LAPACK  3.8.0 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.

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.).

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.).
Date
December 2016

Definition at line 228 of file slarot.f.

228 *
229 * -- LAPACK auxiliary routine (version 3.7.0) --
230 * -- LAPACK is a software package provided by Univ. of Tennessee, --
231 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232 * December 2016
233 *
234 * .. Scalar Arguments ..
235  LOGICAL lleft, lright, lrows
236  INTEGER lda, nl
237  REAL c, s, xleft, xright
238 * ..
239 * .. Array Arguments ..
240  REAL a( * )
241 * ..
242 *
243 * =====================================================================
244 *
245 * .. Local Scalars ..
246  INTEGER iinc, inext, ix, iy, iyt, nt
247 * ..
248 * .. Local Arrays ..
249  REAL xt( 2 ), yt( 2 )
250 * ..
251 * .. External Subroutines ..
252  EXTERNAL srot, xerbla
253 * ..
254 * .. Executable Statements ..
255 *
256 * Set up indices, arrays for ends
257 *
258  IF( lrows ) THEN
259  iinc = lda
260  inext = 1
261  ELSE
262  iinc = 1
263  inext = lda
264  END IF
265 *
266  IF( lleft ) THEN
267  nt = 1
268  ix = 1 + iinc
269  iy = 2 + lda
270  xt( 1 ) = a( 1 )
271  yt( 1 ) = xleft
272  ELSE
273  nt = 0
274  ix = 1
275  iy = 1 + inext
276  END IF
277 *
278  IF( lright ) THEN
279  iyt = 1 + inext + ( nl-1 )*iinc
280  nt = nt + 1
281  xt( nt ) = xright
282  yt( nt ) = a( iyt )
283  END IF
284 *
285 * Check for errors
286 *
287  IF( nl.LT.nt ) THEN
288  CALL xerbla( 'SLAROT', 4 )
289  RETURN
290  END IF
291  IF( lda.LE.0 .OR. ( .NOT.lrows .AND. lda.LT.nl-nt ) ) THEN
292  CALL xerbla( 'SLAROT', 8 )
293  RETURN
294  END IF
295 *
296 * Rotate
297 *
298  CALL srot( nl-nt, a( ix ), iinc, a( iy ), iinc, c, s )
299  CALL srot( nt, xt, 1, yt, 1, c, s )
300 *
301 * Stuff values back into XLEFT, XRIGHT, etc.
302 *
303  IF( lleft ) THEN
304  a( 1 ) = xt( 1 )
305  xleft = yt( 1 )
306  END IF
307 *
308  IF( lright ) THEN
309  xright = xt( nt )
310  a( iyt ) = yt( nt )
311  END IF
312 *
313  RETURN
314 *
315 * End of SLAROT
316 *
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:94
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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