Annotation of rpl/lapack/lapack/dtgsna.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DTGSNA
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DTGSNA + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsna.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsna.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsna.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
! 22: * LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
! 23: * IWORK, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER HOWMNY, JOB
! 27: * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * LOGICAL SELECT( * )
! 31: * INTEGER IWORK( * )
! 32: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
! 33: * $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> DTGSNA estimates reciprocal condition numbers for specified
! 43: *> eigenvalues and/or eigenvectors of a matrix pair (A, B) in
! 44: *> generalized real Schur canonical form (or of any matrix pair
! 45: *> (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
! 46: *> Z**T denotes the transpose of Z.
! 47: *>
! 48: *> (A, B) must be in generalized real Schur form (as returned by DGGES),
! 49: *> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
! 50: *> blocks. B is upper triangular.
! 51: *>
! 52: *> \endverbatim
! 53: *
! 54: * Arguments:
! 55: * ==========
! 56: *
! 57: *> \param[in] JOB
! 58: *> \verbatim
! 59: *> JOB is CHARACTER*1
! 60: *> Specifies whether condition numbers are required for
! 61: *> eigenvalues (S) or eigenvectors (DIF):
! 62: *> = 'E': for eigenvalues only (S);
! 63: *> = 'V': for eigenvectors only (DIF);
! 64: *> = 'B': for both eigenvalues and eigenvectors (S and DIF).
! 65: *> \endverbatim
! 66: *>
! 67: *> \param[in] HOWMNY
! 68: *> \verbatim
! 69: *> HOWMNY is CHARACTER*1
! 70: *> = 'A': compute condition numbers for all eigenpairs;
! 71: *> = 'S': compute condition numbers for selected eigenpairs
! 72: *> specified by the array SELECT.
! 73: *> \endverbatim
! 74: *>
! 75: *> \param[in] SELECT
! 76: *> \verbatim
! 77: *> SELECT is LOGICAL array, dimension (N)
! 78: *> If HOWMNY = 'S', SELECT specifies the eigenpairs for which
! 79: *> condition numbers are required. To select condition numbers
! 80: *> for the eigenpair corresponding to a real eigenvalue w(j),
! 81: *> SELECT(j) must be set to .TRUE.. To select condition numbers
! 82: *> corresponding to a complex conjugate pair of eigenvalues w(j)
! 83: *> and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
! 84: *> set to .TRUE..
! 85: *> If HOWMNY = 'A', SELECT is not referenced.
! 86: *> \endverbatim
! 87: *>
! 88: *> \param[in] N
! 89: *> \verbatim
! 90: *> N is INTEGER
! 91: *> The order of the square matrix pair (A, B). N >= 0.
! 92: *> \endverbatim
! 93: *>
! 94: *> \param[in] A
! 95: *> \verbatim
! 96: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 97: *> The upper quasi-triangular matrix A in the pair (A,B).
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[in] LDA
! 101: *> \verbatim
! 102: *> LDA is INTEGER
! 103: *> The leading dimension of the array A. LDA >= max(1,N).
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[in] B
! 107: *> \verbatim
! 108: *> B is DOUBLE PRECISION array, dimension (LDB,N)
! 109: *> The upper triangular matrix B in the pair (A,B).
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] LDB
! 113: *> \verbatim
! 114: *> LDB is INTEGER
! 115: *> The leading dimension of the array B. LDB >= max(1,N).
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[in] VL
! 119: *> \verbatim
! 120: *> VL is DOUBLE PRECISION array, dimension (LDVL,M)
! 121: *> If JOB = 'E' or 'B', VL must contain left eigenvectors of
! 122: *> (A, B), corresponding to the eigenpairs specified by HOWMNY
! 123: *> and SELECT. The eigenvectors must be stored in consecutive
! 124: *> columns of VL, as returned by DTGEVC.
! 125: *> If JOB = 'V', VL is not referenced.
! 126: *> \endverbatim
! 127: *>
! 128: *> \param[in] LDVL
! 129: *> \verbatim
! 130: *> LDVL is INTEGER
! 131: *> The leading dimension of the array VL. LDVL >= 1.
