1: *> \brief \b ZHETRD
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
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13: *> [ZIP]</a>
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, LWORK, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION D( * ), E( * )
29: * COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZHETRD 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: *> = 'U': Upper triangle of A is stored;
50: *> = 'L': Lower triangle of A is stored.
51: *> \endverbatim
52: *>
53: *> \param[in] N
54: *> \verbatim
55: *> N is INTEGER
56: *> The order of the matrix A. N >= 0.
57: *> \endverbatim
58: *>
59: *> \param[in,out] A
60: *> \verbatim
61: *> A is COMPLEX*16 array, dimension (LDA,N)
62: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
63: *> N-by-N upper triangular part of A contains the upper
64: *> triangular part of the matrix A, and the strictly lower
65: *> triangular part of A is not referenced. If UPLO = 'L', the
66: *> leading N-by-N lower triangular part of A contains the lower
67: *> triangular part of the matrix A, and the strictly upper
68: *> triangular part of A is not referenced.
69: *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
70: *> of A are overwritten by the corresponding elements of the
71: *> tridiagonal matrix T, and the elements above the first
72: *> superdiagonal, with the array TAU, represent the unitary
73: *> matrix Q as a product of elementary reflectors; if UPLO
74: *> = 'L', the diagonal and first subdiagonal of A are over-
75: *> written by the corresponding elements of the tridiagonal
76: *> matrix T, and the elements below the first subdiagonal, with
77: *> the array TAU, represent the unitary matrix Q as a product
78: *> of elementary reflectors. See Further Details.
79: *> \endverbatim
80: *>
81: *> \param[in] LDA
82: *> \verbatim
83: *> LDA is INTEGER
84: *> The leading dimension of the array A. LDA >= max(1,N).
85: *> \endverbatim
86: *>
87: *> \param[out] D
88: *> \verbatim
89: *> D is DOUBLE PRECISION array, dimension (N)
90: *> The diagonal elements of the tridiagonal matrix T:
91: *> D(i) = A(i,i).
92: *> \endverbatim
93: *>
94: *> \param[out] E
95: *> \verbatim
96: *> E is DOUBLE PRECISION array, dimension (N-1)
97: *> The off-diagonal elements of the tridiagonal matrix T:
98: *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
99: *> \endverbatim
100: *>
101: *> \param[out] TAU
102: *> \verbatim
103: *> TAU is COMPLEX*16 array, dimension (N-1)
104: *> The scalar factors of the elementary reflectors (see Further
105: *> Details).
106: *> \endverbatim
107: *>
108: *> \param[out] WORK
109: *> \verbatim
110: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
111: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
112: *> \endverbatim
113: *>
114: *> \param[in] LWORK
115: *> \verbatim
116: *> LWORK is INTEGER
117: *> The dimension of the array WORK. LWORK >= 1.
118: *> For optimum performance LWORK >= N*NB, where NB is the
119: *> optimal blocksize.
120: *>
121: *> If LWORK = -1, then a workspace query is assumed; the routine
122: *> only calculates the optimal size of the WORK array, returns
123: *> this value as the first entry of the WORK array, and no error
124: *> message related to LWORK is issued by XERBLA.
125: *> \endverbatim
126: *>
127: *> \param[out] INFO
128: *> \verbatim
129: *> INFO is INTEGER
130: *> = 0: successful exit
131: *> < 0: if INFO = -i, the i-th argument had an illegal value
132: *> \endverbatim
133: *
134: * Authors:
135: * ========
136: *
137: *> \author Univ. of Tennessee
138: *> \author Univ. of California Berkeley
139: *> \author Univ. of Colorado Denver
140: *> \author NAG Ltd.
141: *
142: *> \ingroup complex16HEcomputational
143: *
144: *> \par Further Details:
145: * =====================
146: *>
147: *> \verbatim
148: *>
149: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
150: *> reflectors
151: *>
152: *> Q = H(n-1) . . . H(2) H(1).
153: *>
154: *> Each H(i) has the form
155: *>
156: *> H(i) = I - tau * v * v**H
157: *>
158: *> where tau is a complex scalar, and v is a complex vector with
159: *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
160: *> A(1:i-1,i+1), and tau in TAU(i).
161: *>
162: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
163: *> reflectors
164: *>
165: *> Q = H(1) H(2) . . . H(n-1).
166: *>
167: *> Each H(i) has the form
168: *>
169: *> H(i) = I - tau * v * v**H
170: *>
171: *> where tau is a complex scalar, and v is a complex vector with
172: *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
173: *> and tau in TAU(i).
174: *>
175: *> The contents of A on exit are illustrated by the following examples
176: *> with n = 5:
177: *>
178: *> if UPLO = 'U': if UPLO = 'L':
179: *>
180: *> ( d e v2 v3 v4 ) ( d )
181: *> ( d e v3 v4 ) ( e d )
182: *> ( d e v4 ) ( v1 e d )
183: *> ( d e ) ( v1 v2 e d )
184: *> ( d ) ( v1 v2 v3 e d )
185: *>
186: *> where d and e denote diagonal and off-diagonal elements of T, and vi
187: *> denotes an element of the vector defining H(i).
188: *> \endverbatim
189: *>
190: * =====================================================================
191: SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
192: *
193: * -- LAPACK computational routine --
194: * -- LAPACK is a software package provided by Univ. of Tennessee, --
195: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196: *
197: * .. Scalar Arguments ..
198: CHARACTER UPLO
199: INTEGER INFO, LDA, LWORK, N
200: * ..
