00001 /* dtgsen.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 static integer c__2 = 2; 00020 static doublereal c_b28 = 1.; 00021 00022 /* Subroutine */ int dtgsen_(integer *ijob, logical *wantq, logical *wantz, 00023 logical *select, integer *n, doublereal *a, integer *lda, doublereal * 00024 b, integer *ldb, doublereal *alphar, doublereal *alphai, doublereal * 00025 beta, doublereal *q, integer *ldq, doublereal *z__, integer *ldz, 00026 integer *m, doublereal *pl, doublereal *pr, doublereal *dif, 00027 doublereal *work, integer *lwork, integer *iwork, integer *liwork, 00028 integer *info) 00029 { 00030 /* System generated locals */ 00031 integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1, 00032 z_offset, i__1, i__2; 00033 doublereal d__1; 00034 00035 /* Builtin functions */ 00036 double sqrt(doublereal), d_sign(doublereal *, doublereal *); 00037 00038 /* Local variables */ 00039 integer i__, k, n1, n2, kk, ks, mn2, ijb; 00040 doublereal eps; 00041 integer kase; 00042 logical pair; 00043 integer ierr; 00044 doublereal dsum; 00045 logical swap; 00046 extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *, 00047 integer *, doublereal *, doublereal *, doublereal *, doublereal *, 00048 doublereal *, doublereal *); 00049 integer isave[3]; 00050 logical wantd; 00051 integer lwmin; 00052 logical wantp; 00053 extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *, 00054 integer *, doublereal *, integer *, integer *); 00055 logical wantd1, wantd2; 00056 extern doublereal dlamch_(char *); 00057 doublereal dscale, rdscal; 00058 extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, 00059 doublereal *, integer *, doublereal *, integer *), 00060 xerbla_(char *, integer *), dtgexc_(logical *, logical *, 00061 integer *, doublereal *, integer *, doublereal *, integer *, 00062 doublereal *, integer *, doublereal *, integer *, integer *, 00063 integer *, doublereal *, integer *, integer *), dlassq_(integer *, 00064 doublereal *, integer *, doublereal *, doublereal *); 00065 integer liwmin; 00066 extern /* Subroutine */ int dtgsyl_(char *, integer *, integer *, integer 00067 *, doublereal *, integer *, doublereal *, integer *, doublereal *, 00068 integer *, doublereal *, integer *, doublereal *, integer *, 00069 doublereal *, integer *, doublereal *, doublereal *, doublereal *, 00070 integer *, integer *, integer *); 00071 doublereal smlnum; 00072 logical lquery; 00073 00074 00075 /* -- LAPACK routine (version 3.2) -- */ 00076 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ 00077 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ 00078 /* January 2007 */ 00079 00080 /* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */ 00081 00082 /* .. Scalar Arguments .. */ 00083 /* .. */ 00084 /* .. Array Arguments .. */ 00085 /* .. */ 00086 00087 /* Purpose */ 00088 /* ======= */ 00089 00090 /* DTGSEN reorders the generalized real Schur decomposition of a real */ 00091 /* matrix pair (A, B) (in terms of an orthonormal equivalence trans- */ 00092 /* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */ 00093 /* appears in the leading diagonal blocks of the upper quasi-triangular */ 00094 /* matrix A and the upper triangular B. The leading columns of Q and */ 00095 /* Z form orthonormal bases of the corresponding left and right eigen- */ 00096 /* spaces (deflating subspaces). (A, B) must be in generalized real */ 00097 /* Schur canonical form (as returned by DGGES), i.e. A is block upper */ 00098 /* triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper */ 00099 /* triangular. */ 00100 00101 /* DTGSEN also computes the generalized eigenvalues */ 