00001 /* ctgsen.f -- translated by f2c (version 20061008). 00002 You must link the resulting object file with libf2c: 00003 on Microsoft Windows system, link with libf2c.lib; 00004 on Linux or Unix systems, link with .../path/to/libf2c.a -lm 00005 or, if you install libf2c.a in a standard place, with -lf2c -lm 00006 -- in that order, at the end of the command line, as in 00007 cc *.o -lf2c -lm 00008 Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., 00009 00010 http://www.netlib.org/f2c/libf2c.zip 00011 */ 00012 00013 #include "f2c.h" 00014 #include "blaswrap.h" 00015 00016 /* Table of constant values */ 00017 00018 static integer c__1 = 1; 00019 00020 /* Subroutine */ int ctgsen_(integer *ijob, logical *wantq, logical *wantz, 00021 logical *select, integer *n, complex *a, integer *lda, complex *b, 00022 integer *ldb, complex *alpha, complex *beta, complex *q, integer *ldq, 00023 complex *z__, integer *ldz, integer *m, real *pl, real *pr, real * 00024 dif, complex *work, integer *lwork, integer *iwork, integer *liwork, 00025 integer *info) 00026 { 00027 /* System generated locals */ 00028 integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1, 00029 z_offset, i__1, i__2, i__3; 00030 complex q__1, q__2; 00031 00032 /* Builtin functions */ 00033 double sqrt(doublereal), c_abs(complex *); 00034 void r_cnjg(complex *, complex *); 00035 00036 /* Local variables */ 00037 integer i__, k, n1, n2, ks, mn2, ijb, kase, ierr; 00038 real dsum; 00039 logical swap; 00040 complex temp1, temp2; 00041 extern /* Subroutine */ int cscal_(integer *, complex *, complex *, 00042 integer *); 00043 integer isave[3]; 00044 logical wantd; 00045 integer lwmin; 00046 logical wantp; 00047 extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real 00048 *, integer *, integer *); 00049 logical wantd1, wantd2; 00050 real dscale; 00051 extern doublereal slamch_(char *); 00052 real rdscal; 00053 extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex 00054 *, integer *, complex *, integer *); 00055 real safmin; 00056 extern /* Subroutine */ int ctgexc_(logical *, logical *, integer *, 00057 complex *, integer *, complex *, integer *, complex *, integer *, 00058 complex *, integer *, integer *, integer *, integer *), xerbla_( 00059 char *, integer *), classq_(integer *, complex *, integer 00060 *, real *, real *); 00061 integer liwmin; 00062 extern /* Subroutine */ int ctgsyl_(char *, integer *, integer *, integer 00063 *, complex *, integer *, complex *, integer *, complex *, integer 00064 *, complex *, integer *, complex *, integer *, complex *, integer 00065 *, real *, real *, complex *, integer *, integer *, integer *); 00066 logical lquery; 00067 00068 00069 /* -- LAPACK routine (version 3.2) -- */ 00070 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00071 /* January 2007 */ 00072 00073 /* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */ 00074 00075 /* .. Scalar Arguments .. */ 00076 /* .. */ 00077 /* .. Array Arguments .. */ 00078 /* .. */ 00079 00080 /* Purpose */ 00081 /* ======= */ 00082 00083 /* CTGSEN reorders the generalized Schur decomposition of a complex */ 00084 /* matrix pair (A, B) (in terms of an unitary equivalence trans- */ 00085 /* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */ 00086 /* appears in the leading diagonal blocks of the pair (A,B). The leading */ 00087 /* columns of Q and Z form unitary bases of the corresponding left and */ 00088 /* right eigenspaces (deflating subspaces). (A, B) must be in */ 00089 /* generalized Schur canonical form, that is, A and B are both upper */ 00090 /* triangular. */ 00091 00092 /* CTGSEN also computes the generalized eigenvalues */ 00093 00094 /* w(j)= ALPHA(j) / BETA(j) */ 00095 00096 /* of the reordered matrix pair (A, B). */ 00097 00098 /* Optionally, the routine computes estimates of reciprocal condition */ 00099 /* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */ 