1 SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
2 $ EPS3, SMLNUM, INFO )
3 *
4 * -- LAPACK auxiliary routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 LOGICAL NOINIT, RIGHTV
11 INTEGER INFO, LDB, LDH, N
12 DOUBLE PRECISION EPS3, SMLNUM
13 COMPLEX*16 W
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION RWORK( * )
17 COMPLEX*16 B( LDB, * ), H( LDH, * ), V( * )
18 * ..
19 *
20 * Purpose
21 * =======
22 *
23 * ZLAEIN uses inverse iteration to find a right or left eigenvector
24 * corresponding to the eigenvalue W of a complex upper Hessenberg
25 * matrix H.
26 *
27 * Arguments
28 * =========
29 *
30 * RIGHTV (input) LOGICAL
31 * = .TRUE. : compute right eigenvector;
32 * = .FALSE.: compute left eigenvector.
33 *
34 * NOINIT (input) LOGICAL
35 * = .TRUE. : no initial vector supplied in V
36 * = .FALSE.: initial vector supplied in V.
37 *
38 * N (input) INTEGER
39 * The order of the matrix H. N >= 0.
40 *
41 * H (input) COMPLEX*16 array, dimension (LDH,N)
42 * The upper Hessenberg matrix H.
43 *
44 * LDH (input) INTEGER
45 * The leading dimension of the array H. LDH >= max(1,N).
46 *
47 * W (input) COMPLEX*16
48 * The eigenvalue of H whose corresponding right or left
49 * eigenvector is to be computed.
50 *
51 * V (input/output) COMPLEX*16 array, dimension (N)
52 * On entry, if NOINIT = .FALSE., V must contain a starting
53 * vector for inverse iteration; otherwise V need not be set.
54 * On exit, V contains the computed eigenvector, normalized so
55 * that the component of largest magnitude has magnitude 1; here
56 * the magnitude of a complex number (x,y) is taken to be
57 * |x| + |y|.
58 *
59 * B (workspace) COMPLEX*16 array, dimension (LDB,N)
60 *
61 * LDB (input) INTEGER
62 * The leading dimension of the array B. LDB >= max(1,N).
63 *
64 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
65 *
66 * EPS3 (input) DOUBLE PRECISION
67 * A small machine-dependent value which is used to perturb
68 * close eigenvalues, and to replace zero pivots.
69 *
70 * SMLNUM (input) DOUBLE PRECISION
71 * A machine-dependent value close to the underflow threshold.
72 *
73 * INFO (output) INTEGER
74 * = 0: successful exit
75 * = 1: inverse iteration did not converge; V is set to the
76 * last iterate.
77 *
78 * =====================================================================
79 *
80 * .. Parameters ..
81 DOUBLE PRECISION ONE, TENTH
82 PARAMETER ( ONE = 1.0D+0, TENTH = 1.0D-1 )
83 COMPLEX*16 ZERO
84 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) )
85 * ..
86 * .. Local Scalars ..
87 CHARACTER NORMIN, TRANS
88 INTEGER I, IERR, ITS, J
89 DOUBLE PRECISION GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
90 COMPLEX*16 CDUM, EI, EJ, TEMP, X
91 * ..
92 * .. External Functions ..
93 INTEGER IZAMAX
94 DOUBLE PRECISION DZASUM, DZNRM2
95 COMPLEX*16 ZLADIV
96 EXTERNAL IZAMAX, DZASUM, DZNRM2, ZLADIV
97 * ..
98 * .. External Subroutines ..
99 EXTERNAL ZDSCAL, ZLATRS
100 * ..
101 * .. Intrinsic Functions ..
102 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
103 * ..
104 * .. Statement Functions ..
105 DOUBLE PRECISION CABS1
106 * ..
107 * .. Statement Function definitions ..
108 CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
109 * ..
110 * .. Executable Statements ..
111 *
112 INFO = 0
113 *
114 * GROWTO is the threshold used in the acceptance test for an
115 * eigenvector.
116 *
117 ROOTN = SQRT( DBLE( N ) )
118 GROWTO = TENTH / ROOTN
119 NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
120 *
121 * Form B = H - W*I (except that the subdiagonal elements are not
122 * stored).
123 *
124 DO 20 J = 1, N
125 DO 10 I = 1, J - 1
126 B( I, J ) = H( I, J )
127 10 CONTINUE
128 B( J, J ) = H( J, J ) - W
129 20 CONTINUE
130 *
131 IF( NOINIT ) THEN
132 *
133 * Initialize V.
