1: *> \brief \b ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAHR2 + dependencies
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAHR2( 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: *> ZLAHR2 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 an 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 auxiliary routine called by ZGEHRD.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] N
50: *> \verbatim
51: *> N is INTEGER
52: *> The order of the matrix A.
53: *> \endverbatim
54: *>
55: *> \param[in] K
56: *> \verbatim
57: *> K is INTEGER
58: *> The offset for the reduction. Elements below the k-th
59: *> subdiagonal in the first NB columns are reduced to zero.
60: *> K < N.
61: *> \endverbatim
62: *>
63: *> \param[in] NB
64: *> \verbatim
65: *> NB is INTEGER
66: *> The number of columns to be reduced.
67: *> \endverbatim
68: *>
69: *> \param[in,out] A
70: *> \verbatim
71: *> A is COMPLEX*16 array, dimension (LDA,N-K+1)
72: *> On entry, the n-by-(n-k+1) general matrix A.
73: *> On exit, the elements on and above the k-th subdiagonal in
74: *> the first NB columns are overwritten with the corresponding
75: *> elements of the reduced matrix; the elements below the k-th
76: *> subdiagonal, with the array TAU, represent the matrix Q as a
77: *> product of elementary reflectors. The other columns of A are
78: *> unchanged. 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] TAU
88: *> \verbatim
89: *> TAU is COMPLEX*16 array, dimension (NB)
90: *> The scalar factors of the elementary reflectors. See Further
91: *> Details.
92: *> \endverbatim
93: *>
94: *> \param[out] T
95: *> \verbatim
96: *> T is COMPLEX*16 array, dimension (LDT,NB)
97: *> The upper triangular matrix T.
98: *> \endverbatim
99: *>
100: *> \param[in] LDT
101: *> \verbatim
102: *> LDT is INTEGER
103: *> The leading dimension of the array T. LDT >= NB.
104: *> \endverbatim
105: *>
106: *> \param[out] Y
107: *> \verbatim
108: *> Y is COMPLEX*16 array, dimension (LDY,NB)
109: *> The n-by-nb matrix Y.
110: *> \endverbatim
111: *>
112: *> \param[in] LDY
113: *> \verbatim
114: *> LDY is INTEGER
115: *> The leading dimension of the array Y. LDY >= N.
116: *> \endverbatim
117: *
118: * Authors:
119: * ========
120: *
121: *> \author Univ. of Tennessee
122: *> \author Univ. of California Berkeley
123: *> \author Univ. of Colorado Denver
124: *> \author NAG Ltd.
125: *
126: *> \ingroup complex16OTHERauxiliary
127: *
128: *> \par Further Details:
129: * =====================
130: *>
131: *> \verbatim
132: *>
133: *> The matrix Q is represented as a product of nb elementary reflectors
134: *>
135: *> Q = H(1) H(2) . . . H(nb).
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(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
143: *> A(i+k+1:n,i), and tau in TAU(i).
144: *>
145: *> The elements of the vectors v together form the (n-k+1)-by-nb matrix
146: *> V which is needed, with T and Y, to apply the transformation to the
147: *> unreduced part of the matrix, using an update of the form:
148: *> A := (I - V*T*V**H) * (A - Y*V**H).
149: *>
150: *> The contents of A on exit are illustrated by the following example
151: *> with n = 7, k = 3 and nb = 2:
152: *>
153: *> ( a a a a a )
154: *> ( a a a a a )
155: *> ( a a a a a )
156: *> ( h h a a a )
157: *> ( v1 h a a a )
158: *> ( v1 v2 a a a )
159: *> ( v1 v2 a a a )
160: *>
161: *> where a denotes an element of the original matrix A, h denotes a
162: *> modified element of the upper Hessenberg matrix H, and vi denotes an
163: *> element of the vector defining H(i).
164: *>
165: *> This subroutine is a slight modification of LAPACK-3.0's ZLAHRD
166: *> incorporating improvements proposed by Quintana-Orti and Van de
167: *> Gejin. Note that the entries of A(1:K,2:NB) differ from those
168: *> returned by the original LAPACK-3.0's ZLAHRD routine. (This
169: *> subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)
170: *> \endverbatim
171: *
172: *> \par References:
173: * ================
174: *>
175: *> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
176: *> performance of reduction to Hessenberg form," ACM Transactions on
177: *> Mathematical Software, 32(2):180-194, June 2006.