00100 /* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */ 00101 /* between the matrix pairs (A11, B11) and (A22,B22) that correspond to */ 00102 /* the selected cluster and the eigenvalues outside the cluster, resp., */ 00103 /* and norms of "projections" onto left and right eigenspaces w.r.t. */ 00104 /* the selected cluster in the (1,1)-block. */ 00105 00106 00107 /* Arguments */ 00108 /* ========= */ 00109 00110 /* IJOB (input) integer */ 00111 /* Specifies whether condition numbers are required for the */ 00112 /* cluster of eigenvalues (PL and PR) or the deflating subspaces */ 00113 /* (Difu and Difl): */ 00114 /* =0: Only reorder w.r.t. SELECT. No extras. */ 00115 /* =1: Reciprocal of norms of "projections" onto left and right */ 00116 /* eigenspaces w.r.t. the selected cluster (PL and PR). */ 00117 /* =2: Upper bounds on Difu and Difl. F-norm-based estimate */ 00118 /* (DIF(1:2)). */ 00119 /* =3: Estimate of Difu and Difl. 1-norm-based estimate */ 00120 /* (DIF(1:2)). */ 00121 /* About 5 times as expensive as IJOB = 2. */ 00122 /* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */ 00123 /* version to get it all. */ 00124 /* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */ 00125 00126 /* WANTQ (input) LOGICAL */ 00127 /* .TRUE. : update the left transformation matrix Q; */ 00128 /* .FALSE.: do not update Q. */ 00129 00130 /* WANTZ (input) LOGICAL */ 00131 /* .TRUE. : update the right transformation matrix Z; */ 00132 /* .FALSE.: do not update Z. */ 00133 00134 /* SELECT (input) LOGICAL array, dimension (N) */ 00135 /* SELECT specifies the eigenvalues in the selected cluster. To */ 00136 /* select an eigenvalue w(j), SELECT(j) must be set to */ 00137 /* .TRUE.. */ 00138 00139 /* N (input) INTEGER */ 00140 /* The order of the matrices A and B. N >= 0. */ 00141 00142 /* A (input/output) COMPLEX array, dimension(LDA,N) */ 00143 /* On entry, the upper triangular matrix A, in generalized */ 00144 /* Schur canonical form. */ 00145 /* On exit, A is overwritten by the reordered matrix A. */ 00146 00147 /* LDA (input) INTEGER */ 00148 /* The leading dimension of the array A. LDA >= max(1,N). */ 00149 00150 /* B (input/output) COMPLEX array, dimension(LDB,N) */ 00151 /* On entry, the upper triangular matrix B, in generalized */ 00152 /* Schur canonical form. */ 00153 /* On exit, B is overwritten by the reordered matrix B. */ 00154 00155 /* LDB (input) INTEGER */ 00156 /* The leading dimension of the array B. LDB >= max(1,N). */ 00157 00158 /* ALPHA (output) COMPLEX array, dimension (N) */ 00159 /* BETA (output) COMPLEX array, dimension (N) */ 00160 /* The diagonal elements of A and B, respectively, */ 00161 /* when the pair (A,B) has been reduced to generalized Schur */ 00162 /* form. ALPHA(i)/BETA(i) i=1,...,N are the generalized */ 00163 /* eigenvalues. */ 00164 00165 /* Q (input/output) COMPLEX array, dimension (LDQ,N) */ 00166 /* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */ 00167 /* On exit, Q has been postmultiplied by the left unitary */ 00168 /* transformation matrix which reorder (A, B); The leading M */ 00169 /* columns of Q form orthonormal bases for the specified pair of */ 00170 /* left eigenspaces (deflating subspaces). */ 00171 /* If WANTQ = .FALSE., Q is not referenced. */ 00172 00173 /* LDQ (input) INTEGER */ 00174 /* The leading dimension of the array Q. LDQ >= 1. */ 00175 /* If WANTQ = .TRUE., LDQ >= N. */ 00176 00177 /* Z (input/output) COMPLEX array, dimension (LDZ,N) */ 00178 /* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */ 00179 /* On exit, Z has been postmultiplied by the left unitary */ 00180 /* transformation matrix which reorder (A, B); The leading M */ 00181 /* columns of Z form orthonormal bases for the specified pair of */ 00182 /* left eigenspaces (deflating subspaces). */ 00183 /* If WANTZ = .FALSE., Z is not referenced. */ 00184 00185 /* LDZ (input) INTEGER */ 