134 *
135 DO 30 I = 1, N
136 V( I ) = EPS3
137 30 CONTINUE
138 ELSE
139 *
140 * Scale supplied initial vector.
141 *
142 VNORM = DZNRM2( N, V, 1 )
143 CALL ZDSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), V, 1 )
144 END IF
145 *
146 IF( RIGHTV ) THEN
147 *
148 * LU decomposition with partial pivoting of B, replacing zero
149 * pivots by EPS3.
150 *
151 DO 60 I = 1, N - 1
152 EI = H( I+1, I )
153 IF( CABS1( B( I, I ) ).LT.CABS1( EI ) ) THEN
154 *
155 * Interchange rows and eliminate.
156 *
157 X = ZLADIV( B( I, I ), EI )
158 B( I, I ) = EI
159 DO 40 J = I + 1, N
160 TEMP = B( I+1, J )
161 B( I+1, J ) = B( I, J ) - X*TEMP
162 B( I, J ) = TEMP
163 40 CONTINUE
164 ELSE
165 *
166 * Eliminate without interchange.
167 *
168 IF( B( I, I ).EQ.ZERO )
169 $ B( I, I ) = EPS3
170 X = ZLADIV( EI, B( I, I ) )
171 IF( X.NE.ZERO ) THEN
172 DO 50 J = I + 1, N
173 B( I+1, J ) = B( I+1, J ) - X*B( I, J )
174 50 CONTINUE
175 END IF
176 END IF
177 60 CONTINUE
178 IF( B( N, N ).EQ.ZERO )
179 $ B( N, N ) = EPS3
180 *
181 TRANS = 'N'
182 *
183 ELSE
184 *
185 * UL decomposition with partial pivoting of B, replacing zero
186 * pivots by EPS3.
187 *
188 DO 90 J = N, 2, -1
189 EJ = H( J, J-1 )
190 IF( CABS1( B( J, J ) ).LT.CABS1( EJ ) ) THEN
191 *
192 * Interchange columns and eliminate.
193 *
194 X = ZLADIV( B( J, J ), EJ )
195 B( J, J ) = EJ
196 DO 70 I = 1, J - 1
197 TEMP = B( I, J-1 )
198 B( I, J-1 ) = B( I, J ) - X*TEMP
199 B( I, J ) = TEMP
200 70 CONTINUE
201 ELSE
202 *
203 * Eliminate without interchange.
204 *
205 IF( B( J, J ).EQ.ZERO )
206 $ B( J, J ) = EPS3
207 X = ZLADIV( EJ, B( J, J ) )
208 IF( X.NE.ZERO ) THEN
209 DO 80 I = 1, J - 1
210 B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
211 80 CONTINUE
212 END IF
213 END IF
214 90 CONTINUE
215 IF( B( 1, 1 ).EQ.ZERO )
216 $ B( 1, 1 ) = EPS3
217 *
218 TRANS = 'C'
219 *
220 END IF
221 *
222 NORMIN = 'N'
223 DO 110 ITS = 1, N
224 *
225 * Solve U*x = scale*v for a right eigenvector
226 * or U**H *x = scale*v for a left eigenvector,
227 * overwriting x on v.
228 *
229 CALL ZLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, V,
230 $ SCALE, RWORK, IERR )
231 NORMIN = 'Y'
232 *
233 * Test for sufficient growth in the norm of v.
234 *
235 VNORM = DZASUM( N, V, 1 )
236 IF( VNORM.GE.GROWTO*SCALE )
237 $ GO TO 120
238 *
239 * Choose new orthogonal starting vector and try again.
240 *
241 RTEMP = EPS3 / ( ROOTN+ONE )
242 V( 1 ) = EPS3
243 DO 100 I = 2, N
244 V( I ) = RTEMP
245 100 CONTINUE
246 V( N-ITS+1 ) = V( N-ITS+1 ) - EPS3*ROOTN
247 110 CONTINUE
248 *
249 * Failure to find eigenvector in N iterations.
250 *
251 INFO = 1
252 *
253 120 CONTINUE
254 *
255 * Normalize eigenvector.
