00001 /* ztgsna.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 doublecomplex c_b19 = {1.,0.}; 00020 static doublecomplex c_b20 = {0.,0.}; 00021 static logical c_false = FALSE_; 00022 static integer c__3 = 3; 00023 00024 /* Subroutine */ int ztgsna_(char *job, char *howmny, logical *select, 00025 integer *n, doublecomplex *a, integer *lda, doublecomplex *b, integer 00026 *ldb, doublecomplex *vl, integer *ldvl, doublecomplex *vr, integer * 00027 ldvr, doublereal *s, doublereal *dif, integer *mm, integer *m, 00028 doublecomplex *work, integer *lwork, integer *iwork, integer *info) 00029 { 00030 /* System generated locals */ 00031 integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, 00032 vr_offset, i__1; 00033 doublereal d__1, d__2; 00034 doublecomplex z__1; 00035 00036 /* Builtin functions */ 00037 double z_abs(doublecomplex *); 00038 00039 /* Local variables */ 00040 integer i__, k, n1, n2, ks; 00041 doublereal eps, cond; 00042 integer ierr, ifst; 00043 doublereal lnrm; 00044 doublecomplex yhax, yhbx; 00045 integer ilst; 00046 doublereal rnrm, scale; 00047 extern logical lsame_(char *, char *); 00048 extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *, 00049 doublecomplex *, integer *, doublecomplex *, integer *); 00050 integer lwmin; 00051 extern /* Subroutine */ int zgemv_(char *, integer *, integer *, 00052 doublecomplex *, doublecomplex *, integer *, doublecomplex *, 00053 integer *, doublecomplex *, doublecomplex *, integer *); 00054 logical wants; 00055 doublecomplex dummy[1]; 00056 extern doublereal dlapy2_(doublereal *, doublereal *); 00057 extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); 00058 doublecomplex dummy1[1]; 00059 extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_( 00060 char *); 00061 extern /* Subroutine */ int xerbla_(char *, integer *); 00062 doublereal bignum; 00063 logical wantbh, wantdf, somcon; 00064 extern /* Subroutine */ int zlacpy_(char *, integer *, integer *, 00065 doublecomplex *, integer *, doublecomplex *, integer *), 00066 ztgexc_(logical *, logical *, integer *, doublecomplex *, integer 00067 *, doublecomplex *, integer *, doublecomplex *, integer *, 00068 doublecomplex *, integer *, integer *, integer *, integer *); 00069 doublereal smlnum; 00070 logical lquery; 00071 extern /* Subroutine */ int ztgsyl_(char *, integer *, integer *, integer 00072 *, doublecomplex *, integer *, doublecomplex *, integer *, 00073 doublecomplex *, integer *, doublecomplex *, integer *, 00074 doublecomplex *, integer *, doublecomplex *, integer *, 00075 doublereal *, doublereal *, doublecomplex *, integer *, integer *, 00076 integer *); 00077 00078 00079 /* -- LAPACK routine (version 3.2) -- */ 00080 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00081 /* November 2006 */ 00082 00083 /* .. Scalar Arguments .. */ 00084 /* .. */ 00085 /* .. Array Arguments .. */ 00086 /* .. */ 00087 00088 /* Purpose */ 00089 /* ======= */ 00090 00091 /* ZTGSNA estimates reciprocal condition numbers for specified */ 00092 /* eigenvalues and/or eigenvectors of a matrix pair (A, B). */ 00093 00094 /* (A, B) must be in generalized Schur canonical form, that is, A and */ 00095 /* B are both upper triangular. */ 00096 00097 /* Arguments */ 00098 /* ========= */ 00099 00100 /* JOB (input) CHARACTER*1 */ 00101 /* Specifies whether condition numbers are required for */ 00102 /* eigenvalues (S) or eigenvectors (DIF): */ 00103 /* = 'E': for eigenvalues only (S); */ 00104 /* = 'V': for eigenvectors only (DIF); */ 00105 /* = 'B': for both eigenvalues and eigenvectors (S and DIF). */ 00106 00107 /* HOWMNY (input) CHARACTER*1 */ 00108 /* = 'A': compute condition numbers for all eigenpairs; */ 00109 /* = 'S': compute condition numbers for selected eigenpairs */ 00110 /* specified by the array SELECT. */ 00111 00112 /* SELECT (input) LOGICAL array, dimension (N) */ 00113 /* If HOWMNY = 'S', SELECT specifies the eigenpairs for which */ 00114 /* condition numbers are required. To select condition numbers */ 00115 /* for the corresponding j-th eigenvalue and/or eigenvector, */ 00116 /* SELECT(j) must be set to .TRUE.. */ 00117 /* If HOWMNY = 'A', SELECT is not referenced. */ 00118 00119 /* N (input) INTEGER */ 00120 /* The order of the square matrix pair (A, B). N >= 0. */ 00121 00122 /* A (input) COMPLEX*16 array, dimension (LDA,N) */ 00123 /* The upper triangular matrix A in the pair (A,B). */ 00124 00125 /* LDA (input) INTEGER */ 00126 /* The leading dimension of the array A. LDA >= max(1,N). */ 00127 00128 /* B (input) COMPLEX*16 array, dimension (LDB,N) */ 00129 /* The upper triangular matrix B in the pair (A, B). */ 00130 00131 /* LDB (input) INTEGER */ 00132 /* The leading dimension of the array B. LDB >= max(1,N). */ 00133 00134 /* VL (input) COMPLEX*16 array, dimension (LDVL,M) */ 00135 /* IF JOB = 'E' or 'B', VL must contain left eigenvectors of */ 00136 /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ 00137 /* and SELECT. The eigenvectors must be stored in consecutive */ 00138 /* columns of VL, as returned by ZTGEVC. */ 00139 /* If JOB = 'V', VL is not referenced. */ 00140 00141 /* LDVL (input) INTEGER */ 00142 /* The leading dimension of the array VL. LDVL >= 1; and */ 00143 /* If JOB = 'E' or 'B', LDVL >= N. */ 00144 00145 /* VR (input) COMPLEX*16 array, dimension (LDVR,M) */ 00146 /* IF JOB = 'E' or 'B', VR must contain right eigenvectors of */ 00147 /* (A, B), corresponding to the eigenpairs specified by HOWMNY */ 00148 /* and SELECT. The eigenvectors must be stored in consecutive */ 00149 /* columns of VR, as returned by ZTGEVC. */ 00150 /* If JOB = 'V', VR is not referenced. */ 00151 00152 /* LDVR (input) INTEGER */ 00153 /* The leading dimension of the array VR. LDVR >= 1; */ 00154 /* If JOB = 'E' or 'B', LDVR >= N. */ 00155 00156 /* S (output) DOUBLE PRECISION array, dimension (MM) */ 00157 /* If JOB = 'E' or 'B', the reciprocal condition numbers of the */ 00158 /* selected eigenvalues, stored in consecutive elements of the */ 00159 /* array. */ 00160 /* If JOB = 'V', S is not referenced. */ 00161 00162 /* DIF (output) DOUBLE PRECISION array, dimension (MM) */ 00163 /* If JOB = 'V' or 'B', the estimated reciprocal condition */ 00164 /* numbers of the selected eigenvectors, stored in consecutive */ 00165 /* elements of the array. */ 00166 /* If the eigenvalues cannot be reordered to compute DIF(j), */ 00167 /* DIF(j) is set to 0; this can only occur when the true value */ 00168 /* would be very small anyway. */ 00169 /* For each eigenvalue/vector specified by SELECT, DIF stores */ 00170 /* a Frobenius