00186 /* The leading dimension of the array Z. LDZ >= 1. */ 00187 /* If WANTZ = .TRUE., LDZ >= N. */ 00188 00189 /* M (output) INTEGER */ 00190 /* The dimension of the specified pair of left and right */ 00191 /* eigenspaces, (deflating subspaces) 0 <= M <= N. */ 00192 00193 /* PL (output) REAL */ 00194 /* PR (output) REAL */ 00195 /* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */ 00196 /* reciprocal of the norm of "projections" onto left and right */ 00197 /* eigenspace with respect to the selected cluster. */ 00198 /* 0 < PL, PR <= 1. */ 00199 /* If M = 0 or M = N, PL = PR = 1. */ 00200 /* If IJOB = 0, 2 or 3 PL, PR are not referenced. */ 00201 00202 /* DIF (output) REAL array, dimension (2). */ 00203 /* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */ 00204 /* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */ 00205 /* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */ 00206 /* estimates of Difu and Difl, computed using reversed */ 00207 /* communication with CLACN2. */ 00208 /* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */ 00209 /* If IJOB = 0 or 1, DIF is not referenced. */ 00210 00211 /* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) */ 00212 /* IF IJOB = 0, WORK is not referenced. Otherwise, */ 00213 /* on exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ 00214 00215 /* LWORK (input) INTEGER */ 00216 /* The dimension of the array WORK. LWORK >= 1 */ 00217 /* If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) */ 00218 /* If IJOB = 3 or 5, LWORK >= 4*M*(N-M) */ 00219 00220 /* If LWORK = -1, then a workspace query is assumed; the routine */ 00221 /* only calculates the optimal size of the WORK array, returns */ 00222 /* this value as the first entry of the WORK array, and no error */ 00223 /* message related to LWORK is issued by XERBLA. */ 00224 00225 /* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */ 00226 /* IF IJOB = 0, IWORK is not referenced. Otherwise, */ 00227 /* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */ 00228 00229 /* LIWORK (input) INTEGER */ 00230 /* The dimension of the array IWORK. LIWORK >= 1. */ 00231 /* If IJOB = 1, 2 or 4, LIWORK >= N+2; */ 00232 /* If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); */ 00233 00234 /* If LIWORK = -1, then a workspace query is assumed; the */ 00235 /* routine only calculates the optimal size of the IWORK array, */ 00236 /* returns this value as the first entry of the IWORK array, and */ 00237 /* no error message related to LIWORK is issued by XERBLA. */ 00238 00239 /* INFO (output) INTEGER */ 00240 /* =0: Successful exit. */ 00241 /* <0: If INFO = -i, the i-th argument had an illegal value. */ 00242 /* =1: Reordering of (A, B) failed because the transformed */ 00243 /* matrix pair (A, B) would be too far from generalized */ 00244 /* Schur form; the problem is very ill-conditioned. */ 00245 /* (A, B) may have been partially reordered. */ 00246 /* If requested, 0 is returned in DIF(*), PL and PR. */ 00247 00248 00249 /* Further Details */ 00250 /* =============== */ 00251 00252 /* CTGSEN first collects the selected eigenvalues by computing unitary */ 00253 /* U and W that move them to the top left corner of (A, B). In other */ 00254 /* words, the selected eigenvalues are the eigenvalues of (A11, B11) in */ 00255 00256 /* U'*(A, B)*W = (A11 A12) (B11 B12) n1 */ 00257 /* ( 0 A22),( 0 B22) n2 */ 00258 /* n1 n2 n1 n2 */ 00259 00260 /* where N = n1+n2 and U' means the conjugate transpose of U. The first */ 00261 /* n1 columns of U and W span the specified pair of left and right */ 00262 /* eigenspaces (deflating subspaces) of (A, B). */ 00263 00264 /* If (A, B) has been obtained from the generalized real Schur */ 00265 /* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */ 00266 /* reordered generalized Schur form of (C, D) is given by */ 00267 00268 /* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', */ 