Annotation of rpl/lapack/lapack/zlahrd.f, revision 1.11
1.9 bertrand 1: *> \brief \b ZLAHRD
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAHRD + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahrd.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahrd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER K, LDA, LDT, LDY, N, NB
25: * ..
26: * .. Array Arguments ..
27: * COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
28: * $ Y( LDY, NB )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1)
38: *> matrix A so that elements below the k-th subdiagonal are zero. The
39: *> reduction is performed by a unitary similarity transformation
40: *> Q**H * A * Q. The routine returns the matrices V and T which determine
41: *> Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
42: *>
43: *> This is an OBSOLETE auxiliary routine.
44: *> This routine will be 'deprecated' in a future release.
45: *> Please use the new routine ZLAHR2 instead.
46: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] N
52: *> \verbatim
53: *> N is INTEGER
54: *> The order of the matrix A.
55: *> \endverbatim
56: *>
57: *> \param[in] K
58: *> \verbatim
59: *> K is INTEGER
60: *> The offset for the reduction. Elements below the k-th
61: *> subdiagonal in the first NB columns are reduced to zero.
62: *> \endverbatim
63: *>
64: *> \param[in] NB
65: *> \verbatim
66: *> NB is INTEGER
67: *> The number of columns to be reduced.
68: *> \endverbatim
69: *>
70: *> \param[in,out] A
71: *> \verbatim
72: *> A is COMPLEX*16 array, dimension (LDA,N-K+1)
73: *> On entry, the n-by-(n-k+1) general matrix A.
74: *> On exit, the elements on and above the k-th subdiagonal in
75: *> the first NB columns are overwritten with the corresponding
76: *> elements of the reduced matrix; the elements below the k-th
77: *> subdiagonal, with the array TAU, represent the matrix Q as a
78: *> product of elementary reflectors. The other columns of A are
79: *> unchanged. See Further Details.
80: *> \endverbatim
81: *>
82: *> \param[in] LDA
83: *> \verbatim
84: *> LDA is INTEGER
85: *> The leading dimension of the array A. LDA >= max(1,N).
86: *> \endverbatim
87: *>
88: *> \param[out] TAU
89: *> \verbatim
90: *> TAU is COMPLEX*16 array, dimension (NB)
91: *> The scalar factors of the elementary reflectors. See Further
92: *> Details.
93: *> \endverbatim
94: *>
95: *> \param[out] T
96: *> \verbatim
97: *> T is COMPLEX*16 array, dimension (LDT,NB)
98: *> The upper triangular matrix T.
99: *> \endverbatim
100: *>
101: *> \param[in] LDT
102: *> \verbatim
103: *> LDT is INTEGER
104: *> The leading dimension of the array T. LDT >= NB.
105: *> \endverbatim
106: *>
107: *> \param[out] Y
108: *> \verbatim
109: *> Y is COMPLEX*16 array, dimension (LDY,NB)
110: *> The n-by-nb matrix Y.
111: *> \endverbatim
112: *>
113: *> \param[in] LDY
114: *> \verbatim
115: *> LDY is INTEGER
116: *> The leading dimension of the array Y. LDY >= max(1,N).
117: *> \endverbatim
118: *
119: * Authors:
120: * ========
121: *
122: *> \author Univ. of Tennessee
123: *> \author Univ. of California Berkeley
124: *> \author Univ. of Colorado Denver
125: *> \author NAG Ltd.
126: *
127: *> \date November 2011
128: *
129: *> \ingroup complex16OTHERauxiliary
130: *
131: *> \par Further Details:
132: * =====================
133: *>
134: *> \verbatim
135: *>
136: *> The matrix Q is represented as a product of nb elementary reflectors
137: *>
138: *> Q = H(1) H(2) . . . H(nb).
139: *>
140: *> Each H(i) has the form
141: *>
142: *> H(i) = I - tau * v * v**H
143: *>
144: *> where tau is a complex scalar, and v is a complex vector with
145: *> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
146: *> A(i+k+1:n,i), and tau in TAU(i).
147: *>
148: *> The elements of the vectors v together form the (n-k+1)-by-nb matrix
149: *> V which is needed, with T and Y, to apply the transformation to the
150: *> unreduced part of the matrix, using an update of the form:
151: *> A := (I - V*T*V**H) * (A - Y*V**H).
