00001 /* stgsen.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 real c_b28 = 1.f; 00021 00022 /* Subroutine */ int stgsen_(integer *ijob, logical *wantq, logical *wantz, 00023 logical *select, integer *n, real *a, integer *lda, real *b, integer * 00024 ldb, real *alphar, real *alphai, real *beta, real *q, integer *ldq, 00025 real *z__, integer *ldz, integer *m, real *pl, real *pr, real *dif, 00026 real *work, integer *lwork, integer *iwork, integer *liwork, integer * 00027 info) 00028 { 00029 /* System generated locals */ 00030 integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1, 00031 z_offset, i__1, i__2; 00032 real r__1; 00033 00034 /* Builtin functions */ 00035 double sqrt(doublereal), r_sign(real *, real *); 00036 00037 /* Local variables */ 00038 integer i__, k, n1, n2, kk, ks, mn2, ijb; 00039 real eps; 00040 integer kase; 00041 logical pair; 00042 integer ierr; 00043 real dsum; 00044 logical swap; 00045 extern /* Subroutine */ int slag2_(real *, integer *, real *, integer *, 00046 real *, real *, real *, real *, real *, real *); 00047 integer isave[3]; 00048 logical wantd; 00049 integer lwmin; 00050 logical wantp; 00051 extern /* Subroutine */ int slacn2_(integer *, real *, real *, integer *, 00052 real *, integer *, integer *); 00053 logical wantd1, wantd2; 00054 real dscale, rdscal; 00055 extern doublereal slamch_(char *); 00056 extern /* Subroutine */ int xerbla_(char *, integer *), slacpy_( 00057 char *, integer *, integer *, real *, integer *, real *, integer * 00058 ), stgexc_(logical *, logical *, integer *, real *, 00059 integer *, real *, integer *, real *, integer *, real *, integer * 00060 , integer *, integer *, real *, integer *, integer *); 00061 integer liwmin; 00062 extern /* Subroutine */ int slassq_(integer *, real *, integer *, real *, 00063 real *); 00064 real smlnum; 00065 logical lquery; 00066 extern /* Subroutine */ int stgsyl_(char *, integer *, integer *, integer 00067 *, real *, integer *, real *, integer *, real *, integer *, real * 00068 , integer *, real *, integer *, real *, integer *, real *, real *, 00069 real *, integer *, integer *, integer *); 00070 00071 00072 /* -- LAPACK routine (version 3.2) -- */ 00073 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ 00074 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ 00075 /* January 2007 */ 00076 00077 /* Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH. */ 00078 00079 /* .. Scalar Arguments .. */ 00080 /* .. */ 00081 /* .. Array Arguments .. */ 00082 /* .. */ 00083 00084 /* Purpose */ 00085 /* ======= */ 00086 00087 /* STGSEN reorders the generalized real Schur decomposition of a real */ 00088 /* matrix pair (A, B) (in terms of an orthonormal equivalence trans- */ 00089 /* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */ 00090 /* appears in the leading diagonal blocks of the upper quasi-triangular */ 00091 /* matrix A and the upper triangular B. The leading columns of Q and */ 00092 /* Z form orthonormal bases of the corresponding left and right eigen- */ 00093 /* spaces (deflating subspaces). (A, B) must be in generalized real */ 00094 /* Schur canonical form (as returned by SGGES), i.e. A is block upper */ 00095 /* triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper */ 00096 /* triangular. */ 00097 00098 /* STGSEN also computes the generalized eigenvalues */ 00099 00100 /* w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j) */ 00101 00102 /* of the reordered matrix pair (A, B). */ 00103 00104 /* Optionally, STGSEN computes the estimates of reciprocal condition */ 00105 /* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */ 00106 /* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */ 00107 /* between the matrix pairs (A11, B11) and (A22,B22) that correspond to */ 00108 /* the selected cluster and the eigenvalues outside the cluster, resp., */ 00109 /* and norms of "projections" onto left and right eigenspaces w.r.t. */ 00110 /* the selected cluster in the (1,1)-block. */ 00111 00112 /* Arguments */ 00113 /* ========= */ 00114 00115 /* IJOB (input) INTEGER */ 00116 /* Specifies whether condition numbers are required for the */ 00117 /* cluster of eigenvalues (PL and PR) or the deflating subspaces */ 00118 /* (Difu and Difl): */ 00119 /* =0: Only reorder w.r.t. SELECT. No extras. */ 00120 /* =1: Reciprocal of norms of "projections" onto left and right */ 00121 /* eigenspaces w.r.t. the selected cluster (PL and PR). */ 00122 /* =2: Upper bounds on Difu and Difl. F-norm-based estimate */ 00123 /* (DIF(1:2)). */ 00124 /* =3: Estimate of Difu and Difl. 1-norm-based estimate */ 00125 /* (DIF(1:2)). */ 00126 /* About 5 times as expensive as IJOB = 2. */ 00127 /* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */ 00128 /* version to get it all. */ 00129 /* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */ 00130 00131 /* WANTQ (input) LOGICAL */ 00132 /* .TRUE. : update the left transformation matrix Q; */ 00133 /* .FALSE.: do not update Q. */ 00134 00135 /* WANTZ (input) LOGICAL */ 00136 /* .TRUE. : update the right transformation matrix Z; */ 00137 /* .FALSE.: do not update Z. */ 00138 00139 /* SELECT (input) LOGICAL array, dimension (N) */ 00140 /* SELECT specifies the eigenvalues in the selected cluster. */ 00141 /* To select a real eigenvalue w(j), SELECT(j) must be set to */ 00142 /* .TRUE.. To select a complex conjugate pair of eigenvalues */ 00143 /* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, */ 00144 /* either SELECT(j) or SELECT(j+1) or both must be set to */ 00145 /* .TRUE.; a complex conjugate pair of eigenvalues must be */ 00146 /* either both included in the cluster or both excluded. */ 00147 00148 /* N (input) INTEGER */ 00149 /* The order of the matrices A and B. N >= 0. */ 00150 00151 /* A (input/output) REAL array, dimension(LDA,N) */ 00152 /* On entry, the upper quasi-triangular matrix A, with (A, B) in */ 00153 /* generalized real Schur canonical form. */ 00154 /* On exit, A is overwritten by the reordered matrix A. */ 00155 00156 /* LDA (input) INTEGER */ 00157 /* The leading dimension of the array A. LDA >= max(1,N). */ 00158 00159 /* B (input/output) REAL array, dimension(LDB,N) */ 00160 /* On entry, the upper triangular matrix B, with (A, B) in */ 00161 /* generalized real Schur canonical form. */ 00162 /* On exit, B is overwritten by the reordered matrix B. */ 00163 00164 /* LDB (input) INTEGER */ 00165 /* The leading dimension of the array B. LDB >= max(1,N). */ 00166 00167 /* ALPHAR (output) REAL array, dimension (N) */ 00168 /* ALPHAI (output) REAL array, dimension (N) */ 00169 /* BETA (output) REAL array, dimension (N) */ 00170 /* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will */ 00171 /* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i */ 00172 /* and BETA(j),j=1,...,N are the diagonals of the complex Schur */ 00173 /* form (S,T) that would result if the 2-by-2 diagonal blocks of */ 00174 /* the real generalized Schur form of (A,B) were further reduced */ 00175 /* to triangular form using complex unitary transformations. */ 00176 /* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if */ 00177 /* positive, then the j-th