! 132: *> If JOB = 'E' or 'B', LDVL >= N.
! 133: *> \endverbatim
! 134: *>
! 135: *> \param[in] VR
! 136: *> \verbatim
! 137: *> VR is DOUBLE PRECISION array, dimension (LDVR,M)
! 138: *> If JOB = 'E' or 'B', VR must contain right eigenvectors of
! 139: *> (A, B), corresponding to the eigenpairs specified by HOWMNY
! 140: *> and SELECT. The eigenvectors must be stored in consecutive
! 141: *> columns ov VR, as returned by DTGEVC.
! 142: *> If JOB = 'V', VR is not referenced.
! 143: *> \endverbatim
! 144: *>
! 145: *> \param[in] LDVR
! 146: *> \verbatim
! 147: *> LDVR is INTEGER
! 148: *> The leading dimension of the array VR. LDVR >= 1.
! 149: *> If JOB = 'E' or 'B', LDVR >= N.
! 150: *> \endverbatim
! 151: *>
! 152: *> \param[out] S
! 153: *> \verbatim
! 154: *> S is DOUBLE PRECISION array, dimension (MM)
! 155: *> If JOB = 'E' or 'B', the reciprocal condition numbers of the
! 156: *> selected eigenvalues, stored in consecutive elements of the
! 157: *> array. For a complex conjugate pair of eigenvalues two
! 158: *> consecutive elements of S are set to the same value. Thus
! 159: *> S(j), DIF(j), and the j-th columns of VL and VR all
! 160: *> correspond to the same eigenpair (but not in general the
! 161: *> j-th eigenpair, unless all eigenpairs are selected).
! 162: *> If JOB = 'V', S is not referenced.
! 163: *> \endverbatim
! 164: *>
! 165: *> \param[out] DIF
! 166: *> \verbatim
! 167: *> DIF is DOUBLE PRECISION array, dimension (MM)
! 168: *> If JOB = 'V' or 'B', the estimated reciprocal condition
! 169: *> numbers of the selected eigenvectors, stored in consecutive
! 170: *> elements of the array. For a complex eigenvector two
! 171: *> consecutive elements of DIF are set to the same value. If
! 172: *> the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
! 173: *> is set to 0; this can only occur when the true value would be
! 174: *> very small anyway.
! 175: *> If JOB = 'E', DIF is not referenced.
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in] MM
! 179: *> \verbatim
! 180: *> MM is INTEGER
! 181: *> The number of elements in the arrays S and DIF. MM >= M.
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[out] M
! 185: *> \verbatim
! 186: *> M is INTEGER
! 187: *> The number of elements of the arrays S and DIF used to store
! 188: *> the specified condition numbers; for each selected real
! 189: *> eigenvalue one element is used, and for each selected complex
! 190: *> conjugate pair of eigenvalues, two elements are used.
! 191: *> If HOWMNY = 'A', M is set to N.
! 192: *> \endverbatim
! 193: *>
! 194: *> \param[out] WORK
! 195: *> \verbatim
! 196: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 197: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 198: *> \endverbatim
! 199: *>
! 200: *> \param[in] LWORK
! 201: *> \verbatim
! 202: *> LWORK is INTEGER
! 203: *> The dimension of the array WORK. LWORK >= max(1,N).
! 204: *> If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
! 205: *>
! 206: *> If LWORK = -1, then a workspace query is assumed; the routine
! 207: *> only calculates the optimal size of the WORK array, returns
! 208: *> this value as the first entry of the WORK array, and no error
! 209: *> message related to LWORK is issued by XERBLA.
! 210: *> \endverbatim
! 211: *>
! 212: *> \param[out] IWORK
! 213: *> \verbatim
! 214: *> IWORK is INTEGER array, dimension (N + 6)
! 215: *> If JOB = 'E', IWORK is not referenced.
