LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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clatrd.f
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1*> \brief \b CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
22*
23* .. Scalar Arguments ..
24* CHARACTER UPLO
25* INTEGER LDA, LDW, N, NB
26* ..
27* .. Array Arguments ..
28* REAL E( * )
29* COMPLEX A( LDA, * ), TAU( * ), W( LDW, * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
39*> Hermitian tridiagonal form by a unitary similarity
40*> transformation Q**H * A * Q, and returns the matrices V and W which are
41*> needed to apply the transformation to the unreduced part of A.
42*>
43*> If UPLO = 'U', CLATRD reduces the last NB rows and columns of a
44*> matrix, of which the upper triangle is supplied;
45*> if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
46*> matrix, of which the lower triangle is supplied.
47*>
48*> This is an auxiliary routine called by CHETRD.
49*> \endverbatim
50*
51* Arguments:
52* ==========
53*
54*> \param[in] UPLO
55*> \verbatim
56*> UPLO is CHARACTER*1
57*> Specifies whether the upper or lower triangular part of the
58*> Hermitian matrix A is stored:
59*> = 'U': Upper triangular
60*> = 'L': Lower triangular
61*> \endverbatim
62*>
63*> \param[in] N
64*> \verbatim
65*> N is INTEGER
66*> The order of the matrix A.
67*> \endverbatim
68*>
69*> \param[in] NB
70*> \verbatim
71*> NB is INTEGER
72*> The number of rows and columns to be reduced.
73*> \endverbatim
74*>
75*> \param[in,out] A
76*> \verbatim
77*> A is COMPLEX array, dimension (LDA,N)
78*> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
79*> n-by-n upper triangular part of A contains the upper
80*> triangular part of the matrix A, and the strictly lower
81*> triangular part of A is not referenced. If UPLO = 'L', the
82*> leading n-by-n lower triangular part of A contains the lower
83*> triangular part of the matrix A, and the strictly upper
84*> triangular part of A is not referenced.
85*> On exit:
86*> if UPLO = 'U', the last NB columns have been reduced to
87*> tridiagonal form, with the diagonal elements overwriting
88*> the diagonal elements of A; the elements above the diagonal
89*> with the array TAU, represent the unitary matrix Q as a
90*> product of elementary reflectors;
91*> if UPLO = 'L', the first NB columns have been reduced to
92*> tridiagonal form, with the diagonal elements overwriting
93*> the diagonal elements of A; the elements below the diagonal
94*> with the array TAU, represent the unitary matrix Q as a
95*> product of elementary reflectors.
96*> See Further Details.
97*> \endverbatim
98*>
99*> \param[in] LDA
100*> \verbatim
101*> LDA is INTEGER
102*> The leading dimension of the array A. LDA >= max(1,N).
103*> \endverbatim
104*>
105*> \param[out] E
106*> \verbatim
107*> E is REAL array, dimension (N-1)
108*> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
109*> elements of the last NB columns of the reduced matrix;
110*> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
111*> the first NB columns of the reduced matrix.
112*> \endverbatim
113*>
114*> \param[out] TAU
115*> \verbatim
116*> TAU is COMPLEX array, dimension (N-1)
117*> The scalar factors of the elementary reflectors, stored in
118*> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
119*> See Further Details.
120*> \endverbatim
121*>
122*> \param[out] W
123*> \verbatim
124*> W is COMPLEX array, dimension (LDW,NB)
125*> The n-by-nb matrix W required to update the unreduced part
126*> of A.
127*> \endverbatim
128*>
129*> \param[in] LDW
130*> \verbatim
131*> LDW is INTEGER
132*> The leading dimension of the array W. LDW >= max(1,N).
133*> \endverbatim
134*
135* Authors:
136* ========
137*
138*> \author Univ. of Tennessee
139*> \author Univ. of California Berkeley
140*> \author Univ. of Colorado Denver
141*> \author NAG Ltd.
142*
143*> \ingroup latrd
144*
145*> \par Further Details:
146* =====================
147*>
148*> \verbatim
149*>
150*> If UPLO = 'U', the matrix Q is represented as a product of elementary
151*> reflectors
152*>
153*> Q = H(n) H(n-1) . . . H(n-nb+1).
154*>
155*> Each H(i) has the form
156*>
157*> H(i) = I - tau * v * v**H
158*>
159*> where tau is a complex scalar, and v is a complex vector with
160*> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
161*> and tau in TAU(i-1).
162*>
163*> If UPLO = 'L', the matrix Q is represented as a product of elementary
164*> reflectors
165*>
166*> Q = H(1) H(2) . . . H(nb).