00102 00103 /* w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j) */ 00104 00105 /* of the reordered matrix pair (A, B). */ 00106 00107 /* Optionally, DTGSEN computes the estimates of reciprocal condition */ 00108 /* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */ 00109 /* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */ 00110 /* between the matrix pairs (A11, B11) and (A22,B22) that correspond to */ 00111 /* the selected cluster and the eigenvalues outside the cluster, resp., */ 00112 /* and norms of "projections" onto left and right eigenspaces w.r.t. */ 00113 /* the selected cluster in the (1,1)-block. */ 00114 00115 /* Arguments */ 00116 /* ========= */ 00117 00118 /* IJOB (input) INTEGER */ 00119 /* Specifies whether condition numbers are required for the */ 00120 /* cluster of eigenvalues (PL and PR) or the deflating subspaces */ 00121 /* (Difu and Difl): */ 00122 /* =0: Only reorder w.r.t. SELECT. No extras. */ 00123 /* =1: Reciprocal of norms of "projections" onto left and right */ 00124 /* eigenspaces w.r.t. the selected cluster (PL and PR). */ 00125 /* =2: Upper bounds on Difu and Difl. F-norm-based estimate */ 00126 /* (DIF(1:2)). */ 00127 /* =3: Estimate of Difu and Difl. 1-norm-based estimate */ 00128 /* (DIF(1:2)). */ 00129 /* About 5 times as expensive as IJOB = 2. */ 00130 /* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */ 00131 /* version to get it all. */ 00132 /* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */ 00133 00134 /* WANTQ (input) LOGICAL */ 00135 /* .TRUE. : update the left transformation matrix Q; */ 00136 /* .FALSE.: do not update Q. */ 00137 00138 /* WANTZ (input) LOGICAL */ 00139 /* .TRUE. : update the right transformation matrix Z; */ 00140 /* .FALSE.: do not update Z. */ 00141 00142 /* SELECT (input) LOGICAL array, dimension (N) */ 00143 /* SELECT specifies the eigenvalues in the selected cluster. */ 00144 /* To select a real eigenvalue w(j), SELECT(j) must be set to */ 00145 /* .TRUE.. To select a complex conjugate pair of eigenvalues */ 00146 /* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, */ 00147 /* either SELECT(j) or SELECT(j+1) or both must be set to */ 00148 /* .TRUE.; a complex conjugate pair of eigenvalues must be */ 00149 /* either both included in the cluster or both excluded. */ 00150 00151 /* N (input) INTEGER */ 00152 /* The order of the matrices A and B. N >= 0. */ 00153 00154 /* A (input/output) DOUBLE PRECISION array, dimension(LDA,N) */ 00155 /* On entry, the upper quasi-triangular matrix A, with (A, B) in */ 00156 /* generalized real Schur canonical form. */ 00157 /* On exit, A is overwritten by the reordered matrix A. */ 00158 00159 /* LDA (input) INTEGER */ 00160 /* The leading dimension of the array A. LDA >= max(1,N). */ 00161 00162 /* B (input/output) DOUBLE PRECISION array, dimension(LDB,N) */ 00163 /* On entry, the upper triangular matrix B, with (A, B) in */ 00164 /* generalized real Schur canonical form. */ 00165 /* On exit, B is overwritten by the reordered matrix B. */ 00166 00167 /* LDB (input) INTEGER */ 00168 /* The leading dimension of the array B. LDB >= max(1,N). */ 00169 00170 /* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */ 00171 /* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */ 00172 /* BETA (output) DOUBLE PRECISION array, dimension (N) */ 00173 /* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will */ 00174 /* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i */ 00175 /* and BETA(j),j=1,...,N are the diagonals of the complex Schur */ 00176 /* form (S,T) that would result if the 2-by-2 diagonal blocks of */ 00177 /* the