! 216: *> \endverbatim
! 217: *>
! 218: *> \param[out] INFO
! 219: *> \verbatim
! 220: *> INFO is INTEGER
! 221: *> =0: Successful exit
! 222: *> <0: If INFO = -i, the i-th argument had an illegal value
! 223: *> \endverbatim
! 224: *
! 225: * Authors:
! 226: * ========
! 227: *
! 228: *> \author Univ. of Tennessee
! 229: *> \author Univ. of California Berkeley
! 230: *> \author Univ. of Colorado Denver
! 231: *> \author NAG Ltd.
! 232: *
! 233: *> \date November 2011
! 234: *
! 235: *> \ingroup doubleOTHERcomputational
! 236: *
! 237: *> \par Further Details:
! 238: * =====================
! 239: *>
! 240: *> \verbatim
! 241: *>
! 242: *> The reciprocal of the condition number of a generalized eigenvalue
! 243: *> w = (a, b) is defined as
! 244: *>
! 245: *> S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
! 246: *>
! 247: *> where u and v are the left and right eigenvectors of (A, B)
! 248: *> corresponding to w; |z| denotes the absolute value of the complex
! 249: *> number, and norm(u) denotes the 2-norm of the vector u.
! 250: *> The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
! 251: *> of the matrix pair (A, B). If both a and b equal zero, then (A B) is
! 252: *> singular and S(I) = -1 is returned.
! 253: *>
! 254: *> An approximate error bound on the chordal distance between the i-th
! 255: *> computed generalized eigenvalue w and the corresponding exact
! 256: *> eigenvalue lambda is
! 257: *>
! 258: *> chord(w, lambda) <= EPS * norm(A, B) / S(I)
! 259: *>
! 260: *> where EPS is the machine precision.
! 261: *>
! 262: *> The reciprocal of the condition number DIF(i) of right eigenvector u
! 263: *> and left eigenvector v corresponding to the generalized eigenvalue w
! 264: *> is defined as follows:
! 265: *>
! 266: *> a) If the i-th eigenvalue w = (a,b) is real
! 267: *>
! 268: *> Suppose U and V are orthogonal transformations such that
! 269: *>
! 270: *> U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
! 271: *> ( 0 S22 ),( 0 T22 ) n-1
! 272: *> 1 n-1 1 n-1
! 273: *>
! 274: *> Then the reciprocal condition number DIF(i) is
! 275: *>
! 276: *> Difl((a, b), (S22, T22)) = sigma-min( Zl ),
! 277: *>
! 278: *> where sigma-min(Zl) denotes the smallest singular value of the
! 279: *> 2(n-1)-by-2(n-1) matrix
! 280: *>
! 281: *> Zl = [ kron(a, In-1) -kron(1, S22) ]
! 282: *> [ kron(b, In-1) -kron(1, T22) ] .
! 283: *>
! 284: *> Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
! 285: *> Kronecker product between the matrices X and Y.
! 286: *>
! 287: *> Note that if the default method for computing DIF(i) is wanted
! 288: *> (see DLATDF), then the parameter DIFDRI (see below) should be
! 289: *> changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
! 290: *> See DTGSYL for more details.
! 291: *>
! 292: *> b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
! 293: *>
! 294: *> Suppose U and V are orthogonal transformations such that
! 295: *>
! 296: *> U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
! 297: *> ( 0 S22 ),( 0 T22) n-2
! 298: *> 2 n-2 2 n-2
! 299: *>
! 300: *> and (S11, T11) corresponds to the complex conjugate eigenvalue
! 301: *> pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
! 302: *> that
! 303: *>
! 304: *> U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
! 305: *> ( 0 s22 ) ( 0 t22 )
! 306: *>
! 307: *> where the generalized eigenvalues w = s11/t11 and
! 308: *> conjg(w) = s22/t22.
! 309: *>
! 310: *> Then the reciprocal condition number DIF(i) is bounded by
! 311: *>
! 312: *> min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
! 313: *>
! 314: *> where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
! 315: *> Z1 is the complex 2-by-2 matrix
! 316: *>
! 317: *> Z1 = [ s11 -s22 ]
! 318: *> [ t11 -t22 ],
! 319: *>
! 320: *> This is done by computing (using real arithmetic) the
! 321: *> roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
! 322: *> where Z1**T denotes the transpose of Z1 and det(X) denotes
! 323: *> the determinant of X.
