LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ cgghrd()

subroutine cgghrd ( character  COMPQ,
character  COMPZ,
integer  N,
integer  ILO,
integer  IHI,
complex, dimension( lda, * )  A,
integer  LDA,
complex, dimension( ldb, * )  B,
integer  LDB,
complex, dimension( ldq, * )  Q,
integer  LDQ,
complex, dimension( ldz, * )  Z,
integer  LDZ,
integer  INFO 
)

CGGHRD

Download CGGHRD + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CGGHRD 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 CGGHRD reduces the original
 problem to generalized Hessenberg form.
Parameters
[in]COMPQ
          COMPQ is 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.
[in]COMPZ
          COMPZ is CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 unitary matrix Z is returned;
          = 'V': Z must contain a unitary matrix Z1 on entry,
                 and the product Z1*Z is returned.
[in]N
          N is INTEGER
          The order of the matrices A and B.  N >= 0.
[in]ILO
          ILO is INTEGER
[in]IHI
          IHI is 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 CGGBAL; 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.
[in,out]A
          A is COMPLEX 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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
[in,out]B
          B is COMPLEX 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.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
[in,out]Q
          Q is COMPLEX 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'.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
[in,out]Z
          Z is COMPLEX 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'.
[in]LDZ
          LDZ is INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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).

Definition at line 202 of file cgghrd.f.

204 *
205 * -- LAPACK computational routine --
206 * -- LAPACK is a software package provided by Univ. of Tennessee, --
207 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208 *
209 * .. Scalar Arguments ..
210  CHARACTER COMPQ, COMPZ
211  INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
212 * ..
213 * .. Array Arguments ..
214  COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
215  $ Z( LDZ, * )
216 * ..
217 *
218 * =====================================================================
219 *
220 * .. Parameters ..
221  COMPLEX CONE, CZERO
222  parameter( cone = ( 1.0e+0, 0.0e+0 ),
223  $ czero = ( 0.0e+0, 0.0e+0 ) )
224 * ..
225 * .. Local Scalars ..
226  LOGICAL ILQ, ILZ
227  INTEGER ICOMPQ, ICOMPZ, JCOL, JROW
228  REAL C
229  COMPLEX CTEMP, S
230 * ..
231 * .. External Functions ..
232  LOGICAL LSAME
233  EXTERNAL lsame
234 * ..
235 * .. External Subroutines ..
236  EXTERNAL clartg, claset, crot, xerbla
237 * ..
238 * .. Intrinsic Functions ..
239  INTRINSIC conjg, max
240 * ..
241 * .. Executable Statements ..
242 *
243 * Decode COMPQ
244 *
245  IF( lsame( compq, 'N' ) ) THEN
246  ilq = .false.
247  icompq = 1
248  ELSE IF( lsame( compq, 'V' ) ) THEN
249  ilq = .true.
250  icompq = 2
251  ELSE IF( lsame( compq, 'I' ) ) THEN
252  ilq = .true.
253  icompq = 3
254  ELSE
255  icompq = 0
256  END IF
257 *
258 * Decode COMPZ
259 *
260  IF( lsame( compz, 'N' ) ) THEN
261  ilz = .false.
262  icompz = 1
263  ELSE IF( lsame( compz, 'V' ) ) THEN
264  ilz = .true.
265  icompz = 2
266  ELSE IF( lsame( compz, 'I' ) ) THEN
267  ilz = .true.
268  icompz = 3
269  ELSE
270  icompz = 0
271  END IF
272 *
273 * Test the input parameters.
274 *
275  info = 0
276  IF( icompq.LE.0 ) THEN
277  info = -1
278  ELSE IF( icompz.LE.0 ) THEN
279  info = -2
280  ELSE IF( n.LT.0 ) THEN
281  info = -3
282  ELSE IF( ilo.LT.1 ) THEN
283  info = -4
284  ELSE IF( ihi.GT.n .OR. ihi.LT.ilo-1 ) THEN
285  info = -5
286  ELSE IF( lda.LT.max( 1, n ) ) THEN
287  info = -7
288  ELSE IF( ldb.LT.max( 1, n ) ) THEN
289  info = -9
290  ELSE IF( ( ilq .AND. ldq.LT.n ) .OR. ldq.LT.1 ) THEN
291  info = -11
292  ELSE IF( ( ilz .AND. ldz.LT.n ) .OR. ldz.LT.1 ) THEN
293  info = -13
294  END IF
295  IF( info.NE.0 ) THEN
296  CALL xerbla( 'CGGHRD', -info )
297  RETURN
298  END IF
299 *
300 * Initialize Q and Z if desired.
301 *
302  IF( icompq.EQ.3 )
303  $ CALL claset( 'Full', n, n, czero, cone, q, ldq )
304  IF( icompz.EQ.3 )
305  $ CALL claset( 'Full', n, n, czero, cone, z, ldz )
306 *
307 * Quick return if possible
308 *
309  IF( n.LE.1 )
310  $ RETURN
311 *
312 * Zero out lower triangle of B
313 *
314  DO 20 jcol = 1, n - 1
315  DO 10 jrow = jcol + 1, n
316  b( jrow, jcol ) = czero
317  10 CONTINUE
318  20 CONTINUE
319 *
320 * Reduce A and B
321 *
322  DO 40 jcol = ilo, ihi - 2
323 *
324  DO 30 jrow = ihi, jcol + 2, -1
325 *
326 * Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
327 *
328  ctemp = a( jrow-1, jcol )
329  CALL clartg( ctemp, a( jrow, jcol ), c, s,
330  $ a( jrow-1, jcol ) )
331  a( jrow, jcol ) = czero
332  CALL crot( n-jcol, a( jrow-1, jcol+1 ), lda,
333  $ a( jrow, jcol+1 ), lda, c, s )
334  CALL crot( n+2-jrow, b( jrow-1, jrow-1 ), ldb,
335  $ b( jrow, jrow-1 ), ldb, c, s )
336  IF( ilq )
337  $ CALL crot( n, q( 1, jrow-1 ), 1, q( 1, jrow ), 1, c,
338  $ conjg( s ) )
339 *
340 * Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
341 *
342  ctemp = b( jrow, jrow )
343  CALL clartg( ctemp, b( jrow, jrow-1 ), c, s,
344  $ b( jrow, jrow ) )
345  b( jrow, jrow-1 ) = czero
346  CALL crot( ihi, a( 1, jrow ), 1, a( 1, jrow-1 ), 1, c, s )
347  CALL crot( jrow-1, b( 1, jrow ), 1, b( 1, jrow-1 ), 1, c,
348  $ s )
349  IF( ilz )
350  $ CALL crot( n, z( 1, jrow ), 1, z( 1, jrow-1 ), 1, c, s )
351  30 CONTINUE
352  40 CONTINUE
353 *
354  RETURN
355 *
356 * End of CGGHRD
357 *
subroutine clartg(f, g, c, s, r)
CLARTG generates a plane rotation with real cosine and complex sine.
Definition: clartg.f90:118
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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:106
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:103
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