LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ zlarot()

subroutine zlarot ( logical  LROWS,
logical  LLEFT,
logical  LRIGHT,
integer  NL,
complex*16  C,
complex*16  S,
complex*16, dimension( * )  A,
integer  LDA,
complex*16  XLEFT,
complex*16  XRIGHT 
)

ZLAROT

Purpose:
    ZLAROT applies a (Givens) rotation to two adjacent rows or
    columns, where one element of the first and/or last column/row
    for use on matrices stored in some format other than GE, so
    that elements of the matrix may be used or modified for which
    no array element is provided.

    One example is a symmetric matrix in SB format (bandwidth=4), for
    which UPLO='L':  Two adjacent rows will have the format:

    row j:     C> C> C> C> C> .  .  .  .
    row j+1:      C> C> C> C> C> .  .  .  .

    '*' indicates elements for which storage is provided,
    '.' indicates elements for which no storage is provided, but
    are not necessarily zero; their values are determined by
    symmetry.  ' ' indicates elements which are necessarily zero,
     and have no storage provided.

    Those columns which have two '*'s can be handled by DROT.
    Those columns which have no '*'s can be ignored, since as long
    as the Givens rotations are carefully applied to preserve
    symmetry, their values are determined.
    Those columns which have one '*' have to be handled separately,
    by using separate variables "p" and "q":

    row j:     C> C> C> C> C> p  .  .  .
    row j+1:   q  C> C> C> C> C> .  .  .  .

    The element p would have to be set correctly, then that column
    is rotated, setting p to its new value.  The next call to
    ZLAROT would rotate columns j and j+1, using p, and restore
    symmetry.  The element q would start out being zero, and be
    made non-zero by the rotation.  Later, rotations would presumably
    be chosen to zero q out.

    Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
    ------- ------- ---------

      General dense matrix:

              CALL ZLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
                      A(i,1),LDA, DUMMY, DUMMY)

      General banded matrix in GB format:

              j = MAX(1, i-KL )
              NL = MIN( N, i+KU+1 ) + 1-j
              CALL ZLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
                      A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )

              [ note that i+1-j is just MIN(i,KL+1) ]

      Symmetric banded matrix in SY format, bandwidth K,
      lower triangle only:

              j = MAX(1, i-K )
              NL = MIN( K+1, i ) + 1
              CALL ZLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
                      A(i,j), LDA, XLEFT, XRIGHT )

      Same, but upper triangle only:

              NL = MIN( K+1, N-i ) + 1
              CALL ZLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
                      A(i,i), LDA, XLEFT, XRIGHT )

      Symmetric banded matrix in SB format, bandwidth K,
      lower triangle only:

              [ same as for SY, except:]
                  . . . .
                      A(i+1-j,j), LDA-1, XLEFT, XRIGHT )

              [ note that i+1-j is just MIN(i,K+1) ]

      Same, but upper triangle only:
                  . . .
                      A(K+1,i), LDA-1, XLEFT, XRIGHT )

      Rotating columns is just the transpose of rotating rows, except
      for GB and SB: (rotating columns i and i+1)

      GB:
              j = MAX(1, i-KU )
              NL = MIN( N, i+KL+1 ) + 1-j
              CALL ZLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
                      A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )

              [note that KU+j+1-i is just MAX(1,KU+2-i)]

      SB: (upper triangle)

                   . . . . . .
                      A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )

      SB: (lower triangle)

                   . . . . . .
                      A(1,i),LDA-1, XTOP, XBOTTM )
  LROWS  - LOGICAL
           If .TRUE., then ZLAROT will rotate two rows.  If .FALSE.,
           then it will rotate two columns.
           Not modified.

  LLEFT  - LOGICAL
           If .TRUE., then XLEFT will be used instead of the
           corresponding element of A for the first element in the
           second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.)
           If .FALSE., then the corresponding element of A will be
           used.
           Not modified.

  LRIGHT - LOGICAL
           If .TRUE., then XRIGHT will be used instead of the
           corresponding element of A for the last element in the
           first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If
           .FALSE., then the corresponding element of A will be used.
           Not modified.

  NL     - INTEGER
           The length of the rows (if LROWS=.TRUE.) or columns (if
           LROWS=.FALSE.) to be rotated.  If XLEFT and/or XRIGHT are
           used, the columns/rows they are in should be included in
           NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at
           least 2.  The number of rows/columns to be rotated
           exclusive of those involving XLEFT and/or XRIGHT may
           not be negative, i.e., NL minus how many of LLEFT and
           LRIGHT are .TRUE. must be at least zero; if not, XERBLA
           will be called.
           Not modified.

  C, S   - COMPLEX*16
           Specify the Givens rotation to be applied.  If LROWS is
           true, then the matrix ( c  s )
                                 ( _  _ )
                                 (-s  c )  is applied from the left;
           if false, then the transpose (not conjugated) thereof is
           applied from the right.  Note that in contrast to the
           output of ZROTG or to most versions of ZROT, both C and S
           are complex.  For a Givens rotation, |C|**2 + |S|**2 should
           be 1, but this is not checked.
           Not modified.

