 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

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.
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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