1: SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
2: *
3: * -- LAPACK auxiliary routine (version 3.2.1) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * -- April 2009 --
7: *
8: * .. Scalar Arguments ..
9: INTEGER K, LDA, LDT, LDY, N, NB
10: * ..
11: * .. Array Arguments ..
12: COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
13: $ Y( LDY, NB )
14: * ..
15: *
16: * Purpose
17: * =======
18: *
19: * ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
20: * matrix A so that elements below the k-th subdiagonal are zero. The
21: * reduction is performed by an unitary similarity transformation
22: * Q' * A * Q. The routine returns the matrices V and T which determine
23: * Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.
24: *
25: * This is an auxiliary routine called by ZGEHRD.
26: *
27: * Arguments
28: * =========
29: *
30: * N (input) INTEGER
31: * The order of the matrix A.
32: *
33: * K (input) INTEGER
34: * The offset for the reduction. Elements below the k-th
35: * subdiagonal in the first NB columns are reduced to zero.
36: * K < N.
37: *
38: * NB (input) INTEGER
39: * The number of columns to be reduced.
40: *
41: * A (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
42: * On entry, the n-by-(n-k+1) general matrix A.
43: * On exit, the elements on and above the k-th subdiagonal in
44: * the first NB columns are overwritten with the corresponding
45: * elements of the reduced matrix; the elements below the k-th
46: * subdiagonal, with the array TAU, represent the matrix Q as a
47: * product of elementary reflectors. The other columns of A are
48: * unchanged. See Further Details.
49: *
50: * LDA (input) INTEGER
51: * The leading dimension of the array A. LDA >= max(1,N).
52: *
53: * TAU (output) COMPLEX*16 array, dimension (NB)
54: * The scalar factors of the elementary reflectors. See Further
55: * Details.
56: *
57: * T (output) COMPLEX*16 array, dimension (LDT,NB)
58: * The upper triangular matrix T.
59: *
60: * LDT (input) INTEGER
61: * The leading dimension of the array T. LDT >= NB.
62: *
63: * Y (output) COMPLEX*16 array, dimension (LDY,NB)
64: * The n-by-nb matrix Y.
65: *
66: * LDY (input) INTEGER
67: * The leading dimension of the array Y. LDY >= N.
68: *
69: * Further Details
70: * ===============
71: *
72: * The matrix Q is represented as a product of nb elementary reflectors
73: *
74: * Q = H(1) H(2) . . . H(nb).
75: *
76: * Each H(i) has the form
77: *
78: * H(i) = I - tau * v * v'
79: *
80: * where tau is a complex scalar, and v is a complex vector with
81: * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
82: * A(i+k+1:n,i), and tau in TAU(i).
83: *
84: * The elements of the vectors v together form the (n-k+1)-by-nb matrix
85: * V which is needed, with T and Y, to apply the transformation to the
86: * unreduced part of the matrix, using an update of the form:
87: * A := (I - V*T*V') * (A - Y*V').
88: *
89: * The contents of A on exit are illustrated by the following example
90: * with n = 7, k = 3 and nb = 2:
91: *
92: * ( a a a a a )
93: * ( a a a a a )
94: * ( a a a a a )
95: * ( h h a a a )
96: * ( v1 h a a a )
97: * ( v1 v2 a a a )
98: * ( v1 v2 a a a )
99: *
100: * where a denotes an element of the original matrix A, h denotes a
101: * modified element of the upper Hessenberg matrix H, and vi denotes an
102: * element of the vector defining H(i).
103: *
104: * This subroutine is a slight modification of LAPACK-3.0's DLAHRD
105: * incorporating improvements proposed by Quintana-Orti and Van de
106: * Gejin. Note that the entries of A(1:K,2:NB) differ from those
107: * returned by the original LAPACK-3.0's DLAHRD routine. (This
108: * subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
109: *
110: * References
111: * ==========
112: *
113: * Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
114: * performance of reduction to Hessenberg form," ACM Transactions on
115: * Mathematical Software, 32(2):180-194, June 2006.
116: *
117: * =====================================================================
118: *
119: * .. Parameters ..
120: COMPLEX*16 ZERO, ONE
121: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
122: $ ONE = ( 1.0D+0, 0.0D+0 ) )
123: * ..
124: * .. Local Scalars ..
125: INTEGER I
126: COMPLEX*16 EI
127: * ..
128: * .. External Subroutines ..
129: EXTERNAL ZAXPY, ZCOPY, ZGEMM, ZGEMV, ZLACPY,
130: $ ZLARFG, ZSCAL, ZTRMM, ZTRMV, ZLACGV
131: * ..
132: * .. Intrinsic Functions ..
133: INTRINSIC MIN
134: * ..
135: * .. Executable Statements ..
136: *
137: * Quick return if possible
138: *
139: IF( N.LE.1 )
140: $ RETURN
141: *
142: DO 10 I = 1, NB
143: IF( I.GT.1 ) THEN
144: *
145: * Update A(K+1:N,I)
146: *
147: * Update I-th column of A - Y * V'
148: *
149: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
150: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
151: $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
152: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
153: *
154: * Apply I - V * T' * V' to this column (call it b) from the
155: * left, using the last column of T as workspace
156: *
157: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
158: * ( V2 ) ( b2 )
159: *
160: * where V1 is unit lower triangular
161: *
162: * w := V1' * b1
163: *
164: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
165: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT',
166: $ I-1, A( K+1, 1 ),
167: $ LDA, T( 1, NB ), 1 )
168: *
169: * w := w + V2'*b2
170: *
171: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
172: $ ONE, A( K+I, 1 ),
173: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
174: *
175: * w := T'*w
176: *
177: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',
178: $ I-1, T, LDT,
179: $ T( 1, NB ), 1 )
180: *
181: * b2 := b2 - V2*w
182: *
183: CALL ZGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
184: $ A( K+I, 1 ),
185: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
186: *
187: * b1 := b1 - V1*w
188: *
189: CALL ZTRMV( 'Lower', 'NO TRANSPOSE',
190: $ 'UNIT', I-1,
191: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
192: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
193: *
194: A( K+I-1, I-1 ) = EI
195: END IF
196: *
197: * Generate the elementary reflector H(I) to annihilate
198: * A(K+I+1:N,I)
199: *
200: CALL ZLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
201: $ TAU( I ) )
202: EI = A( K+I, I )
203: A( K+I, I ) = ONE
204: *
205: * Compute Y(K+1:N,I)
206: *
207: CALL ZGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
208: $ ONE, A( K+1, I+1 ),
209: $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
210: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
211: $ ONE, A( K+I, 1 ), LDA,
212: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
213: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
214: $ Y( K+1, 1 ), LDY,
215: $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
216: CALL ZSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
217: *
218: * Compute T(1:I,I)
219: *
220: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
221: CALL ZTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
222: $ I-1, T, LDT,
223: $ T( 1, I ), 1 )
224: T( I, I ) = TAU( I )
225: *
226: 10 CONTINUE
227: A( K+NB, NB ) = EI
228: *
229: * Compute Y(1:K,1:NB)
230: *
231: CALL ZLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
232: CALL ZTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
233: $ 'UNIT', K, NB,
234: $ ONE, A( K+1, 1 ), LDA, Y, LDY )
235: IF( N.GT.K+NB )
236: $ CALL ZGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
237: $ NB, N-K-NB, ONE,
238: $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
239: $ LDY )
240: CALL ZTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
241: $ 'NON-UNIT', K, NB,
242: $ ONE, T, LDT, Y, LDY )
243: *
244: RETURN
245: *
246: * End of ZLAHR2
247: *
248: END
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