and (j+1)-st eigenvalues are a */ 00178 /* complex conjugate pair, with ALPHAI(j+1) negative. */ 00179 00180 /* Q (input/output) REAL array, dimension (LDQ,N) */ 00181 /* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */ 00182 /* On exit, Q has been postmultiplied by the left orthogonal */ 00183 /* transformation matrix which reorder (A, B); The leading M */ 00184 /* columns of Q form orthonormal bases for the specified pair of */ 00185 /* left eigenspaces (deflating subspaces). */ 00186 /* If WANTQ = .FALSE., Q is not referenced. */ 00187 00188 /* LDQ (input) INTEGER */ 00189 /* The leading dimension of the array Q. LDQ >= 1; */ 00190 /* and if WANTQ = .TRUE., LDQ >= N. */ 00191 00192 /* Z (input/output) REAL array, dimension (LDZ,N) */ 00193 /* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */ 00194 /* On exit, Z has been postmultiplied by the left orthogonal */ 00195 /* transformation matrix which reorder (A, B); The leading M */ 00196 /* columns of Z form orthonormal bases for the specified pair of */ 00197 /* left eigenspaces (deflating subspaces). */ 00198 /* If WANTZ = .FALSE., Z is not referenced. */ 00199 00200 /* LDZ (input) INTEGER */ 00201 /* The leading dimension of the array Z. LDZ >= 1; */ 00202 /* If WANTZ = .TRUE., LDZ >= N. */ 00203 00204 /* M (output) INTEGER */ 00205 /* The dimension of the specified pair of left and right eigen- */ 00206 /* spaces (deflating subspaces). 0 <= M <= N. */ 00207 00208 /* PL (output) REAL */ 00209 /* PR (output) REAL */ 00210 /* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */ 00211 /* reciprocal of the norm of "projections" onto left and right */ 00212 /* eigenspaces with respect to the selected cluster. */ 00213 /* 0 < PL, PR <= 1. */ 00214 /* If M = 0 or M = N, PL = PR = 1. */ 00215 /* If IJOB = 0, 2 or 3, PL and PR are not referenced. */ 00216 00217 /* DIF (output) REAL array, dimension (2). */ 00218 /* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */ 00219 /* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */ 00220 /* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */ 00221 /* estimates of Difu and Difl. */ 00222 /* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */ 00223 /* If IJOB = 0 or 1, DIF is not referenced. */ 00224 00225 /* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) */ 00226 /* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ 00227 00228 /* LWORK (input) INTEGER */ 00229 /* The dimension of the array WORK. LWORK >= 4*N+16. */ 00230 /* If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). */ 00231 /* If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)). */ 00232 00233 /* If LWORK = -1, then a workspace query is assumed; the routine */ 00234 /* only calculates the optimal size of the WORK array, returns */ 00235 /* this value as the first entry of the WORK array, and no error */ 00236 /* message related to LWORK is issued by XERBLA. */ 00237 00238 /* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */ 00239 /* IF IJOB = 0, IWORK is not referenced. Otherwise, */ 00240 /* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */ 00241 00242 /* LIWORK (input) INTEGER */ 00243 /* The dimension of the array IWORK. LIWORK >= 1. */ 00244 /* If IJOB = 1, 2 or 4, LIWORK >= N+6. */ 00245 /* If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). */ 00246 00247 /* If LIWORK = -1, then a workspace query is assumed; the */ 00248 /* routine only calculates the optimal size of the IWORK array, */ 00249 /* returns this value as the first entry of the IWORK array, and */ 00250 /* no error message related to