! 324: *>
! 325: *> and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
! 326: *> upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
! 327: *>
! 328: *> Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ]
! 329: *> [ kron(T11**T, In-2) -kron(I2, T22) ]
! 330: *>
! 331: *> Note that if the default method for computing DIF is wanted (see
! 332: *> DLATDF), then the parameter DIFDRI (see below) should be changed
! 333: *> from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
! 334: *> for more details.
! 335: *>
! 336: *> For each eigenvalue/vector specified by SELECT, DIF stores a
! 337: *> Frobenius norm-based estimate of Difl.
! 338: *>
! 339: *> An approximate error bound for the i-th computed eigenvector VL(i) or
! 340: *> VR(i) is given by
! 341: *>
! 342: *> EPS * norm(A, B) / DIF(i).
! 343: *>
! 344: *> See ref. [2-3] for more details and further references.
! 345: *> \endverbatim
! 346: *
! 347: *> \par Contributors:
! 348: * ==================
! 349: *>
! 350: *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
! 351: *> Umea University, S-901 87 Umea, Sweden.
! 352: *
! 353: *> \par References:
! 354: * ================
! 355: *>
! 356: *> \verbatim
! 357: *>
! 358: *> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
! 359: *> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
! 360: *> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
! 361: *> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
! 362: *>
! 363: *> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
! 364: *> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
! 365: *> Estimation: Theory, Algorithms and Software,
! 366: *> Report UMINF - 94.04, Department of Computing Science, Umea
! 367: *> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
! 368: *> Note 87. To appear in Numerical Algorithms, 1996.
! 369: *>
! 370: *> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
! 371: *> for Solving the Generalized Sylvester Equation and Estimating the
! 372: *> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
! 373: *> Department of Computing Science, Umea University, S-901 87 Umea,
! 374: *> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
! 375: *> Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
! 376: *> No 1, 1996.
! 377: *> \endverbatim
! 378: *>
! 379: * =====================================================================
1.1 bertrand 380: SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
381: $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
382: $ IWORK, INFO )
383: *
1.9 ! bertrand 384: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 385: * -- LAPACK is a software package provided by Univ. of Tennessee, --
386: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 387: * November 2011
1.1 bertrand 388: *
389: * .. Scalar Arguments ..
390: CHARACTER HOWMNY, JOB
391: INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
392: * ..
393: * .. Array Arguments ..
394: LOGICAL SELECT( * )
395: INTEGER IWORK( * )
396: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
397: $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
398: * ..
399: *
400: * =====================================================================
401: *
402: * .. Parameters ..
403: INTEGER DIFDRI
404: PARAMETER ( DIFDRI = 3 )
405: DOUBLE PRECISION ZERO, ONE, TWO, FOUR
406: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
407: $ FOUR = 4.0D+0 )
408: * ..
409: * .. Local Scalars ..
410: LOGICAL LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
411: INTEGER I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2
412: DOUBLE PRECISION ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND,
413: $ EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM,
414: $ TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV,
415: $ UHBVI
416: * ..
417: * .. Local Arrays ..
418: DOUBLE PRECISION DUMMY( 1 ), DUMMY1( 1 )
419: * ..
420: * .. External Functions ..
421: LOGICAL LSAME
422: DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2
423: EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
424: * ..
425: * .. External Subroutines ..
426: EXTERNAL DGEMV, DLACPY, DLAG2, DTGEXC, DTGSYL, XERBLA
427: * ..
428: * .. Intrinsic Functions ..
429: INTRINSIC MAX, MIN, SQRT
430: * ..
431: * .. Executable Statements ..