167*>
168*> Each H(i) has the form
169*>
170*> H(i) = I - tau * v * v**H
171*>
172*> where tau is a complex scalar, and v is a complex vector with
173*> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
174*> and tau in TAU(i).
175*>
176*> The elements of the vectors v together form the n-by-nb matrix V
177*> which is needed, with W, to apply the transformation to the unreduced
178*> part of the matrix, using a Hermitian rank-2k update of the form:
179*> A := A - V*W**H - W*V**H.
180*>
181*> The contents of A on exit are illustrated by the following examples
182*> with n = 5 and nb = 2:
183*>
184*> if UPLO = 'U': if UPLO = 'L':
185*>
186*> ( a a a v4 v5 ) ( d )
187*> ( a a v4 v5 ) ( 1 d )
188*> ( a 1 v5 ) ( v1 1 a )
189*> ( d 1 ) ( v1 v2 a a )
190*> ( d ) ( v1 v2 a a a )
191*>
192*> where d denotes a diagonal element of the reduced matrix, a denotes
193*> an element of the original matrix that is unchanged, and vi denotes
194*> an element of the vector defining H(i).
195*> \endverbatim
196*>
197* =====================================================================
198 SUBROUTINE clatrd( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
199*
200* -- LAPACK auxiliary routine --
201* -- LAPACK is a software package provided by Univ. of Tennessee, --
202* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
203*
204* .. Scalar Arguments ..
205 CHARACTER UPLO
206 INTEGER LDA, LDW, N, NB
207* ..
208* .. Array Arguments ..
209 REAL E( * )
210 COMPLEX A( LDA, * ), TAU( * ), W( LDW, * )
211* ..
212*
213* =====================================================================
214*
215* .. Parameters ..
216 COMPLEX ZERO, ONE, HALF
217 parameter( zero = ( 0.0e+0, 0.0e+0 ),
218 \$ one = ( 1.0e+0, 0.0e+0 ),
219 \$ half = ( 0.5e+0, 0.0e+0 ) )
220* ..
221* .. Local Scalars ..
222 INTEGER I, IW
223 COMPLEX ALPHA
224* ..
225* .. External Subroutines ..
226 EXTERNAL caxpy, cgemv, chemv, clacgv, clarfg, cscal
227* ..
228* .. External Functions ..
229 LOGICAL LSAME
230 COMPLEX CDOTC
231 EXTERNAL lsame, cdotc
232* ..
233* .. Intrinsic Functions ..
234 INTRINSIC min, real
235* ..
236* .. Executable Statements ..
237*
238* Quick return if possible
239*
240 IF( n.LE.0 )
241 \$ RETURN
242*
243 IF( lsame( uplo, 'U' ) ) THEN
244*
245* Reduce last NB columns of upper triangle
246*
247 DO 10 i = n, n - nb + 1, -1
248 iw = i - n + nb
249 IF( i.LT.n ) THEN
250*
251* Update A(1:i,i)
252*
253 a( i, i ) = real( a( i, i ) )
254 CALL clacgv( n-i, w( i, iw+1 ), ldw )
255 CALL cgemv( 'No transpose', i, n-i, -one, a( 1, i+1 ),
256 \$ lda, w( i, iw+1 ), ldw, one, a( 1, i ), 1 )
257 CALL clacgv( n-i, w( i, iw+1 ), ldw )
258 CALL clacgv( n-i, a( i, i+1 ), lda )
259 CALL cgemv( 'No transpose', i, n-i, -one, w( 1, iw+1 ),
260 \$ ldw, a( i, i+1 ), lda, one, a( 1, i ), 1 )
261 CALL clacgv( n-i, a( i, i+1 ), lda )
262 a( i, i ) = real( a( i, i ) )
263 END IF
264 IF( i.GT.1 ) THEN
265*
266* Generate elementary reflector H(i) to annihilate
267* A(1:i-2,i)
268*
269 alpha = a( i-1, i )
270 CALL clarfg( i-1, alpha, a( 1, i ), 1, tau( i-1 ) )
271 e( i-1 ) = real( alpha )
272 a( i-1, i ) = one
273*
274* Compute W(1:i-1,i)
275*