real generalized Schur form of (A,B) were further reduced */ 00178 /* to triangular form using complex unitary transformations. */ 00179 /* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if */ 00180 /* positive, then the j-th and (j+1)-st eigenvalues are a */ 00181 /* complex conjugate pair, with ALPHAI(j+1) negative. */ 00182 00183 /* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) */ 00184 /* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */ 00185 /* On exit, Q has been postmultiplied by the left orthogonal */ 00186 /* transformation matrix which reorder (A, B); The leading M */ 00187 /* columns of Q form orthonormal bases for the specified pair of */ 00188 /* left eigenspaces (deflating subspaces). */ 00189 /* If WANTQ = .FALSE., Q is not referenced. */ 00190 00191 /* LDQ (input) INTEGER */ 00192 /* The leading dimension of the array Q. LDQ >= 1; */ 00193 /* and if WANTQ = .TRUE., LDQ >= N. */ 00194 00195 /* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) */ 00196 /* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */ 00197 /* On exit, Z has been postmultiplied by the left orthogonal */ 00198 /* transformation matrix which reorder (A, B); The leading M */ 00199 /* columns of Z form orthonormal bases for the specified pair of */ 00200 /* left eigenspaces (deflating subspaces). */ 00201 /* If WANTZ = .FALSE., Z is not referenced. */ 00202 00203 /* LDZ (input) INTEGER */ 00204 /* The leading dimension of the array Z. LDZ >= 1; */ 00205 /* If WANTZ = .TRUE., LDZ >= N. */ 00206 00207 /* M (output) INTEGER */ 00208 /* The dimension of the specified pair of left and right eigen- */ 00209 /* spaces (deflating subspaces). 0 <= M <= N. */ 00210 00211 /* PL (output) DOUBLE PRECISION */ 00212 /* PR (output) DOUBLE PRECISION */ 00213 /* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */ 00214 /* reciprocal of the norm of "projections" onto left and right */ 00215 /* eigenspaces with respect to the selected cluster. */ 00216 /* 0 < PL, PR <= 1. */ 00217 /* If M = 0 or M = N, PL = PR = 1. */ 00218 /* If IJOB = 0, 2 or 3, PL and PR are not referenced. */ 00219 00220 /* DIF (output) DOUBLE PRECISION array, dimension (2). */ 00221 /* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */ 00222 /* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */ 00223 /* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */ 00224 /* estimates of Difu and Difl. */ 00225 /* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */ 00226 /* If IJOB = 0 or 1, DIF is not referenced. */ 00227 00228 /* WORK (workspace/output) DOUBLE PRECISION array, */ 00229 /* dimension (MAX(1,LWORK)) */ 00230 /* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ 00231 00232 /* LWORK (input) INTEGER */ 00233 /* The dimension of the array WORK. LWORK >= 4*N+16. */ 00234 /* If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). */ 00235 /* If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)). */ 00236 00237 /* If LWORK = -1, then a workspace query is assumed; the routine */ 00238 /* only calculates the optimal size of the WORK array, returns */ 00239 /* this value as the first entry of the WORK array, and no error */ 00240 /* message related to LWORK is issued by XERBLA. */ 00241 00242 /* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */ 00243 /* IF IJOB = 0, IWORK is not referenced. Otherwise, */ 00244 /* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */ 00245 00246 /* LIWORK (input) INTEGER */ 00247 /* The dimension of the array IWORK. LIWORK >= 1. */ 00248 /* If IJOB = 1, 2 or 4, LIWORK >= N+6. */ 00249 /* If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). */ 