0.; } else { /* Reordering successful, solve generalized Sylvester */ /* equation for R and L, */ /* A22 * R - L * A11 = A12 */ /* B22 * R - L * B11 = B12, */ /* and compute estimate of Difl((A11,B11), (A22, B22)). */ n1 = 1; if (work[2] != 0.) { n1 = 2; } n2 = *n - n1; if (n2 == 0) { dif[ks] = cond; } else { i__ = *n * *n + 1; iz = (*n << 1) * *n + 1; i__2 = *lwork - (*n << 1) * *n; dtgsyl_("N", &c__3, &n2, &n1, &work[*n * n1 + n1 + 1], n, &work[1], n, &work[n1 + 1], n, &work[*n * n1 + n1 + i__], n, &work[i__], n, &work[n1 + i__], n, & scale, &dif[ks], &work[iz + 1], &i__2, &iwork[1], &ierr); if (pair) { /* Computing MIN */ d__1 = max(1.,alprqt) * dif[ks]; dif[ks] = min(d__1,cond); } } } if (pair) { dif[ks + 1] = dif[ks]; } } if (pair) { ++ks; } L20: ; } work[1] = (doublereal) lwmin; return 0; /* End of DTGSNA */ } /* dtgsna_ */
/*< >*/ /* 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 *); logical wantd; integer lwmin; logical wantp, wantd1, wantd2; extern doublereal dlamch_(char *, ftnlen); doublereal dscale; extern /* Subroutine */ int dlacon_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); doublereal rdscal; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, ftnlen), xerbla_(char *, integer *, ftnlen), 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 *, ftnlen); doublereal smlnum; logical lquery; /* -- LAPACK routine (version 3.0) -- */ /* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */ /* Courant Institute, Argonne National Lab, and Rice University */ /* June 30, 1999 */ /* .. Scalar Arguments .. */ /*< LOGICAL WANTQ, WANTZ >*/ /*< >*/ /*< DOUBLE PRECISION PL, PR >*/ /* .. */ /* .. Array Arguments .. */ /*< LOGICAL SELECT( * ) >*/ /*< INTEGER IWORK( * ) >*/ /*< >*/ /* .. */ /* 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, 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 (LWORK) */ /* IF IJOB = 0, WORK is not referenced. Otherwise, */ /* 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 (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 .. */ /*< INTEGER IDIFJB >*/ /*< PARAMETER ( IDIFJB = 3 ) >*/ /*< DOUBLE PRECISION ZERO, ONE >*/ /*< PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) >*/ /* .. */ /* .. Local Scalars .. */ /*< >*/ /*< >*/ /*< DOUBLE PRECISION DSCALE, DSUM, EPS, RDSCAL, SMLNUM >*/ /* .. */ /* .. External Subroutines .. */ /*< >*/ /* .. */ /* .. External Functions .. */ /*< DOUBLE PRECISION DLAMCH >*/ /*< EXTERNAL DLAMCH >*/ /* .. */ /* .. Intrinsic Functions .. */ /*< INTRINSIC MAX, SIGN, SQRT >*/ /* .. */ /* .. Executable Statements .. */ /* Decode and test the input parameters */ /*< INFO = 0 >*/ /* 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.EQ.-1 .OR. LIWORK.EQ.