256 *
257 I = IZAMAX( N, V, 1 )
258 CALL ZDSCAL( N, ONE / CABS1( V( I ) ), V, 1 )
259 *
260 RETURN
261 *
262 * End of ZLAEIN
263 *
264 END
2 $ EPS3, SMLNUM, INFO )
3 *
4 * -- LAPACK auxiliary routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 LOGICAL NOINIT, RIGHTV
11 INTEGER INFO, LDB, LDH, N
12 DOUBLE PRECISION EPS3, SMLNUM
13 COMPLEX*16 W
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION RWORK( * )
17 COMPLEX*16 B( LDB, * ), H( LDH, * ), V( * )
18 * ..
19 *
20 * Purpose
21 * =======
22 *
23 * ZLAEIN uses inverse iteration to find a right or left eigenvector
24 * corresponding to the eigenvalue W of a complex upper Hessenberg
25 * matrix H.
26 *
27 * Arguments
28 * =========
29 *
30 * RIGHTV (input) LOGICAL
31 * = .TRUE. : compute right eigenvector;
32 * = .FALSE.: compute left eigenvector.
33 *
34 * NOINIT (input) LOGICAL
35 * = .TRUE. : no initial vector supplied in V
36 * = .FALSE.: initial vector supplied in V.
37 *
38 * N (input) INTEGER
39 * The order of the matrix H. N >= 0.
40 *
41 * H (input) COMPLEX*16 array, dimension (LDH,N)
42 * The upper Hessenberg matrix H.
43 *
44 * LDH (input) INTEGER
45 * The leading dimension of the array H. LDH >= max(1,N).
46 *
47 * W (input) COMPLEX*16
48 * The eigenvalue of H whose corresponding right or left
49 * eigenvector is to be computed.
50 *
51 * V (input/output) COMPLEX*16 array, dimension (N)
52 * On entry, if NOINIT = .FALSE., V must contain a starting
53 * vector for inverse iteration; otherwise V need not be set.
54 * On exit, V contains the computed eigenvector, normalized so
55 * that the component of largest magnitude has magnitude 1; here
56 * the magnitude of a complex number (x,y) is taken to be
57 * |x| + |y|.
58 *
59 * B (workspace) COMPLEX*16 array, dimension (LDB,N)
60 *
61 * LDB (input) INTEGER
62 * The leading dimension of the array B. LDB >= max(1,N).
63 *
64 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
65 *
66 * EPS3 (input) DOUBLE PRECISION
67 * A small machine-dependent value which is used to perturb
68 * close eigenvalues, and to replace zero pivots.
69 *
70 * SMLNUM (input) DOUBLE PRECISION
71 * A machine-dependent value close to the underflow threshold.
72 *
73 * INFO (output) INTEGER
74 * = 0: successful exit
75 * = 1: inverse iteration did not converge; V is set to the
76 * last iterate.
77 *
78 * =====================================================================
79 *
80 * .. Parameters ..
81 DOUBLE PRECISION ONE, TENTH
82 PARAMETER ( ONE = 1.0D+0, TENTH = 1.0D-1 )
83 COMPLEX*16 ZERO
84 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) )
85 * ..
86 * .. Local Scalars ..
87 CHARACTER NORMIN, TRANS
88 INTEGER I, IERR, ITS, J
89 DOUBLE PRECISION GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
90 COMPLEX*16 CDUM, EI, EJ, TEMP, X
91 * ..
92 * .. External Functions ..
93 INTEGER IZAMAX
94 DOUBLE PRECISION DZASUM, DZNRM2
95 COMPLEX*16 ZLADIV
96 EXTERNAL IZAMAX, DZASUM, DZNRM2, ZLADIV
97 * ..
98 * .. External Subroutines ..
99 EXTERNAL ZDSCAL, ZLATRS
100 * ..
101 * .. Intrinsic Functions ..
102 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
103 * ..
104 * .. Statement Functions ..
105 DOUBLE PRECISION CABS1
106 * ..
107 * .. Statement Function definitions ..
108 CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
109 * ..
110 * .. Executable Statements ..
111 *
112 INFO = 0
113 *
114 * GROWTO is the threshold used in the acceptance test for an
115 * eigenvector.
116 *
117 ROOTN = SQRT( DBLE( N ) )
118 GROWTO = TENTH / ROOTN
119 NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
120 *
121 * Form B = H - W*I (except that the subdiagonal elements are not
122 * stored).
123 *
124 DO 20 J = 1, N
125 DO 10 I = 1, J - 1
126 B( I, J ) = H( I, J )
127 10 CONTINUE
128 B( J, J ) = H( J, J ) - W
129 20 CONTINUE
130 *
131 IF( NOINIT ) THEN
132 *
133 * Initialize V.
