1 SUBROUTINE ZGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR,
2 $ WORK, LWORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBVL, JOBVR
11 INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION RWORK( * )
15 COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
16 $ W( * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the
23 * eigenvalues and, optionally, the left and/or right eigenvectors.
24 *
25 * The right eigenvector v(j) of A satisfies
26 * A * v(j) = lambda(j) * v(j)
27 * where lambda(j) is its eigenvalue.
28 * The left eigenvector u(j) of A satisfies
29 * u(j)**H * A = lambda(j) * u(j)**H
30 * where u(j)**H denotes the conjugate transpose of u(j).
31 *
32 * The computed eigenvectors are normalized to have Euclidean norm
33 * equal to 1 and largest component real.
34 *
35 * Arguments
36 * =========
37 *
38 * JOBVL (input) CHARACTER*1
39 * = 'N': left eigenvectors of A are not computed;
40 * = 'V': left eigenvectors of are computed.
41 *
42 * JOBVR (input) CHARACTER*1
43 * = 'N': right eigenvectors of A are not computed;
44 * = 'V': right eigenvectors of A are computed.
45 *
46 * N (input) INTEGER
47 * The order of the matrix A. N >= 0.
48 *
49 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
50 * On entry, the N-by-N matrix A.
51 * On exit, A has been overwritten.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,N).
55 *
56 * W (output) COMPLEX*16 array, dimension (N)
57 * W contains the computed eigenvalues.
58 *
59 * VL (output) COMPLEX*16 array, dimension (LDVL,N)
60 * If JOBVL = 'V', the left eigenvectors u(j) are stored one
61 * after another in the columns of VL, in the same order
62 * as their eigenvalues.
63 * If JOBVL = 'N', VL is not referenced.
64 * u(j) = VL(:,j), the j-th column of VL.
65 *
66 * LDVL (input) INTEGER
67 * The leading dimension of the array VL. LDVL >= 1; if
68 * JOBVL = 'V', LDVL >= N.
69 *
70 * VR (output) COMPLEX*16 array, dimension (LDVR,N)
71 * If JOBVR = 'V', the right eigenvectors v(j) are stored one
72 * after another in the columns of VR, in the same order
73 * as their eigenvalues.
74 * If JOBVR = 'N', VR is not referenced.
75 * v(j) = VR(:,j), the j-th column of VR.
76 *
77 * LDVR (input) INTEGER
78 * The leading dimension of the array VR. LDVR >= 1; if
79 * JOBVR = 'V', LDVR >= N.
80 *
81 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
82 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
83 *
84 * LWORK (input) INTEGER
85 * The dimension of the array WORK. LWORK >= max(1,2*N).
86 * For good performance, LWORK must generally be larger.
87 *
88 * If LWORK = -1, then a workspace query is assumed; the routine
89 * only calculates the optimal size of the WORK array, returns
90 * this value as the first entry of the WORK array, and no error
91 * message related to LWORK is issued by XERBLA.
92 *
93 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
94 *
95 * INFO (output) INTEGER
96 * = 0: successful exit
97 * < 0: if INFO = -i, the i-th argument had an illegal value.
98 * > 0: if INFO = i, the QR algorithm failed to compute all the
99 * eigenvalues, and no eigenvectors have been computed;
100 * elements and i+1:N of W contain eigenvalues which have
101 * converged.
102 *
103 * =====================================================================
104 *
105 * .. Parameters ..
106 DOUBLE PRECISION ZERO, ONE
107 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
108 * ..
109 * .. Local Scalars ..
110 LOGICAL LQUERY, SCALEA, WANTVL, WANTVR
111 CHARACTER SIDE
112 INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU,
113 $ IWRK, K, MAXWRK, MINWRK, NOUT
114 DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
115 COMPLEX*16 TMP
116 * ..
117 * .. Local Arrays ..
118 LOGICAL SELECT( 1 )
119 DOUBLE PRECISION DUM( 1 )
120 * ..
121 * .. External Subroutines ..
122 EXTERNAL DLABAD, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL, ZGEHRD,
123 $ ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC, ZUNGHR
124 * ..
125 * .. External Functions ..
126 LOGICAL LSAME
127 INTEGER IDAMAX, ILAENV
128 DOUBLE PRECISION DLAMCH, DZNRM2, ZLANGE
129 EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DZNRM2, ZLANGE
130 * ..
131 * .. Intrinsic Functions ..
132 INTRINSIC DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT
133 * ..
134 * .. Executable Statements ..
