/* 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_ */
