00001 /* dtgsna.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 doublereal c_b19 = 1.; 00020 static doublereal c_b21 = 0.; 00021 static integer c__2 = 2; 00022 static logical c_false = FALSE_; 00023 static integer c__3 = 3; 00024 00025 /* Subroutine */ int dtgsna_(char *job, char *howmny, logical *select, 00026 integer *n, doublereal *a, integer *lda, doublereal *b, integer *ldb, 00027 doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr, 00028 doublereal *s, doublereal *dif, integer *mm, integer *m, doublereal * 00029 work, integer *lwork, integer *iwork, integer *info) 00030 { 00031 /* System generated locals */ 00032 integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, 00033 vr_offset, i__1, i__2; 00034 doublereal d__1, d__2; 00035 00036 /* Builtin functions */ 00037 double sqrt(doublereal); 00038 00039 /* Local variables */ 00040 integer i__, k; 00041 doublereal c1, c2; 00042 integer n1, n2, ks, iz; 00043 doublereal eps, beta, cond; 00044 extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, 00045 integer *); 00046 logical pair; 00047 integer ierr; 00048 doublereal uhav, uhbv; 00049 integer ifst; 00050 doublereal lnrm; 00051 integer ilst; 00052 doublereal rnrm; 00053 extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *, 00054 integer *, doublereal *, doublereal *, doublereal *, doublereal *, 00055 doublereal *, doublereal *); 00056 extern doublereal dnrm2_(integer *, doublereal *, integer *); 00057 doublereal root1, root2, scale; 00058 extern logical lsame_(char *, char *); 00059 extern /* Subroutine */ int dgemv_(char *, integer *, integer *, 00060 doublereal *, doublereal *, integer *, doublereal *, integer *, 00061 doublereal *, doublereal *, integer *); 00062 doublereal uhavi, uhbvi, tmpii; 00063 integer lwmin; 00064 logical wants; 00065 doublereal tmpir, tmpri, dummy[1], tmprr; 00066 extern doublereal dlapy2_(doublereal *, doublereal *); 00067 doublereal dummy1[1]; 00068 extern doublereal dlamch_(char *); 00069 doublereal alphai, alphar; 00070 extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, 00071 doublereal *, integer *, doublereal *, integer *), 00072 xerbla_(char *, integer *), dtgexc_(logical *, logical *, 00073 integer *, doublereal *, integer *, doublereal *, integer *, 00074 doublereal *, integer *, doublereal *, integer *, integer *, 00075 integer *, doublereal *, integer *, integer *); 00076 logical wantbh, wantdf, somcon; 00077 doublereal alprqt; 00078 extern /* Subroutine */ int dtgsyl_(char *, integer *, integer *, integer 00079 *, doublereal *, integer *, doublereal *, integer *, doublereal *, 00080 integer *, doublereal *, integer *, doublereal *, integer *, 00081 doublereal *, integer *, doublereal *, doublereal *, doublereal *, 00082 integer *, integer *, integer *); 00083 doublereal smlnum; 00084 logical lquery; 00085 00086 00087 /* -- LAPACK routine (version 3.2) -- */ 00088 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00089 /* November 2006 */ 00090 00091 /* .. Scalar Arguments .. */ 00092 /* .. */ 00093 /* .. Array Arguments .. */ 00094 /* .. */ 00095 00096 /* Purpose */ 00097 /* ======= */ 00098 00099 /* DTGSNA estimates reciprocal condition numbers for specified */ 00100 /* eigenvalues and/or eigenvectors of a matrix pair (A, B) in */ 00101 /* generalized real Schur canonical form (or of any matrix pair */ 00102 /* (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where */ 00103 /* Z' denotes the transpose of Z. */ 00104 00105 /* (A, B) must be in generalized real Schur form (as returned by DGGES), */ 00106 /* i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal */ 00107 /* blocks. B is upper triangular. */ 00108 00109 00110 /* Arguments */ 00111 /* ========= */ 00112 00113 /* JOB (input) CHARACTER*1 */ 00114 /* Specifies whether condition numbers are required for */ 00115 /* eigenvalues (S) or eigenvectors (DIF): */ 00116 /* = 'E': for eigenvalues only (S); */ 00117 /* = 'V': for eigenvectors only (DIF); */ 