1: *> \brief \b ZLATRD 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: *
8: *> \htmlonly
9: *> Download ZLATRD + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlatrd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLATRD( 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: * DOUBLE PRECISION E( * )
29: * COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZLATRD 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', ZLATRD reduces the last NB rows and columns of a
44: *> matrix, of which the upper triangle is supplied;
45: *> if UPLO = 'L', ZLATRD 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 ZHETRD.
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*16 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 DOUBLE PRECISION 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*16 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*16 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 complex16OTHERauxiliary
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 ZLATRD( 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: DOUBLE PRECISION E( * )
210: COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
211: * ..
212: *
213: * =====================================================================
214: *
215: * .. Parameters ..
216: COMPLEX*16 ZERO, ONE, HALF
217: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
218: $ ONE = ( 1.0D+0, 0.0D+0 ),
219: $ HALF = ( 0.5D+0, 0.0D+0 ) )
220: * ..
221: * .. Local Scalars ..
222: INTEGER I, IW
223: COMPLEX*16 ALPHA
224: * ..
225: * .. External Subroutines ..
226: EXTERNAL ZAXPY, ZGEMV, ZHEMV, ZLACGV, ZLARFG, ZSCAL
227: * ..
228: * .. External Functions ..
229: LOGICAL LSAME
230: COMPLEX*16 ZDOTC
231: EXTERNAL LSAME, ZDOTC
232: * ..
233: * .. Intrinsic Functions ..
234: INTRINSIC DBLE, MIN
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 ) = DBLE( A( I, I ) )
254: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
255: CALL ZGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
256: $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
257: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
258: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
259: CALL ZGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
260: $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
261: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
262: A( I, I ) = DBLE( 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 ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) )
271: E( I-1 ) = DBLE( ALPHA )
272: A( I-1, I ) = ONE
273: *
274: * Compute W(1:i-1,i)
275: *
276: CALL ZHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
277: $ ZERO, W( 1, IW ), 1 )
278: IF( I.LT.N ) THEN
279: CALL ZGEMV( '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 ZGEMV( '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 ZGEMV( '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 ZGEMV( '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 ZSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
293: ALPHA = -HALF*TAU( I-1 )*ZDOTC( I-1, W( 1, IW ), 1,
294: $ A( 1, I ), 1 )
295: CALL ZAXPY( 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 ) = DBLE( A( I, I ) )
308: CALL ZLACGV( I-1, W( I, 1 ), LDW )
309: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
310: $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
311: CALL ZLACGV( I-1, W( I, 1 ), LDW )
312: CALL ZLACGV( I-1, A( I, 1 ), LDA )
313: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
314: $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
315: CALL ZLACGV( I-1, A( I, 1 ), LDA )
316: A( I, I ) = DBLE( 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 ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1,
324: $ TAU( I ) )
325: E( I ) = DBLE( ALPHA )
326: A( I+1, I ) = ONE
327: *
328: * Compute W(i+1:n,i)
329: *
330: CALL ZHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
331: $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
332: CALL ZGEMV( '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 ZGEMV( '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 ZGEMV( '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 ZGEMV( '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 ZSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
343: ALPHA = -HALF*TAU( I )*ZDOTC( N-I, W( I+1, I ), 1,
344: $ A( I+1, I ), 1 )
345: CALL ZAXPY( 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 ZLATRD
354: *
355: END
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