00250 00251 /* If LIWORK = -1, then a workspace query is assumed; the */ 00252 /* routine only calculates the optimal size of the IWORK array, */ 00253 /* returns this value as the first entry of the IWORK array, and */ 00254 /* no error message related to LIWORK is issued by XERBLA. */ 00255 00256 /* INFO (output) INTEGER */ 00257 /* =0: Successful exit. */ 00258 /* <0: If INFO = -i, the i-th argument had an illegal value. */ 00259 /* =1: Reordering of (A, B) failed because the transformed */ 00260 /* matrix pair (A, B) would be too far from generalized */ 00261 /* Schur form; the problem is very ill-conditioned. */ 00262 /* (A, B) may have been partially reordered. */ 00263 /* If requested, 0 is returned in DIF(*), PL and PR. */ 00264 00265 /* Further Details */ 00266 /* =============== */ 00267 00268 /* DTGSEN first collects the selected eigenvalues by computing */ 00269 /* orthogonal U and W that move them to the top left corner of (A, B). */ 00270 /* In other words, the selected eigenvalues are the eigenvalues of */ 00271 /* (A11, B11) in: */ 00272 00273 /* U'*(A, B)*W = (A11 A12) (B11 B12) n1 */ 00274 /* ( 0 A22),( 0 B22) n2 */ 00275 /* n1 n2 n1 n2 */ 00276 00277 /* where N = n1+n2 and U' means the transpose of U. The first n1 columns */ 00278 /* of U and W span the specified pair of left and right eigenspaces */ 00279 /* (deflating subspaces) of (A, B). */ 00280 00281 /* If (A, B) has been obtained from the generalized real Schur */ 00282 /* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */ 00283 /* reordered generalized real Schur form of (C, D) is given by */ 00284 00285 /* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', */ 00286 00287 /* and the first n1 columns of Q*U and Z*W span the corresponding */ 00288 /* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */ 00289 00290 /* Note that if the selected eigenvalue is sufficiently ill-conditioned, */ 00291 /* then its value may differ significantly from its value before */ 00292 /* reordering. */ 00293 00294 /* The reciprocal condition numbers of the left and right eigenspaces */ 00295 /* spanned by the first n1 columns of U and W (or Q*U and Z*W) may */ 00296 /* be returned in DIF(1:2), corresponding to Difu and Difl, resp. */ 00297 00298 /* The Difu and Difl are defined as: */ 00299 00300 /* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) */ 00301 /* and */ 00302 /* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */ 00303 00304 /* where sigma-min(Zu) is the smallest singular value of the */ 00305 /* (2*n1*n2)-by-(2*n1*n2) matrix */ 00306 00307 /* Zu = [ kron(In2, A11) -kron(A22', In1) ] */ 00308 /* [ kron(In2, B11) -kron(B22', In1) ]. */ 00309 00310 /* Here, Inx is the identity matrix of size nx and A22' is the */ 00311 /* transpose of A22. kron(X, Y) is the Kronecker product between */ 00312 /* the matrices X and Y. */ 00313 00314 /* When DIF(2) is small, small changes in (A, B) can cause large changes */ 00315 /* in the deflating subspace. An approximate (asymptotic) bound on the */ 00316 /* maximum angular error in the computed deflating subspaces is */ 00317 00318 /* EPS * norm((A, B)) / DIF(2), */ 00319 00320 /* where EPS is the machine precision. */ 00321 00322 /* The reciprocal norm of the projectors on the left and right */ 00323 /* eigenspaces associated with (A11, B11) may be returned in PL and PR. */ 00324 /* They are computed as follows. First we compute L and R so that */ 00325 /* P*(A, B)*Q is block diagonal, where */ 00326 00327 /* P = ( I -L ) n1 Q = ( I R ) n1 */ 00328 /* ( 0 I ) n2 and ( 0 I ) n2 */ 00329 /* n1 n2 n1 n2 */ 00330 00331 /* and (L, R) is the solution to the generalized Sylvester equation */ 00332 00333 /* A11*R - L*A22 = -A12 */ 00334 /* B11*R - L*B22 = -B12 */ 00335 00336 /* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */ 00337 /* An approximate (asymptotic) bound on the average absolute error of */ 00338 /* the selected eigenvalues is */ 00339 00340 /* EPS * norm((A, B)) / PL. */ 00341 00342 /* There are also global error bounds which valid for perturbations up */ 00343 /* to a certain restriction: A lower bound (x) on the smallest */ 00344 /* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */ 00345 /* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */ 00346 /* (i.e. (A + E, B + F), is */ 00347 00348 /* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). */ 00349 00350 /* An approximate bound on x can be computed from DIF(1:2), PL and PR. */ 00351 00352 /* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */ 00353 /* (L', R') and unperturbed (L, R) left and right deflating subspaces */ 00354 /* associated with the selected cluster in the (1,1)-blocks can be */ 00355 /* bounded as */ 00356 00357 /* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */ 00358 /* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */ 00359 00360 /* See LAPACK User's Guide section 4.11 or the following references */ 00361 /* for more information. */ 00362 00363 /* Note that if the default method for computing the Frobenius-norm- */ 00364 /* based estimate DIF is not wanted (see DLATDF), then the parameter */ 00365 /* IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF */ 00366 /* (IJOB = 2 will be used)). See DTGSYL for more details. */ 00367 00368 /* Based on contributions by */ 00369 /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ 00370 /* Umea University, S-901 87 Umea, Sweden. */ 00371 00372 /* References */ 00373 /* ========== */ 00374 00375 /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ 00376 /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ 00377 /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ 00378 /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ 00379 00380 /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ 00381 /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ 00382 /* Estimation: Theory, Algorithms and Software, */ 00383 /* Report UMINF - 94.04, Department of Computing Science, Umea */ 00384 /* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working */ 00385 /* Note 87. To appear in Numerical Algorithms, 1996. */ 00386 00387 /* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */ 00388 /* for Solving the Generalized Sylvester Equation and Estimating the */ 00389 /* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */ 00390 /* Department of Computing Science, Umea University, S-901 87 Umea, */ 00391 /* Sweden, December 1993, Revised April 1994, Also as LAPACK Working */ 00392 /* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */ 00393 /* 1996. */ 00394 00395 /* ===================================================================== */ 00396 00397 /* .. Parameters .. */ 00398 /* .. */ 00399 /* .. Local Scalars .. */ 00400 /* .. */ 00401 /* .. Local Arrays .. */ 00402 /* .. */ 00403 /* .. External Subroutines .. */ 00404 /* .. */ 00405 /* .. External Functions .. */ 00406 /* .. */ 00407 /* .. Intrinsic Functions .. */ 00408 /* .. */ 00409 /* .. Executable Statements .. */ 00410 00411 /* Decode and test the input parameters */ 00412 00413 /* Parameter adjustments */ 00414 --select; 00415 a_dim1 = *lda; 00416 a_offset = 1 + a_dim1; 00417 a -= a_offset; 00418 b_dim1 = *ldb; 00419 b_offset = 1 + b_dim1; 00420 b -= b_offset; 