LIWORK is issued by XERBLA. */ 00251 00252 /* INFO (output) INTEGER */ 00253 /* =0: Successful exit. */ 00254 /* <0: If INFO = -i, the i-th argument had an illegal value. */ 00255 /* =1: Reordering of (A, B) failed because the transformed */ 00256 /* matrix pair (A, B) would be too far from generalized */ 00257 /* Schur form; the problem is very ill-conditioned. */ 00258 /* (A, B) may have been partially reordered. */ 00259 /* If requested, 0 is returned in DIF(*), PL and PR. */ 00260 00261 /* Further Details */ 00262 /* =============== */ 00263 00264 /* STGSEN first collects the selected eigenvalues by computing */ 00265 /* orthogonal U and W that move them to the top left corner of (A, B). */ 00266 /* In other words, the selected eigenvalues are the eigenvalues of */ 00267 /* (A11, B11) in: */ 00268 00269 /* U'*(A, B)*W = (A11 A12) (B11 B12) n1 */ 00270 /* ( 0 A22),( 0 B22) n2 */ 00271 /* n1 n2 n1 n2 */ 00272 00273 /* where N = n1+n2 and U' means the transpose of U. The first n1 columns */ 00274 /* of U and W span the specified pair of left and right eigenspaces */ 00275 /* (deflating subspaces) of (A, B). */ 00276 00277 /* If (A, B) has been obtained from the generalized real Schur */ 00278 /* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */ 00279 /* reordered generalized real Schur form of (C, D) is given by */ 00280 00281 /* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', */ 00282 00283 /* and the first n1 columns of Q*U and Z*W span the corresponding */ 00284 /* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */ 00285 00286 /* Note that if the selected eigenvalue is sufficiently ill-conditioned, */ 00287 /* then its value may differ significantly from its value before */ 00288 /* reordering. */ 00289 00290 /* The reciprocal condition numbers of the left and right eigenspaces */ 00291 /* spanned by the first n1 columns of U and W (or Q*U and Z*W) may */ 00292 /* be returned in DIF(1:2), corresponding to Difu and Difl, resp. */ 00293 00294 /* The Difu and Difl are defined as: */ 00295 00296 /* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) */ 00297 /* and */ 00298 /* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */ 00299 00300 /* where sigma-min(Zu) is the smallest singular value of the */ 00301 /* (2*n1*n2)-by-(2*n1*n2) matrix */ 00302 00303 /* Zu = [ kron(In2, A11) -kron(A22', In1) ] */ 00304 /* [ kron(In2, B11) -kron(B22', In1) ]. */ 00305 00306 /* Here, Inx is the identity matrix of size nx and A22' is the */ 00307 /* transpose of A22. kron(X, Y) is the Kronecker product between */ 00308 /* the matrices X and Y. */ 00309 00310 /* When DIF(2) is small, small changes in (A, B) can cause large changes */ 00311 /* in the deflating subspace. An approximate (asymptotic) bound on the */ 00312 /* maximum angular error in the computed deflating subspaces is */ 00313 00314 /* EPS * norm((A, B)) / DIF(2), */ 00315 00316 /* where EPS is the machine precision. */ 00317 00318 /* The reciprocal norm of the projectors on the left and right */ 00319 /* eigenspaces associated with (A11, B11) may be returned in PL and PR. */ 00320 /* They are computed as follows. First we compute L and R so that */ 00321 /* P*(A, B)*Q is block diagonal, where */ 00322 00323 /* P = ( I -L ) n1 Q = ( I R ) n1 */ 00324 /* ( 0 I ) n2 and ( 0 I ) n2 */ 00325 /* n1 n2 n1 n2 */ 00326 00327 /* and (L, R) is the solution to the generalized Sylvester equation */ 00328 00329 /* A11*R - L*A22 = -A12 */ 00330 /* B11*R - L*B22 = -B12 */ 00331 00332 /* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */ 00333 /* An approximate (asymptotic) bound on the average absolute error of */ 00334 /* the selected eigenvalues is */ 00335 00336 /* EPS * norm((A, B)) / PL. */ 00337 00338 /* There are also global error bounds which valid for perturbations up */ 00339 /* to a certain restriction: A lower bound (x) on the smallest */ 00340 /* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */ 00341 /* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */ 00342 /* (i.e. (A + E, B + F), is */ 00343 00344 /* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). */ 00345 00346 /* An approximate bound on x can be computed from DIF(1:2), PL and PR. */ 00347 00348 /* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */ 00349 /* (L', R') and unperturbed (L, R) left and right deflating subspaces */ 00350 /* associated with the selected cluster in the (1,1)-blocks can be */ 00351 /* bounded as */ 00352 00353 /* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */ 00354 /* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */ 00355 00356 /* See LAPACK User's Guide section 4.11 or the following references */ 00357 /* for more information. */ 00358 00359 /* Note that if the default method for computing the Frobenius-norm- */ 00360 /* based estimate DIF is not wanted (see SLATDF), then the parameter */ 00361 /* IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF */ 00362 /* (IJOB = 2 will be used)). See STGSYL for more details. */ 00363 00364 /* Based on contributions by */ 00365 /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ 00366 /* Umea University, S-901 87 Umea, Sweden. */ 00367 00368 /* References */ 00369 /* ========== */ 00370 00371 /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ 00372 /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ 00373 /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ 00374 /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ 00375 00376 /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ 00377 /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ 00378 /* Estimation: Theory, Algorithms and Software, */ 00379 /* Report UMINF - 94.04, Department of Computing Science, Umea */ 00380 /* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working */ 00381 /* Note 87. To appear in Numerical Algorithms, 1996. */ 00382 00383 /* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */ 00384 /* for Solving the Generalized Sylvester Equation and Estimating the */ 00385 /* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */ 00386 /* Department of Computing Science, Umea University, S-901 87 Umea, */ 00387 /* Sweden, December 1993, Revised April 1994, Also as LAPACK Working */ 00388 /* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */ 00389 /* 1996. */ 00390 00391 /* ===================================================================== */ 00392 00393 /* .. Parameters .. */ 00394 /* .. */ 00395 /* .. Local Scalars .. */ 00396 /* .. */ 00397 /* .. Local Arrays .. */ 00398 /* .. */ 00399 /* .. External Subroutines .. */ 00400 /* .. */ 00401 /* .. External Functions .. */ 00402 /* .. */ 00403 /* .. Intrinsic Functions .. */ 00404 /* .. */ 00405 /* .. Executable Statements .. */ 00406 00407 /* Decode and test the input parameters */ 00408 00409 /* Parameter adjustments */ 00410 --select; 00411 a_dim1 = *lda; 00412 a_offset = 1 + a_dim1; 00413 a -= a_offset; 00414 b_dim1 = *ldb; 00415 b_offset = 1 + b_dim1; 00416 b -= b_offset; 00417 --alphar; 00418 --alphai; 00419 --beta; 00420 q_dim1 = *ldq; 00421 q_offset = 1 + q_dim1; 00422 q -= q_offset; 00423 z_dim1 = *ldz; 00424 z_offset = 1 + z_dim1; 00425 