1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
|
SUBROUTINE ZGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
$ LDQ, Z, LDZ, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper
* Hessenberg form using unitary transformations, where A is a
* general matrix and B is upper triangular. The form of the
* generalized eigenvalue problem is
* A*x = lambda*B*x,
* and B is typically made upper triangular by computing its QR
* factorization and moving the unitary matrix Q to the left side
* of the equation.
*
* This subroutine simultaneously reduces A to a Hessenberg matrix H:
* Q**H*A*Z = H
* and transforms B to another upper triangular matrix T:
* Q**H*B*Z = T
* in order to reduce the problem to its standard form
* H*y = lambda*T*y
* where y = Z**H*x.
*
* The unitary matrices Q and Z are determined as products of Givens
* rotations. They may either be formed explicitly, or they may be
* postmultiplied into input matrices Q1 and Z1, so that
* Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
* Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
* If Q1 is the unitary matrix from the QR factorization of B in the
* original equation A*x = lambda*B*x, then ZGGHRD reduces the original
* problem to generalized Hessenberg form.
*
* Arguments
* =========
*
* COMPQ (input) CHARACTER*1
* = 'N': do not compute Q;
* = 'I': Q is initialized to the unit matrix, and the
* unitary matrix Q is returned;
* = 'V': Q must contain a unitary matrix Q1 on entry,
* and the product Q1*Q is returned.
*
* COMPZ (input) CHARACTER*1
* = 'N': do not compute Q;
* = 'I': Q is initialized to the unit matrix, and the
* unitary matrix Q is returned;
* = 'V': Q must contain a unitary matrix Q1 on entry,
* and the product Q1*Q is returned.
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* ILO and IHI mark the rows and columns of A which are to be
* reduced. It is assumed that A is already upper triangular
* in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
* normally set by a previous call to ZGGBAL; otherwise they
* should be set to 1 and N respectively.
* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
* A (input/output) COMPLEX*16 array, dimension (LDA, N)
* On entry, the N-by-N general matrix to be reduced.
* On exit, the upper triangle and the first subdiagonal of A
* are overwritten with the upper Hessenberg matrix H, and the
* rest is set to zero.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) COMPLEX*16 array, dimension (LDB, N)
* On entry, the N-by-N upper triangular matrix B.
* On exit, the upper triangular matrix T = Q**H B Z. The
* elements below the diagonal are set to zero.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
* On entry, if COMPQ = 'V', the unitary matrix Q1, typically
* from the QR factorization of B.
* On exit, if COMPQ='I', the unitary matrix Q, and if
* COMPQ = 'V', the product Q1*Q.
* Not referenced if COMPQ='N'.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q.
* LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
*
* Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
* On entry, if COMPZ = 'V', the unitary matrix Z1.
* On exit, if COMPZ='I', the unitary matrix Z, and if
* COMPZ = 'V', the product Z1*Z.
* Not referenced if COMPZ='N'.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z.
* LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* This routine reduces A to Hessenberg and B to triangular form by
* an unblocked reduction, as described in _Matrix_Computations_,
* by Golub and van Loan (Johns Hopkins Press).
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 CONE, CZERO
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
$ CZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL ILQ, ILZ
INTEGER ICOMPQ, ICOMPZ, JCOL, JROW
DOUBLE PRECISION C
COMPLEX*16 CTEMP, S
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZLARTG, ZLASET, ZROT
* ..
* .. Intrinsic Functions ..
INTRINSIC DCONJG, MAX
* ..
* .. Executable Statements ..
*
* Decode COMPQ
*
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
*
* Decode COMPZ
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
*
* Test the input parameters.
*
INFO = 0
IF( ICOMPQ.LE.0 ) THEN
INFO = -1
ELSE IF( ICOMPZ.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 ) THEN
INFO = -4
ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN
INFO = -11
ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGGHRD', -INFO )
RETURN
END IF
*
* Initialize Q and Z if desired.
*
IF( ICOMPQ.EQ.3 )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
* Zero out lower triangle of B
*
DO 20 JCOL = 1, N - 1
DO 10 JROW = JCOL + 1, N
B( JROW, JCOL ) = CZERO
10 CONTINUE
20 CONTINUE
*
* Reduce A and B
*
DO 40 JCOL = ILO, IHI - 2
*
DO 30 JROW = IHI, JCOL + 2, -1
*
* Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
*
CTEMP = A( JROW-1, JCOL )
CALL ZLARTG( CTEMP, A( JROW, JCOL ), C, S,
$ A( JROW-1, JCOL ) )
A( JROW, JCOL ) = CZERO
CALL ZROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA,
$ A( JROW, JCOL+1 ), LDA, C, S )
CALL ZROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB,
$ B( JROW, JROW-1 ), LDB, C, S )
IF( ILQ )
$ CALL ZROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C,
$ DCONJG( S ) )
*
* Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
*
CTEMP = B( JROW, JROW )
CALL ZLARTG( CTEMP, B( JROW, JROW-1 ), C, S,
$ B( JROW, JROW ) )
B( JROW, JROW-1 ) = CZERO
CALL ZROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C, S )
CALL ZROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C,
$ S )
IF( ILZ )
$ CALL ZROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C, S )
30 CONTINUE
40 CONTINUE
*
RETURN
*
* End of ZGGHRD
*
END
|