135 *
136 * Test the input arguments
137 *
138 INFO = 0
139 LQUERY = ( LWORK.EQ.-1 )
140 WANTVL = LSAME( JOBVL, 'V' )
141 WANTVR = LSAME( JOBVR, 'V' )
142 IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
143 INFO = -1
144 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
145 INFO = -2
146 ELSE IF( N.LT.0 ) THEN
147 INFO = -3
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -5
150 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
151 INFO = -8
152 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
153 INFO = -10
154 END IF
155 *
156 * Compute workspace
157 * (Note: Comments in the code beginning "Workspace:" describe the
158 * minimal amount of workspace needed at that point in the code,
159 * as well as the preferred amount for good performance.
160 * CWorkspace refers to complex workspace, and RWorkspace to real
161 * workspace. NB refers to the optimal block size for the
162 * immediately following subroutine, as returned by ILAENV.
163 * HSWORK refers to the workspace preferred by ZHSEQR, as
164 * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
165 * the worst case.)
166 *
167 IF( INFO.EQ.0 ) THEN
168 IF( N.EQ.0 ) THEN
169 MINWRK = 1
170 MAXWRK = 1
171 ELSE
172 MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 )
173 MINWRK = 2*N
174 IF( WANTVL ) THEN
175 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
176 $ ' ', N, 1, N, -1 ) )
177 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL,
178 $ WORK, -1, INFO )
179 ELSE IF( WANTVR ) THEN
180 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
181 $ ' ', N, 1, N, -1 ) )
182 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR,
183 $ WORK, -1, INFO )
184 ELSE
185 CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR,
186 $ WORK, -1, INFO )
187 END IF
188 HSWORK = WORK( 1 )
189 MAXWRK = MAX( MAXWRK, HSWORK, MINWRK )
190 END IF
191 WORK( 1 ) = MAXWRK
192 *
193 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
194 INFO = -12
195 END IF
196 END IF
197 *
198 IF( INFO.NE.0 ) THEN
199 CALL XERBLA( 'ZGEEV ', -INFO )
200 RETURN
201 ELSE IF( LQUERY ) THEN
202 RETURN
203 END IF
204 *
205 * Quick return if possible
206 *
207 IF( N.EQ.0 )
208 $ RETURN
209 *
210 * Get machine constants
211 *
212 EPS = DLAMCH( 'P' )
213 SMLNUM = DLAMCH( 'S' )
214 BIGNUM = ONE / SMLNUM
215 CALL DLABAD( SMLNUM, BIGNUM )
216 SMLNUM = SQRT( SMLNUM ) / EPS
217 BIGNUM = ONE / SMLNUM
218 *
219 * Scale A if max element outside range [SMLNUM,BIGNUM]
220 *
221 ANRM = ZLANGE( 'M', N, N, A, LDA, DUM )
222 SCALEA = .FALSE.
223 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
224 SCALEA = .TRUE.
225 CSCALE = SMLNUM
226 ELSE IF( ANRM.GT.BIGNUM ) THEN
227 SCALEA = .TRUE.
228 CSCALE = BIGNUM
229 END IF
230 IF( SCALEA )
231 $ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
232 *
233 * Balance the matrix
234 * (CWorkspace: none)
235 * (RWorkspace: need N)
236 *
237 IBAL = 1
238 CALL ZGEBAL( 'B', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
239 *
240 * Reduce to upper Hessenberg form
241 * (CWorkspace: need 2*N, prefer N+N*NB)
242 * (RWorkspace: none)
243 *
244 ITAU = 1
245 IWRK = ITAU + N
246 CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
247 $ LWORK-IWRK+1, IERR )
248 *
249 IF( WANTVL ) THEN
250 *
251 * Want left eigenvectors
252 * Copy Householder vectors to VL
253 *
254 SIDE = 'L'
255 CALL ZLACPY( 'L', N, N, A, LDA, VL, LDVL )
256 *
257 * Generate unitary matrix in VL
258 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
259 * (RWorkspace: none)
260 *
261 CALL ZUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
262 $ LWORK-IWRK+1, IERR )
263 *
264 * Perform QR iteration, accumulating Schur vectors in VL
265 * (CWorkspace: need 1, prefer HSWORK (see comments) )
266 * (RWorkspace: none)
267 *
268 IWRK = ITAU
269 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL,
