 LAPACK  3.9.1 LAPACK: Linear Algebra PACKage

## ◆ slarot()

 subroutine slarot ( logical LROWS, logical LLEFT, logical LRIGHT, integer NL, real C, real S, real, dimension( * ) A, integer LDA, real XLEFT, real XRIGHT )

SLAROT

Purpose:
```    SLAROT 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
SLAROT 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 SLAROT(.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 SLAROT( .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 SLAROT( .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 SLAROT( .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 SLAROT( .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 SLAROT 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   - REAL
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 thereof is applied from the
right.  For a Givens rotation, C**2 + S**2 should be 1,
but this is not checked.
Not modified.

A      - REAL 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 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 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 SLAROT, 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  - REAL
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 - REAL
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.```

Definition at line 224 of file slarot.f.

226 *
227 * -- LAPACK auxiliary routine --
228 * -- LAPACK is a software package provided by Univ. of Tennessee, --
229 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
230 *
231 * .. Scalar Arguments ..
232  LOGICAL LLEFT, LRIGHT, LROWS
233  INTEGER LDA, NL
234  REAL C, S, XLEFT, XRIGHT
235 * ..
236 * .. Array Arguments ..
237  REAL A( * )
238 * ..
239 *
240 * =====================================================================
241 *
242 * .. Local Scalars ..
243  INTEGER IINC, INEXT, IX, IY, IYT, NT
244 * ..
245 * .. Local Arrays ..
246  REAL XT( 2 ), YT( 2 )
247 * ..
248 * .. External Subroutines ..
249  EXTERNAL srot, xerbla
250 * ..
251 * .. Executable Statements ..
252 *
253 * Set up indices, arrays for ends
254 *
255  IF( lrows ) THEN
256  iinc = lda
257  inext = 1
258  ELSE
259  iinc = 1
260  inext = lda
261  END IF
262 *
263  IF( lleft ) THEN
264  nt = 1
265  ix = 1 + iinc
266  iy = 2 + lda
267  xt( 1 ) = a( 1 )
268  yt( 1 ) = xleft
269  ELSE
270  nt = 0
271  ix = 1
272  iy = 1 + inext
273  END IF
274 *
275  IF( lright ) THEN
276  iyt = 1 + inext + ( nl-1 )*iinc
277  nt = nt + 1
278  xt( nt ) = xright
279  yt( nt ) = a( iyt )
280  END IF
281 *
282 * Check for errors
283 *
284  IF( nl.LT.nt ) THEN
285  CALL xerbla( 'SLAROT', 4 )
286  RETURN
287  END IF
288  IF( lda.LE.0 .OR. ( .NOT.lrows .AND. lda.LT.nl-nt ) ) THEN
289  CALL xerbla( 'SLAROT', 8 )
290  RETURN
291  END IF
292 *
293 * Rotate
294 *
295  CALL srot( nl-nt, a( ix ), iinc, a( iy ), iinc, c, s )
296  CALL srot( nt, xt, 1, yt, 1, c, s )
297 *
298 * Stuff values back into XLEFT, XRIGHT, etc.
299 *
300  IF( lleft ) THEN
301  a( 1 ) = xt( 1 )
302  xleft = yt( 1 )
303  END IF
304 *
305  IF( lright ) THEN
306  xright = xt( nt )
307  a( iyt ) = yt( nt )
308  END IF
309 *
310  RETURN
311 *
312 * End of SLAROT
313 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:92
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