152: *>
153: *> The contents of A on exit are illustrated by the following example
154: *> with n = 7, k = 3 and nb = 2:
155: *>
156: *> ( a h a a a )
157: *> ( a h a a a )
158: *> ( a h a a a )
159: *> ( h h a a a )
160: *> ( v1 h a a a )
161: *> ( v1 v2 a a a )
162: *> ( v1 v2 a a a )
163: *>
164: *> where a denotes an element of the original matrix A, h denotes a
165: *> modified element of the upper Hessenberg matrix H, and vi denotes an
166: *> element of the vector defining H(i).
167: *> \endverbatim
168: *>
169: * =====================================================================
1.1 bertrand 170: SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
171: *
1.9 bertrand 172: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 173: * -- LAPACK is a software package provided by Univ. of Tennessee, --
174: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 bertrand 175: * November 2011
1.1 bertrand 176: *
177: * .. Scalar Arguments ..
178: INTEGER K, LDA, LDT, LDY, N, NB
179: * ..
180: * .. Array Arguments ..
181: COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
182: $ Y( LDY, NB )
183: * ..
184: *
185: * =====================================================================
186: *
187: * .. Parameters ..
188: COMPLEX*16 ZERO, ONE
189: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
190: $ ONE = ( 1.0D+0, 0.0D+0 ) )
191: * ..
192: * .. Local Scalars ..
193: INTEGER I
194: COMPLEX*16 EI
195: * ..
196: * .. External Subroutines ..
197: EXTERNAL ZAXPY, ZCOPY, ZGEMV, ZLACGV, ZLARFG, ZSCAL,
198: $ ZTRMV
199: * ..
200: * .. Intrinsic Functions ..
201: INTRINSIC MIN
202: * ..
203: * .. Executable Statements ..
204: *
205: * Quick return if possible
206: *
207: IF( N.LE.1 )
208: $ RETURN
209: *
210: DO 10 I = 1, NB
211: IF( I.GT.1 ) THEN
212: *
213: * Update A(1:n,i)
214: *
1.8 bertrand 215: * Compute i-th column of A - Y * V**H
1.1 bertrand 216: *
217: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
218: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
219: $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
220: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
221: *
1.8 bertrand 222: * Apply I - V * T**H * V**H to this column (call it b) from the
1.1 bertrand 223: * left, using the last column of T as workspace
224: *
225: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
226: * ( V2 ) ( b2 )
227: *
228: * where V1 is unit lower triangular
229: *
1.8 bertrand 230: * w := V1**H * b1
1.1 bertrand 231: *
232: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
233: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1,
234: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
235: *
1.8 bertrand 236: * w := w + V2**H *b2
1.1 bertrand 237: *
238: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
239: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ONE,
240: $ T( 1, NB ), 1 )
241: *
1.8 bertrand 242: * w := T**H *w
1.1 bertrand 243: *
244: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1,
245: $ T, LDT, T( 1, NB ), 1 )
246: *
247: * b2 := b2 - V2*w
248: *
249: CALL ZGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
250: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
251: *
252: * b1 := b1 - V1*w
253: *
254: CALL ZTRMV( 'Lower', 'No transpose', 'Unit', I-1,
255: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
256: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
257: *
258: A( K+I-1, I-1 ) = EI
259: END IF
260: *
261: * Generate the elementary reflector H(i) to annihilate
262: * A(k+i+1:n,i)
263: *
264: EI = A( K+I, I )
265: CALL ZLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1,
266: $ TAU( I ) )
267: A( K+I, I ) = ONE
268: *
269: * Compute Y(1:n,i)
270: *
271: CALL ZGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
272: $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
273: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
274: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ),
275: $ 1 )
276: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
277: $ ONE, Y( 1, I ), 1 )
278: CALL ZSCAL( N, TAU( I ), Y( 1, I ), 1 )
279: *
280: * Compute T(1:i,i)
281: *
282: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
283: CALL ZTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
284: $ T( 1, I ), 1 )
285: T( I, I ) = TAU( I )
286: *
287: 10 CONTINUE
288: A( K+NB, NB ) = EI
289: *
290: RETURN
291: *
292: * End of ZLAHRD
293: *
294: END
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