270 $ WORK( IWRK ), LWORK-IWRK+1, INFO )
271 *
272 IF( WANTVR ) THEN
273 *
274 * Want left and right eigenvectors
275 * Copy Schur vectors to VR
276 *
277 SIDE = 'B'
278 CALL ZLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
279 END IF
280 *
281 ELSE IF( WANTVR ) THEN
282 *
283 * Want right eigenvectors
284 * Copy Householder vectors to VR
285 *
286 SIDE = 'R'
287 CALL ZLACPY( 'L', N, N, A, LDA, VR, LDVR )
288 *
289 * Generate unitary matrix in VR
290 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
291 * (RWorkspace: none)
292 *
293 CALL ZUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
294 $ LWORK-IWRK+1, IERR )
295 *
296 * Perform QR iteration, accumulating Schur vectors in VR
297 * (CWorkspace: need 1, prefer HSWORK (see comments) )
298 * (RWorkspace: none)
299 *
300 IWRK = ITAU
301 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR,
302 $ WORK( IWRK ), LWORK-IWRK+1, INFO )
303 *
304 ELSE
305 *
306 * Compute eigenvalues only
307 * (CWorkspace: need 1, prefer HSWORK (see comments) )
308 * (RWorkspace: none)
309 *
310 IWRK = ITAU
311 CALL ZHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
312 $ WORK( IWRK ), LWORK-IWRK+1, INFO )
313 END IF
314 *
315 * If INFO > 0 from ZHSEQR, then quit
316 *
317 IF( INFO.GT.0 )
318 $ GO TO 50
319 *
320 IF( WANTVL .OR. WANTVR ) THEN
321 *
322 * Compute left and/or right eigenvectors
323 * (CWorkspace: need 2*N)
324 * (RWorkspace: need 2*N)
325 *
326 IRWORK = IBAL + N
327 CALL ZTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
328 $ N, NOUT, WORK( IWRK ), RWORK( IRWORK ), IERR )
329 END IF
330 *
331 IF( WANTVL ) THEN
332 *
333 * Undo balancing of left eigenvectors
334 * (CWorkspace: none)
335 * (RWorkspace: need N)
336 *
337 CALL ZGEBAK( 'B', 'L', N, ILO, IHI, RWORK( IBAL ), N, VL, LDVL,
338 $ IERR )
339 *
340 * Normalize left eigenvectors and make largest component real
341 *
342 DO 20 I = 1, N
343 SCL = ONE / DZNRM2( N, VL( 1, I ), 1 )
344 CALL ZDSCAL( N, SCL, VL( 1, I ), 1 )
345 DO 10 K = 1, N
346 RWORK( IRWORK+K-1 ) = DBLE( VL( K, I ) )**2 +
347 $ DIMAG( VL( K, I ) )**2
348 10 CONTINUE
349 K = IDAMAX( N, RWORK( IRWORK ), 1 )
350 TMP = DCONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
351 CALL ZSCAL( N, TMP, VL( 1, I ), 1 )
352 VL( K, I ) = DCMPLX( DBLE( VL( K, I ) ), ZERO )
353 20 CONTINUE
354 END IF
355 *
356 IF( WANTVR ) THEN
357 *
358 * Undo balancing of right eigenvectors
359 * (CWorkspace: none)
360 * (RWorkspace: need N)
361 *
362 CALL ZGEBAK( 'B', 'R', N, ILO, IHI, RWORK( IBAL ), N, VR, LDVR,
363 $ IERR )
364 *
365 * Normalize right eigenvectors and make largest component real
366 *
367 DO 40 I = 1, N
368 SCL = ONE / DZNRM2( N, VR( 1, I ), 1 )
369 CALL ZDSCAL( N, SCL, VR( 1, I ), 1 )
370 DO 30 K = 1, N
371 RWORK( IRWORK+K-1 ) = DBLE( VR( K, I ) )**2 +
372 $ DIMAG( VR( K, I ) )**2
373 30 CONTINUE
374 K = IDAMAX( N, RWORK( IRWORK ), 1 )
375 TMP = DCONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
376 CALL ZSCAL( N, TMP, VR( 1, I ), 1 )
377 VR( K, I ) = DCMPLX( DBLE( VR( K, I ) ), ZERO )
378 40 CONTINUE
379 END IF
380 *
381 * Undo scaling if necessary
382 *
383 50 CONTINUE
384 IF( SCALEA ) THEN
385 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ),
386 $ MAX( N-INFO, 1 ), IERR )
387 IF( INFO.GT.0 ) THEN
388 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
389 END IF
390 END IF
391 *
392 WORK( 1 ) = MAXWRK
393 RETURN
394 *
395 * End of ZGEEV
396 *
397 END
2 $ WORK, LWORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBVL, JOBVR
11 INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION RWORK( * )
15 COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
16 $ W( * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the
23 * eigenvalues and, optionally, the left and/or right eigenvectors.
