LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
cgghrd.f
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1 *> \brief \b CGGHRD
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
22 * LDQ, Z, LDZ, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER COMPQ, COMPZ
26 * INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
27 * ..
28 * .. Array Arguments ..
29 * COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
30 * $ Z( LDZ, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CGGHRD reduces a pair of complex matrices (A,B) to generalized upper
40 *> Hessenberg form using unitary transformations, where A is a
41 *> general matrix and B is upper triangular. The form of the generalized
42 *> eigenvalue problem is
43 *> A*x = lambda*B*x,
44 *> and B is typically made upper triangular by computing its QR
45 *> factorization and moving the unitary matrix Q to the left side
46 *> of the equation.
47 *>
48 *> This subroutine simultaneously reduces A to a Hessenberg matrix H:
49 *> Q**H*A*Z = H
50 *> and transforms B to another upper triangular matrix T:
51 *> Q**H*B*Z = T
52 *> in order to reduce the problem to its standard form
53 *> H*y = lambda*T*y
54 *> where y = Z**H*x.
55 *>
56 *> The unitary matrices Q and Z are determined as products of Givens
57 *> rotations. They may either be formed explicitly, or they may be
58 *> postmultiplied into input matrices Q1 and Z1, so that
59 *> Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
60 *> Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
61 *> If Q1 is the unitary matrix from the QR factorization of B in the
62 *> original equation A*x = lambda*B*x, then CGGHRD reduces the original
63 *> problem to generalized Hessenberg form.
64 *> \endverbatim
65 *
66 * Arguments:
67 * ==========
68 *
69 *> \param[in] COMPQ
70 *> \verbatim
71 *> COMPQ is CHARACTER*1
72 *> = 'N': do not compute Q;
73 *> = 'I': Q is initialized to the unit matrix, and the
74 *> unitary matrix Q is returned;
75 *> = 'V': Q must contain a unitary matrix Q1 on entry,
76 *> and the product Q1*Q is returned.
77 *> \endverbatim
78 *>
79 *> \param[in] COMPZ
80 *> \verbatim
81 *> COMPZ is CHARACTER*1
82 *> = 'N': do not compute Z;
83 *> = 'I': Z is initialized to the unit matrix, and the
84 *> unitary matrix Z is returned;
85 *> = 'V': Z must contain a unitary matrix Z1 on entry,
86 *> and the product Z1*Z is returned.
87 *> \endverbatim
88 *>
89 *> \param[in] N
90 *> \verbatim
91 *> N is INTEGER
92 *> The order of the matrices A and B. N >= 0.
93 *> \endverbatim
94 *>
95 *> \param[in] ILO
96 *> \verbatim
97 *> ILO is INTEGER
98 *> \endverbatim
99 *>
100 *> \param[in] IHI
101 *> \verbatim
102 *> IHI is INTEGER
103 *>
104 *> ILO and IHI mark the rows and columns of A which are to be
105 *> reduced. It is assumed that A is already upper triangular
106 *> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
107 *> normally set by a previous call to CGGBAL; otherwise they
108 *> should be set to 1 and N respectively.
109 *> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
110 *> \endverbatim
111 *>
112 *> \param[in,out] A
113 *> \verbatim
114 *> A is COMPLEX array, dimension (LDA, N)
115 *> On entry, the N-by-N general matrix to be reduced.
116 *> On exit, the upper triangle and the first subdiagonal of A
117 *> are overwritten with the upper Hessenberg matrix H, and the
118 *> rest is set to zero.
119 *> \endverbatim
120 *>
121 *> \param[in] LDA
122 *> \verbatim
123 *> LDA is INTEGER
124 *> The leading dimension of the array A. LDA >= max(1,N).
125 *> \endverbatim
126 *>
127 *> \param[in,out] B
128 *> \verbatim
129 *> B is COMPLEX array, dimension (LDB, N)
130 *> On entry, the N-by-N upper triangular matrix B.
131 *> On exit, the upper triangular matrix T = Q**H B Z. The
132 *> elements below the diagonal are set to zero.
133 *> \endverbatim
134 *>
135 *> \param[in] LDB
136 *> \verbatim
137 *> LDB is INTEGER
138 *> The leading dimension of the array B. LDB >= max(1,N).
