1 SUBROUTINE ZLAHR2( 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 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**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 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**H
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**H) * (A - Y*V**H).
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**H
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**H * V**H 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**H * 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**H * 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**H * 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
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 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**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 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**H
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**H) * (A - Y*V**H).
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**H
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**H * V**H 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**H * 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**H * 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**H * 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