 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.

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.).

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.).

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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