norm-based estimate of Difl. */ 00171 /* If JOB = 'E', DIF is not referenced. */ 00172 00173 /* MM (input) INTEGER */ 00174 /* The number of elements in the arrays S and DIF. MM >= M. */ 00175 00176 /* M (output) INTEGER */ 00177 /* The number of elements of the arrays S and DIF used to store */ 00178 /* the specified condition numbers; for each selected eigenvalue */ 00179 /* one element is used. If HOWMNY = 'A', M is set to N. */ 00180 00181 /* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) */ 00182 /* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ 00183 00184 /* LWORK (input) INTEGER */ 00185 /* The dimension of the array WORK. LWORK >= max(1,N). */ 00186 /* If JOB = 'V' or 'B', LWORK >= max(1,2*N*N). */ 00187 00188 /* IWORK (workspace) INTEGER array, dimension (N+2) */ 00189 /* If JOB = 'E', IWORK is not referenced. */ 00190 00191 /* INFO (output) INTEGER */ 00192 /* = 0: Successful exit */ 00193 /* < 0: If INFO = -i, the i-th argument had an illegal value */ 00194 00195 /* Further Details */ 00196 /* =============== */ 00197 00198 /* The reciprocal of the condition number of the i-th generalized */ 00199 /* eigenvalue w = (a, b) is defined as */ 00200 00201 /* S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v)) */ 00202 00203 /* where u and v are the right and left eigenvectors of (A, B) */ 00204 /* corresponding to w; |z| denotes the absolute value of the complex */ 00205 /* number, and norm(u) denotes the 2-norm of the vector u. The pair */ 00206 /* (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the */ 00207 /* matrix pair (A, B). If both a and b equal zero, then (A,B) is */ 00208 /* singular and S(I) = -1 is returned. */ 00209 00210 /* An approximate error bound on the chordal distance between the i-th */ 00211 /* computed generalized eigenvalue w and the corresponding exact */ 00212 /* eigenvalue lambda is */ 00213 00214 /* chord(w, lambda) <= EPS * norm(A, B) / S(I), */ 00215 00216 /* where EPS is the machine precision. */ 00217 00218 /* The reciprocal of the condition number of the right eigenvector u */ 00219 /* and left eigenvector v corresponding to the generalized eigenvalue w */ 00220 /* is defined as follows. Suppose */ 00221 00222 /* (A, B) = ( a * ) ( b * ) 1 */ 00223 /* ( 0 A22 ),( 0 B22 ) n-1 */ 00224 /* 1 n-1 1 n-1 */ 00225 00226 /* Then the reciprocal condition number DIF(I) is */ 00227 00228 /* Difl[(a, b), (A22, B22)] = sigma-min( Zl ) */ 00229 00230 /* where sigma-min(Zl) denotes the smallest singular value of */ 00231 00232 /* Zl = [ kron(a, In-1) -kron(1, A22) ] */ 00233 /* [ kron(b, In-1) -kron(1, B22) ]. */ 00234 00235 /* Here In-1 is the identity matrix of size n-1 and X' is the conjugate */ 00236 /* transpose of X. kron(X, Y) is the Kronecker product between the */ 00237 /* matrices X and Y. */ 00238 00239 /* We approximate the smallest singular value of Zl with an upper */ 00240 /* bound. This is done by ZLATDF. */ 00241 00242 /* An approximate error bound for a computed eigenvector VL(i) or */ 00243 /* VR(i) is given by */ 00244 00245 /* EPS * norm(A, B) / DIF(i). */ 00246 00247 /* See ref. [2-3] for more details and further references. */ 00248 00249 /* Based on contributions by */ 00250 /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ 00251 /* Umea University, S-901 