139 *> \endverbatim
140 *>
141 *> \param[in,out] Q
142 *> \verbatim
143 *> Q is COMPLEX array, dimension (LDQ, N)
144 *> On entry, if COMPQ = 'V', the unitary matrix Q1, typically
145 *> from the QR factorization of B.
146 *> On exit, if COMPQ='I', the unitary matrix Q, and if
147 *> COMPQ = 'V', the product Q1*Q.
148 *> Not referenced if COMPQ='N'.
149 *> \endverbatim
150 *>
151 *> \param[in] LDQ
152 *> \verbatim
153 *> LDQ is INTEGER
154 *> The leading dimension of the array Q.
155 *> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
156 *> \endverbatim
157 *>
158 *> \param[in,out] Z
159 *> \verbatim
160 *> Z is COMPLEX array, dimension (LDZ, N)
161 *> On entry, if COMPZ = 'V', the unitary matrix Z1.
162 *> On exit, if COMPZ='I', the unitary matrix Z, and if
163 *> COMPZ = 'V', the product Z1*Z.
164 *> Not referenced if COMPZ='N'.
165 *> \endverbatim
166 *>
167 *> \param[in] LDZ
168 *> \verbatim
169 *> LDZ is INTEGER
170 *> The leading dimension of the array Z.
171 *> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
172 *> \endverbatim
173 *>
174 *> \param[out] INFO
175 *> \verbatim
176 *> INFO is INTEGER
177 *> = 0: successful exit.
178 *> < 0: if INFO = -i, the i-th argument had an illegal value.
179 *> \endverbatim
180 *
181 * Authors:
182 * ========
183 *
184 *> \author Univ. of Tennessee
185 *> \author Univ. of California Berkeley
186 *> \author Univ. of Colorado Denver
187 *> \author NAG Ltd.
188 *
189 *> \date December 2016
190 *
191 *> \ingroup complexOTHERcomputational
192 *
193 *> \par Further Details:
194 * =====================
195 *>
196 *> \verbatim
197 *>
198 *> This routine reduces A to Hessenberg and B to triangular form by
199 *> an unblocked reduction, as described in _Matrix_Computations_,
200 *> by Golub and van Loan (Johns Hopkins Press).
201 *> \endverbatim
202 *>
203 * =====================================================================
204  SUBROUTINE cgghrd( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
205  $ LDQ, Z, LDZ, INFO )
206 *
207 * -- LAPACK computational routine (version 3.7.0) --
208 * -- LAPACK is a software package provided by Univ. of Tennessee, --
209 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
210 * December 2016
211 *
212 * .. Scalar Arguments ..
213  CHARACTER COMPQ, COMPZ
214  INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
215 * ..
216 * .. Array Arguments ..
217  COMPLEX A( lda, * ), B( ldb, * ), Q( ldq, * ),
218  $ z( ldz, * )
219 * ..
220 *
221 * =====================================================================
222 *
223 * .. Parameters ..
224  COMPLEX CONE, CZERO
225  parameter( cone = ( 1.0e+0, 0.0e+0 ),
226  $ czero = ( 0.0e+0, 0.0e+0 ) )
227 * ..
228 * .. Local Scalars ..
229  LOGICAL ILQ, ILZ
230  INTEGER ICOMPQ, ICOMPZ, JCOL, JROW
231  REAL C
232  COMPLEX CTEMP, S
233 * ..
234 * .. External Functions ..
235  LOGICAL LSAME
236  EXTERNAL lsame
237 * ..
238 * .. External Subroutines ..
239  EXTERNAL clartg, claset, crot, xerbla
240 * ..
241 * .. Intrinsic Functions ..
242  INTRINSIC conjg, max
243 * ..
244 * .. Executable Statements ..
245 *
246 * Decode COMPQ
247 *
248  IF( lsame( compq, 'N' ) ) THEN
249  ilq = .false.
250  icompq = 1
251  ELSE IF( lsame( compq, 'V' ) ) THEN
252  ilq = .true.
253  icompq = 2
254  ELSE IF( lsame( compq, 'I' ) ) THEN
255  ilq = .true.
