1: SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
2: $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
3: $ IWORK, INFO )
4: *
5: * -- LAPACK routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: CHARACTER HOWMNY, JOB
12: INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
13: * ..
14: * .. Array Arguments ..
15: LOGICAL SELECT( * )
16: INTEGER IWORK( * )
17: DOUBLE PRECISION DIF( * ), S( * )
18: COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
19: $ VR( LDVR, * ), WORK( * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * ZTGSNA estimates reciprocal condition numbers for specified
26: * eigenvalues and/or eigenvectors of a matrix pair (A, B).
27: *
28: * (A, B) must be in generalized Schur canonical form, that is, A and
29: * B are both upper triangular.
30: *
31: * Arguments
32: * =========
33: *
34: * JOB (input) CHARACTER*1
35: * Specifies whether condition numbers are required for
36: * eigenvalues (S) or eigenvectors (DIF):
37: * = 'E': for eigenvalues only (S);
38: * = 'V': for eigenvectors only (DIF);
39: * = 'B': for both eigenvalues and eigenvectors (S and DIF).
40: *
41: * HOWMNY (input) CHARACTER*1
42: * = 'A': compute condition numbers for all eigenpairs;
43: * = 'S': compute condition numbers for selected eigenpairs
44: * specified by the array SELECT.
45: *
46: * SELECT (input) LOGICAL array, dimension (N)
47: * If HOWMNY = 'S', SELECT specifies the eigenpairs for which
48: * condition numbers are required. To select condition numbers
49: * for the corresponding j-th eigenvalue and/or eigenvector,
50: * SELECT(j) must be set to .TRUE..
51: * If HOWMNY = 'A', SELECT is not referenced.
52: *
53: * N (input) INTEGER
54: * The order of the square matrix pair (A, B). N >= 0.
55: *
56: * A (input) COMPLEX*16 array, dimension (LDA,N)
57: * The upper triangular matrix A in the pair (A,B).
58: *
59: * LDA (input) INTEGER
60: * The leading dimension of the array A. LDA >= max(1,N).
61: *
62: * B (input) COMPLEX*16 array, dimension (LDB,N)
63: * The upper triangular matrix B in the pair (A, B).
64: *
65: * LDB (input) INTEGER
66: * The leading dimension of the array B. LDB >= max(1,N).
67: *
68: * VL (input) COMPLEX*16 array, dimension (LDVL,M)
69: * IF JOB = 'E' or 'B', VL must contain left eigenvectors of
70: * (A, B), corresponding to the eigenpairs specified by HOWMNY
71: * and SELECT. The eigenvectors must be stored in consecutive
72: * columns of VL, as returned by ZTGEVC.
73: * If JOB = 'V', VL is not referenced.
74: *
75: * LDVL (input) INTEGER
76: * The leading dimension of the array VL. LDVL >= 1; and
77: * If JOB = 'E' or 'B', LDVL >= N.
78: *
79: * VR (input) COMPLEX*16 array, dimension (LDVR,M)
80: * IF JOB = 'E' or 'B', VR must contain right eigenvectors of
81: * (A, B), corresponding to the eigenpairs specified by HOWMNY
82: * and SELECT. The eigenvectors must be stored in consecutive
83: * columns of VR, as returned by ZTGEVC.
84: * If JOB = 'V', VR is not referenced.
85: *
86: * LDVR (input) INTEGER
87: * The leading dimension of the array VR. LDVR >= 1;
88: * If JOB = 'E' or 'B', LDVR >= N.
89: *
90: * S (output) DOUBLE PRECISION array, dimension (MM)
91: * If JOB = 'E' or 'B', the reciprocal condition numbers of the
92: * selected eigenvalues, stored in consecutive elements of the
93: * array.
94: * If JOB = 'V', S is not referenced.
95: *
96: * DIF (output) DOUBLE PRECISION array, dimension (MM)
97: * If JOB = 'V' or 'B', the estimated reciprocal condition
98: * numbers of the selected eigenvectors, stored in consecutive
99: * elements of the array.
100: * If the eigenvalues cannot be reordered to compute DIF(j),
101: * DIF(j) is set to 0; this can only occur when the true value
102: * would be very small anyway.
103: * For each eigenvalue/vector specified by SELECT, DIF stores
104: * a Frobenius norm-based estimate of Difl.
105: * If JOB = 'E', DIF is not referenced.
106: *
107: * MM (input) INTEGER
108: * The number of elements in the arrays S and DIF. MM >= M.
109: *
110: * M (output) INTEGER
111: * The number of elements of the arrays S and DIF used to store
112: * the specified condition numbers; for each selected eigenvalue
113: * one element is used. If HOWMNY = 'A', M is set to N.