-1 ) >*/ lquery = *lwork == -1 || *liwork == -1; /*< IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN >*/ if (*ijob < 0 || *ijob > 5) { /*< INFO = -1 >*/ *info = -1; /*< ELSE IF( N.LT.0 ) THEN >*/ } else if (*n < 0) { /*< INFO = -5 >*/ *info = -5; /*< ELSE IF( LDA.LT.MAX( 1, N ) ) THEN >*/ } else if (*lda < max(1,*n)) { /*< INFO = -7 >*/ *info = -7; /*< ELSE IF( LDB.LT.MAX( 1, N ) ) THEN >*/ } else if (*ldb < max(1,*n)) { /*< INFO = -9 >*/ *info = -9; /*< ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN >*/ } else if (*ldq < 1 || (*wantq && *ldq < *n)) { /*< INFO = -14 >*/ *info = -14; /*< ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN >*/ } else if (*ldz < 1 || (*wantz && *ldz < *n)) { /*< INFO = -16 >*/ *info = -16; /*< END IF >*/ } /*< IF( INFO.NE.0 ) THEN >*/ if (*info != 0) { /*< CALL XERBLA( 'DTGSEN', -INFO ) >*/ i__1 = -(*info); xerbla_("DTGSEN", &i__1, (ftnlen)6); /*< RETURN >*/ return 0; /*< END IF >*/ } /* Get machine constants */ /*< EPS = DLAMCH( 'P' ) >*/ eps = dlamch_("P", (ftnlen)1); /*< SMLNUM = DLAMCH( 'S' ) / EPS >*/ smlnum = dlamch_("S", (ftnlen)1) / eps; /*< IERR = 0 >*/ ierr = 0; /*< WANTP = IJOB.EQ.1 .OR. IJOB.GE.4 >*/ wantp = *ijob == 1 || *ijob >= 4; /*< WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4 >*/ wantd1 = *ijob == 2 || *ijob == 4; /*< WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5 >*/ wantd2 = *ijob == 3 || *ijob == 5; /*< WANTD = WANTD1 .OR. WANTD2 >*/ wantd = wantd1 || wantd2; /* Set M to the dimension of the specified pair of deflating */ /* subspaces. */ /*< M = 0 >*/ *m = 0; /*< PAIR = .FALSE. >*/ pair = FALSE_; /*< DO 10 K = 1, N >*/ i__1 = *n; for (k = 1; k <= i__1; ++k) { /*< IF( PAIR ) THEN >*/ if (pair) { /*< PAIR = .FALSE. >*/ pair = FALSE_; /*< ELSE >*/ } else { /*< IF( K.LT.N ) THEN >*/ if (k < *n) { /*< IF( A( K+1, K ).EQ.ZERO ) THEN >*/ if (a[k + 1 + k * a_dim1] == 0.) { /*< >*/ if (select[k]) { ++(*m); } /*< ELSE >*/ } else { /*< PAIR = .TRUE. >*/ pair = TRUE_; /*< >*/ if (select[k] || select[k + 1]) { *m += 2; } /*< END IF >*/ } /*< ELSE >*/ } else { /*< >*/ if (select[*n]) { ++(*m); } /*< END IF >*/ } /*< END IF >*/ } /*< 10 CONTINUE >*/ /* L10: */ } /*< IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN >*/ if (*ijob == 1 || *ijob == 2 || *ijob == 4) { /*< LWMIN = MAX( 1, 4*N+16, 2*M*( N-M ) ) >*/ /* 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); /*< LIWMIN = MAX( 1, N+6 ) >*/ /* Computing MAX */ i__1 = 1, i__2 = *n + 6; liwmin = max(i__1,i__2); /*< ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN >*/ } else if (*ijob == 3 || *ijob == 5) { /*< LWMIN = MAX( 1, 4*N+16, 4*M*( N-M ) ) >*/ /* 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); /*< LIWMIN = MAX( 1, 2*M*( N-M ), N+6 ) >*/ /* 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 >*/ } else { /*< LWMIN = MAX( 1, 4*N+16 ) >*/ /* Computing MAX */ i__1 = 1, i__2 = (*n << 2) + 16; lwmin = max(i__1,i__2); /*< LIWMIN = 1 >*/ liwmin = 1; /*< END IF >*/ } /*< WORK( 1 ) = LWMIN >*/ work[1] = (doublereal) lwmin; /*< IWORK( 1 ) = LIWMIN >*/ iwork[1] = liwmin; /*< IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN >*/ if (*lwork < lwmin && ! lquery) { /*< INFO = -22 >*/ *info = -22; /*< ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN >*/ } else if (*liwork < liwmin && ! lquery) { /*< INFO = -24 >*/ *info = -24; /*< END IF >*/ } /*< IF( INFO.NE.0 ) THEN >*/ if (*info != 0) { /*< CALL XERBLA( 'DTGSEN', -INFO ) >*/ i__1 = -(*info); xerbla_("DTGSEN", &i__1, (ftnlen)6); /*< RETURN >*/ return 0; /*< ELSE IF( LQUERY ) THEN >*/ } else if (lquery) { /*< RETURN >*/ return 0; /*< END IF >*/ } /* Quick return if possible. */ /*< IF( M.EQ.N .OR. M.EQ.0 ) THEN >*/ if (*m == *n || *m == 0) { /*< IF( WANTP ) THEN >*/ if (wantp) { /*< PL = ONE >*/ *pl = 1.; /*< PR = ONE >*/ *pr = 1.; /*< END IF >*/ } /*< IF( WANTD ) THEN >*/ if (wantd) { /*< DSCALE = ZERO >*/ dscale = 0.; /*< DSUM = ONE >*/ dsum = 1.; /*< DO 20 I = 1, N >*/ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { /*< CALL DLASSQ( N, A( 1, I ), 1, DSCALE, DSUM ) >*/ dlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum); /*< CALL DLASSQ( N, B( 1, I ), 1, DSCALE, DSUM ) >*/ dlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum); /*< 20 CONTINUE >*/ /* L20: */ } /*< DIF( 1 ) = DSCALE*SQRT( DSUM ) >*/ dif[1] = dscale * sqrt(dsum); /*< DIF( 2 ) = DIF( 1 ) >*/ dif[2] = dif[1]; /*< END IF >*/ } /*< GO TO 60 >*/ goto L60; /*< END IF >*/ } /* Collect the selected blocks at the top-left corner of (A, B). */ /*< KS = 0 >*/ ks = 0; /*< PAIR = .FALSE. >*/ pair = FALSE_; /*< DO 30 K = 1, N >*/ i__1 = *n; for (k = 1; k <= i__1; ++k) { /*< IF( PAIR ) THEN >*/ if (pair) { /*< PAIR = .FALSE. >*/ pair = FALSE_; /*< ELSE >*/ } else { /*< SWAP = SELECT( K ) >*/ swap = select[k]; /*< IF( K.LT.N ) THEN >*/ if (k < *n) { /*< IF( A( K+1, K ).NE.ZERO ) THEN >*/ if (a[k + 1 + k * a_dim1] != 0.) { /*< PAIR = .TRUE. >*/ pair = TRUE_; /*< SWAP = SWAP .OR. SELECT( K+1 ) >*/ swap = swap || select[k + 1]; /*< END IF >*/ } /*< END IF >*/ } /*< IF( SWAP ) THEN >*/ if (swap) { /*< KS = KS + 1 >*/ ++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 >*/ 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.GT.0 ) THEN >*/ if (ierr > 0) { /* Swap is rejected: exit. */ /*< INFO = 1 >*/ *info = 1; /*< IF( WANTP ) THEN >*/ if (wantp) { /*< PL = ZERO >*/ *pl = 0.; /*< PR = ZERO >*/ *pr = 0.; /*< END IF >*/ } /*< IF( WANTD ) THEN >*/ if (wantd) { /*< DIF( 1 ) = ZERO >*/ dif[1] = 0.; /*< DIF( 2 ) = ZERO >*/ dif[2] = 0.; /*< END IF >*/ } /*< GO TO 60 >*/ goto L60; /*< END IF >*/ } /*< >*/ if (pair) { ++ks; } /*< END IF >*/ } /*< END IF >*/ } /*< 30 CONTINUE >*/ /* L30: */ } /*< IF( WANTP ) THEN >*/ if (wantp) { /* Solve generalized Sylvester equation for R and L */ /* and compute PL and PR. */ /*< N1 = M >*/ n1 = *m; /*< N2 = N - M >*/ n2 = *n - *m; /*< I = N1 + 1 >*/ i__ = n1 + 1; /*< IJB = 0 >*/ ijb = 0; /*< CALL DLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 ) >*/ dlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1, ( ftnlen)4); /*< >*/ dlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 + 1], &n1, (ftnlen)4); /*< >*/ 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, (ftnlen)1); /* Estimate the reciprocal of norms of "projections" onto left */ /* and right eigenspaces. */ /*< RDSCAL = ZERO >*/ rdscal = 0.; /*< DSUM = ONE >*/ dsum = 1.; /*< CALL DLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM ) >*/ i__1 = n1 * n2; dlassq_(&i__1, &work[1], &c__1, &rdscal, &dsum); /*< PL = RDSCAL*SQRT( DSUM ) >*/ *pl = rdscal * sqrt(dsum); /*< IF( PL.EQ.ZERO ) THEN >*/ if (*pl == 0.) { /*< PL = ONE >*/ *pl = 1.; /*< ELSE >*/ } else { /*< PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) ) >*/ *pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl)); /*< END IF >*/ } /*< RDSCAL = ZERO >*/ rdscal = 0.; /*< DSUM = ONE >*/ dsum = 1.; /*< CALL DLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM ) >*/ i__1 = n1 * n2; dlassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum); /*< PR = RDSCAL*SQRT( DSUM ) >*/ *pr = rdscal * sqrt(dsum); /*< IF( PR.EQ.ZERO ) THEN >*/ if (*pr == 0.) { /*< PR = ONE >*/ *pr = 1.; /*< ELSE >*/ } else { /*< PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) ) >*/ *pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr)); /*< END IF >*/ } /*< END IF >*/ } /*< IF( WANTD ) THEN >*/ if (wantd) { /* Compute estimates of Difu and Difl. */ /*< IF( WANTD1 ) THEN >*/ if (wantd1) { /*< N1 = M >*/ n1 = *m; /*< N2 = N - M >*/ n2 = *n - *m; /*< I = N1 + 1 >*/ i__ = n1 + 1; /*< IJB = IDIFJB >*/ 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, (ftnlen)1); /* 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, (ftnlen)1); /*< ELSE >*/ } else { /* Compute 1-norm-based estimates of Difu and Difl using */ /* reversed communication with DLACON. In each step a */ /* generalized Sylvester equation or a transposed variant */ /* is solved. */ /*< KASE = 0 >*/ kase = 0; /*< N1 = M >*/ n1 = *m; /*< N2 = N - M >*/ n2 = *n - *m; /*< I = N1 + 1 >*/ i__ = n1 + 1; /*< IJB = 0 >*/ ijb = 0; /*< MN2 = 2*N1*N2 >*/ mn2 = (n1 << 1) * n2; /* 1-norm-based estimate of Difu. */ /*< 40 CONTINUE >*/ L40: /*< >*/ dlacon_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[1], &kase) ; /*< IF( KASE.NE.0 ) THEN >*/ if (kase != 0) { /*< IF( KASE.EQ.1 ) THEN >*/ 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, (ftnlen)1); /*< ELSE >*/ } 