LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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zlarot.f
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1*> \brief \b ZLAROT
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8* Definition:
9* ===========
10*
11* SUBROUTINE ZLAROT( LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT,
12* XRIGHT )
13*
14* .. Scalar Arguments ..
15* LOGICAL LLEFT, LRIGHT, LROWS
16* INTEGER LDA, NL
17* COMPLEX*16 C, S, XLEFT, XRIGHT
18* ..
19* .. Array Arguments ..
20* COMPLEX*16 A( * )
21* ..
22*
23*
24*> \par Purpose:
25* =============
26*>
27*> \verbatim
28*>
29*> ZLAROT applies a (Givens) rotation to two adjacent rows or
30*> columns, where one element of the first and/or last column/row
31*> for use on matrices stored in some format other than GE, so
32*> that elements of the matrix may be used or modified for which
33*> no array element is provided.
34*>
35*> One example is a symmetric matrix in SB format (bandwidth=4), for
36*> which UPLO='L': Two adjacent rows will have the format:
37*>
38*> row j: C> C> C> C> C> . . . .
39*> row j+1: C> C> C> C> C> . . . .
40*>
41*> '*' indicates elements for which storage is provided,
42*> '.' indicates elements for which no storage is provided, but
43*> are not necessarily zero; their values are determined by
44*> symmetry. ' ' indicates elements which are necessarily zero,
45*> and have no storage provided.
46*>
47*> Those columns which have two '*'s can be handled by DROT.
48*> Those columns which have no '*'s can be ignored, since as long
49*> as the Givens rotations are carefully applied to preserve
50*> symmetry, their values are determined.
51*> Those columns which have one '*' have to be handled separately,
52*> by using separate variables "p" and "q":
53*>
54*> row j: C> C> C> C> C> p . . .
55*> row j+1: q C> C> C> C> C> . . . .
56*>
57*> The element p would have to be set correctly, then that column
58*> is rotated, setting p to its new value. The next call to
59*> ZLAROT would rotate columns j and j+1, using p, and restore
60*> symmetry. The element q would start out being zero, and be
61*> made non-zero by the rotation. Later, rotations would presumably
62*> be chosen to zero q out.
63*>
64*> Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
65*> ------- ------- ---------
66*>
67*> General dense matrix:
68*>
69*> CALL ZLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
70*> A(i,1),LDA, DUMMY, DUMMY)
71*>
72*> General banded matrix in GB format:
73*>
74*> j = MAX(1, i-KL )
75*> NL = MIN( N, i+KU+1 ) + 1-j
76*> CALL ZLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
77*> A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )
78*>
79*> [ note that i+1-j is just MIN(i,KL+1) ]
80*>
81*> Symmetric banded matrix in SY format, bandwidth K,
82*> lower triangle only:
83*>
84*> j = MAX(1, i-K )
85*> NL = MIN( K+1, i ) + 1
86*> CALL ZLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
87*> A(i,j), LDA, XLEFT, XRIGHT )
88*>
89*> Same, but upper triangle only:
90*>
91*> NL = MIN( K+1, N-i ) + 1
92*> CALL ZLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
93*> A(i,i), LDA, XLEFT, XRIGHT )
94*>
95*> Symmetric banded matrix in SB format, bandwidth K,
96*> lower triangle only:
97*>
98*> [ same as for SY, except:]
99*> . . . .
100*> A(i+1-j,j), LDA-1, XLEFT, XRIGHT )
101*>
102*> [ note that i+1-j is just MIN(i,K+1) ]
103*>
104*> Same, but upper triangle only:
105*> . . .
106*> A(K+1,i), LDA-1, XLEFT, XRIGHT )
107*>
108*> Rotating columns is just the transpose of rotating rows, except
109*> for GB and SB: (rotating columns i and i+1)
110*>
111*> GB:
112*> j = MAX(1, i-KU )
113*> NL = MIN( N, i+KL+1 ) + 1-j
114*> CALL ZLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
115*> A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )
116*>
117*> [note that KU+j+1-i is just MAX(1,KU+2-i)]
118*>
119*> SB: (upper triangle)
120*>
121*> . . . . . .
122*> A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )
123*>
124*> SB: (lower triangle)
125*>
126*> . . . . . .