87 Umea, Sweden. */ 00252 00253 /* References */ 00254 /* ========== */ 00255 00256 /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ 00257 /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ 00258 /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ 00259 /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ 00260 00261 /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ 00262 /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ 00263 /* Estimation: Theory, Algorithms and Software, Report */ 00264 /* UMINF - 94.04, Department of Computing Science, Umea University, */ 00265 /* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. */ 00266 /* To appear in Numerical Algorithms, 1996. */ 00267 00268 /* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */ 00269 /* for Solving the Generalized Sylvester Equation and Estimating the */ 00270 /* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */ 00271 /* Department of Computing Science, Umea University, S-901 87 Umea, */ 00272 /* Sweden, December 1993, Revised April 1994, Also as LAPACK Working */ 00273 /* Note 75. */ 00274 /* To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996. */ 00275 00276 /* ===================================================================== */ 00277 00278 /* .. Parameters .. */ 00279 /* .. */ 00280 /* .. Local Scalars .. */ 00281 /* .. */ 00282 /* .. Local Arrays .. */ 00283 /* .. */ 00284 /* .. External Functions .. */ 00285 /* .. */ 00286 /* .. External Subroutines .. */ 00287 /* .. */ 00288 /* .. Intrinsic Functions .. */ 00289 /* .. */ 00290 /* .. Executable Statements .. */ 00291 00292 /* Decode and test the input parameters */ 00293 00294 /* Parameter adjustments */ 00295 --select; 00296 a_dim1 = *lda; 00297 a_offset = 1 + a_dim1; 00298 a -= a_offset; 00299 b_dim1 = *ldb; 00300 b_offset = 1 + b_dim1; 00301 b -= b_offset; 00302 vl_dim1 = *ldvl; 00303 vl_offset = 1 + vl_dim1; 00304 vl -= vl_offset; 00305 vr_dim1 = *ldvr; 00306 vr_offset = 1 + vr_dim1; 00307 vr -= vr_offset; 00308 --s; 00309 --dif; 00310 --work; 00311 --iwork; 00312 00313 /* Function Body */ 00314 wantbh = lsame_(job, "B"); 00315 wants = lsame_(job, "E") || wantbh; 00316 wantdf = lsame_(job, "V") || wantbh; 00317 00318 somcon = lsame_(howmny, "S"); 00319 00320 *info = 0; 00321 lquery = *lwork == -1; 00322 00323 if (! wants && ! wantdf) { 00324 *info = -1; 00325 } else if (! lsame_(howmny, "A") && ! somcon) { 00326 *info = -2; 00327 } else if (*n < 0) { 00328 *info = -4; 00329 } else if (*lda < max(1,*n)) { 00330 *info = -6; 00331 } else if (*ldb < max(1,*n)) { 00332 *info = -8; 00333 } else if (wants && *ldvl < *n) { 00334 *info = -10; 00335 } else if (wants && *ldvr < *n) { 00336 *info = -12; 00337 } else { 00338 00339 /* Set M to the number of eigenpairs for which condition numbers */ 00340 /* are required, and test MM. */ 00341 00342 if (somcon) { 00343 *m = 0; 00344 i__1 = *n; 00345 for (k = 1; k <= i__1; ++k) { 00346 if (select[k]) { 00347 ++(*m); 00348 } 00349 /* L10: */ 00350 } 00351 } else { 00352 *m = *n; 00353 } 00354 00355 if (*n == 0) { 00356 lwmin = 1; 00357 } else if (lsame_(job, "V") || lsame_(job, 00358 "B")) { 00359 lwmin = (*n << 1) * *n; 00360 } else { 00361 lwmin = *n; 00362 } 00363 work[1].r = (doublereal) lwmin, work[1].i = 0.; 00364 00365 