1: *> \brief \b DLAHR2 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 DLAHR2 + dependencies
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahr2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLAHR2( 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: * DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
28: * $ Y( LDY, NB )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DLAHR2 reduces the first NB columns of A real 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 orthogonal similarity transformation
40: *> Q**T * A * Q. The routine returns the matrices V and T which determine
41: *> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
42: *>
43: *> This is an auxiliary routine called by DGEHRD.
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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: *> \date September 2012
127: *
128: *> \ingroup doubleOTHERauxiliary
129: *
130: *> \par Further Details:
131: * =====================
132: *>
133: *> \verbatim
134: *>
135: *> The matrix Q is represented as a product of nb elementary reflectors
136: *>
137: *> Q = H(1) H(2) . . . H(nb).
138: *>
139: *> Each H(i) has the form
140: *>
141: *> H(i) = I - tau * v * v**T
142: *>
143: *> where tau is a real scalar, and v is a real vector with
144: *> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
145: *> A(i+k+1:n,i), and tau in TAU(i).
146: *>
147: *> The elements of the vectors v together form the (n-k+1)-by-nb matrix
148: *> V which is needed, with T and Y, to apply the transformation to the
149: *> unreduced part of the matrix, using an update of the form:
150: *> A := (I - V*T*V**T) * (A - Y*V**T).
151: *>
152: *> The contents of A on exit are illustrated by the following example
153: *> with n = 7, k = 3 and nb = 2:
154: *>
155: *> ( a a a a a )
156: *> ( a a a a a )
157: *> ( a a a a a )
158: *> ( h h a a a )
159: *> ( v1 h a a a )
160: *> ( v1 v2 a a a )
161: *> ( v1 v2 a a a )
162: *>
163: *> where a denotes an element of the original matrix A, h denotes a
164: *> modified element of the upper Hessenberg matrix H, and vi denotes an
165: *> element of the vector defining H(i).
166: *>
167: *> This subroutine is a slight modification of LAPACK-3.0's DLAHRD
168: *> incorporating improvements proposed by Quintana-Orti and Van de
169: *> Gejin. Note that the entries of A(1:K,2:NB) differ from those
170: *> returned by the original LAPACK-3.0's DLAHRD routine. (This
171: *> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
172: *> \endverbatim
173: *
174: *> \par References:
175: * ================
176: *>
177: *> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
178: *> performance of reduction to Hessenberg form," ACM Transactions on
179: *> Mathematical Software, 32(2):180-194, June 2006.
180: *>
181: * =====================================================================
182: SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
183: *
184: * -- LAPACK auxiliary routine (version 3.4.2) --
185: * -- LAPACK is a software package provided by Univ. of Tennessee, --
186: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187: * September 2012
188: *
189: * .. Scalar Arguments ..
190: INTEGER K, LDA, LDT, LDY, N, NB
191: * ..
192: * .. Array Arguments ..
193: DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
194: $ Y( LDY, NB )
195: * ..
196: *
197: * =====================================================================
198: *
199: * .. Parameters ..
200: DOUBLE PRECISION ZERO, ONE
201: PARAMETER ( ZERO = 0.0D+0,
202: $ ONE = 1.0D+0 )
203: * ..
204: * .. Local Scalars ..
205: INTEGER I
206: DOUBLE PRECISION EI
207: * ..
208: * .. External Subroutines ..
209: EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
210: $ DLARFG, DSCAL, DTRMM, DTRMV
211: * ..
212: * .. Intrinsic Functions ..
213: INTRINSIC MIN
214: * ..
215: * .. Executable Statements ..
216: *
217: * Quick return if possible
218: *
219: IF( N.LE.1 )
220: $ RETURN
221: *
222: DO 10 I = 1, NB
223: IF( I.GT.1 ) THEN
224: *
225: * Update A(K+1:N,I)
226: *
227: * Update I-th column of A - Y * V**T
228: *
229: CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
230: $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
231: *
232: * Apply I - V * T**T * V**T to this column (call it b) from the
233: * left, using the last column of T as workspace
234: *
235: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
236: * ( V2 ) ( b2 )
237: *
238: * where V1 is unit lower triangular
239: *
240: * w := V1**T * b1
241: *
242: CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
243: CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
244: $ I-1, A( K+1, 1 ),
245: $ LDA, T( 1, NB ), 1 )
246: *
247: * w := w + V2**T * b2
248: *
249: CALL DGEMV( 'Transpose', N-K-I+1, I-1,
250: $ ONE, A( K+I, 1 ),
251: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
252: *
253: * w := T**T * w
254: *
255: CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
256: $ I-1, T, LDT,
257: $ T( 1, NB ), 1 )
258: *
259: * b2 := b2 - V2*w
260: *
261: CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
262: $ A( K+I, 1 ),
263: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
264: *
265: * b1 := b1 - V1*w
266: *
267: CALL DTRMV( 'Lower', 'NO TRANSPOSE',
268: $ 'UNIT', I-1,
269: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
270: CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
271: *
272: A( K+I-1, I-1 ) = EI
273: END IF
274: *
275: * Generate the elementary reflector H(I) to annihilate
276: * A(K+I+1:N,I)
277: *
278: CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
279: $ TAU( I ) )
280: EI = A( K+I, I )
281: A( K+I, I ) = ONE
282: *
283: * Compute Y(K+1:N,I)
284: *
285: CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
286: $ ONE, A( K+1, I+1 ),
287: $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
288: CALL DGEMV( 'Transpose', N-K-I+1, I-1,
289: $ ONE, A( K+I, 1 ), LDA,
290: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
291: CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
292: $ Y( K+1, 1 ), LDY,
293: $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
294: CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
295: *
296: * Compute T(1:I,I)
297: *
298: CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
299: CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
300: $ I-1, T, LDT,
301: $ T( 1, I ), 1 )
302: T( I, I ) = TAU( I )
303: *
304: 10 CONTINUE
305: A( K+NB, NB ) = EI
306: *
307: * Compute Y(1:K,1:NB)
308: *
309: CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
310: CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
311: $ 'UNIT', K, NB,
312: $ ONE, A( K+1, 1 ), LDA, Y, LDY )
313: IF( N.GT.K+NB )
314: $ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
315: $ NB, N-K-NB, ONE,
316: $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
317: $ LDY )
318: CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
319: $ 'NON-UNIT', K, NB,
320: $ ONE, T, LDT, Y, LDY )
321: *
322: RETURN
323: *
324: * End of DLAHR2
325: *
326: END
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