114: *
115: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
116: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
117: *
118: * LWORK (input) INTEGER
119: * The dimension of the array WORK. LWORK >= max(1,N).
120: * If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
121: *
122: * IWORK (workspace) INTEGER array, dimension (N+2)
123: * If JOB = 'E', IWORK is not referenced.
124: *
125: * INFO (output) INTEGER
126: * = 0: Successful exit
127: * < 0: If INFO = -i, the i-th argument had an illegal value
128: *
129: * Further Details
130: * ===============
131: *
132: * The reciprocal of the condition number of the i-th generalized
133: * eigenvalue w = (a, b) is defined as
134: *
135: * S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))
136: *
137: * where u and v are the right and left eigenvectors of (A, B)
138: * corresponding to w; |z| denotes the absolute value of the complex
139: * number, and norm(u) denotes the 2-norm of the vector u. The pair
140: * (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the
141: * matrix pair (A, B). If both a and b equal zero, then (A,B) is
142: * singular and S(I) = -1 is returned.
143: *
144: * An approximate error bound on the chordal distance between the i-th
145: * computed generalized eigenvalue w and the corresponding exact
146: * eigenvalue lambda is
147: *
148: * chord(w, lambda) <= EPS * norm(A, B) / S(I),
149: *
150: * where EPS is the machine precision.
151: *
152: * The reciprocal of the condition number of the right eigenvector u
153: * and left eigenvector v corresponding to the generalized eigenvalue w
154: * is defined as follows. Suppose
155: *
156: * (A, B) = ( a * ) ( b * ) 1
157: * ( 0 A22 ),( 0 B22 ) n-1
158: * 1 n-1 1 n-1
159: *
160: * Then the reciprocal condition number DIF(I) is
161: *
162: * Difl[(a, b), (A22, B22)] = sigma-min( Zl )
163: *
164: * where sigma-min(Zl) denotes the smallest singular value of
165: *
166: * Zl = [ kron(a, In-1) -kron(1, A22) ]
167: * [ kron(b, In-1) -kron(1, B22) ].
168: *
169: * Here In-1 is the identity matrix of size n-1 and X' is the conjugate
170: * transpose of X. kron(X, Y) is the Kronecker product between the
171: * matrices X and Y.
172: *
173: * We approximate the smallest singular value of Zl with an upper
174: * bound. This is done by ZLATDF.
175: *
176: * An approximate error bound for a computed eigenvector VL(i) or
177: * VR(i) is given by
178: *
179: * EPS * norm(A, B) / DIF(i).
180: *
181: * See ref. [2-3] for more details and further references.
182: *
183: * Based on contributions by
184: * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
185: * Umea University, S-901 87 Umea, Sweden.
186: *
187: * References
188: * ==========
189: *
190: * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
191: * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
192: * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
193: * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
194: *
195: * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
196: * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
197: * Estimation: Theory, Algorithms and Software, Report
198: * UMINF - 94.04, Department of Computing Science, Umea University,
199: * S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
200: * To appear in Numerical Algorithms, 1996.
201: *
202: * [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
203: * for Solving the Generalized Sylvester Equation and Estimating the
204: * Separation between Regular Matrix Pairs, Report UMINF - 93.23,
205: * Department of Computing Science, Umea University, S-901 87 Umea,
206: * Sweden, December 1993, Revised April 1994, Also as LAPACK Working
207: * Note 75.
208: * To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
209: *
210: * =====================================================================
211: *
212: * .. Parameters ..
213: DOUBLE PRECISION ZERO, ONE
214: INTEGER IDIFJB
215: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, IDIFJB = 3 )
216: * ..
217: * .. Local Scalars ..
218: LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS
219: INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
220: DOUBLE PRECISION BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
221: COMPLEX*16 YHAX, YHBX
222: * ..
223: * .. Local Arrays ..
224: COMPLEX*16 DUMMY( 1 ), DUMMY1( 1 )
225: * ..
226: * .. External Functions ..
227: LOGICAL LSAME
228: DOUBLE PRECISION DLAMCH, DLAPY2, DZNRM2
229: COMPLEX*16 ZDOTC
230: EXTERNAL LSAME, DLAMCH, DLAPY2, DZNRM2, ZDOTC
231: * ..
232: * .. External Subroutines ..
233: EXTERNAL DLABAD, XERBLA, ZGEMV, ZLACPY, ZTGEXC, ZTGSYL
234: * ..
235: * .. Intrinsic Functions ..
236: INTRINSIC ABS, DCMPLX, MAX
237: * ..