00269 00270 /* and the first n1 columns of Q*U and Z*W span the corresponding */ 00271 /* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */ 00272 00273 /* Note that if the selected eigenvalue is sufficiently ill-conditioned, */ 00274 /* then its value may differ significantly from its value before */ 00275 /* reordering. */ 00276 00277 /* The reciprocal condition numbers of the left and right eigenspaces */ 00278 /* spanned by the first n1 columns of U and W (or Q*U and Z*W) may */ 00279 /* be returned in DIF(1:2), corresponding to Difu and Difl, resp. */ 00280 00281 /* The Difu and Difl are defined as: */ 00282 00283 /* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) */ 00284 /* and */ 00285 /* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */ 00286 00287 /* where sigma-min(Zu) is the smallest singular value of the */ 00288 /* (2*n1*n2)-by-(2*n1*n2) matrix */ 00289 00290 /* Zu = [ kron(In2, A11) -kron(A22', In1) ] */ 00291 /* [ kron(In2, B11) -kron(B22', In1) ]. */ 00292 00293 /* Here, Inx is the identity matrix of size nx and A22' is the */ 00294 /* transpose of A22. kron(X, Y) is the Kronecker product between */ 00295 /* the matrices X and Y. */ 00296 00297 /* When DIF(2) is small, small changes in (A, B) can cause large changes */ 00298 /* in the deflating subspace. An approximate (asymptotic) bound on the */ 00299 /* maximum angular error in the computed deflating subspaces is */ 00300 00301 /* EPS * norm((A, B)) / DIF(2), */ 00302 00303 /* where EPS is the machine precision. */ 00304 00305 /* The reciprocal norm of the projectors on the left and right */ 00306 /* eigenspaces associated with (A11, B11) may be returned in PL and PR. */ 00307 /* They are computed as follows. First we compute L and R so that */ 00308 /* P*(A, B)*Q is block diagonal, where */ 00309 00310 /* P = ( I -L ) n1 Q = ( I R ) n1 */ 00311 /* ( 0 I ) n2 and ( 0 I ) n2 */ 00312 /* n1 n2 n1 n2 */ 00313 00314 /* and (L, R) is the solution to the generalized Sylvester equation */ 00315 00316 /* A11*R - L*A22 = -A12 */ 00317 /* B11*R - L*B22 = -B12 */ 00318 00319 /* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */ 00320 /* An approximate (asymptotic) bound on the average absolute error of */ 00321 /* the selected eigenvalues is */ 00322 00323 /* EPS * norm((A, B)) / PL. */ 00324 00325 /* There are also global error bounds which valid for perturbations up */ 00326 /* to a certain restriction: A lower bound (x) on the smallest */ 00327 /* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */ 00328 /* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */ 00329 /* (i.e. (A + E, B + F), is */ 00330 00331 /* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). */ 00332 00333 /* An approximate bound on x can be computed from DIF(1:2), PL and PR. */ 00334 00335 /* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */ 00336 /* (L', R') and unperturbed (L, R) left and right deflating subspaces */ 00337 /* associated with the selected cluster in the (1,1)-blocks can be */ 00338 /* bounded as */ 00339 00340 /* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */ 00341 /* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */ 00342 00343 /* See LAPACK User's Guide section 4.11 or the following references */ 00344 /* for more information. */ 00345 00346 /* Note that if the default method for computing the Frobenius-norm- */ 00347 /* based estimate DIF is not wanted (see CLATDF), then the parameter */ 00348 /* IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF */ 00349 /* (IJOB = 2 will be used)). See CTGSYL for more details. */ 00350 00351 /* Based on contributions by */ 00352 /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ 00353 /* Umea University, S-901 87 Umea, Sweden. */ 00354 00355 /* References */ 00356 /* ========== */ 00357 00358 /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ 00359 /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ 00360 /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ 00361 /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ 00362 00363 /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ 00364 /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ 00365 /* Estimation: Theory, Algorithms and Software, Report */ 00366 /* UMINF - 94.04, Department of Computing Science, Umea University, */ 00367 /* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. */ 00368 /* To appear in Numerical Algorithms, 1996. */ 00369 00370 /* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */ 00371 /* for Solving the Generalized Sylvester Equation and Estimating the */ 00372 /* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */ 00373 /* Department of Computing Science, Umea University, S-901 87 Umea, */ 00374 /* Sweden, December 1993, Revised April 1994, Also as LAPACK working */ 00375 /* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */ 00376 /* 1996. */ 00377 00378 /* ===================================================================== */ 00379 00380 /* .. Parameters .. */ 00381 /* .. */ 00382 /* .. Local Scalars .. */ 00383 /* .. */ 00384 /* .. Local Arrays .. */ 00385 /* .. */ 00386 /* .. External Subroutines .. */ 00387 /* .. */ 00388 /* .. Intrinsic Functions .. */ 00389 /* .. */ 00390 /* .. Executable Statements .. */ 00391 00392 /* Decode and test the input parameters */ 00393 00394 /* Parameter adjustments */ 00395 --select; 00396 a_dim1 = *lda; 00397 a_offset = 1 + a_dim1; 00398 a -= a_offset; 00399 b_dim1 = *ldb; 00400 b_offset = 1 + b_dim1; 00401 b -= b_offset; 00402 --alpha; 00403 --beta; 00404 q_dim1 = *ldq; 00405 q_offset = 1 + q_dim1; 00406 q -= q_offset; 00407 z_dim1 = *ldz; 00408 z_offset = 1 + z_dim1; 00409 z__ -= z_offset; 00410 --dif; 00411 --work; 00412 --iwork; 00413 00414 /* Function Body */ 00415 *info = 0; 00416 lquery = *lwork == -1 || *liwork == -1; 00417 00418 if (*ijob < 0 || *ijob > 5) { 00419 *info = -1; 00420 } else if (*n < 0) { 00421 *info = -5; 00422 } else if (*lda < max(1,*n)) { 00423 *info = -7; 00424 } else if (*ldb < max(1,*n)) { 00425 *info = -9; 00426 } else if (*ldq < 1 || *wantq && *ldq < *n) { 00427 *info = -13; 00428 } else if (*ldz < 1 || *wantz && *ldz < *n) { 00429 *info = -15; 00430 } 00431 00432 if (*info != 0) { 00433 i__1 = -(*info); 00434 xerbla_("CTGSEN", &i__1); 00435 return 0; 00436 } 00437 00438 ierr = 0; 00439 00440 wantp = *ijob == 1 || *ijob >= 4; 00441 wantd1 = *ijob == 2 || *ijob == 4; 00442 wantd2 = *ijob == 3 || *ijob == 5; 00443 wantd = wantd1 || wantd2; 00444 00445 /* Set M to the dimension of the specified pair of deflating */ 00446 /* subspaces. */ 00447 00448 *m = 0; 00449 i__1 = *n; 00450 for (k = 1; k <= i__1; ++k) { 00451 i__2 = k; 00452 i__3 = k + k * a_dim1; 00453 alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i; 00454 i__2 = k; 00455 i__3 = k + k * b_dim1; 00456 beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i; 00457 if (k < *n) { 00458 if (select[k]) { 00459 ++(*m); 00460 } 00461 } else { 00462 if (select[*n]) { 00463 ++(*m); 00464 } 00465 } 00466 /* L10: */ 00467 } 00468 00469 if (*ijob == 1 || *ijob == 2 || *ijob == 4) { 00470 /* Computing MAX */ 00471 i__1 = 1, i__2 = (*m << 1) * (*n - *m); 00472 lwmin = max(i__1,i__2); 00473 /* Computing MAX */ 00474 i__1 = 1, i__2 = *n + 2; 00475 liwmin = max(i__1,i__2); 00476 } else if (*ijob == 3 || *ijob == 5) { 00477 /* Computing MAX */ 00478 i__1 = 1, i__2 = (*m << 2) * (*n - *m); 00479 lwmin = max(i__1,i__2); 00480 /* Computing MAX */ 00481 i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = max(i__1,i__2), i__2 = 00482 *n + 2; 00483 liwmin = max(i__1,i__2); 00484 } else { 00485 lwmin = 1; 00486 liwmin = 1; 00487 } 00488 00489 work[1].r = (real) lwmin, work[1].i = 0.f; 00490 iwork[1] = liwmin; 00491 00492 if (*lwork < lwmin && ! lquery) { 00493 *info = -21; 00494 } else if (*liwork < liwmin && ! lquery) { 00495 *info = -23; 00496 } 00497 00498 if (*info != 0) { 00499 i__1 = -(*info); 00500 xerbla_("CTGSEN", &i__1); 00501 return 0; 00502 } else if (lquery) { 00503 return 0; 00504 } 00505 00506 /* Quick return if possible. */ 00507 00508 if (*m == *n || *m == 0) { 00509 if (wantp) { 00510 *pl = 1.f; 00511 *pr = 1.f; 00512 } 00513 if (wantd) { 00514 dscale = 0.f; 00515 dsum = 1.f; 00516 i__1 = *n; 00517 for (i__ = 1; i__ <= i__1; ++i__) { 00518 classq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum); 00519 classq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum); 00520 /* L20: */ 00521 } 00522 dif[1] = dscale * sqrt(dsum); 00523 dif[2] = dif[1]; 00524 } 00525 goto L70; 00526 } 00527 00528 /* Get machine constant */ 00529 00530 safmin = slamch_("S"); 00531 00532 /* Collect the selected blocks at the top-left corner of (A, B). */ 00533 00534 ks = 0; 00535 i__1 = *n; 00536 for (k = 1; k <= i__1; ++k) { 00537 swap = select[k]; 00538 if (swap) { 00539 ++ks; 00540 00541 /* Swap the K-th block to position KS. Compute unitary Q */ 00542 /* and Z that will swap adjacent diagonal blocks in (A, B). */ 00543 00544 if (k != ks) { 00545 ctgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb, 00546 &q[q_offset], ldq, &z__[z_offset], ldz, &k, &ks, & 00547 ierr); 00548 } 00549 00550 if (ierr > 0) { 00551 00552 /* Swap is rejected: exit. */ 00553 00554 *info = 1; 00555 if (wantp) { 00556 *pl = 0.f; 00557 *pr = 0.f; 00558 } 00559 if (wantd) { 00560 dif[1] = 0.f; 00561 dif[2] = 0.f; 00562 } 00563 goto L70; 00564 } 00565 } 00566 /* L30: */ 00567 } 00568 if (wantp) { 00569 00570 /* Solve generalized Sylvester equation for R and L: */ 00571 /* A11 * R - L * A22 = A12 */ 00572 /* B11 * R - L * B22 = B12 */ 00573 00574 n1 = *m; 00575 n2 = *n - *m; 00576 i__ = n1 + 1; 00577 clacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1); 00578 clacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 + 00579 1], &n1); 00580 ijb = 0; 00581 i__1 = *lwork - (n1 << 1) * n2; 00582 ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1] 00583 , lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * 00584 b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], & 00585 work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr); 00586 00587 /* Estimate the reciprocal of norms of "projections" onto */ 00588 /* left and right eigenspaces */ 00589 00590 rdscal = 0.f; 00591 dsum = 1.f; 00592 i__1 = n1 * n2; 00593 classq_(&i__1, &work[1], &c__1, &rdscal, &dsum); 00594 *pl = rdscal * sqrt(dsum); 00595 if (*pl == 0.f) { 00596 *pl = 1.f; 00597 } else { 00598 *pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl)); 00599 } 00600 rdscal = 0.f; 00601 dsum = 1.f; 00602 i__1 = n1 * n2; 00603 classq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum); 00604 *pr = rdscal * sqrt(dsum); 00605 if (*pr == 0.f) { 00606 *pr = 1.f; 00607 } else { 00608 *pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr)); 00609 } 00610 } 00611 if (wantd) { 00612 00613 /* Compute estimates Difu and Difl. */ 00614 00615 if (wantd1) { 00616 n1 = *m; 00617 n2 = *n - *m; 00618 i__ = n1 + 1; 00619 ijb = 3; 00620 00621 /* Frobenius norm-based Difu estimate. */ 00622 00623 i__1 = *lwork - (n1 << 1) * n2; 00624 ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * 00625 a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + 00626 i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, & 00627 dif[1], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], & 00628 ierr); 00629 00630 /* Frobenius norm-based Difl estimate. */ 00631 00632 i__1 = *lwork - (n1 << 1) * n2; 00633 ctgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[ 00634 a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], 00635 ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, 00636 &dif[2], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], & 00637 ierr); 00638 } else { 00639 