 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dgghrd()

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

DGGHRD

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Purpose:
``` DGGHRD reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal 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 orthogonal matrix Q to the left side
of the equation.

This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.

The orthogonal 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**T = (Q1*Q) * H * (Z1*Z)**T

Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then DGGHRD 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 orthogonal matrix Q is returned; = 'V': Q must contain an orthogonal 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 orthogonal matrix Z is returned; = 'V': Z must contain an orthogonal 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 DGGBAL; 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**T 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 DOUBLE PRECISION array, dimension (LDQ, N) On entry, if COMPQ = 'V', the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ, N) On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit, if COMPZ='I', the orthogonal 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 205 of file dgghrd.f.

207 *
208 * -- LAPACK computational routine --
209 * -- LAPACK is a software package provided by Univ. of Tennessee, --
210 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
211 *
212 * .. Scalar Arguments ..
213  CHARACTER COMPQ, COMPZ
214  INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
215 * ..
216 * .. Array Arguments ..
217  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
218  \$ Z( LDZ, * )
219 * ..
220 *
221 * =====================================================================
222 *
223 * .. Parameters ..
224  DOUBLE PRECISION ONE, ZERO
225  parameter( one = 1.0d+0, zero = 0.0d+0 )
226 * ..
227 * .. Local Scalars ..
228  LOGICAL ILQ, ILZ
229  INTEGER ICOMPQ, ICOMPZ, JCOL, JROW
230  DOUBLE PRECISION C, S, TEMP
231 * ..
232 * .. External Functions ..
233  LOGICAL LSAME
234  EXTERNAL lsame
235 * ..
236 * .. External Subroutines ..
237  EXTERNAL dlartg, dlaset, drot, xerbla
238 * ..
239 * .. Intrinsic Functions ..
240  INTRINSIC max
241 * ..
242 * .. Executable Statements ..
243 *
244 * Decode COMPQ
245 *
246  IF( lsame( compq, 'N' ) ) THEN
247  ilq = .false.
248  icompq = 1
249  ELSE IF( lsame( compq, 'V' ) ) THEN
250  ilq = .true.
251  icompq = 2
252  ELSE IF( lsame( compq, 'I' ) ) THEN
253  ilq = .true.
254  icompq = 3
255  ELSE
256  icompq = 0
257  END IF
258 *
259 * Decode COMPZ
260 *
261  IF( lsame( compz, 'N' ) ) THEN
262  ilz = .false.
263  icompz = 1
264  ELSE IF( lsame( compz, 'V' ) ) THEN
265  ilz = .true.
266  icompz = 2
267  ELSE IF( lsame( compz, 'I' ) ) THEN
268  ilz = .true.
269  icompz = 3
270  ELSE
271  icompz = 0
272  END IF
273 *
274 * Test the input parameters.
275 *
276  info = 0
277  IF( icompq.LE.0 ) THEN
278  info = -1
279  ELSE IF( icompz.LE.0 ) THEN
280  info = -2
281  ELSE IF( n.LT.0 ) THEN
282  info = -3
283  ELSE IF( ilo.LT.1 ) THEN
284  info = -4
285  ELSE IF( ihi.GT.n .OR. ihi.LT.ilo-1 ) THEN
286  info = -5
287  ELSE IF( lda.LT.max( 1, n ) ) THEN
288  info = -7
289  ELSE IF( ldb.LT.max( 1, n ) ) THEN
290  info = -9
291  ELSE IF( ( ilq .AND. ldq.LT.n ) .OR. ldq.LT.1 ) THEN
292  info = -11
293  ELSE IF( ( ilz .AND. ldz.LT.n ) .OR. ldz.LT.1 ) THEN
294  info = -13
295  END IF
296  IF( info.NE.0 ) THEN
297  CALL xerbla( 'DGGHRD', -info )
298  RETURN
299  END IF
300 *
301 * Initialize Q and Z if desired.
302 *
303  IF( icompq.EQ.3 )
304  \$ CALL dlaset( 'Full', n, n, zero, one, q, ldq )
305  IF( icompz.EQ.3 )
306  \$ CALL dlaset( 'Full', n, n, zero, one, z, ldz )
307 *
308 * Quick return if possible
309 *
310  IF( n.LE.1 )
311  \$ RETURN
312 *
313 * Zero out lower triangle of B
314 *
315  DO 20 jcol = 1, n - 1
316  DO 10 jrow = jcol + 1, n
317  b( jrow, jcol ) = zero
318  10 CONTINUE
319  20 CONTINUE
320 *
321 * Reduce A and B
322 *
323  DO 40 jcol = ilo, ihi - 2
324 *
325  DO 30 jrow = ihi, jcol + 2, -1
326 *
327 * Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL)
328 *
329  temp = a( jrow-1, jcol )
330  CALL dlartg( temp, a( jrow, jcol ), c, s,
331  \$ a( jrow-1, jcol ) )
332  a( jrow, jcol ) = zero
333  CALL drot( n-jcol, a( jrow-1, jcol+1 ), lda,
334  \$ a( jrow, jcol+1 ), lda, c, s )
335  CALL drot( n+2-jrow, b( jrow-1, jrow-1 ), ldb,
336  \$ b( jrow, jrow-1 ), ldb, c, s )
337  IF( ilq )
338  \$ CALL drot( n, q( 1, jrow-1 ), 1, q( 1, jrow ), 1, c, s )
339 *
340 * Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1)
341 *
342  temp = b( jrow, jrow )
343  CALL dlartg( temp, b( jrow, jrow-1 ), c, s,
344  \$ b( jrow, jrow ) )
345  b( jrow, jrow-1 ) = zero
346  CALL drot( ihi, a( 1, jrow ), 1, a( 1, jrow-1 ), 1, c, s )
347  CALL drot( jrow-1, b( 1, jrow ), 1, b( 1, jrow-1 ), 1, c,
348  \$ s )
349  IF( ilz )
350  \$ CALL drot( 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 DGGHRD
357 *
subroutine dlartg(f, g, c, s, r)
DLARTG generates a plane rotation with real cosine and real sine.
Definition: dlartg.f90:113
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:92
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