00421 --alphar; 00422 --alphai; 00423 --beta; 00424 q_dim1 = *ldq; 00425 q_offset = 1 + q_dim1; 00426 q -= q_offset; 00427 z_dim1 = *ldz; 00428 z_offset = 1 + z_dim1; 00429 z__ -= z_offset; 00430 --dif; 00431 --work; 00432 --iwork; 00433 00434 /* Function Body */ 00435 *info = 0; 00436 lquery = *lwork == -1 || *liwork == -1; 00437 00438 if (*ijob < 0 || *ijob > 5) { 00439 *info = -1; 00440 } else if (*n < 0) { 00441 *info = -5; 00442 } else if (*lda < max(1,*n)) { 00443 *info = -7; 00444 } else if (*ldb < max(1,*n)) { 00445 *info = -9; 00446 } else if (*ldq < 1 || *wantq && *ldq < *n) { 00447 *info = -14; 00448 } else if (*ldz < 1 || *wantz && *ldz < *n) { 00449 *info = -16; 00450 } 00451 00452 if (*info != 0) { 00453 i__1 = -(*info); 00454 xerbla_("DTGSEN", &i__1); 00455 return 0; 00456 } 00457 00458 /* Get machine constants */ 00459 00460 eps = dlamch_("P"); 00461 smlnum = dlamch_("S") / eps; 00462 ierr = 0; 00463 00464 wantp = *ijob == 1 || *ijob >= 4; 00465 wantd1 = *ijob == 2 || *ijob == 4; 00466 wantd2 = *ijob == 3 || *ijob == 5; 00467 wantd = wantd1 || wantd2; 00468 00469 /* Set M to the dimension of the specified pair of deflating */ 00470 /* subspaces. */ 00471 00472 *m = 0; 00473 pair = FALSE_; 00474 i__1 = *n; 00475 for (k = 1; k <= i__1; ++k) { 00476 if (pair) { 00477 pair = FALSE_; 00478 } else { 00479 if (k < *n) { 00480 if (a[k + 1 + k * a_dim1] == 0.) { 00481 if (select[k]) { 00482 ++(*m); 00483 } 00484 } else { 00485 pair = TRUE_; 00486 if (select[k] || select[k + 1]) { 00487 *m += 2; 00488 } 00489 } 00490 } else { 00491 if (select[*n]) { 00492 ++(*m); 00493 } 00494 } 00495 } 00496 /* L10: */ 00497 } 00498 00499 if (*ijob == 1 || *ijob == 2 || *ijob == 4) { 00500 /* Computing MAX */ 00501 i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), i__2 = (*m << 00502 1) * (*n - *m); 00503 lwmin = max(i__1,i__2); 00504 /* Computing MAX */ 00505 i__1 = 1, i__2 = *n + 6; 00506 liwmin = max(i__1,i__2); 00507 } else if (*ijob == 3 || *ijob == 5) { 00508 /* Computing MAX */ 00509 i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), i__2 = (*m << 00510 2) * (*n - *m); 00511 lwmin = max(i__1,i__2); 00512 /* Computing MAX */ 00513 i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = max(i__1,i__2), i__2 = 00514 *n + 6; 00515 liwmin = max(i__1,i__2); 00516 } else { 00517 /* Computing MAX */ 00518 i__1 = 1, i__2 = (*n << 2) + 16; 00519 lwmin = max(i__1,i__2); 00520 liwmin = 1; 00521 } 00522 00523 work[1] = (doublereal) lwmin; 00524 iwork[1] = liwmin; 00525 00526 if (*lwork < lwmin && ! lquery) { 00527 *info = -22; 00528 } else if (*liwork < liwmin && ! lquery) { 00529 *info = -24; 00530 } 00531 00532 if (*info != 0) { 00533 i__1 = -(*info); 00534 xerbla_("DTGSEN", &i__1); 00535 return 0; 00536 } else if (lquery) { 00537 return 0; 00538 } 00539 00540 /* Quick return if possible. */ 00541 00542 if (*m == *n || *m == 0) { 00543 if (wantp) { 00544 *pl = 1.; 00545 *pr = 1.; 00546 } 00547 if (wantd) { 00548 dscale = 0.; 00549 dsum = 1.; 00550 i__1 = *n; 00551 for (i__ = 1; i__ <= i__1; ++i__) { 00552 dlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum); 00553 dlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum); 00554 /* L20: */ 00555 } 00556 dif[1] = dscale * sqrt(dsum); 00557 dif[2] = dif[1]; 00558 } 00559 goto L60; 00560 } 00561 00562 /* Collect the selected blocks at the top-left corner of (A, B). */ 00563 00564 ks = 0; 00565 pair = FALSE_; 00566 i__1 = *n; 00567 for (k = 1; k <= i__1; ++k) { 00568 if (pair) { 00569 pair = FALSE_; 00570 } else { 00571 00572 swap = select[k]; 00573 if (k < *n) { 00574 if (a[k + 1 + k * a_dim1] != 0.) { 00575 