00640 /* Compute 1-norm-based estimates of Difu and Difl using */ 00641 /* reversed communication with CLACN2. In each step a */ 00642 /* generalized Sylvester equation or a transposed variant */ 00643 /* is solved. */ 00644 00645 kase = 0; 00646 n1 = *m; 00647 n2 = *n - *m; 00648 i__ = n1 + 1; 00649 ijb = 0; 00650 mn2 = (n1 << 1) * n2; 00651 00652 /* 1-norm-based estimate of Difu. */ 00653 00654 L40: 00655 clacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[1], &kase, isave); 00656 if (kase != 0) { 00657 if (kase == 1) { 00658 00659 /* Solve generalized Sylvester equation */ 00660 00661 i__1 = *lwork - (n1 << 1) * n2; 00662 ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + 00663 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], 00664 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 00665 1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) + 00666 1], &i__1, &iwork[1], &ierr); 00667 } else { 00668 00669 /* Solve the transposed variant. */ 00670 00671 i__1 = *lwork - (n1 << 1) * n2; 00672 ctgsyl_("C", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + 00673 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], 00674 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 00675 1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) + 00676 1], &i__1, &iwork[1], &ierr); 00677 } 00678 goto L40; 00679 } 00680 dif[1] = dscale / dif[1]; 00681 00682 /* 1-norm-based estimate of Difl. */ 00683 00684 L50: 00685 clacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[2], &kase, isave); 00686 if (kase != 0) { 00687 if (kase == 1) { 00688 00689 /* Solve generalized Sylvester equation */ 00690 00691 i__1 = *lwork - (n1 << 1) * n2; 00692 ctgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, 00693 &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * 00694 b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 00695 1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) + 00696 1], &i__1, &iwork[1], &ierr); 00697 } else { 00698 00699 /* Solve the transposed variant. */ 00700 00701 i__1 = *lwork - (n1 << 1) * n2; 00702 ctgsyl_("C", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, 00703 &a[a_offset], lda, &work[1], &n2, &b[b_offset], 00704 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 00705 1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) + 00706 1], &i__1, &iwork[1], &ierr); 00707 } 00708 goto L50; 00709 } 00710 dif[2] = dscale / dif[2]; 00711 } 00712 } 00713 00714 /* If B(K,K) is complex, make it real and positive (normalization */ 00715 /* of the generalized Schur form) and Store the generalized */ 00716 /* eigenvalues of reordered pair (A, B) */ 00717 00718 i__1 = *n; 00719 for (k = 1; k <= i__1; ++k) { 00720 dscale = c_abs(&b[k + k * b_dim1]); 00721 if (dscale > safmin) { 00722 i__2 = k + k * b_dim1; 00723 q__2.r = b[i__2].r / dscale, q__2.i = b[i__2].i / dscale; 00724 r_cnjg(&q__1, &q__2); 00725 temp1.r = q__1.r, temp1.i = q__1.i; 00726 i__2 = k + k * b_dim1; 00727 q__1.r = b[i__2].r / dscale, q__1.i = b[i__2].i / dscale; 00728 temp2.r = q__1.r, temp2.i = q__1.i; 00729 i__2 = k + k * b_dim1; 00730 b[i__2].r = dscale, b[i__2].i = 0.f; 00731 i__2 = *n - k; 00732 cscal_(&i__2, &temp1, &b[k + (k + 1) * b_dim1], ldb); 00733 i__2 = *n - k + 1; 00734 cscal_(&i__2, &temp1, &a[k + k * a_dim1], lda); 00735 if (*wantq) { 00736 cscal_(n, &temp2, &q[k * q_dim1 + 1], &c__1); 00737 } 00738 } else { 00739 i__2 = k + k * b_dim1; 00740 b[i__2].r = 0.f, b[i__2].i = 0.f; 00741 } 00742 00743 i__2 = k; 00744 i__3 = k + k * a_dim1; 00745 alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i; 00746 i__2 = k; 00747 i__3 = k + k * b_dim1; 00748 beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i; 00749 00750 /* L60: */ 00751 } 00752 00753 L70: 00754 00755 work[1].r = (real) lwmin, work[1].i = 0.f; 00756 iwork[1] = liwmin; 00757 00758 return 0; 00759 00760 /* End of CTGSEN */ 00761 00762 } /* ctgsen_ */