256  icompq = 3
257  ELSE
258  icompq = 0
259  END IF
260 *
261 * Decode COMPZ
262 *
263  IF( lsame( compz, 'N' ) ) THEN
264  ilz = .false.
265  icompz = 1
266  ELSE IF( lsame( compz, 'V' ) ) THEN
267  ilz = .true.
268  icompz = 2
269  ELSE IF( lsame( compz, 'I' ) ) THEN
270  ilz = .true.
271  icompz = 3
272  ELSE
273  icompz = 0
274  END IF
275 *
276 * Test the input parameters.
277 *
278  info = 0
279  IF( icompq.LE.0 ) THEN
280  info = -1
281  ELSE IF( icompz.LE.0 ) THEN
282  info = -2
283  ELSE IF( n.LT.0 ) THEN
284  info = -3
285  ELSE IF( ilo.LT.1 ) THEN
286  info = -4
287  ELSE IF( ihi.GT.n .OR. ihi.LT.ilo-1 ) THEN
288  info = -5
289  ELSE IF( lda.LT.max( 1, n ) ) THEN
290  info = -7
291  ELSE IF( ldb.LT.max( 1, n ) ) THEN
292  info = -9
293  ELSE IF( ( ilq .AND. ldq.LT.n ) .OR. ldq.LT.1 ) THEN
294  info = -11
295  ELSE IF( ( ilz .AND. ldz.LT.n ) .OR. ldz.LT.1 ) THEN
296  info = -13
297  END IF
298  IF( info.NE.0 ) THEN
299  CALL xerbla( 'CGGHRD', -info )
300  RETURN
301  END IF
302 *
303 * Initialize Q and Z if desired.
304 *
305  IF( icompq.EQ.3 )
306  $ CALL claset( 'Full', n, n, czero, cone, q, ldq )
307  IF( icompz.EQ.3 )
308  $ CALL claset( 'Full', n, n, czero, cone, z, ldz )
309 *
310 * Quick return if possible
311 *
312  IF( n.LE.1 )
313  $ RETURN
314 *
315 * Zero out lower triangle of B
316 *
317  DO 20 jcol = 1, n - 1
318  DO 10 jrow = jcol + 1, n
319  b( jrow, jcol ) = czero
320  10 CONTINUE
321  20 CONTINUE
322 *
323 * Reduce A and B
324 *
325  DO 40 jcol = ilo, ihi - 2
326 *
327  DO 30 jrow = ihi, jcol + 2, -1
328 *
329 * Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
330 *
331  ctemp = a( jrow-1, jcol )
332  CALL clartg( ctemp, a( jrow, jcol ), c, s,
333  $ a( jrow-1, jcol ) )
334  a( jrow, jcol ) = czero
335  CALL crot( n-jcol, a( jrow-1, jcol+1 ), lda,
336  $ a( jrow, jcol+1 ), lda, c, s )
337  CALL crot( n+2-jrow, b( jrow-1, jrow-1 ), ldb,
338  $ b( jrow, jrow-1 ), ldb, c, s )
339  IF( ilq )
340  $ CALL crot( n, q( 1, jrow-1 ), 1, q( 1, jrow ), 1, c,
341  $ conjg( s ) )
342 *
343 * Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
344 *
345  ctemp = b( jrow, jrow )
346  CALL clartg( ctemp, b( jrow, jrow-1 ), c, s,
347  $ b( jrow, jrow ) )
348  b( jrow, jrow-1 ) = czero
349  CALL crot( ihi, a( 1, jrow ), 1, a( 1, jrow-1 ), 1, c, s )
350  CALL crot( jrow-1, b( 1, jrow ), 1, b( 1, jrow-1 ), 1, c,
351  $ s )
352  IF( ilz )
353  $ CALL crot( n, z( 1, jrow ), 1, z( 1, jrow-1 ), 1, c, s )
354  30 CONTINUE
355  40 CONTINUE
356 *
357  RETURN
358 *
359 * End of CGGHRD
360 *
361  END
subroutine crot(N, CX, INCX, CY, INCY, C, S)
CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors...
Definition: crot.f:105
subroutine clartg(F, G, CS, SN, R)
CLARTG generates a plane rotation with real cosine and complex sine.
Definition: clartg.f:105
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: claset.f:108
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
CGGHRD
Definition: cgghrd.f:206