z__ -= z_offset; 00426 --dif; 00427 --work; 00428 --iwork; 00429 00430 /* Function Body */ 00431 *info = 0; 00432 lquery = *lwork == -1 || *liwork == -1; 00433 00434 if (*ijob < 0 || *ijob > 5) { 00435 *info = -1; 00436 } else if (*n < 0) { 00437 *info = -5; 00438 } else if (*lda < max(1,*n)) { 00439 *info = -7; 00440 } else if (*ldb < max(1,*n)) { 00441 *info = -9; 00442 } else if (*ldq < 1 || *wantq && *ldq < *n) { 00443 *info = -14; 00444 } else if (*ldz < 1 || *wantz && *ldz < *n) { 00445 *info = -16; 00446 } 00447 00448 if (*info != 0) { 00449 i__1 = -(*info); 00450 xerbla_("STGSEN", &i__1); 00451 return 0; 00452 } 00453 00454 /* Get machine constants */ 00455 00456 eps = slamch_("P"); 00457 smlnum = slamch_("S") / eps; 00458 ierr = 0; 00459 00460 wantp = *ijob == 1 || *ijob >= 4; 00461 wantd1 = *ijob == 2 || *ijob == 4; 00462 wantd2 = *ijob == 3 || *ijob == 5; 00463 wantd = wantd1 || wantd2; 00464 00465 /* Set M to the dimension of the specified pair of deflating */ 00466 /* subspaces. */ 00467 00468 *m = 0; 00469 pair = FALSE_; 00470 i__1 = *n; 00471 for (k = 1; k <= i__1; ++k) { 00472 if (pair) { 00473 pair = FALSE_; 00474 } else { 00475 if (k < *n) { 00476 if (a[k + 1 + k * a_dim1] == 0.f) { 00477 if (select[k]) { 00478 ++(*m); 00479 } 00480 } else { 00481 pair = TRUE_; 00482 if (select[k] || select[k + 1]) { 00483 *m += 2; 00484 } 00485 } 00486 } else { 00487 if (select[*n]) { 00488 ++(*m); 00489 } 00490 } 00491 } 00492 /* L10: */ 00493 } 00494 00495 if (*ijob == 1 || *ijob == 2 || *ijob == 4) { 00496 /* Computing MAX */ 00497 i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), i__2 = (*m << 00498 1) * (*n - *m); 00499 lwmin = max(i__1,i__2); 00500 /* Computing MAX */ 00501 i__1 = 1, i__2 = *n + 6; 00502 liwmin = max(i__1,i__2); 00503 } else if (*ijob == 3 || *ijob == 5) { 00504 /* Computing MAX */ 00505 i__1 = 1, i__2 = (*n << 2) + 16, i__1 = max(i__1,i__2), i__2 = (*m << 00506 2) * (*n - *m); 00507 lwmin = max(i__1,i__2); 00508 /* Computing MAX */ 00509 i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = max(i__1,i__2), i__2 = 00510 *n + 6; 00511 liwmin = max(i__1,i__2); 00512 } else { 00513 /* Computing MAX */ 00514 i__1 = 1, i__2 = (*n << 2) + 16; 00515 lwmin = max(i__1,i__2); 00516 liwmin = 1; 00517 } 00518 00519 work[1] = (real) lwmin; 00520 iwork[1] = liwmin; 00521 00522 if (*lwork < lwmin && ! lquery) { 00523 *info = -22; 00524 } else if (*liwork < liwmin && ! lquery) { 00525 *info = -24; 00526 } 00527 00528 if (*info != 0) { 00529 i__1 = -(*info); 00530 xerbla_("STGSEN", &i__1); 00531 return 0; 00532 } else if (lquery) { 00533 return 0; 00534 } 00535 00536 /* Quick return if possible. */ 00537 00538 if (*m == *n || *m == 0) { 00539 if (wantp) { 00540 *pl = 1.f; 00541 *pr = 1.f; 00542 } 00543 if (wantd) { 00544 dscale = 0.f; 00545 dsum = 1.f; 00546 i__1 = *n; 00547 for (i__ = 1; i__ <= i__1; ++i__) { 00548 slassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum); 00549 slassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum); 00550 /* L20: */ 00551 } 00552 dif[1] = dscale * sqrt(dsum); 00553 dif[2] = dif[1]; 00554 } 00555 goto L60; 00556 } 00557 00558 /* Collect the selected blocks at the top-left corner of (A, B). */ 00559 00560 ks = 0; 00561 pair = FALSE_; 00562 i__1 = *n; 00563 for (k = 1; k <= i__1; ++k) { 00564 if (pair) { 00565 pair = FALSE_; 00566 } else { 00567 00568 swap = select[k]; 00569 if (k < *n) { 00570 if (a[k + 1 + k * a_dim1] != 0.f) { 00571 pair = TRUE_; 00572 swap = swap || select[k + 1]; 00573 } 00574 } 00575 00576 if (swap) { 00577 ++ks; 00578 00579 /* Swap