24 *
25 * The right eigenvector v(j) of A satisfies
26 * A * v(j) = lambda(j) * v(j)
27 * where lambda(j) is its eigenvalue.
28 * The left eigenvector u(j) of A satisfies
29 * u(j)**H * A = lambda(j) * u(j)**H
30 * where u(j)**H denotes the conjugate transpose of u(j).
31 *
32 * The computed eigenvectors are normalized to have Euclidean norm
33 * equal to 1 and largest component real.
34 *
35 * Arguments
36 * =========
37 *
38 * JOBVL (input) CHARACTER*1
39 * = 'N': left eigenvectors of A are not computed;
40 * = 'V': left eigenvectors of are computed.
41 *
42 * JOBVR (input) CHARACTER*1
43 * = 'N': right eigenvectors of A are not computed;
44 * = 'V': right eigenvectors of A are computed.
45 *
46 * N (input) INTEGER
47 * The order of the matrix A. N >= 0.
48 *
49 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
50 * On entry, the N-by-N matrix A.
51 * On exit, A has been overwritten.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,N).
55 *
56 * W (output) COMPLEX*16 array, dimension (N)
57 * W contains the computed eigenvalues.
58 *
59 * VL (output) COMPLEX*16 array, dimension (LDVL,N)
60 * If JOBVL = 'V', the left eigenvectors u(j) are stored one
61 * after another in the columns of VL, in the same order
62 * as their eigenvalues.
63 * If JOBVL = 'N', VL is not referenced.
64 * u(j) = VL(:,j), the j-th column of VL.
65 *
66 * LDVL (input) INTEGER
67 * The leading dimension of the array VL. LDVL >= 1; if
68 * JOBVL = 'V', LDVL >= N.
69 *
70 * VR (output) COMPLEX*16 array, dimension (LDVR,N)
71 * If JOBVR = 'V', the right eigenvectors v(j) are stored one
72 * after another in the columns of VR, in the same order
73 * as their eigenvalues.
74 * If JOBVR = 'N', VR is not referenced.
75 * v(j) = VR(:,j), the j-th column of VR.
76 *
77 * LDVR (input) INTEGER
78 * The leading dimension of the array VR. LDVR >= 1; if
79 * JOBVR = 'V', LDVR >= N.
80 *
81 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
82 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
83 *
84 * LWORK (input) INTEGER
85 * The dimension of the array WORK. LWORK >= max(1,2*N).
86 * For good performance, LWORK must generally be larger.
87 *
88 * If LWORK = -1, then a workspace query is assumed; the routine
89 * only calculates the optimal size of the WORK array, returns
90 * this value as the first entry of the WORK array, and no error
91 * message related to LWORK is issued by XERBLA.
92 *
93 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
94 *
95 * INFO (output) INTEGER
96 * = 0: successful exit
97 * < 0: if INFO = -i, the i-th argument had an illegal value.
98 * > 0: if INFO = i, the QR algorithm failed to compute all the
99 * eigenvalues, and no eigenvectors have been computed;
100 * elements and i+1:N of W contain eigenvalues which have
101 * converged.
102 *
103 * =====================================================================
104 *
105 * .. Parameters ..
106 DOUBLE PRECISION ZERO, ONE
107 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
108 * ..
109 * .. Local Scalars ..
110 LOGICAL LQUERY, SCALEA, WANTVL, WANTVR
111 CHARACTER SIDE
112 INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU,
113 $ IWRK, K, MAXWRK, MINWRK, NOUT
114 DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
115 COMPLEX*16 TMP
116 * ..
117 * .. Local Arrays ..
118 LOGICAL SELECT( 1 )
119 DOUBLE PRECISION DUM( 1 )
120 * ..
121 * .. External Subroutines ..
122 EXTERNAL DLABAD, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL, ZGEHRD,
123 $ ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC, ZUNGHR
124 * ..
125 * .. External Functions ..
126 LOGICAL LSAME
127 INTEGER IDAMAX, ILAENV
128 DOUBLE PRECISION DLAMCH, DZNRM2, ZLANGE
129 EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DZNRM2, ZLANGE
130 * ..
131 * .. Intrinsic Functions ..
132 INTRINSIC DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT
133 * ..
134 * .. Executable Statements ..