if (*mm < *m) { 00366 *info = -15; 00367 } else if (*lwork < lwmin && ! lquery) { 00368 *info = -18; 00369 } 00370 } 00371 00372 if (*info != 0) { 00373 i__1 = -(*info); 00374 xerbla_("ZTGSNA", &i__1); 00375 return 0; 00376 } else if (lquery) { 00377 return 0; 00378 } 00379 00380 /* Quick return if possible */ 00381 00382 if (*n == 0) { 00383 return 0; 00384 } 00385 00386 /* Get machine constants */ 00387 00388 eps = dlamch_("P"); 00389 smlnum = dlamch_("S") / eps; 00390 bignum = 1. / smlnum; 00391 dlabad_(&smlnum, &bignum); 00392 ks = 0; 00393 i__1 = *n; 00394 for (k = 1; k <= i__1; ++k) { 00395 00396 /* Determine whether condition numbers are required for the k-th */ 00397 /* eigenpair. */ 00398 00399 if (somcon) { 00400 if (! select[k]) { 00401 goto L20; 00402 } 00403 } 00404 00405 ++ks; 00406 00407 if (wants) { 00408 00409 /* Compute the reciprocal condition number of the k-th */ 00410 /* eigenvalue. */ 00411 00412 rnrm = dznrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); 00413 lnrm = dznrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); 00414 zgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 + 1] 00415 , &c__1, &c_b20, &work[1], &c__1); 00416 zdotc_(&z__1, n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1); 00417 yhax.r = z__1.r, yhax.i = z__1.i; 00418 zgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 + 1] 00419 , &c__1, &c_b20, &work[1], &c__1); 00420 zdotc_(&z__1, n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1); 00421 yhbx.r = z__1.r, yhbx.i = z__1.i; 00422 d__1 = z_abs(&yhax); 00423 d__2 = z_abs(&yhbx); 00424 cond = dlapy2_(&d__1, &d__2); 00425 if (cond == 0.) { 00426 s[ks] = -1.; 00427 } else { 00428 s[ks] = cond / (rnrm * lnrm); 00429 } 00430 } 00431 00432 if (wantdf) { 00433 if (*n == 1) { 00434 d__1 = z_abs(&a[a_dim1 + 1]); 00435 d__2 = z_abs(&b[b_dim1 + 1]); 00436 dif[ks] = dlapy2_(&d__1, &d__2); 00437 } else { 00438 00439 /* Estimate the reciprocal condition number of the k-th */ 00440 /* eigenvectors. */ 00441 00442 /* Copy the matrix (A, B) to the array WORK and move the */ 00443 /* (k,k)th pair to the (1,1) position. */ 00444 00445 zlacpy_("Full", n, n, &a[a_offset], lda, &work[1], n); 00446 zlacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1], 00447 n); 00448 ifst = k; 00449 ilst = 1; 00450 00451 ztgexc_(&c_false, &c_false, n, &work[1], n, &work[*n * *n + 1] 00452 , n, dummy, &c__1, dummy1, &c__1, &ifst, &ilst, &ierr) 00453 ; 00454 00455 if (ierr > 0) { 00456 00457 /* Ill-conditioned problem - swap rejected. */ 00458 00459 dif[ks] = 0.; 00460 } else { 00461 00462 /* Reordering successful, solve generalized Sylvester */ 00463 /* equation for R and L, */ 00464 /* A22 * R - L * A11 = A12 */ 00465 /* B22 * R - L * B11 = B12, */ 00466 /* and compute estimate of Difl[(A11,B11), (A22, B22)]. */ 00467 00468 n1 = 1; 00469 n2 = *n - n1; 00470 i__ = *n * *n + 1; 00471 ztgsyl_("N", &c__3, &n2, &n1, &work[*n * n1 + n1 + 1], n, 00472 &work[1], n, &work[n1 + 1], n, &work[*n * n1 + n1 00473 + i__], n, &work[i__], n, &work[n1 + i__], n, & 00474 scale, &dif[ks], dummy, &c__1, &iwork[1], &ierr); 00475 } 00476 } 00477 } 00478 00479 L20: 00480 ; 00481 } 00482 work[1].r = (doublereal) lwmin, work[1].i = 0.; 00483 return 0; 00484 00485 /* End of ZTGSNA */ 00486 00487 } /* ztgsna_ */