the K-th block to position KS. */ 00580 /* Perform the reordering of diagonal blocks in (A, B) */ 00581 /* by orthogonal transformation matrices and update */ 00582 /* Q and Z accordingly (if requested): */ 00583 00584 kk = k; 00585 if (k != ks) { 00586 stgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], 00587 ldb, &q[q_offset], ldq, &z__[z_offset], ldz, &kk, 00588 &ks, &work[1], lwork, &ierr); 00589 } 00590 00591 if (ierr > 0) { 00592 00593 /* Swap is rejected: exit. */ 00594 00595 *info = 1; 00596 if (wantp) { 00597 *pl = 0.f; 00598 *pr = 0.f; 00599 } 00600 if (wantd) { 00601 dif[1] = 0.f; 00602 dif[2] = 0.f; 00603 } 00604 goto L60; 00605 } 00606 00607 if (pair) { 00608 ++ks; 00609 } 00610 } 00611 } 00612 /* L30: */ 00613 } 00614 if (wantp) { 00615 00616 /* Solve generalized Sylvester equation for R and L */ 00617 /* and compute PL and PR. */ 00618 00619 n1 = *m; 00620 n2 = *n - *m; 00621 i__ = n1 + 1; 00622 ijb = 0; 00623 slacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1); 00624 slacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 + 00625 1], &n1); 00626 i__1 = *lwork - (n1 << 1) * n2; 00627 stgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1] 00628 , lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * 00629 b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], & 00630 work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr); 00631 00632 /* Estimate the reciprocal of norms of "projections" onto left */ 00633 /* and right eigenspaces. */ 00634 00635 rdscal = 0.f; 00636 dsum = 1.f; 00637 i__1 = n1 * n2; 00638 slassq_(&i__1, &work[1], &c__1, &rdscal, &dsum); 00639 *pl = rdscal * sqrt(dsum); 00640 if (*pl == 0.f) { 00641 *pl = 1.f; 00642 } else { 00643 *pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl)); 00644 } 00645 rdscal = 0.f; 00646 dsum = 1.f; 00647 i__1 = n1 * n2; 00648 slassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum); 00649 *pr = rdscal * sqrt(dsum); 00650 if (*pr == 0.f) { 00651 *pr = 1.f; 00652 } else { 00653 *pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr)); 00654 } 00655 } 00656 00657 if (wantd) { 00658 00659 /* Compute estimates of Difu and Difl. */ 00660 00661 if (wantd1) { 00662 n1 = *m; 00663 n2 = *n - *m; 00664 i__ = n1 + 1; 00665 ijb = 3; 00666 00667 /* Frobenius norm-based Difu-estimate. */ 00668 00669 i__1 = *lwork - (n1 << 1) * n2; 00670 stgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * 00671 a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + 00672 i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, & 00673 dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & 00674 ierr); 00675 00676 /* Frobenius norm-based Difl-estimate. */ 00677 00678 i__1 = *lwork - (n1 << 1) * n2; 00679 stgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[ 00680 a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], 00681 ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, 00682 &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & 00683 ierr); 00684 } else { 00685 00686 00687 /* Compute 1-norm-based estimates of Difu and Difl using */ 00688 /* reversed communication with SLACN2. In each step a */ 00689 /* generalized Sylvester equation or a transposed variant */ 00690 /* is solved. */ 00691 00692 kase = 0; 00693 n1 = *m; 00694 n2 = *n - *m; 00695 i__ = n1 + 1; 00696 ijb = 0; 00697 mn2 = (n1 << 1) * n2; 00698 00699 /* 1-norm-based estimate of Difu. */ 00700 00701 L40: 00702 slacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[1], &kase, 00703 isave); 00704 if (kase != 0) { 00705 if (kase == 1) { 00706 00707 /* Solve