178: *>
179: * =====================================================================
180: SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
181: *
182: * -- LAPACK auxiliary routine --
183: * -- LAPACK is a software package provided by Univ. of Tennessee, --
184: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
185: *
186: * .. Scalar Arguments ..
187: INTEGER K, LDA, LDT, LDY, N, NB
188: * ..
189: * .. Array Arguments ..
190: COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
191: $ Y( LDY, NB )
192: * ..
193: *
194: * =====================================================================
195: *
196: * .. Parameters ..
197: COMPLEX*16 ZERO, ONE
198: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
199: $ ONE = ( 1.0D+0, 0.0D+0 ) )
200: * ..
201: * .. Local Scalars ..
202: INTEGER I
203: COMPLEX*16 EI
204: * ..
205: * .. External Subroutines ..
206: EXTERNAL ZAXPY, ZCOPY, ZGEMM, ZGEMV, ZLACPY,
207: $ ZLARFG, ZSCAL, ZTRMM, ZTRMV, ZLACGV
208: * ..
209: * .. Intrinsic Functions ..
210: INTRINSIC MIN
211: * ..
212: * .. Executable Statements ..
213: *
214: * Quick return if possible
215: *
216: IF( N.LE.1 )
217: $ RETURN
218: *
219: DO 10 I = 1, NB
220: IF( I.GT.1 ) THEN
221: *
222: * Update A(K+1:N,I)
223: *
224: * Update I-th column of A - Y * V**H
225: *
226: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
227: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
228: $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
229: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
230: *
231: * Apply I - V * T**H * V**H to this column (call it b) from the
232: * left, using the last column of T as workspace
233: *
234: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
235: * ( V2 ) ( b2 )
236: *
237: * where V1 is unit lower triangular
238: *
239: * w := V1**H * b1
240: *
241: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
242: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT',
243: $ I-1, A( K+1, 1 ),
244: $ LDA, T( 1, NB ), 1 )
245: *
246: * w := w + V2**H * b2
247: *
248: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
249: $ ONE, A( K+I, 1 ),
250: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
251: *
252: * w := T**H * w
253: *
254: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',
255: $ I-1, T, LDT,
256: $ T( 1, NB ), 1 )
257: *
258: * b2 := b2 - V2*w
259: *
260: CALL ZGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
261: $ A( K+I, 1 ),
262: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
263: *
264: * b1 := b1 - V1*w
265: *
266: CALL ZTRMV( 'Lower', 'NO TRANSPOSE',
267: $ 'UNIT', I-1,
268: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
269: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
270: *
271: A( K+I-1, I-1 ) = EI
272: END IF
273: *
274: * Generate the elementary reflector H(I) to annihilate
275: * A(K+I+1:N,I)
276: *
277: CALL ZLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
278: $ TAU( I ) )
279: EI = A( K+I, I )
280: A( K+I, I ) = ONE
281: *
282: * Compute Y(K+1:N,I)
283: *
284: CALL ZGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
285: $ ONE, A( K+1, I+1 ),
286: $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
287: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
288: $ ONE, A( K+I, 1 ), LDA,
289: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
290: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
291: $ Y( K+1, 1 ), LDY,
292: $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
293: CALL ZSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
294: *
295: * Compute T(1:I,I)
296: *
297: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
298: CALL ZTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
299: $ I-1, T, LDT,
300: $ T( 1, I ), 1 )
301: T( I, I ) = TAU( I )
302: *
303: 10 CONTINUE
304: A( K+NB, NB ) = EI
305: *
306: * Compute Y(1:K,1:NB)
307: *
308: CALL ZLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
309: CALL ZTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
310: $ 'UNIT', K, NB,
311: $ ONE, A( K+1, 1 ), LDA, Y, LDY )
312: IF( N.GT.K+NB )
313: $ CALL ZGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
314: $ NB, N-K-NB, ONE,
315: $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
316: $ LDY )
317: CALL ZTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
318: $ 'NON-UNIT', K, NB,
319: $ ONE, T, LDT, Y, LDY )
320: *
321: RETURN
322: *
323: * End of ZLAHR2
324: *
325: END
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