00118 /* = 'B': for both eigenvalues and eigenvectors (S and DIF). */ 00119 00120 /* HOWMNY (input) CHARACTER*1 */ 00121 /* = 'A': compute condition numbers for all eigenpairs; */ 00122 /* = 'S': compute condition numbers for selected eigenpairs */ 00123 /* specified by the array SELECT. */ 00124 00125 /* SELECT (input) LOGICAL array, dimension (N) */ 00126 /* If HOWMNY = 'S', SELECT specifies the eigenpairs for which */ 00127 /* condition numbers are required. To select condition numbers */ 00128 /* for the eigenpair corresponding to a real eigenvalue w(j), */ 00129 /* SELECT(j) must be set to .TRUE.. To select condition numbers */ 00130 /* corresponding to a complex conjugate pair of eigenvalues w(j) */ 00131 /* and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be */ 00132 /* set to .TRUE.. */ 00133 /* If HOWMNY = 'A', SELECT is not referenced. */ 00134 00135 /* N (input) INTEGER */ 00136 /* The order of the square matrix pair (A, B). N >= 0. */ 00137 00138 /* A (input) DOUBLE PRECISION array, dimension (LDA,N) */ 00139 /* The upper quasi-triangular matrix A in the pair (A,B). */ 00140 00141 /* LDA (input) INTEGER */ 00142 /* The leading dimension of the array A. LDA >= max(1,N). */ 00143 00144 /* B (input) DOUBLE PRECISION array, dimension (LDB,N) */ 00145 /* The upper triangular matrix B in the pair (A,B). */ 00146 00147 /* LDB (input) INTEGER */ 00148 /* The leading dimension of the array B. LDB >= max(1,N). */ 00149 00150 /* VL (input) DOUBLE PRECISION array, dimension (LDVL,M) */ 00151 /* If JOB = 'E' or 'B', VL must contain left eigenvectors of */ 00152 /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ 00153 /* and SELECT. The eigenvectors must be stored in consecutive */ 00154 /* columns of VL, as returned by DTGEVC. */ 00155 /* If JOB = 'V', VL is not referenced. */ 00156 00157 /* LDVL (input) INTEGER */ 00158 /* The leading dimension of the array VL. LDVL >= 1. */ 00159 /* If JOB = 'E' or 'B', LDVL >= N. */ 00160 00161 /* VR (input) DOUBLE PRECISION array, dimension (LDVR,M) */ 00162 /* If JOB = 'E' or 'B', VR must contain right eigenvectors of */ 00163 /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ 00164 /* and SELECT. The eigenvectors must be stored in consecutive */ 00165 /* columns ov VR, as returned by DTGEVC. */ 00166 /* If JOB = 'V', VR is not referenced. */ 00167 00168 /* LDVR (input) INTEGER */ 00169 /* The leading dimension of the array VR. LDVR >= 1. */ 00170 /* If JOB = 'E' or 'B', LDVR >= N. */ 00171 00172 /* S (output) DOUBLE PRECISION array, dimension (MM) */ 00173 /* If JOB = 'E' or 'B', the reciprocal condition numbers of the */ 00174 /* selected eigenvalues, stored in consecutive elements of the */ 00175 /* array. For a complex conjugate pair of eigenvalues two */ 00176 /* consecutive elements of S are set to the same value. Thus */ 00177 /* S(j), DIF(j), and the j-th columns of VL and VR all */ 00178 /* correspond to the same eigenpair (but not in general the */ 00179 /* j-th eigenpair, unless all eigenpairs are selected). */ 00180 /* If JOB = 'V', S is not referenced. */ 00181 00182 /* DIF (output) DOUBLE PRECISION array, dimension (MM) */ 00183 /* If JOB = 'V' or 'B', the estimated reciprocal condition */ 00184 /* numbers of the selected eigenvectors, stored in consecutive */ 00185 /* elements of the array. For a complex eigenvector two */ 00186 /* consecutive elements of DIF are set to the same value. If */ 00187 /* the eigenvalues cannot be reordered to compute DIF(j), DIF(j) */ 00188 /* is set to 0; this can only occur when the true value would be */ 00189 /* very small anyway. */ 00190 /* If JOB = 'E', DIF is not referenced. */ 00191 00192 /* MM (input) INTEGER */ 00193 /* The number of elements in the arrays S and DIF. MM >= M. */ 00194 00195 /* M (output) INTEGER */ 00196 /* The number of elements of the arrays S and DIF used to store */ 00197 /* the specified condition numbers; for each selected real */ 00198 /* eigenvalue one