LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ clarot()

 subroutine clarot ( logical lrows, logical lleft, logical lright, integer nl, complex c, complex s, complex, dimension( * ) a, integer lda, complex xleft, complex xright )

CLAROT

Purpose:
```    CLAROT 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 SROT.
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
CLAROT 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 CLAROT(.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 CLAROT( .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 CLAROT( .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 CLAROT( .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 CLAROT( .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 CLAROT 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
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 CROTG or to most versions of CROT, 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 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 CLAROT, 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
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
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 clarot.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 C, S, XLEFT, XRIGHT
238* ..
239* .. Array Arguments ..
240 COMPLEX A( * )
241* ..
242*
243* =====================================================================
244*
245* .. Local Scalars ..
246 INTEGER IINC, INEXT, IX, IY, IYT, J, NT
247 COMPLEX TEMPX
248* ..
249* .. Local Arrays ..
250 COMPLEX XT( 2 ), YT( 2 )
251* ..
252* .. External Subroutines ..
253 EXTERNAL xerbla
254* ..
255* .. Intrinsic Functions ..
256 INTRINSIC conjg
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( 'CLAROT', 4 )
293 RETURN
294 END IF
295 IF( lda.LE.0 .OR. ( .NOT.lrows .AND. lda.LT.nl-nt ) ) THEN
296 CALL xerbla( 'CLAROT', 8 )
297 RETURN
298 END IF
299*
300* Rotate
301*
302* CROT( 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 ) = -conjg( s )*a( ix+j*iinc ) +
307 \$ conjg( c )*a( iy+j*iinc )
308 a( ix+j*iinc ) = tempx
309 10 CONTINUE
310*
311* CROT( 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 ) = -conjg( s )*xt( j ) + conjg( 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 CLAROT
334*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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