LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
zhetd2.f
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1 *> \brief \b ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, LDA, N
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION D( * ), E( * )
29 * COMPLEX*16 A( LDA, * ), TAU( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> ZHETD2 reduces a complex Hermitian matrix A to real symmetric
39 *> tridiagonal form T by a unitary similarity transformation:
40 *> Q**H * A * Q = T.
41 *> \endverbatim
42 *
43 * Arguments:
44 * ==========
45 *
46 *> \param[in] UPLO
47 *> \verbatim
48 *> UPLO is CHARACTER*1
49 *> Specifies whether the upper or lower triangular part of the
50 *> Hermitian matrix A is stored:
51 *> = 'U': Upper triangular
52 *> = 'L': Lower triangular
53 *> \endverbatim
54 *>
55 *> \param[in] N
56 *> \verbatim
57 *> N is INTEGER
58 *> The order of the matrix A. N >= 0.
59 *> \endverbatim
60 *>
61 *> \param[in,out] A
62 *> \verbatim
63 *> A is COMPLEX*16 array, dimension (LDA,N)
64 *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
65 *> n-by-n upper triangular part of A contains the upper
66 *> triangular part of the matrix A, and the strictly lower
67 *> triangular part of A is not referenced. If UPLO = 'L', the
68 *> leading n-by-n lower triangular part of A contains the lower
69 *> triangular part of the matrix A, and the strictly upper
70 *> triangular part of A is not referenced.
71 *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
72 *> of A are overwritten by the corresponding elements of the
73 *> tridiagonal matrix T, and the elements above the first
74 *> superdiagonal, with the array TAU, represent the unitary
75 *> matrix Q as a product of elementary reflectors; if UPLO
76 *> = 'L', the diagonal and first subdiagonal of A are over-
77 *> written by the corresponding elements of the tridiagonal
78 *> matrix T, and the elements below the first subdiagonal, with
79 *> the array TAU, represent the unitary matrix Q as a product
80 *> of elementary reflectors. See Further Details.
81 *> \endverbatim
82 *>
83 *> \param[in] LDA
84 *> \verbatim
85 *> LDA is INTEGER
86 *> The leading dimension of the array A. LDA >= max(1,N).
87 *> \endverbatim
88 *>
89 *> \param[out] D
90 *> \verbatim
91 *> D is DOUBLE PRECISION array, dimension (N)
92 *> The diagonal elements of the tridiagonal matrix T:
93 *> D(i) = A(i,i).
94 *> \endverbatim
95 *>
96 *> \param[out] E
97 *> \verbatim
98 *> E is DOUBLE PRECISION array, dimension (N-1)
99 *> The off-diagonal elements of the tridiagonal matrix T:
100 *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
101 *> \endverbatim
102 *>
103 *> \param[out] TAU
104 *> \verbatim
105 *> TAU is COMPLEX*16 array, dimension (N-1)
106 *> The scalar factors of the elementary reflectors (see Further
107 *> Details).
108 *> \endverbatim
109 *>
110 *> \param[out] INFO
111 *> \verbatim
112 *> INFO is INTEGER
113 *> = 0: successful exit
114 *> < 0: if INFO = -i, the i-th argument had an illegal value.
115 *> \endverbatim
116 *
117 * Authors:
118 * ========
119 *
120 *> \author Univ. of Tennessee
121 *> \author Univ. of California Berkeley
122 *> \author Univ. of Colorado Denver
123 *> \author NAG Ltd.
124 *
125 *> \ingroup complex16HEcomputational
126 *
127 *> \par Further Details:
128 * =====================
129 *>
130 *> \verbatim
131 *>
132 *> If UPLO = 'U', the matrix Q is represented as a product of elementary
133 *> reflectors
134 *>
135 *> Q = H(n-1) . . . H(2) H(1).
136 *>
137 *> Each H(i) has the form
138 *>
139 *> H(i) = I - tau * v * v**H
140 *>
141 *> where tau is a complex scalar, and v is a complex vector with
142 *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
143 *> A(1:i-1,i+1), and tau in TAU(i).
144 *>
145 *> If UPLO = 'L', the matrix Q is represented as a product of elementary
146 *> reflectors
147 *>
148 *> Q = H(1) H(2) . . . H(n-1).
149 *>
150 *> Each H(i) has the form
151 *>
152 *> H(i) = I - tau * v * v**H
153 *>
154 *> where tau is a complex scalar, and v is a complex vector with
155 *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
156 *> and tau in TAU(i).
157 *>
158 *> The contents of A on exit are illustrated by the following examples
159 *> with n = 5:
160 *>
161 *> if UPLO = 'U': if UPLO = 'L':
162 *>
163 *> ( d e v2 v3 v4 ) ( d )
164 *> ( d e v3 v4 ) ( e d )
165 *> ( d e v4 ) ( v1 e d )
166 *> ( d e ) ( v1 v2 e d )
167 *> ( d ) ( v1 v2 v3 e d )
168 *>
169 *> where d and e denote diagonal and off-diagonal elements of T, and vi
170 *> denotes an element of the vector defining H(i).
171 *> \endverbatim
172 *>
173 * =====================================================================
174  SUBROUTINE zhetd2( UPLO, N, A, LDA, D, E, TAU, INFO )
175 *
176 * -- LAPACK computational routine --
177 * -- LAPACK is a software package provided by Univ. of Tennessee, --
178 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
179 *
180 * .. Scalar Arguments ..
181  CHARACTER UPLO
182  INTEGER INFO, LDA, N
183 * ..
184 * .. Array Arguments ..
185  DOUBLE PRECISION D( * ), E( * )
186  COMPLEX*16 A( LDA, * ), TAU( * )
187 * ..