134 *
135 DO 30 I = 1, N
136 V( I ) = EPS3
137 30 CONTINUE
138 ELSE
139 *
140 * Scale supplied initial vector.
141 *
142 VNORM = DZNRM2( N, V, 1 )
143 CALL ZDSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), V, 1 )
144 END IF
145 *
146 IF( RIGHTV ) THEN
147 *
148 * LU decomposition with partial pivoting of B, replacing zero
149 * pivots by EPS3.
150 *
151 DO 60 I = 1, N - 1
152 EI = H( I+1, I )
153 IF( CABS1( B( I, I ) ).LT.CABS1( EI ) ) THEN
154 *
155 * Interchange rows and eliminate.
156 *
157 X = ZLADIV( B( I, I ), EI )
158 B( I, I ) = EI
159 DO 40 J = I + 1, N
160 TEMP = B( I+1, J )
161 B( I+1, J ) = B( I, J ) - X*TEMP
162 B( I, J ) = TEMP
163 40 CONTINUE
164 ELSE
165 *
166 * Eliminate without interchange.
167 *
168 IF( B( I, I ).EQ.ZERO )
169 $ B( I, I ) = EPS3
170 X = ZLADIV( EI, B( I, I ) )
171 IF( X.NE.ZERO ) THEN
172 DO 50 J = I + 1, N
173 B( I+1, J ) = B( I+1, J ) - X*B( I, J )
174 50 CONTINUE
175 END IF
176 END IF
177 60 CONTINUE
178 IF( B( N, N ).EQ.ZERO )
179 $ B( N, N ) = EPS3
180 *
181 TRANS = 'N'
182 *
183 ELSE
184 *
185 * UL decomposition with partial pivoting of B, replacing zero
186 * pivots by EPS3.
187 *
188 DO 90 J = N, 2, -1
189 EJ = H( J, J-1 )
190 IF( CABS1( B( J, J ) ).LT.CABS1( EJ ) ) THEN
191 *
192 * Interchange columns and eliminate.
193 *
194 X = ZLADIV( B( J, J ), EJ )
195 B( J, J ) = EJ
196 DO 70 I = 1, J - 1
197 TEMP = B( I, J-1 )
198 B( I, J-1 ) = B( I, J ) - X*TEMP
199 B( I, J ) = TEMP
200 70 CONTINUE
201 ELSE
202 *
203 * Eliminate without interchange.
204 *
205 IF( B( J, J ).EQ.ZERO )
206 $ B( J, J ) = EPS3
207 X = ZLADIV( EJ, B( J, J ) )
208 IF( X.NE.ZERO ) THEN
209 DO 80 I = 1, J - 1
210 B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
211 80 CONTINUE
212 END IF
213 END IF
214 90 CONTINUE
215 IF( B( 1, 1 ).EQ.ZERO )
216 $ B( 1, 1 ) = EPS3
217 *
218 TRANS = 'C'
219 *
220 END IF
221 *
222 NORMIN = 'N'
223 DO 110 ITS = 1, N
224 *
225 * Solve U*x = scale*v for a right eigenvector
226 * or U**H *x = scale*v for a left eigenvector,
227 * overwriting x on v.
228 *
229 CALL ZLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, V,
230 $ SCALE, RWORK, IERR )
231 NORMIN = 'Y'
232 *
233 * Test for sufficient growth in the norm of v.
234 *
235 VNORM = DZASUM( N, V, 1 )
236 IF( VNORM.GE.GROWTO*SCALE )
237 $ GO TO 120
238 *
239 * Choose new orthogonal starting vector and try again.
240 *
241 RTEMP = EPS3 / ( ROOTN+ONE )
242 V( 1 ) = EPS3
243 DO 100 I = 2, N
244 V( I ) = RTEMP
245 100 CONTINUE
246 V( N-ITS+1 ) = V( N-ITS+1 ) - EPS3*ROOTN
247 110 CONTINUE
248 *
249 * Failure to find eigenvector in N iterations.
250 *
251 INFO = 1
252 *
253 120 CONTINUE
254 *
255 * Normalize eigenvector.
256 *
257 I = IZAMAX( N, V, 1 )
258 CALL ZDSCAL( N, ONE / CABS1( V( I ) ), V, 1 )
259 *
260 RETURN
261 *
262 * End of ZLAEIN
263 *
264 END