135 *
136 * Test the input arguments
137 *
138 INFO = 0
139 LQUERY = ( LWORK.EQ.-1 )
140 WANTVL = LSAME( JOBVL, 'V' )
141 WANTVR = LSAME( JOBVR, 'V' )
142 IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
143 INFO = -1
144 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
145 INFO = -2
146 ELSE IF( N.LT.0 ) THEN
147 INFO = -3
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -5
150 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
151 INFO = -8
152 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
153 INFO = -10
154 END IF
155 *
156 * Compute workspace
157 * (Note: Comments in the code beginning "Workspace:" describe the
158 * minimal amount of workspace needed at that point in the code,
159 * as well as the preferred amount for good performance.
160 * CWorkspace refers to complex workspace, and RWorkspace to real
161 * workspace. NB refers to the optimal block size for the
162 * immediately following subroutine, as returned by ILAENV.
163 * HSWORK refers to the workspace preferred by ZHSEQR, as
164 * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
165 * the worst case.)
166 *
167 IF( INFO.EQ.0 ) THEN
168 IF( N.EQ.0 ) THEN
169 MINWRK = 1
170 MAXWRK = 1
171 ELSE
172 MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 )
173 MINWRK = 2*N
174 IF( WANTVL ) THEN
175 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
176 $ ' ', N, 1, N, -1 ) )
177 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL,
178 $ WORK, -1, INFO )
179 ELSE IF( WANTVR ) THEN
180 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
181 $ ' ', N, 1, N, -1 ) )
182 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR,
183 $ WORK, -1, INFO )
184 ELSE
185 CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR,
186 $ WORK, -1, INFO )
187 END IF
188 HSWORK = WORK( 1 )
189 MAXWRK = MAX( MAXWRK, HSWORK, MINWRK )
190 END IF
191 WORK( 1 ) = MAXWRK
192 *
193 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
194 INFO = -12
195 END IF
196 END IF
197 *
198 IF( INFO.NE.0 ) THEN
199 CALL XERBLA( 'ZGEEV ', -INFO )
200 RETURN
201 ELSE IF( LQUERY ) THEN
202 RETURN
203 END IF
204 *
205 * Quick return if possible
206 *
207 IF( N.EQ.0 )
208 $ RETURN
209 *
210 * Get machine constants
211 *
212 EPS = DLAMCH( 'P' )
213 SMLNUM = DLAMCH( 'S' )
214 BIGNUM = ONE / SMLNUM
215 CALL DLABAD( SMLNUM, BIGNUM )
216 SMLNUM = SQRT( SMLNUM ) / EPS
217 BIGNUM = ONE / SMLNUM
218 *
219 * Scale A if max element outside range [SMLNUM,BIGNUM]
220 *
221 ANRM = ZLANGE( 'M', N, N, A, LDA, DUM )
222 SCALEA = .FALSE.
223 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
224 SCALEA = .TRUE.
225 CSCALE = SMLNUM
226 ELSE IF( ANRM.GT.BIGNUM ) THEN
227 SCALEA = .TRUE.
228 CSCALE = BIGNUM
229 END IF
230 IF( SCALEA )
231 $ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
232 *
233 * Balance the matrix
234 * (CWorkspace: none)
235 * (RWorkspace: need N)
236 *
237 IBAL = 1
238 CALL ZGEBAL( 'B', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
239 *
240 * Reduce to upper Hessenberg form
241 * (CWorkspace: need 2*N, prefer N+N*NB)
242 * (RWorkspace: none)
243 *
244 ITAU = 1
245 IWRK = ITAU + N
246 CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
247 $ LWORK-IWRK+1, IERR )
248 *
249 IF( WANTVL ) THEN
250 *
251 * Want left eigenvectors
252 * Copy Householder vectors to VL
253 *
254 SIDE = 'L'
255 CALL ZLACPY( 'L', N, N, A, LDA, VL, LDVL )
256 *
257 * Generate unitary matrix in VL
258 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
259 * (RWorkspace: none)
260 *
261 CALL ZUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
262 $ LWORK-IWRK+1, IERR )
263 *