  A      - COMPLEX*16 array.
           The array containing the rows/columns to be rotated.  The
           first element of A should be the upper left element to
           be rotated.
           Read and modified.

  LDA    - INTEGER
           The "effective" leading dimension of A.  If A contains
           a matrix stored in GE, HE, or SY format, then this is just
           the leading dimension of A as dimensioned in the calling
           routine.  If A contains a matrix stored in band (GB, HB, or
           SB) format, then this should be *one less* than the leading
           dimension used in the calling routine.  Thus, if A were
           dimensioned A(LDA,*) in ZLAROT, then A(1,j) would be the
           j-th element in the first of the two rows to be rotated,
           and A(2,j) would be the j-th in the second, regardless of
           how the array may be stored in the calling routine.  [A
           cannot, however, actually be dimensioned thus, since for
           band format, the row number may exceed LDA, which is not
           legal FORTRAN.]
           If LROWS=.TRUE., then LDA must be at least 1, otherwise
           it must be at least NL minus the number of .TRUE. values
           in XLEFT and XRIGHT.
           Not modified.

  XLEFT  - COMPLEX*16
           If LLEFT is .TRUE., then XLEFT will be used and modified
           instead of A(2,1) (if LROWS=.TRUE.) or A(1,2)
           (if LROWS=.FALSE.).
           Read and modified.

  XRIGHT - COMPLEX*16
           If LRIGHT is .TRUE., then XRIGHT will be used and modified
           instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1)
           (if LROWS=.FALSE.).
           Read and modified.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 227 of file zlarot.f.

229 *
230 * -- LAPACK auxiliary routine --
231 * -- LAPACK is a software package provided by Univ. of Tennessee, --
232 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
233 *
234 * .. Scalar Arguments ..
235  LOGICAL LLEFT, LRIGHT, LROWS
236  INTEGER LDA, NL
237  COMPLEX*16 C, S, XLEFT, XRIGHT
238 * ..
239 * .. Array Arguments ..
240  COMPLEX*16 A( * )
241 * ..
242 *
243 * =====================================================================
244 *
245 * .. Local Scalars ..
246  INTEGER IINC, INEXT, IX, IY, IYT, J, NT
247  COMPLEX*16 TEMPX
248 * ..
249 * .. Local Arrays ..
250  COMPLEX*16 XT( 2 ), YT( 2 )
251 * ..
252 * .. External Subroutines ..
253  EXTERNAL xerbla
254 * ..
255 * .. Intrinsic Functions ..
256  INTRINSIC dconjg
257 * ..
258 * .. Executable Statements ..
259 *
260 * Set up indices, arrays for ends
261 *
262  IF( lrows ) THEN
263  iinc = lda
264  inext = 1
265  ELSE
266  iinc = 1
267  inext = lda
268  END IF
269 *
270  IF( lleft ) THEN
271  nt = 1
272  ix = 1 + iinc
273  iy = 2 + lda
274  xt( 1 ) = a( 1 )
275  yt( 1 ) = xleft
276  ELSE
277  nt = 0
278  ix = 1
279  iy = 1 + inext
280  END IF
281 *
282  IF( lright ) THEN
283  iyt = 1 + inext + ( nl-1 )*iinc
284  nt = nt + 1
285  xt( nt ) = xright
286  yt( nt ) = a( iyt )
287  END IF
288 *
289 * Check for errors
290 *
291  IF( nl.LT.nt ) THEN
292  CALL xerbla( 'ZLAROT', 4 )
293  RETURN
294  END IF
295  IF( lda.LE.0 .OR. ( .NOT.lrows .AND. lda.LT.nl-nt ) ) THEN
296  CALL xerbla( 'ZLAROT', 8 )
297  RETURN
298  END IF
299 *
300 * Rotate
301 *
302 * ZROT( NL-NT, A(IX),IINC, A(IY),IINC, C, S ) with complex C, S
303 *
304  DO 10 j = 0, nl - nt - 1
305  tempx = c*a( ix+j*iinc ) + s*a( iy+j*iinc )
306  a( iy+j*iinc ) = -dconjg( s )*a( ix+j*iinc ) +
307  $ dconjg( c )*a( iy+j*iinc )
308  a( ix+j*iinc ) = tempx
309  10 CONTINUE
310 *
311 * ZROT( NT, XT,1, YT,1, C, S ) with complex C, S
312 *
313  DO 20 j = 1, nt
314  tempx = c*xt( j ) + s*yt( j )
315  yt( j ) = -dconjg( s )*xt( j ) + dconjg( c )*yt( j )
316  xt( j ) = tempx
317  20 CONTINUE
318 *
319 * Stuff values back into XLEFT, XRIGHT, etc.
320 *
321  IF( lleft ) THEN
322  a( 1 ) = xt( 1 )
323  xleft = yt( 1 )
324  END IF
325 *
326  IF( lright ) THEN
327  xright = xt( nt )
328  a( iyt ) = yt( nt )
329  END IF
330 *
331  RETURN
332 *
333 * End of ZLAROT
334 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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