pair = TRUE_; 00576 swap = swap || select[k + 1]; 00577 } 00578 } 00579 00580 if (swap) { 00581 ++ks; 00582 00583 /* Swap the K-th block to position KS. */ 00584 /* Perform the reordering of diagonal blocks in (A, B) */ 00585 /* by orthogonal transformation matrices and update */ 00586 /* Q and Z accordingly (if requested): */ 00587 00588 kk = k; 00589 if (k != ks) { 00590 dtgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], 00591 ldb, &q[q_offset], ldq, &z__[z_offset], ldz, &kk, 00592 &ks, &work[1], lwork, &ierr); 00593 } 00594 00595 if (ierr > 0) { 00596 00597 /* Swap is rejected: exit. */ 00598 00599 *info = 1; 00600 if (wantp) { 00601 *pl = 0.; 00602 *pr = 0.; 00603 } 00604 if (wantd) { 00605 dif[1] = 0.; 00606 dif[2] = 0.; 00607 } 00608 goto L60; 00609 } 00610 00611 if (pair) { 00612 ++ks; 00613 } 00614 } 00615 } 00616 /* L30: */ 00617 } 00618 if (wantp) { 00619 00620 /* Solve generalized Sylvester equation for R and L */ 00621 /* and compute PL and PR. */ 00622 00623 n1 = *m; 00624 n2 = *n - *m; 00625 i__ = n1 + 1; 00626 ijb = 0; 00627 dlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1); 00628 dlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 + 00629 1], &n1); 00630 i__1 = *lwork - (n1 << 1) * n2; 00631 dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1] 00632 , lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * 00633 b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], & 00634 work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr); 00635 00636 /* Estimate the reciprocal of norms of "projections" onto left */ 00637 /* and right eigenspaces. */ 00638 00639 rdscal = 0.; 00640 dsum = 1.; 00641 i__1 = n1 * n2; 00642 dlassq_(&i__1, &work[1], &c__1, &rdscal, &dsum); 00643 *pl = rdscal * sqrt(dsum); 00644 if (*pl == 0.) { 00645 *pl = 1.; 00646 } else { 00647 *pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl)); 00648 } 00649 rdscal = 0.; 00650 dsum = 1.; 00651 i__1 = n1 * n2; 00652 dlassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum); 00653 *pr = rdscal * sqrt(dsum); 00654 if (*pr == 0.) { 00655 *pr = 1.; 00656 } else { 00657 *pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr)); 00658 } 00659 } 00660 00661 if (wantd) { 00662 00663 /* Compute estimates of Difu and Difl. */ 00664 00665 if (wantd1) { 00666 n1 = *m; 00667 n2 = *n - *m; 00668 i__ = n1 + 1; 00669 ijb = 3; 00670 00671 /* Frobenius norm-based Difu-estimate. */ 00672 00673 i__1 = *lwork - (n1 << 1) * n2; 00674 dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * 00675 a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + 00676 i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, & 00677 dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & 00678 ierr); 00679 00680 /* Frobenius norm-based Difl-estimate. */ 00681 00682 i__1 = *lwork - (n1 << 1) * n2; 00683 dtgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[ 00684 a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], 00685 ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, 00686 &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & 00687 ierr); 00688 } else { 00689 00690 00691 /* Compute 1-norm-based estimates of Difu and Difl using */ 00692 /* reversed communication with DLACN2. In each step a */ 00693 /* generalized Sylvester equation or a transposed variant */ 00694 /* is solved. */ 00695 00696 kase = 0; 00697 n1 = *m; 00698 n2 = *n - *m; 00699 i__ = n1 + 1; 00700 ijb = 0; 00701 mn2 = (n1 << 1) * n2; 00702 00703 /* 1-norm-based estimate of Difu. */ 00704 00705 L40: 00706 dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[1], &kase, 00707 isave); 00708 if (kase != 