188 *
189 * =====================================================================
190 *
191 * .. Parameters ..
192  COMPLEX*16 ONE, ZERO, HALF
193  parameter( one = ( 1.0d+0, 0.0d+0 ),
194  $ zero = ( 0.0d+0, 0.0d+0 ),
195  $ half = ( 0.5d+0, 0.0d+0 ) )
196 * ..
197 * .. Local Scalars ..
198  LOGICAL UPPER
199  INTEGER I
200  COMPLEX*16 ALPHA, TAUI
201 * ..
202 * .. External Subroutines ..
203  EXTERNAL xerbla, zaxpy, zhemv, zher2, zlarfg
204 * ..
205 * .. External Functions ..
206  LOGICAL LSAME
207  COMPLEX*16 ZDOTC
208  EXTERNAL lsame, zdotc
209 * ..
210 * .. Intrinsic Functions ..
211  INTRINSIC dble, max, min
212 * ..
213 * .. Executable Statements ..
214 *
215 * Test the input parameters
216 *
217  info = 0
218  upper = lsame( uplo, 'U')
219  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
220  info = -1
221  ELSE IF( n.LT.0 ) THEN
222  info = -2
223  ELSE IF( lda.LT.max( 1, n ) ) THEN
224  info = -4
225  END IF
226  IF( info.NE.0 ) THEN
227  CALL xerbla( 'ZHETD2', -info )
228  RETURN
229  END IF
230 *
231 * Quick return if possible
232 *
233  IF( n.LE.0 )
234  $ RETURN
235 *
236  IF( upper ) THEN
237 *
238 * Reduce the upper triangle of A
239 *
240  a( n, n ) = dble( a( n, n ) )
241  DO 10 i = n - 1, 1, -1
242 *
243 * Generate elementary reflector H(i) = I - tau * v * v**H
244 * to annihilate A(1:i-1,i+1)
245 *
246  alpha = a( i, i+1 )
247  CALL zlarfg( i, alpha, a( 1, i+1 ), 1, taui )
248  e( i ) = dble( alpha )
249 *
250  IF( taui.NE.zero ) THEN
251 *
252 * Apply H(i) from both sides to A(1:i,1:i)
253 *
254  a( i, i+1 ) = one
255 *
256 * Compute x := tau * A * v storing x in TAU(1:i)
257 *
258  CALL zhemv( uplo, i, taui, a, lda, a( 1, i+1 ), 1, zero,
259  $ tau, 1 )
260 *
261 * Compute w := x - 1/2 * tau * (x**H * v) * v
262 *
263  alpha = -half*taui*zdotc( i, tau, 1, a( 1, i+1 ), 1 )
264  CALL zaxpy( i, alpha, a( 1, i+1 ), 1, tau, 1 )
265 *
266 * Apply the transformation as a rank-2 update:
267 * A := A - v * w**H - w * v**H
268 *
269  CALL zher2( uplo, i, -one, a( 1, i+1 ), 1, tau, 1, a,
270  $ lda )
271 *
272  ELSE
273  a( i, i ) = dble( a( i, i ) )
274  END IF
275  a( i, i+1 ) = e( i )
276  d( i+1 ) = dble( a( i+1, i+1 ) )
277  tau( i ) = taui
278  10 CONTINUE
279  d( 1 ) = dble( a( 1, 1 ) )
280  ELSE
281 *
282 * Reduce the lower triangle of A
283 *
284  a( 1, 1 ) = dble( a( 1, 1 ) )
285  DO 20 i = 1, n - 1
286 *
287 * Generate elementary reflector H(i) = I - tau * v * v**H
288 * to annihilate A(i+2:n,i)
289 *
290  alpha = a( i+1, i )
291  CALL zlarfg( n-i, alpha, a( min( i+2, n ), i ), 1, taui )
292  e( i ) = dble( alpha )
293 *
294  IF( taui.NE.zero ) THEN
295 *
296 * Apply H(i) from both sides to A(i+1:n,i+1:n)
297 *
298  a( i+1, i ) = one
299 *
300 * Compute x := tau * A * v storing y in TAU(i:n-1)
301 *
302  CALL zhemv( uplo, n-i, taui, a( i+1, i+1 ), lda,
303  $ a( i+1, i ), 1, zero, tau( i ), 1 )
304 *
305 * Compute w := x - 1/2 * tau * (x**H * v) * v
306 *
307  alpha = -half*taui*zdotc( n-i, tau( i ), 1, a( i+1, i ),
308  $ 1 )
309  CALL zaxpy( n-i, alpha, a( i+1, i ), 1, tau( i ), 1 )
310 *
311 * Apply the transformation as a rank-2 update:
312 * A := A - v * w**H - w * v**H
313 *
314  CALL zher2( uplo, n-i, -one, a( i+1, i ), 1, tau( i ), 1,
315  $ a( i+1, i+1 ), lda )
316 *
317  ELSE
318  a( i+1, i+1 ) = dble( a( i+1, i+1 ) )
319  END IF
320  a( i+1, i ) = e( i )
321  d( i ) = dble( a( i, i ) )
322  tau( i ) = taui
323  20 CONTINUE
324  d( n ) = dble( a( n, n ) )
325  END IF
326 *
327  RETURN
328 *
329 * End of ZHETD2
330 *
331  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zhemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZHEMV
Definition: zhemv.f:154
subroutine zher2(UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
ZHER2
Definition: zher2.f:150
subroutine zhetd2(UPLO, N, A, LDA, D, E, TAU, INFO)
ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transfo...
Definition: zhetd2.f:175
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106