201: * .. Array Arguments ..
202: DOUBLE PRECISION D( * ), E( * )
203: COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
204: * ..
205: *
206: * =====================================================================
207: *
208: * .. Parameters ..
209: DOUBLE PRECISION ONE
210: PARAMETER ( ONE = 1.0D+0 )
211: COMPLEX*16 CONE
212: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
213: * ..
214: * .. Local Scalars ..
215: LOGICAL LQUERY, UPPER
216: INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
217: $ NBMIN, NX
218: * ..
219: * .. External Subroutines ..
220: EXTERNAL XERBLA, ZHER2K, ZHETD2, ZLATRD
221: * ..
222: * .. Intrinsic Functions ..
223: INTRINSIC MAX
224: * ..
225: * .. External Functions ..
226: LOGICAL LSAME
227: INTEGER ILAENV
228: EXTERNAL LSAME, ILAENV
229: * ..
230: * .. Executable Statements ..
231: *
232: * Test the input parameters
233: *
234: INFO = 0
235: UPPER = LSAME( UPLO, 'U' )
236: LQUERY = ( LWORK.EQ.-1 )
237: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
238: INFO = -1
239: ELSE IF( N.LT.0 ) THEN
240: INFO = -2
241: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
242: INFO = -4
243: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
244: INFO = -9
245: END IF
246: *
247: IF( INFO.EQ.0 ) THEN
248: *
249: * Determine the block size.
250: *
251: NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
252: LWKOPT = N*NB
253: WORK( 1 ) = LWKOPT
254: END IF
255: *
256: IF( INFO.NE.0 ) THEN
257: CALL XERBLA( 'ZHETRD', -INFO )
258: RETURN
259: ELSE IF( LQUERY ) THEN
260: RETURN
261: END IF
262: *
263: * Quick return if possible
264: *
265: IF( N.EQ.0 ) THEN
266: WORK( 1 ) = 1
267: RETURN
268: END IF
269: *
270: NX = N
271: IWS = 1
272: IF( NB.GT.1 .AND. NB.LT.N ) THEN
273: *
274: * Determine when to cross over from blocked to unblocked code
275: * (last block is always handled by unblocked code).
276: *
277: NX = MAX( NB, ILAENV( 3, 'ZHETRD', UPLO, N, -1, -1, -1 ) )
278: IF( NX.LT.N ) THEN
279: *
280: * Determine if workspace is large enough for blocked code.
281: *
282: LDWORK = N
283: IWS = LDWORK*NB
284: IF( LWORK.LT.IWS ) THEN
285: *
286: * Not enough workspace to use optimal NB: determine the
287: * minimum value of NB, and reduce NB or force use of
288: * unblocked code by setting NX = N.
289: *
290: NB = MAX( LWORK / LDWORK, 1 )
291: NBMIN = ILAENV( 2, 'ZHETRD', UPLO, N, -1, -1, -1 )
292: IF( NB.LT.NBMIN )
293: $ NX = N
294: END IF
295: ELSE
296: NX = N
297: END IF
298: ELSE
299: NB = 1
300: END IF
301: *
302: IF( UPPER ) THEN
303: *
304: * Reduce the upper triangle of A.
305: * Columns 1:kk are handled by the unblocked method.
306: *
307: KK = N - ( ( N-NX+NB-1 ) / NB )*NB
308: DO 20 I = N - NB + 1, KK + 1, -NB
309: *
310: * Reduce columns i:i+nb-1 to tridiagonal form and form the
311: * matrix W which is needed to update the unreduced part of
312: * the matrix
313: *
314: CALL ZLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
315: $ LDWORK )
316: *
317: * Update the unreduced submatrix A(1:i-1,1:i-1), using an
318: * update of the form: A := A - V*W**H - W*V**H
319: *
320: CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE,
321: $ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA )
322: *
323: * Copy superdiagonal elements back into A, and diagonal
324: * elements into D
325: *
326: DO 10 J = I, I + NB - 1
327: A( J-1, J ) = E( J-1 )
328: D( J ) = DBLE( A( J, J ) )
329: 10 CONTINUE
330: 20 CONTINUE
331: *
332: * Use unblocked code to reduce the last or only block
333: *
334: CALL ZHETD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
335: ELSE
336: *
337: * Reduce the lower triangle of A
338: *
339: DO 40 I = 1, N - NX, NB
340: *
341: * Reduce columns i:i+nb-1 to tridiagonal form and form the
342: * matrix W which is needed to update the unreduced part of
343: * the matrix
344: *
345: CALL ZLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
346: $ TAU( I ), WORK, LDWORK )
347: *
348: * Update the unreduced submatrix A(i+nb:n,i+nb:n), using
349: * an update of the form: A := A - V*W**H - W*V**H
350: *
351: CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE,
352: $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
353: $ A( I+NB, I+NB ), LDA )
354: *
355: * Copy subdiagonal elements back into A, and diagonal
356: * elements into D
357: *
358: DO 30 J = I, I + NB - 1
359: A( J+1, J ) = E( J )
360: D( J ) = DBLE( A( J, J ) )
361: 30 CONTINUE
362: 40 CONTINUE
363: *
364: * Use unblocked code to reduce the last or only block
365: *
366: CALL ZHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
367: $ TAU( I ), IINFO )
368: END IF
369: *
370: WORK( 1 ) = LWKOPT
371: RETURN
372: *
373: * End of ZHETRD
374: *
375: END
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