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
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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 December 2016
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: * =====================================================================
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: *
384: * -- LAPACK computational routine (version 3.7.0) --
385: * -- LAPACK is a software package provided by Univ. of Tennessee, --
386: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
387: * December 2016
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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