127*> A(1,i),LDA-1, XTOP, XBOTTM )
128*> \endverbatim
129*
130* Arguments:
131* ==========
132*
133*> \verbatim
134*> LROWS - LOGICAL
135*> If .TRUE., then ZLAROT will rotate two rows. If .FALSE.,
136*> then it will rotate two columns.
137*> Not modified.
138*>
139*> LLEFT - LOGICAL
140*> If .TRUE., then XLEFT will be used instead of the
141*> corresponding element of A for the first element in the
142*> second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.)
143*> If .FALSE., then the corresponding element of A will be
144*> used.
145*> Not modified.
146*>
147*> LRIGHT - LOGICAL
148*> If .TRUE., then XRIGHT will be used instead of the
149*> corresponding element of A for the last element in the
150*> first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If
151*> .FALSE., then the corresponding element of A will be used.
152*> Not modified.
153*>
154*> NL - INTEGER
155*> The length of the rows (if LROWS=.TRUE.) or columns (if
156*> LROWS=.FALSE.) to be rotated. If XLEFT and/or XRIGHT are
157*> used, the columns/rows they are in should be included in
158*> NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at
159*> least 2. The number of rows/columns to be rotated
160*> exclusive of those involving XLEFT and/or XRIGHT may
161*> not be negative, i.e., NL minus how many of LLEFT and
162*> LRIGHT are .TRUE. must be at least zero; if not, XERBLA
163*> will be called.
164*> Not modified.
165*>
166*> C, S - COMPLEX*16
167*> Specify the Givens rotation to be applied. If LROWS is
168*> true, then the matrix ( c s )
169*> ( _ _ )
170*> (-s c ) is applied from the left;
171*> if false, then the transpose (not conjugated) thereof is
172*> applied from the right. Note that in contrast to the
173*> output of ZROTG or to most versions of ZROT, both C and S
174*> are complex. For a Givens rotation, |C|**2 + |S|**2 should
175*> be 1, but this is not checked.
176*> Not modified.
177*>
178*> A - COMPLEX*16 array.
179*> The array containing the rows/columns to be rotated. The
180*> first element of A should be the upper left element to
181*> be rotated.
182*> Read and modified.
183*>
184*> LDA - INTEGER
185*> The "effective" leading dimension of A. If A contains
186*> a matrix stored in GE, HE, or SY format, then this is just
187*> the leading dimension of A as dimensioned in the calling
188*> routine. If A contains a matrix stored in band (GB, HB, or
189*> SB) format, then this should be *one less* than the leading
190*> dimension used in the calling routine. Thus, if A were
191*> dimensioned A(LDA,*) in ZLAROT, then A(1,j) would be the
192*> j-th element in the first of the two rows to be rotated,
193*> and A(2,j) would be the j-th in the second, regardless of
194*> how the array may be stored in the calling routine. [A
195*> cannot, however, actually be dimensioned thus, since for
196*> band format, the row number may exceed LDA, which is not
197*> legal FORTRAN.]
198*> If LROWS=.TRUE., then LDA must be at least 1, otherwise
199*> it must be at least NL minus the number of .TRUE. values
200*> in XLEFT and XRIGHT.
201*> Not modified.
202*>
203*> XLEFT - COMPLEX*16
204*> If LLEFT is .TRUE., then XLEFT will be used and modified
205*> instead of A(2,1) (if LROWS=.TRUE.) or A(1,2)
206*> (if LROWS=.FALSE.).
207*> Read and modified.
208*>
209*> XRIGHT - COMPLEX*16
210*> If LRIGHT is .TRUE., then XRIGHT will be used and modified
211*> instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1)
212*> (if LROWS=.FALSE.).
213*> Read and modified.
214*> \endverbatim
215*
216* Authors:
217* ========
218*
219*> \author Univ. of Tennessee
220*> \author Univ. of California Berkeley
221*> \author Univ. of Colorado Denver
222*> \author NAG Ltd.
223*
224*> \ingroup complex16_matgen
225*
226* =====================================================================
227 SUBROUTINE zlarot( LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT,
228 $ XRIGHT )
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*
335 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zlarot(lrows, lleft, lright, nl, c, s, a, lda, xleft, xright)
ZLAROT
Definition zlarot.f:229