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, (ftnlen)1); /*< END IF >*/ } /*< GO TO 40 >*/ goto L40; /*< END IF >*/ } /*< DIF( 1 ) = DSCALE / DIF( 1 ) >*/ dif[1] = dscale / dif[1]; /* 1-norm-based estimate of Difl. */ /*< 50 CONTINUE >*/ L50: /*< >*/ dlacon_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[2], &kase) ; /*< IF( KASE.NE.0 ) THEN >*/ if (kase != 0) { /*< IF( KASE.EQ.1 ) THEN >*/ 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, (ftnlen)1); /*< ELSE >*/ } 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, (ftnlen)1); /*< END IF >*/ } /*< GO TO 50 >*/ goto L50; /*< END IF >*/ } /*< DIF( 2 ) = DSCALE / DIF( 2 ) >*/ dif[2] = dscale / dif[2]; /*< END IF >*/ } /*< END IF >*/ } /*< 60 CONTINUE >*/ L60: /* Compute generalized eigenvalues of reordered pair (A, B) and */ /* normalize the generalized Schur form. */ /*< PAIR = .FALSE. >*/ pair = FALSE_; /*< DO 80 K = 1, N >*/ i__1 = *n; for (k = 1; k <= i__1; ++k) { /*< IF( PAIR ) THEN >*/ if (pair) { /*< PAIR = .FALSE. >*/ pair = FALSE_; /*< ELSE >*/ } else { /*< IF( K.LT.N ) THEN >*/ if (k < *n) { /*< IF( A( K+1, K ).NE.ZERO ) THEN >*/ if (a[k + 1 + k * a_dim1] != 0.) { /*< PAIR = .TRUE. >*/ pair = TRUE_; /*< END IF >*/ } /*< END IF >*/ } /*< IF( PAIR ) THEN >*/ if (pair) { /* Compute the eigenvalue(s) at position K. */ /*< WORK( 1 ) = A( K, K ) >*/ work[1] = a[k + k * a_dim1]; /*< WORK( 2 ) = A( K+1, K ) >*/ work[2] = a[k + 1 + k * a_dim1]; /*< WORK( 3 ) = A( K, K+1 ) >*/ work[3] = a[k + (k + 1) * a_dim1]; /*< WORK( 4 ) = A( K+1, K+1 ) >*/ work[4] = a[k + 1 + (k + 1) * a_dim1]; /*< WORK( 5 ) = B( K, K ) >*/ work[5] = b[k + k * b_dim1]; /*< WORK( 6 ) = B( K+1, K ) >*/ work[6] = b[k + 1 + k * b_dim1]; /*< WORK( 7 ) = B( K, K+1 ) >*/ work[7] = b[k + (k + 1) * b_dim1]; /*< WORK( 8 ) = B( K+1, K+1 ) >*/ 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 ) >*/ alphai[k + 1] = -alphai[k]; /*< ELSE >*/ } else { /*< IF( SIGN( ONE, B( K, K ) ).LT.ZERO ) THEN >*/ if (d_sign(&c_b28, &b[k + k * b_dim1]) < 0.) { /* If B(K,K) is negative, make it positive */ /*< DO 70 I = 1, N >*/ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { /*< A( K, I ) = -A( K, I ) >*/ a[k + i__ * a_dim1] = -a[k + i__ * a_dim1]; /*< B( K, I ) = -B( K, I ) >*/ b[k + i__ * b_dim1] = -b[k + i__ * b_dim1]; /*< Q( I, K ) = -Q( I, K ) >*/ q[i__ + k * q_dim1] = -q[i__ + k * q_dim1]; /*< 70 CONTINUE >*/ /* L70: */ } /*< END IF >*/ } /*< ALPHAR( K ) = A( K, K ) >*/ alphar[k] = a[k + k * a_dim1]; /*< ALPHAI( K ) = ZERO >*/ alphai[k] = 0.; /*< BETA( K ) = B( K, K ) >*/ beta[k] = b[k + k * b_dim1]; /*< END IF >*/ } /*< END IF >*/ } /*< 80 CONTINUE >*/ /* L80: */ } /*< WORK( 1 ) = LWMIN >*/ work[1] = (doublereal) lwmin; /*< IWORK( 1 ) = LIWMIN >*/ iwork[1] = liwmin; /*< RETURN >*/ return 0; /* End of DTGSEN */ /*< END >*/ } /* dtgsen_ */