264 * Perform QR iteration, accumulating Schur vectors in VL
265 * (CWorkspace: need 1, prefer HSWORK (see comments) )
266 * (RWorkspace: none)
267 *
268 IWRK = ITAU
269 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL,
270 $ WORK( IWRK ), LWORK-IWRK+1, INFO )
271 *
272 IF( WANTVR ) THEN
273 *
274 * Want left and right eigenvectors
275 * Copy Schur vectors to VR
276 *
277 SIDE = 'B'
278 CALL ZLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
279 END IF
280 *
281 ELSE IF( WANTVR ) THEN
282 *
283 * Want right eigenvectors
284 * Copy Householder vectors to VR
285 *
286 SIDE = 'R'
287 CALL ZLACPY( 'L', N, N, A, LDA, VR, LDVR )
288 *
289 * Generate unitary matrix in VR
290 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
291 * (RWorkspace: none)
292 *
293 CALL ZUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
294 $ LWORK-IWRK+1, IERR )
295 *
296 * Perform QR iteration, accumulating Schur vectors in VR
297 * (CWorkspace: need 1, prefer HSWORK (see comments) )
298 * (RWorkspace: none)
299 *
300 IWRK = ITAU
301 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR,
302 $ WORK( IWRK ), LWORK-IWRK+1, INFO )
303 *
304 ELSE
305 *
306 * Compute eigenvalues only
307 * (CWorkspace: need 1, prefer HSWORK (see comments) )
308 * (RWorkspace: none)
309 *
310 IWRK = ITAU
311 CALL ZHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
312 $ WORK( IWRK ), LWORK-IWRK+1, INFO )
313 END IF
314 *
315 * If INFO > 0 from ZHSEQR, then quit
316 *
317 IF( INFO.GT.0 )
318 $ GO TO 50
319 *
320 IF( WANTVL .OR. WANTVR ) THEN
321 *
322 * Compute left and/or right eigenvectors
323 * (CWorkspace: need 2*N)
324 * (RWorkspace: need 2*N)
325 *
326 IRWORK = IBAL + N
327 CALL ZTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
328 $ N, NOUT, WORK( IWRK ), RWORK( IRWORK ), IERR )
329 END IF
330 *
331 IF( WANTVL ) THEN
332 *
333 * Undo balancing of left eigenvectors
334 * (CWorkspace: none)
335 * (RWorkspace: need N)
336 *
337 CALL ZGEBAK( 'B', 'L', N, ILO, IHI, RWORK( IBAL ), N, VL, LDVL,
338 $ IERR )
339 *
340 * Normalize left eigenvectors and make largest component real
341 *
342 DO 20 I = 1, N
343 SCL = ONE / DZNRM2( N, VL( 1, I ), 1 )
344 CALL ZDSCAL( N, SCL, VL( 1, I ), 1 )
345 DO 10 K = 1, N
346 RWORK( IRWORK+K-1 ) = DBLE( VL( K, I ) )**2 +
347 $ DIMAG( VL( K, I ) )**2
348 10 CONTINUE
349 K = IDAMAX( N, RWORK( IRWORK ), 1 )
350 TMP = DCONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
351 CALL ZSCAL( N, TMP, VL( 1, I ), 1 )
352 VL( K, I ) = DCMPLX( DBLE( VL( K, I ) ), ZERO )
353 20 CONTINUE
354 END IF
355 *
356 IF( WANTVR ) THEN
357 *
358 * Undo balancing of right eigenvectors
359 * (CWorkspace: none)
360 * (RWorkspace: need N)
361 *
362 CALL ZGEBAK( 'B', 'R', N, ILO, IHI, RWORK( IBAL ), N, VR, LDVR,
363 $ IERR )
364 *
365 * Normalize right eigenvectors and make largest component real
366 *
367 DO 40 I = 1, N
368 SCL = ONE / DZNRM2( N, VR( 1, I ), 1 )
369 CALL ZDSCAL( N, SCL, VR( 1, I ), 1 )
370 DO 30 K = 1, N
371 RWORK( IRWORK+K-1 ) = DBLE( VR( K, I ) )**2 +
372 $ DIMAG( VR( K, I ) )**2
373 30 CONTINUE
374 K = IDAMAX( N, RWORK( IRWORK ), 1 )
375 TMP = DCONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
376 CALL ZSCAL( N, TMP, VR( 1, I ), 1 )
377 VR( K, I ) = DCMPLX( DBLE( VR( K, I ) ), ZERO )
378 40 CONTINUE
379 END IF
380 *
381 * Undo scaling if necessary
382 *
383 50 CONTINUE
384 IF( SCALEA ) THEN
385 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ),
386 $ MAX( N-INFO, 1 ), IERR )
387 IF( INFO.GT.0 ) THEN
388 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
389 END IF
390 END IF
391 *
392 WORK( 1 ) = MAXWRK
393 RETURN
394 *
395 * End of ZGEEV
396 *
397 END