generalized Sylvester equation. */ 00708 00709 i__1 = *lwork - (n1 << 1) * n2; 00710 stgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + 00711 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], 00712 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 00713 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 00714 1], &i__1, &iwork[1], &ierr); 00715 } else { 00716 00717 /* Solve the transposed variant. */ 00718 00719 i__1 = *lwork - (n1 << 1) * n2; 00720 stgsyl_("T", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + 00721 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], 00722 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 00723 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 00724 1], &i__1, &iwork[1], &ierr); 00725 } 00726 goto L40; 00727 } 00728 dif[1] = dscale / dif[1]; 00729 00730 /* 1-norm-based estimate of Difl. */ 00731 00732 L50: 00733 slacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[2], &kase, 00734 isave); 00735 if (kase != 0) { 00736 if (kase == 1) { 00737 00738 /* Solve generalized Sylvester equation. */ 00739 00740 i__1 = *lwork - (n1 << 1) * n2; 00741 stgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, 00742 &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * 00743 b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 00744 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 00745 1], &i__1, &iwork[1], &ierr); 00746 } else { 00747 00748 /* Solve the transposed variant. */ 00749 00750 i__1 = *lwork - (n1 << 1) * n2; 00751 stgsyl_("T", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, 00752 &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * 00753 b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 00754 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 00755 1], &i__1, &iwork[1], &ierr); 00756 } 00757 goto L50; 00758 } 00759 dif[2] = dscale / dif[2]; 00760 00761 } 00762 } 00763 00764 L60: 00765 00766 /* Compute generalized eigenvalues of reordered pair (A, B) and */ 00767 /* normalize the generalized Schur form. */ 00768 00769 pair = FALSE_; 00770 i__1 = *n; 00771 for (k = 1; k <= i__1; ++k) { 00772 if (pair) { 00773 pair = FALSE_; 00774 } else { 00775 00776 if (k < *n) { 00777 if (a[k + 1 + k * a_dim1] != 0.f) { 00778 pair = TRUE_; 00779 } 00780 } 00781 00782 if (pair) { 00783 00784 /* Compute the eigenvalue(s) at position K. */ 00785 00786 work[1] = a[k + k * a_dim1]; 00787 work[2] = a[k + 1 + k * a_dim1]; 00788 work[3] = a[k + (k + 1) * a_dim1]; 00789 work[4] = a[k + 1 + (k + 1) * a_dim1]; 00790 work[5] = b[k + k * b_dim1]; 00791 work[6] = b[k + 1 + k * b_dim1]; 00792 work[7] = b[k + (k + 1) * b_dim1]; 00793 work[8] = b[k + 1 + (k + 1) * b_dim1]; 00794 r__1 = smlnum * eps; 00795 slag2_(&work[1], &c__2, &work[5], &c__2, &r__1, &beta[k], & 00796 beta[k + 1], &alphar[k], &alphar[k + 1], &alphai[k]); 00797 alphai[k + 1] = -alphai[k]; 00798 00799 } else { 00800 00801 if (r_sign(&c_b28, &b[k + k * b_dim1]) < 0.f) { 00802 00803 /* If B(K,K) is negative, make it positive */ 00804 00805 i__2 = *n; 00806 for (i__ = 1; i__ <= i__2; ++i__) { 00807 a[k + i__ * a_dim1] = -a[k + i__ * a_dim1]; 00808 b[k + i__ * b_dim1] = -b[k + i__ * b_dim1]; 00809 if (*wantq) { 00810 q[i__ + k * q_dim1] = -q[i__ + k * q_dim1]; 00811 } 00812 /* L80: */ 00813 } 00814 } 00815 00816 alphar[k] = a[k + k * a_dim1]; 00817 alphai[k] = 0.f; 00818 beta[k] = b[k + k * b_dim1]; 00819 00820 } 00821 } 00822 /* L70: */ 00823 } 00824 00825 work[1] = (real) lwmin; 00826 iwork[1] = liwmin; 00827 00828 return 0; 00829 00830 /* End of STGSEN */ 00831 00832 } /* stgsen_ */