276 CALL chemv( 'Upper', i-1, one, a, lda, a( 1, i ), 1,
277 \$ zero, w( 1, iw ), 1 )
278 IF( i.LT.n ) THEN
279 CALL cgemv( 'Conjugate transpose', i-1, n-i, one,
280 \$ w( 1, iw+1 ), ldw, a( 1, i ), 1, zero,
281 \$ w( i+1, iw ), 1 )
282 CALL cgemv( 'No transpose', i-1, n-i, -one,
283 \$ a( 1, i+1 ), lda, w( i+1, iw ), 1, one,
284 \$ w( 1, iw ), 1 )
285 CALL cgemv( 'Conjugate transpose', i-1, n-i, one,
286 \$ a( 1, i+1 ), lda, a( 1, i ), 1, zero,
287 \$ w( i+1, iw ), 1 )
288 CALL cgemv( 'No transpose', i-1, n-i, -one,
289 \$ w( 1, iw+1 ), ldw, w( i+1, iw ), 1, one,
290 \$ w( 1, iw ), 1 )
291 END IF
292 CALL cscal( i-1, tau( i-1 ), w( 1, iw ), 1 )
293 alpha = -half*tau( i-1 )*cdotc( i-1, w( 1, iw ), 1,
294 \$ a( 1, i ), 1 )
295 CALL caxpy( i-1, alpha, a( 1, i ), 1, w( 1, iw ), 1 )
296 END IF
297*
298 10 CONTINUE
299 ELSE
300*
301* Reduce first NB columns of lower triangle
302*
303 DO 20 i = 1, nb
304*
305* Update A(i:n,i)
306*
307 a( i, i ) = real( a( i, i ) )
308 CALL clacgv( i-1, w( i, 1 ), ldw )
309 CALL cgemv( 'No transpose', n-i+1, i-1, -one, a( i, 1 ),
310 \$ lda, w( i, 1 ), ldw, one, a( i, i ), 1 )
311 CALL clacgv( i-1, w( i, 1 ), ldw )
312 CALL clacgv( i-1, a( i, 1 ), lda )
313 CALL cgemv( 'No transpose', n-i+1, i-1, -one, w( i, 1 ),
314 \$ ldw, a( i, 1 ), lda, one, a( i, i ), 1 )
315 CALL clacgv( i-1, a( i, 1 ), lda )
316 a( i, i ) = real( a( i, i ) )
317 IF( i.LT.n ) THEN
318*
319* Generate elementary reflector H(i) to annihilate
320* A(i+2:n,i)
321*
322 alpha = a( i+1, i )
323 CALL clarfg( n-i, alpha, a( min( i+2, n ), i ), 1,
324 \$ tau( i ) )
325 e( i ) = real( alpha )
326 a( i+1, i ) = one
327*
328* Compute W(i+1:n,i)
329*
330 CALL chemv( 'Lower', n-i, one, a( i+1, i+1 ), lda,
331 \$ a( i+1, i ), 1, zero, w( i+1, i ), 1 )
332 CALL cgemv( 'Conjugate transpose', n-i, i-1, one,
333 \$ w( i+1, 1 ), ldw, a( i+1, i ), 1, zero,
334 \$ w( 1, i ), 1 )
335 CALL cgemv( 'No transpose', n-i, i-1, -one, a( i+1, 1 ),
336 \$ lda, w( 1, i ), 1, one, w( i+1, i ), 1 )
337 CALL cgemv( 'Conjugate transpose', n-i, i-1, one,
338 \$ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
339 \$ w( 1, i ), 1 )
340 CALL cgemv( 'No transpose', n-i, i-1, -one, w( i+1, 1 ),
341 \$ ldw, w( 1, i ), 1, one, w( i+1, i ), 1 )
342 CALL cscal( n-i, tau( i ), w( i+1, i ), 1 )
343 alpha = -half*tau( i )*cdotc( n-i, w( i+1, i ), 1,
344 \$ a( i+1, i ), 1 )
345 CALL caxpy( n-i, alpha, a( i+1, i ), 1, w( i+1, i ), 1 )
346 END IF
347*
348 20 CONTINUE
349 END IF
350*
351 RETURN
352*
353* End of CLATRD
354*
355 END
subroutine caxpy(n, ca, cx, incx, cy, incy)
CAXPY
Definition caxpy.f:88
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
subroutine chemv(uplo, n, alpha, a, lda, x, incx, beta, y, incy)
CHEMV
Definition chemv.f:154
subroutine clacgv(n, x, incx)
CLACGV conjugates a complex vector.
Definition clacgv.f:74
subroutine clarfg(n, alpha, x, incx, tau)
CLARFG generates an elementary reflector (Householder matrix).
Definition clarfg.f:106
subroutine clatrd(uplo, n, nb, a, lda, e, tau, w, ldw)
CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal fo...
Definition clatrd.f:199
subroutine cscal(n, ca, cx, incx)
CSCAL
Definition cscal.f:78