238: * .. Executable Statements ..
239: *
240: * Decode and test the input parameters
241: *
242: WANTBH = LSAME( JOB, 'B' )
243: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
244: WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
245: *
246: SOMCON = LSAME( HOWMNY, 'S' )
247: *
248: INFO = 0
249: LQUERY = ( LWORK.EQ.-1 )
250: *
251: IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
252: INFO = -1
253: ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
254: INFO = -2
255: ELSE IF( N.LT.0 ) THEN
256: INFO = -4
257: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
258: INFO = -6
259: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
260: INFO = -8
261: ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
262: INFO = -10
263: ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
264: INFO = -12
265: ELSE
266: *
267: * Set M to the number of eigenpairs for which condition numbers
268: * are required, and test MM.
269: *
270: IF( SOMCON ) THEN
271: M = 0
272: DO 10 K = 1, N
273: IF( SELECT( K ) )
274: $ M = M + 1
275: 10 CONTINUE
276: ELSE
277: M = N
278: END IF
279: *
280: IF( N.EQ.0 ) THEN
281: LWMIN = 1
282: ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
283: LWMIN = 2*N*N
284: ELSE
285: LWMIN = N
286: END IF
287: WORK( 1 ) = LWMIN
288: *
289: IF( MM.LT.M ) THEN
290: INFO = -15
291: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
292: INFO = -18
293: END IF
294: END IF
295: *
296: IF( INFO.NE.0 ) THEN
297: CALL XERBLA( 'ZTGSNA', -INFO )
298: RETURN
299: ELSE IF( LQUERY ) THEN
300: RETURN
301: END IF
302: *
303: * Quick return if possible
304: *
305: IF( N.EQ.0 )
306: $ RETURN
307: *
308: * Get machine constants
309: *
310: EPS = DLAMCH( 'P' )
311: SMLNUM = DLAMCH( 'S' ) / EPS
312: BIGNUM = ONE / SMLNUM
313: CALL DLABAD( SMLNUM, BIGNUM )
314: KS = 0
315: DO 20 K = 1, N
316: *
317: * Determine whether condition numbers are required for the k-th
318: * eigenpair.
319: *
320: IF( SOMCON ) THEN
321: IF( .NOT.SELECT( K ) )
322: $ GO TO 20
323: END IF
324: *
325: KS = KS + 1
326: *
327: IF( WANTS ) THEN
328: *
329: * Compute the reciprocal condition number of the k-th
330: * eigenvalue.
331: *
332: RNRM = DZNRM2( N, VR( 1, KS ), 1 )
333: LNRM = DZNRM2( N, VL( 1, KS ), 1 )
334: CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), A, LDA,
335: $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
336: YHAX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
337: CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), B, LDB,
338: $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
339: YHBX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
340: COND = DLAPY2( ABS( YHAX ), ABS( YHBX ) )
341: IF( COND.EQ.ZERO ) THEN
342: S( KS ) = -ONE
343: ELSE
344: S( KS ) = COND / ( RNRM*LNRM )
345: END IF
346: END IF
347: *
348: IF( WANTDF ) THEN
349: IF( N.EQ.1 ) THEN
350: DIF( KS ) = DLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) )
351: ELSE
352: *
353: * Estimate the reciprocal condition number of the k-th
354: * eigenvectors.
355: *
356: * Copy the matrix (A, B) to the array WORK and move the
357: * (k,k)th pair to the (1,1) position.
358: *
359: CALL ZLACPY( 'Full', N, N, A, LDA, WORK, N )
360: CALL ZLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
361: IFST = K
362: ILST = 1
363: *
364: CALL ZTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ),
365: $ N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR )
366: *
367: IF( IERR.GT.0 ) THEN
368: *
369: * Ill-conditioned problem - swap rejected.
370: *
371: DIF( KS ) = ZERO
372: ELSE
373: *
374: * Reordering successful, solve generalized Sylvester
375: * equation for R and L,
376: * A22 * R - L * A11 = A12
377: * B22 * R - L * B11 = B12,
378: * and compute estimate of Difl[(A11,B11), (A22, B22)].
379: *
380: N1 = 1
381: N2 = N - N1
382: I = N*N + 1
383: CALL ZTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ),
384: $ N, WORK, N, WORK( N1+1 ), N,
385: $ WORK( N*N1+N1+I ), N, WORK( I ), N,
386: $ WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY,
387: $ 1, IWORK, IERR )
388: END IF
389: END IF
390: END IF
391: *
392: 20 CONTINUE
393: WORK( 1 ) = LWMIN
394: RETURN
395: *
396: * End of ZTGSNA
397: *
398: END
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