432: *
433: * Decode and test the input parameters
434: *
435: WANTBH = LSAME( JOB, 'B' )
436: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
437: WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
438: *
439: SOMCON = LSAME( HOWMNY, 'S' )
440: *
441: INFO = 0
442: LQUERY = ( LWORK.EQ.-1 )
443: *
444: IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
445: INFO = -1
446: ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
447: INFO = -2
448: ELSE IF( N.LT.0 ) THEN
449: INFO = -4
450: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
451: INFO = -6
452: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
453: INFO = -8
454: ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
455: INFO = -10
456: ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
457: INFO = -12
458: ELSE
459: *
460: * Set M to the number of eigenpairs for which condition numbers
461: * are required, and test MM.
462: *
463: IF( SOMCON ) THEN
464: M = 0
465: PAIR = .FALSE.
466: DO 10 K = 1, N
467: IF( PAIR ) THEN
468: PAIR = .FALSE.
469: ELSE
470: IF( K.LT.N ) THEN
471: IF( A( K+1, K ).EQ.ZERO ) THEN
472: IF( SELECT( K ) )
473: $ M = M + 1
474: ELSE
475: PAIR = .TRUE.
476: IF( SELECT( K ) .OR. SELECT( K+1 ) )
477: $ M = M + 2
478: END IF
479: ELSE
480: IF( SELECT( N ) )
481: $ M = M + 1
482: END IF
483: END IF
484: 10 CONTINUE
485: ELSE
486: M = N
487: END IF
488: *
489: IF( N.EQ.0 ) THEN
490: LWMIN = 1
491: ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
492: LWMIN = 2*N*( N + 2 ) + 16
493: ELSE
494: LWMIN = N
495: END IF
496: WORK( 1 ) = LWMIN
497: *
498: IF( MM.LT.M ) THEN
499: INFO = -15
500: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
501: INFO = -18
502: END IF
503: END IF
504: *
505: IF( INFO.NE.0 ) THEN
506: CALL XERBLA( 'DTGSNA', -INFO )
507: RETURN
508: ELSE IF( LQUERY ) THEN
509: RETURN
510: END IF
511: *
512: * Quick return if possible
513: *
514: IF( N.EQ.0 )
515: $ RETURN
516: *
517: * Get machine constants
518: *
519: EPS = DLAMCH( 'P' )
520: SMLNUM = DLAMCH( 'S' ) / EPS
521: KS = 0
522: PAIR = .FALSE.
523: *
524: DO 20 K = 1, N
525: *
526: * Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block.
527: *
528: IF( PAIR ) THEN
529: PAIR = .FALSE.
530: GO TO 20
531: ELSE
532: IF( K.LT.N )
533: $ PAIR = A( K+1, K ).NE.ZERO
534: END IF
535: *
536: * Determine whether condition numbers are required for the k-th
537: * eigenpair.
538: *
539: IF( SOMCON ) THEN
540: IF( PAIR ) THEN
541: IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
542: $ GO TO 20
543: ELSE
544: IF( .NOT.SELECT( K ) )
545: $ GO TO 20
546: END IF
547: END IF
548: *
549: KS = KS + 1
550: *
551: IF( WANTS ) THEN
552: *
553: * Compute the reciprocal condition number of the k-th
554: * eigenvalue.
555: *
556: IF( PAIR ) THEN
557: *
558: * Complex eigenvalue pair.
559: *
560: RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ),
561: $ DNRM2( N, VR( 1, KS+1 ), 1 ) )
562: LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ),
563: $ DNRM2( N, VL( 1, KS+1 ), 1 ) )
564: CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
565: $ WORK, 1 )
566: TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
567: TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
568: CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1,
569: $ ZERO, WORK, 1 )
570: TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
571: TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
572: UHAV = TMPRR + TMPII
573: UHAVI = TMPIR - TMPRI
574: CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
575: $ WORK, 1 )
576: TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
577: TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
578: CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1,
579: $ ZERO, WORK, 1 )
580: TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
581: TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
582: UHBV = TMPRR + TMPII
583: UHBVI = TMPIR - TMPRI
584: UHAV = DLAPY2( UHAV, UHAVI )
585: UHBV = DLAPY2( UHBV, UHBVI )
586: COND = DLAPY2( UHAV, UHBV )
587: S( KS ) = COND / ( RNRM*LNRM )
588: S( KS+1 ) = S( KS )
589: *
590: ELSE
591: *
592: * Real eigenvalue.