element is used, and for each selected complex */ 00199 /* conjugate pair of eigenvalues, two elements are used. */ 00200 /* If HOWMNY = 'A', M is set to N. */ 00201 00202 /* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */ 00203 /* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ 00204 00205 /* LWORK (input) INTEGER */ 00206 /* The dimension of the array WORK. LWORK >= max(1,N). */ 00207 /* If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. */ 00208 00209 /* If LWORK = -1, then a workspace query is assumed; the routine */ 00210 /* only calculates the optimal size of the WORK array, returns */ 00211 /* this value as the first entry of the WORK array, and no error */ 00212 /* message related to LWORK is issued by XERBLA. */ 00213 00214 /* IWORK (workspace) INTEGER array, dimension (N + 6) */ 00215 /* If JOB = 'E', IWORK is not referenced. */ 00216 00217 /* INFO (output) INTEGER */ 00218 /* =0: Successful exit */ 00219 /* <0: If INFO = -i, the i-th argument had an illegal value */ 00220 00221 00222 /* Further Details */ 00223 /* =============== */ 00224 00225 /* The reciprocal of the condition number of a generalized eigenvalue */ 00226 /* w = (a, b) is defined as */ 00227 00228 /* S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v)) */ 00229 00230 /* where u and v are the left and right eigenvectors of (A, B) */ 00231 /* corresponding to w; |z| denotes the absolute value of the complex */ 00232 /* number, and norm(u) denotes the 2-norm of the vector u. */ 00233 /* The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv) */ 00234 /* of the matrix pair (A, B). If both a and b equal zero, then (A B) is */ 00235 /* singular and S(I) = -1 is returned. */ 00236 00237 /* An approximate error bound on the chordal distance between the i-th */ 00238 /* computed generalized eigenvalue w and the corresponding exact */ 00239 /* eigenvalue lambda is */ 00240 00241 /* chord(w, lambda) <= EPS * norm(A, B) / S(I) */ 00242 00243 /* where EPS is the machine precision. */ 00244 00245 /* The reciprocal of the condition number DIF(i) of right eigenvector u */ 00246 /* and left eigenvector v corresponding to the generalized eigenvalue w */ 00247 /* is defined as follows: */ 00248 00249 /* a) If the i-th eigenvalue w = (a,b) is real */ 00250 00251 /* Suppose U and V are orthogonal transformations such that */ 00252 00253 /* U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 */ 00254 /* ( 0 S22 ),( 0 T22 ) n-1 */ 00255 /* 1 n-1 1 n-1 */ 00256 00257 /* Then the reciprocal condition number DIF(i) is */ 00258 00259 /* Difl((a, b), (S22, T22)) = sigma-min( Zl ), */ 00260 00261 /* where sigma-min(Zl) denotes the smallest singular value of the */ 00262 /* 2(n-1)-by-2(n-1) matrix */ 00263 00264 /* Zl = [ kron(a, In-1) -kron(1, S22) ] */ 00265 /* [ kron(b, In-1) -kron(1, T22) ] . */ 00266 00267 /* Here In-1 is the identity matrix of size n-1. kron(X, Y) is the */ 00268 /* Kronecker product between the matrices X and Y. */ 00269 00270 /* Note that if the default method for computing DIF(i) is wanted */ 00271 /* (see DLATDF), then the parameter DIFDRI (see below) should be */ 00272 /* changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). */ 00273 /* See DTGSYL for more details. */ 00274 00275 /* b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair, */ 00276 00277 /* Suppose U and V are orthogonal transformations such that */ 00278 00279 /* U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 */ 00280 /* ( 0 S22 ),( 0 T22) n-2 */ 00281 /* 2 n-2 2 n-2 */ 00282 00283 /* and (S11, T11) corresponds to the complex conjugate eigenvalue */ 00284 /* pair (w, conjg(w)). There exist unitary matrices U1 and V1 such */ 00285 /* that */ 00286 00287 /* U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 ) */ 00288 /* ( 0 s22 ) ( 0 t22 ) */ 00289 00290 /* where the generalized eigenvalues w = s11/t11 and */ 00291 /* conjg(w) = s22/t22. */ 00292 00293 /* Then the reciprocal condition number