1: SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
2: *
3: * -- LAPACK auxiliary routine (version 3.3.1) --
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 2011 --
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: * ZLAHRD 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 a unitary similarity transformation
22: * Q**H * A * Q. The routine returns the matrices V and T which determine
23: * Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
24: *
25: * This is an OBSOLETE auxiliary routine.
26: * This routine will be 'deprecated' in a future release.
27: * Please use the new routine ZLAHR2 instead.
28: *
29: * Arguments
30: * =========
31: *
32: * N (input) INTEGER
33: * The order of the matrix A.
34: *
35: * K (input) INTEGER
36: * The offset for the reduction. Elements below the k-th
37: * subdiagonal in the first NB columns are reduced to zero.
38: *
39: * NB (input) INTEGER
40: * The number of columns to be reduced.
41: *
42: * A (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
43: * On entry, the n-by-(n-k+1) general matrix A.
44: * On exit, the elements on and above the k-th subdiagonal in
45: * the first NB columns are overwritten with the corresponding
46: * elements of the reduced matrix; the elements below the k-th
47: * subdiagonal, with the array TAU, represent the matrix Q as a
48: * product of elementary reflectors. The other columns of A are
49: * unchanged. See Further Details.
50: *
51: * LDA (input) INTEGER
52: * The leading dimension of the array A. LDA >= max(1,N).
53: *
54: * TAU (output) COMPLEX*16 array, dimension (NB)
55: * The scalar factors of the elementary reflectors. See Further
56: * Details.
57: *
58: * T (output) COMPLEX*16 array, dimension (LDT,NB)
59: * The upper triangular matrix T.
60: *
61: * LDT (input) INTEGER
62: * The leading dimension of the array T. LDT >= NB.
63: *
64: * Y (output) COMPLEX*16 array, dimension (LDY,NB)
65: * The n-by-nb matrix Y.
66: *
67: * LDY (input) INTEGER
68: * The leading dimension of the array Y. LDY >= max(1,N).
69: *
70: * Further Details
71: * ===============
72: *
73: * The matrix Q is represented as a product of nb elementary reflectors
74: *
75: * Q = H(1) H(2) . . . H(nb).
76: *
77: * Each H(i) has the form
78: *
79: * H(i) = I - tau * v * v**H
80: *
81: * where tau is a complex scalar, and v is a complex vector with
82: * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
83: * A(i+k+1:n,i), and tau in TAU(i).
84: *
85: * The elements of the vectors v together form the (n-k+1)-by-nb matrix
86: * V which is needed, with T and Y, to apply the transformation to the
87: * unreduced part of the matrix, using an update of the form:
88: * A := (I - V*T*V**H) * (A - Y*V**H).
89: *
90: * The contents of A on exit are illustrated by the following example
91: * with n = 7, k = 3 and nb = 2:
92: *
93: * ( a h a a a )
94: * ( a h a a a )
95: * ( a h a a a )
96: * ( h h a a a )
97: * ( v1 h a a a )
98: * ( v1 v2 a a a )
99: * ( v1 v2 a a a )
100: *
101: * where a denotes an element of the original matrix A, h denotes a
102: * modified element of the upper Hessenberg matrix H, and vi denotes an
103: * element of the vector defining H(i).
104: *
105: * =====================================================================
106: *
107: * .. Parameters ..
108: COMPLEX*16 ZERO, ONE
109: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
110: $ ONE = ( 1.0D+0, 0.0D+0 ) )
111: * ..
112: * .. Local Scalars ..
113: INTEGER I
114: COMPLEX*16 EI
115: * ..
116: * .. External Subroutines ..
117: EXTERNAL ZAXPY, ZCOPY, ZGEMV, ZLACGV, ZLARFG, ZSCAL,
118: $ ZTRMV
119: * ..
120: * .. Intrinsic Functions ..
121: INTRINSIC MIN
122: * ..
123: * .. Executable Statements ..
124: *
125: * Quick return if possible
126: *
127: IF( N.LE.1 )
128: $ RETURN
129: *
130: DO 10 I = 1, NB
131: IF( I.GT.1 ) THEN
132: *
133: * Update A(1:n,i)
134: *
135: * Compute i-th column of A - Y * V**H
136: *
137: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
138: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
139: $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
140: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
141: *
142: * Apply I - V * T**H * V**H to this column (call it b) from the
143: * left, using the last column of T as workspace
144: *
145: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
146: * ( V2 ) ( b2 )
147: *
148: * where V1 is unit lower triangular
149: *
150: * w := V1**H * b1
151: *
152: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
153: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1,
154: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
155: *
156: * w := w + V2**H *b2
157: *
158: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
159: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ONE,
160: $ T( 1, NB ), 1 )
161: *
162: * w := T**H *w
163: *
164: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1,
165: $ T, LDT, T( 1, NB ), 1 )
166: *
167: * b2 := b2 - V2*w
168: *
169: CALL ZGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
170: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
171: *
172: * b1 := b1 - V1*w
173: *
174: CALL ZTRMV( 'Lower', 'No transpose', 'Unit', I-1,
175: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
176: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
177: *
178: A( K+I-1, I-1 ) = EI
179: END IF
180: *
181: * Generate the elementary reflector H(i) to annihilate
182: * A(k+i+1:n,i)
183: *
184: EI = A( K+I, I )
185: CALL ZLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1,
186: $ TAU( I ) )
187: A( K+I, I ) = ONE
188: *
189: * Compute Y(1:n,i)
190: *
191: CALL ZGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
192: $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
193: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
194: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ),
195: $ 1 )
196: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
197: $ ONE, Y( 1, I ), 1 )
198: CALL ZSCAL( N, TAU( I ), Y( 1, I ), 1 )
199: *
200: * Compute T(1:i,i)
201: *
202: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
203: CALL ZTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
204: $ T( 1, I ), 1 )
205: T( I, I ) = TAU( I )
206: *
207: 10 CONTINUE
208: A( K+NB, NB ) = EI
209: *
210: RETURN
211: *
212: * End of ZLAHRD
213: *
214: END
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