0) { 00709 if (kase == 1) { 00710 00711 /* Solve generalized Sylvester equation. */ 00712 00713 i__1 = *lwork - (n1 << 1) * n2; 00714 dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + 00715 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], 00716 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 00717 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 00718 1], &i__1, &iwork[1], &ierr); 00719 } else { 00720 00721 /* Solve the transposed variant. */ 00722 00723 i__1 = *lwork - (n1 << 1) * n2; 00724 dtgsyl_("T", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + 00725 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], 00726 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 00727 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 00728 1], &i__1, &iwork[1], &ierr); 00729 } 00730 goto L40; 00731 } 00732 dif[1] = dscale / dif[1]; 00733 00734 /* 1-norm-based estimate of Difl. */ 00735 00736 L50: 00737 dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[2], &kase, 00738 isave); 00739 if (kase != 0) { 00740 if (kase == 1) { 00741 00742 /* Solve generalized Sylvester equation. */ 00743 00744 i__1 = *lwork - (n1 << 1) * n2; 00745 dtgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, 00746 &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * 00747 b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 00748 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 00749 1], &i__1, &iwork[1], &ierr); 00750 } else { 00751 00752 /* Solve the transposed variant. */ 00753 00754 i__1 = *lwork - (n1 << 1) * n2; 00755 dtgsyl_("T", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, 00756 &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * 00757 b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 00758 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 00759 1], &i__1, &iwork[1], &ierr); 00760 } 00761 goto L50; 00762 } 00763 dif[2] = dscale / dif[2]; 00764 00765 } 00766 } 00767 00768 L60: 00769 00770 /* Compute generalized eigenvalues of reordered pair (A, B) and */ 00771 /* normalize the generalized Schur form. */ 00772 00773 pair = FALSE_; 00774 i__1 = *n; 00775 for (k = 1; k <= i__1; ++k) { 00776 if (pair) { 00777 pair = FALSE_; 00778 } else { 00779 00780 if (k < *n) { 00781 if (a[k + 1 + k * a_dim1] != 0.) { 00782 pair = TRUE_; 00783 } 00784 } 00785 00786 if (pair) { 00787 00788 /* Compute the eigenvalue(s) at position K. */ 00789 00790 work[1] = a[k + k * a_dim1]; 00791 work[2] = a[k + 1 + k * a_dim1]; 00792 work[3] = a[k + (k + 1) * a_dim1]; 00793 work[4] = a[k + 1 + (k + 1) * a_dim1]; 00794 work[5] = b[k + k * b_dim1]; 00795 work[6] = b[k + 1 + k * b_dim1]; 00796 work[7] = b[k + (k + 1) * b_dim1]; 00797 work[8] = b[k + 1 + (k + 1) * b_dim1]; 00798 d__1 = smlnum * eps; 00799 dlag2_(&work[1], &c__2, &work[5], &c__2, &d__1, &beta[k], & 00800 beta[k + 1], &alphar[k], &alphar[k + 1], &alphai[k]); 00801 alphai[k + 1] = -alphai[k]; 00802 00803 } else { 00804 00805 if (d_sign(&c_b28, &b[k + k * b_dim1]) < 0.) { 00806 00807 /* If B(K,K) is negative, make it positive */ 00808 00809 i__2 = *n; 00810 for (i__ = 1; i__ <= i__2; ++i__) { 00811 a[k + i__ * a_dim1] = -a[k + i__ * a_dim1]; 00812 b[k + i__ * b_dim1] = -b[k + i__ * b_dim1]; 00813 if (*wantq) { 00814 q[i__ + k * q_dim1] = -q[i__ + k * q_dim1]; 00815 } 00816 /* L70: */ 00817 } 00818 } 00819 00820 alphar[k] = a[k + k * a_dim1]; 00821 alphai[k] = 0.; 00822 beta[k] = b[k + k * b_dim1]; 00823 00824 } 00825 } 00826 /* L80: */ 00827 } 00828 00829 work[1] = (doublereal) lwmin; 00830 iwork[1] = liwmin; 00831 00832 return 0; 00833 00834 /* End of DTGSEN */ 00835 00836 } /* dtgsen_ */