DIF(i) is bounded by */ 00294 00295 /* min( d1, max( 1, |real(s11)/real(s22)| )*d2 ) */ 00296 00297 /* where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where */ 00298 /* Z1 is the complex 2-by-2 matrix */ 00299 00300 /* Z1 = [ s11 -s22 ] */ 00301 /* [ t11 -t22 ], */ 00302 00303 /* This is done by computing (using real arithmetic) the */ 00304 /* roots of the characteristical polynomial det(Z1' * Z1 - lambda I), */ 00305 /* where Z1' denotes the conjugate transpose of Z1 and det(X) denotes */ 00306 /* the determinant of X. */ 00307 00308 /* and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an */ 00309 /* upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) */ 00310 00311 /* Z2 = [ kron(S11', In-2) -kron(I2, S22) ] */ 00312 /* [ kron(T11', In-2) -kron(I2, T22) ] */ 00313 00314 /* Note that if the default method for computing DIF is wanted (see */ 00315 /* DLATDF), then the parameter DIFDRI (see below) should be changed */ 00316 /* from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL */ 00317 /* for more details. */ 00318 00319 /* For each eigenvalue/vector specified by SELECT, DIF stores a */ 00320 /* Frobenius norm-based estimate of Difl. */ 00321 00322 /* An approximate error bound for the i-th computed eigenvector VL(i) or */ 00323 /* VR(i) is given by */ 00324 00325 /* EPS * norm(A, B) / DIF(i). */ 00326 00327 /* See ref. [2-3] for more details and further references. */ 00328 00329 /* Based on contributions by */ 00330 /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ 00331 /* Umea University, S-901 87 Umea, Sweden. */ 00332 00333 /* References */ 00334 /* ========== */ 00335 00336 /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ 00337 /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ 00338 /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ 00339 /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ 00340 00341 /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ 00342 /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ 00343 /* Estimation: Theory, Algorithms and Software, */ 00344 /* Report UMINF - 94.04, Department of Computing Science, Umea */ 00345 /* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working */ 00346 /* Note 87. To appear in Numerical Algorithms, 1996. */ 00347 00348 /* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */ 00349 /* for Solving the Generalized Sylvester Equation and Estimating the */ 00350 /* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */ 00351 /* Department of Computing Science, Umea University, S-901 87 Umea, */ 00352 /* Sweden, December 1993, Revised April 1994, Also as LAPACK Working */ 00353 /* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, */ 00354 /* No 1, 1996. */ 00355 00356 /* ===================================================================== */ 00357 00358 /* .. Parameters .. */ 00359 /* .. */ 00360 /* .. Local Scalars .. */ 00361 /* .. */ 00362 /* .. Local Arrays .. */ 00363 /* .. */ 00364 /* .. External Functions .. */ 00365 /* .. */ 00366 /* .. External Subroutines .. */ 00367 /* .. */ 00368 /* .. Intrinsic Functions .. */ 00369 /* .. */ 00370 /* .. Executable Statements .. */ 00371 00372 /* Decode and test the input parameters */ 00373 00374 /* Parameter adjustments */ 00375 --select; 00376 a_dim1 = *lda; 00377 a_offset = 1 + a_dim1; 00378 a -= a_offset; 00379 b_dim1 = *ldb; 00380 b_offset = 1 + b_dim1; 00381 b -= b_offset; 00382 vl_dim1 = *ldvl; 00383 vl_offset = 1 + vl_dim1; 00384 vl -= vl_offset; 00385 vr_dim1 = *ldvr; 00386 vr_offset = 1 + vr_dim1; 00387 vr -= vr_offset; 00388 --s; 00389 --dif; 00390 --work; 00391 --iwork; 00392 00393 /* Function Body */ 00394 wantbh = lsame_(job, "B"); 00395 wants = lsame_(job, "E") || wantbh; 00396 wantdf = lsame_(job, "V") || wantbh; 00397 00398 somcon = lsame_(howmny, "S"); 00399 00400 *info = 0; 00401 lquery = *lwork == -1; 00402 00403 if (! wants && ! wantdf) { 00404 *info = -1; 00405 } else if (! lsame_(howmny, "A") && ! somcon) { 