1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
|
SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* ZHETRD reduces a complex Hermitian matrix A to real symmetric
* tridiagonal form T by a unitary similarity transformation:
* Q**H * A * Q = T.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX*16 array, dimension (LDA,N)
* On entry, the Hermitian matrix A. If UPLO = 'U', the leading
* N-by-N upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced.
* On exit, if UPLO = 'U', the diagonal and first superdiagonal
* of A are overwritten by the corresponding elements of the
* tridiagonal matrix T, and the elements above the first
* superdiagonal, with the array TAU, represent the unitary
* matrix Q as a product of elementary reflectors; if UPLO
* = 'L', the diagonal and first subdiagonal of A are over-
* written by the corresponding elements of the tridiagonal
* matrix T, and the elements below the first subdiagonal, with
* the array TAU, represent the unitary matrix Q as a product
* of elementary reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* D (output) DOUBLE PRECISION array, dimension (N)
* The diagonal elements of the tridiagonal matrix T:
* D(i) = A(i,i).
*
* E (output) DOUBLE PRECISION array, dimension (N-1)
* The off-diagonal elements of the tridiagonal matrix T:
* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*
* TAU (output) COMPLEX*16 array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 1.
* For optimum performance LWORK >= N*NB, where NB is the
* optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* If UPLO = 'U', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(n-1) . . . H(2) H(1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
* A(1:i-1,i+1), and tau in TAU(i).
*
* If UPLO = 'L', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(1) H(2) . . . H(n-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
* and tau in TAU(i).
*
* The contents of A on exit are illustrated by the following examples
* with n = 5:
*
* if UPLO = 'U': if UPLO = 'L':
*
* ( d e v2 v3 v4 ) ( d )
* ( d e v3 v4 ) ( e d )
* ( d e v4 ) ( v1 e d )
* ( d e ) ( v1 v2 e d )
* ( d ) ( v1 v2 v3 e d )
*
* where d and e denote diagonal and off-diagonal elements of T, and vi
* denotes an element of the vector defining H(i).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZHER2K, ZHETD2, ZLATRD
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size.
*
NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
LWKOPT = N*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHETRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NX = N
IWS = 1
IF( NB.GT.1 .AND. NB.LT.N ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code).
*
NX = MAX( NB, ILAENV( 3, 'ZHETRD', UPLO, N, -1, -1, -1 ) )
IF( NX.LT.N ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code by setting NX = N.
*
NB = MAX( LWORK / LDWORK, 1 )
NBMIN = ILAENV( 2, 'ZHETRD', UPLO, N, -1, -1, -1 )
IF( NB.LT.NBMIN )
$ NX = N
END IF
ELSE
NX = N
END IF
ELSE
NB = 1
END IF
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* Columns 1:kk are handled by the unblocked method.
*
KK = N - ( ( N-NX+NB-1 ) / NB )*NB
DO 20 I = N - NB + 1, KK + 1, -NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL ZLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
$ LDWORK )
*
* Update the unreduced submatrix A(1:i-1,1:i-1), using an
* update of the form: A := A - V*W' - W*V'
*
CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE,
$ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA )
*
* Copy superdiagonal elements back into A, and diagonal
* elements into D
*
DO 10 J = I, I + NB - 1
A( J-1, J ) = E( J-1 )
D( J ) = A( J, J )
10 CONTINUE
20 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL ZHETD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
ELSE
*
* Reduce the lower triangle of A
*
DO 40 I = 1, N - NX, NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL ZLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
$ TAU( I ), WORK, LDWORK )
*
* Update the unreduced submatrix A(i+nb:n,i+nb:n), using
* an update of the form: A := A - V*W' - W*V'
*
CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE,
$ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
$ A( I+NB, I+NB ), LDA )
*
* Copy subdiagonal elements back into A, and diagonal
* elements into D
*
DO 30 J = I, I + NB - 1
A( J+1, J ) = E( J )
D( J ) = A( J, J )
30 CONTINUE
40 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL ZHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAU( I ), IINFO )
END IF
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of ZHETRD
*
END
|
