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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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/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: *> \ingroup doubleOTHERauxiliary
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**T
140: *>
141: *> where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 DLAHRD
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 DLAHRD routine. (This
169: *> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
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 DLAHR2( 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: DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
191: $ Y( LDY, NB )
192: * ..
193: *
194: * =====================================================================
195: *
196: * .. Parameters ..
197: DOUBLE PRECISION ZERO, ONE
198: PARAMETER ( ZERO = 0.0D+0,
199: $ ONE = 1.0D+0 )
200: * ..
201: * .. Local Scalars ..
202: INTEGER I
203: DOUBLE PRECISION EI
204: * ..
205: * .. External Subroutines ..
206: EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
207: $ DLARFG, DSCAL, DTRMM, DTRMV
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**T
225: *
226: CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
227: $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
228: *
229: * Apply I - V * T**T * V**T to this column (call it b) from the
230: * left, using the last column of T as workspace
231: *
232: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
233: * ( V2 ) ( b2 )
234: *
235: * where V1 is unit lower triangular
236: *
237: * w := V1**T * b1
238: *
239: CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
240: CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
241: $ I-1, A( K+1, 1 ),
242: $ LDA, T( 1, NB ), 1 )
243: *
244: * w := w + V2**T * b2
245: *
246: CALL DGEMV( 'Transpose', N-K-I+1, I-1,
247: $ ONE, A( K+I, 1 ),
248: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
249: *
250: * w := T**T * w
251: *
252: CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
253: $ I-1, T, LDT,
254: $ T( 1, NB ), 1 )
255: *
256: * b2 := b2 - V2*w
257: *
258: CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
259: $ A( K+I, 1 ),
260: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
261: *
262: * b1 := b1 - V1*w
263: *
264: CALL DTRMV( 'Lower', 'NO TRANSPOSE',
265: $ 'UNIT', I-1,
266: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
267: CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
268: *
269: A( K+I-1, I-1 ) = EI
270: END IF
271: *
272: * Generate the elementary reflector H(I) to annihilate
273: * A(K+I+1:N,I)
274: *
275: CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
276: $ TAU( I ) )
277: EI = A( K+I, I )
278: A( K+I, I ) = ONE
279: *
280: * Compute Y(K+1:N,I)
281: *
282: CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
283: $ ONE, A( K+1, I+1 ),
284: $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
285: CALL DGEMV( 'Transpose', N-K-I+1, I-1,
286: $ ONE, A( K+I, 1 ), LDA,
287: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
288: CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
289: $ Y( K+1, 1 ), LDY,
290: $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
291: CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
292: *
293: * Compute T(1:I,I)
294: *
295: CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
296: CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
297: $ I-1, T, LDT,
298: $ T( 1, I ), 1 )
299: T( I, I ) = TAU( I )
300: *
301: 10 CONTINUE
302: A( K+NB, NB ) = EI
303: *
304: * Compute Y(1:K,1:NB)
305: *
306: CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
307: CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
308: $ 'UNIT', K, NB,
309: $ ONE, A( K+1, 1 ), LDA, Y, LDY )
310: IF( N.GT.K+NB )
311: $ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
312: $ NB, N-K-NB, ONE,
313: $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
314: $ LDY )
315: CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
316: $ 'NON-UNIT', K, NB,
317: $ ONE, T, LDT, Y, LDY )
318: *
319: RETURN
320: *
321: * End of DLAHR2
322: *
323: END
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