00406 *info = -2; 00407 } else if (*n < 0) { 00408 *info = -4; 00409 } else if (*lda < max(1,*n)) { 00410 *info = -6; 00411 } else if (*ldb < max(1,*n)) { 00412 *info = -8; 00413 } else if (wants && *ldvl < *n) { 00414 *info = -10; 00415 } else if (wants && *ldvr < *n) { 00416 *info = -12; 00417 } else { 00418 00419 /* Set M to the number of eigenpairs for which condition numbers */ 00420 /* are required, and test MM. */ 00421 00422 if (somcon) { 00423 *m = 0; 00424 pair = FALSE_; 00425 i__1 = *n; 00426 for (k = 1; k <= i__1; ++k) { 00427 if (pair) { 00428 pair = FALSE_; 00429 } else { 00430 if (k < *n) { 00431 if (a[k + 1 + k * a_dim1] == 0.) { 00432 if (select[k]) { 00433 ++(*m); 00434 } 00435 } else { 00436 pair = TRUE_; 00437 if (select[k] || select[k + 1]) { 00438 *m += 2; 00439 } 00440 } 00441 } else { 00442 if (select[*n]) { 00443 ++(*m); 00444 } 00445 } 00446 } 00447 /* L10: */ 00448 } 00449 } else { 00450 *m = *n; 00451 } 00452 00453 if (*n == 0) { 00454 lwmin = 1; 00455 } else if (lsame_(job, "V") || lsame_(job, 00456 "B")) { 00457 lwmin = (*n << 1) * (*n + 2) + 16; 00458 } else { 00459 lwmin = *n; 00460 } 00461 work[1] = (doublereal) lwmin; 00462 00463 if (*mm < *m) { 00464 *info = -15; 00465 } else if (*lwork < lwmin && ! lquery) { 00466 *info = -18; 00467 } 00468 } 00469 00470 if (*info != 0) { 00471 i__1 = -(*info); 00472 xerbla_("DTGSNA", &i__1); 00473 return 0; 00474 } else if (lquery) { 00475 return 0; 00476 } 00477 00478 /* Quick return if possible */ 00479 00480 if (*n == 0) { 00481 return 0; 00482 } 00483 00484 /* Get machine constants */ 00485 00486 eps = dlamch_("P"); 00487 smlnum = dlamch_("S") / eps; 00488 ks = 0; 00489 pair = FALSE_; 00490 00491 i__1 = *n; 00492 for (k = 1; k <= i__1; ++k) { 00493 00494 /* Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block. */ 00495 00496 if (pair) { 00497 pair = FALSE_; 00498 goto L20; 00499 } else { 00500 if (k < *n) { 00501 pair = a[k + 1 + k * a_dim1] != 0.; 00502 } 00503 } 00504 00505 /* Determine whether condition numbers are required for the k-th */ 00506 /* eigenpair. */ 00507 00508 if (somcon) { 00509 if (pair) { 00510 if (! select[k] && ! select[k + 1]) { 00511 goto L20; 00512 } 00513 } else { 00514 if (! select[k]) { 00515 goto L20; 00516 } 00517 } 00518 } 00519 00520 ++ks; 00521 00522 if (wants) { 00523 00524 /* Compute the reciprocal condition number of the k-th */ 00525 /* eigenvalue. */ 00526 00527 if (pair) { 00528 00529 /* Complex eigenvalue pair. */ 00530 00531 d__1 = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); 00532 d__2 = dnrm2_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1); 00533 rnrm = dlapy2_(&d__1, &d__2); 00534 d__1 = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); 00535 d__2 = dnrm2_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1); 00536 lnrm = dlapy2_(&d__1, &d__2); 00537 dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 00538 + 1], &c__1, &c_b21, &work[1], &c__1); 00539 tmprr = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & 00540 c__1); 00541 tmpri = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], 00542 &c__1); 00543 dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[(ks + 1) * 00544 vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); 00545 tmpii = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], 00546 &c__1); 00547 tmpir = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & 00548 c__1); 00549 uhav = tmprr + tmpii; 00550 uhavi = tmpir - tmpri; 00551 dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 00552 + 1], &c__1, &c_b21, &work[1], &c__1); 00553 tmprr = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & 00554 c__1); 00555 tmpri = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], 00556 &c__1); 00557 dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[(ks + 1) * 00558 vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); 00559 tmpii = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], 00560 &c__1); 00561 tmpir = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & 00562 c__1); 00563 uhbv = tmprr + tmpii; 00564 uhbvi = tmpir - tmpri; 00565 uhav = dlapy2_(&uhav, &uhavi); 00566 uhbv = dlapy2_(&uhbv, &uhbvi); 00567 cond = dlapy2_(&uhav, &uhbv); 00568 s[ks] = cond / (rnrm * lnrm); 00569 s[ks + 1] = s[ks]; 00570 00571 } else { 00572 00573 /* Real eigenvalue. */ 00574 00575 rnrm = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); 00576 lnrm = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); 00577 dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 00578 + 1], &c__1, &c_b21, &work[1], &c__1); 00579 uhav = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1) 00580 ; 00581 dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 00582 + 1], &c__1, &c_b21, &work[1], &c__1); 00583 uhbv = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1) 00584 ; 00585 cond = dlapy2_(&uhav, &uhbv); 00586 if (cond == 0.) { 00587 s[ks] = -1.; 00588 } else { 00589 s[ks] = cond / (rnrm * lnrm); 00590 } 00591 } 00592 } 00593 00594 if (wantdf) { 00595 if (*n == 1) { 00596 dif[ks] = dlapy2_(&a[a_dim1 + 1], &b[b_dim1 + 1]); 00597 goto L20; 00598 } 00599 00600 /* Estimate the reciprocal condition number of the k-th */ 00601 /* eigenvectors. */ 00602 if (pair) { 00603 00604 /* Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)). */ 00605 /* Compute the eigenvalue(s) at position K. */ 00606 00607 work[1] = a[k + k * a_dim1]; 00608 work[2] = a[k + 1 + k * a_dim1]; 00609 work[3] = a[k + (k + 1) * a_dim1]; 00610 work[4] = a[k + 1 + (k + 1) * a_dim1]; 00611 work[5] = b[k + k * b_dim1]; 00612 work[6] = b[k + 1 + k * b_dim1]; 00613 work[7] = b[k + (k + 1) * b_dim1]; 00614 work[8] = b[k + 1 + (k + 1) * b_dim1]; 00615 d__1 = smlnum * eps; 00616 dlag2_(&work[1], &c__2, &work[5], &c__2, &d__1, &beta, dummy1, 00617 &alphar, dummy, &alphai); 00618 alprqt = 1.; 00619 c1 = (alphar * alphar + alphai * alphai + beta * beta) * 2.; 00620 c2 = beta * 4. * beta * alphai * alphai; 00621 root1 = c1 + sqrt(c1 * c1 - c2 * 4.); 00622 root2 = c2 / root1; 00623 root1 /= 2.; 00624 /* Computing MIN */ 00625 d__1 = sqrt(root1), d__2 = sqrt(root2); 00626 cond = min(d__1,d__2); 00627 } 00628 00629 /* Copy the matrix (A, B) to the array WORK and swap the */ 00630 /* diagonal block beginning at A(k,k) to the (1,1) position. */ 00631 00632 dlacpy_("Full", n, n, &a[a_offset], lda, &work[1], n); 00633 dlacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1], n); 00634 ifst = k; 00635 ilst = 1; 00636 00637 i__2 = *lwork - (*n << 1) * *n; 00638 dtgexc_(&c_false, &c_false, n, &work[1], n, &work[*n * *n + 1], n, 00639 dummy, &c__1, dummy1, &c__1, &ifst, &ilst, &work[(*n * * 00640 n << 1) + 1], &i__2, &ierr); 00641 00642 if (ierr > 0) { 00643 00644 /* Ill-conditioned problem - swap rejected. */ 00645 00646 dif[ks] = 0.; 00647 } else { 00648 00649 /* Reordering successful, solve generalized Sylvester */ 00650 /* equation for R and L, */ 00651 /* A22 * R - L * A11 = A12 */ 00652 /* B22 * R - L * B11 = B12, */ 00653 /* and compute estimate of Difl((A11,B11), (A22, B22)). */ 00654 00655 n1 = 1; 00656 if (work[2] != 0.) { 00657 n1 = 2; 00658 } 00659 n2 = *n - n1; 00660 if (n2 == 0) { 00661 dif[ks] = cond; 00662 } else { 00663 i__ = *n * *n + 1; 00664 iz = (*n << 1) * *n + 1; 00665 i__2 = *lwork - (*n << 1) * *n; 00666 dtgsyl_("N", &c__3, &n2, &n1, &work[*n * n1 + n1 + 1], n, 00667 &work[1], n, &work[n1 + 1], n, &work[*n * n1 + n1 00668 + i__], n, &work[i__], n, &work[n1 + i__], n, & 00669 scale, &dif[ks], &work[iz + 1], &i__2, &iwork[1], 00670 &ierr); 00671 00672 if (pair) { 00673 /* Computing MIN */ 00674 d__1 = max(1.,alprqt) * dif[ks]; 00675 dif[ks] = min(d__1,cond); 00676 } 00677 } 00678 } 00679 if (pair) { 00680 dif[ks + 1] = dif[ks]; 00681 } 00682 } 00683 if (pair) { 00684 ++ks; 00685 } 00686 00687 L20: 00688 ; 00689 } 00690 work[1] = (doublereal) lwmin; 00691 return 0; 00692 00693 /* End of DTGSNA */ 00694 00695 } /* dtgsna_ */