593: *
594: RNRM = DNRM2( N, VR( 1, KS ), 1 )
595: LNRM = DNRM2( N, VL( 1, KS ), 1 )
596: CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
597: $ WORK, 1 )
598: UHAV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
599: CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
600: $ WORK, 1 )
601: UHBV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
602: COND = DLAPY2( UHAV, UHBV )
603: IF( COND.EQ.ZERO ) THEN
604: S( KS ) = -ONE
605: ELSE
606: S( KS ) = COND / ( RNRM*LNRM )
607: END IF
608: END IF
609: END IF
610: *
611: IF( WANTDF ) THEN
612: IF( N.EQ.1 ) THEN
613: DIF( KS ) = DLAPY2( A( 1, 1 ), B( 1, 1 ) )
614: GO TO 20
615: END IF
616: *
617: * Estimate the reciprocal condition number of the k-th
618: * eigenvectors.
619: IF( PAIR ) THEN
620: *
621: * Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)).
622: * Compute the eigenvalue(s) at position K.
623: *
624: WORK( 1 ) = A( K, K )
625: WORK( 2 ) = A( K+1, K )
626: WORK( 3 ) = A( K, K+1 )
627: WORK( 4 ) = A( K+1, K+1 )
628: WORK( 5 ) = B( K, K )
629: WORK( 6 ) = B( K+1, K )
630: WORK( 7 ) = B( K, K+1 )
631: WORK( 8 ) = B( K+1, K+1 )
632: CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA,
633: $ DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI )
634: ALPRQT = ONE
635: C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA )
636: C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI
637: ROOT1 = C1 + SQRT( C1*C1-4.0D0*C2 )
638: ROOT2 = C2 / ROOT1
639: ROOT1 = ROOT1 / TWO
640: COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) )
641: END IF
642: *
643: * Copy the matrix (A, B) to the array WORK and swap the
644: * diagonal block beginning at A(k,k) to the (1,1) position.
645: *
646: CALL DLACPY( 'Full', N, N, A, LDA, WORK, N )
647: CALL DLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
648: IFST = K
649: ILST = 1
650: *
651: CALL DTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), N,
652: $ DUMMY, 1, DUMMY1, 1, IFST, ILST,
653: $ WORK( N*N*2+1 ), LWORK-2*N*N, IERR )
654: *
655: IF( IERR.GT.0 ) THEN
656: *
657: * Ill-conditioned problem - swap rejected.
658: *
659: DIF( KS ) = ZERO
660: ELSE
661: *
662: * Reordering successful, solve generalized Sylvester
663: * equation for R and L,
664: * A22 * R - L * A11 = A12
665: * B22 * R - L * B11 = B12,
666: * and compute estimate of Difl((A11,B11), (A22, B22)).
667: *
668: N1 = 1
669: IF( WORK( 2 ).NE.ZERO )
670: $ N1 = 2
671: N2 = N - N1
672: IF( N2.EQ.0 ) THEN
673: DIF( KS ) = COND
674: ELSE
675: I = N*N + 1
676: IZ = 2*N*N + 1
677: CALL DTGSYL( 'N', DIFDRI, N2, N1, WORK( N*N1+N1+1 ),
678: $ N, WORK, N, WORK( N1+1 ), N,
679: $ WORK( N*N1+N1+I ), N, WORK( I ), N,
680: $ WORK( N1+I ), N, SCALE, DIF( KS ),
681: $ WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR )
682: *
683: IF( PAIR )
684: $ DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ),
685: $ COND )
686: END IF
687: END IF
688: IF( PAIR )
689: $ DIF( KS+1 ) = DIF( KS )
690: END IF
691: IF( PAIR )
692: $ KS = KS + 1
693: *
694: 20 CONTINUE
695: WORK( 1 ) = LWMIN
696: RETURN
697: *
698: * End of DTGSNA
699: *
700: END
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