1 SUBROUTINE DLAHR2( 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 DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
13 $ Y( LDY, NB )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DLAHR2 reduces the first NB columns of A real 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 orthogonal similarity transformation
22 * Q**T * A * Q. The routine returns the matrices V and T which determine
23 * Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
24 *
25 * This is an auxiliary routine called by DGEHRD.
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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)
54 * The scalar factors of the elementary reflectors. See Further
55 * Details.
56 *
57 * T (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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**T
79 *
80 * where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 DOUBLE PRECISION ZERO, ONE
121 PARAMETER ( ZERO = 0.0D+0,
122 $ ONE = 1.0D+0 )
123 * ..
124 * .. Local Scalars ..
125 INTEGER I
126 DOUBLE PRECISION EI
127 * ..
128 * .. External Subroutines ..
129 EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
130 $ DLARFG, DSCAL, DTRMM, DTRMV
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**T
148 *
149 CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
150 $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
151 *
152 * Apply I - V * T**T * V**T to this column (call it b) from the
153 * left, using the last column of T as workspace
154 *
155 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
156 * ( V2 ) ( b2 )
157 *
158 * where V1 is unit lower triangular
159 *
160 * w := V1**T * b1
161 *
162 CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
163 CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
164 $ I-1, A( K+1, 1 ),
165 $ LDA, T( 1, NB ), 1 )
166 *
167 * w := w + V2**T * b2
168 *
169 CALL DGEMV( 'Transpose', N-K-I+1, I-1,
170 $ ONE, A( K+I, 1 ),
171 $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
172 *
173 * w := T**T * w
174 *
175 CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
176 $ I-1, T, LDT,
177 $ T( 1, NB ), 1 )
178 *
179 * b2 := b2 - V2*w
180 *
181 CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
182 $ A( K+I, 1 ),
183 $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
184 *
185 * b1 := b1 - V1*w
186 *
187 CALL DTRMV( 'Lower', 'NO TRANSPOSE',
188 $ 'UNIT', I-1,
189 $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
190 CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
191 *
192 A( K+I-1, I-1 ) = EI
193 END IF
194 *
195 * Generate the elementary reflector H(I) to annihilate
196 * A(K+I+1:N,I)
197 *
198 CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
199 $ TAU( I ) )
200 EI = A( K+I, I )
201 A( K+I, I ) = ONE
202 *
203 * Compute Y(K+1:N,I)
204 *
205 CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
206 $ ONE, A( K+1, I+1 ),
207 $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
208 CALL DGEMV( 'Transpose', N-K-I+1, I-1,
209 $ ONE, A( K+I, 1 ), LDA,
210 $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
211 CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
212 $ Y( K+1, 1 ), LDY,
213 $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
214 CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
215 *
216 * Compute T(1:I,I)
217 *
218 CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
219 CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
220 $ I-1, T, LDT,
221 $ T( 1, I ), 1 )
222 T( I, I ) = TAU( I )
223 *
224 10 CONTINUE
225 A( K+NB, NB ) = EI
226 *
227 * Compute Y(1:K,1:NB)
228 *
229 CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
230 CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
231 $ 'UNIT', K, NB,
232 $ ONE, A( K+1, 1 ), LDA, Y, LDY )
233 IF( N.GT.K+NB )
234 $ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
235 $ NB, N-K-NB, ONE,
236 $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
237 $ LDY )
238 CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
239 $ 'NON-UNIT', K, NB,
240 $ ONE, T, LDT, Y, LDY )
241 *
242 RETURN
243 *
244 * End of DLAHR2
245 *
246 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 DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
13 $ Y( LDY, NB )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DLAHR2 reduces the first NB columns of A real 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 orthogonal similarity transformation
22 * Q**T * A * Q. The routine returns the matrices V and T which determine
23 * Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
24 *
25 * This is an auxiliary routine called by DGEHRD.
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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)
54 * The scalar factors of the elementary reflectors. See Further
55 * Details.
56 *
57 * T (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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**T
79 *
80 * where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 DOUBLE PRECISION ZERO, ONE
121 PARAMETER ( ZERO = 0.0D+0,
122 $ ONE = 1.0D+0 )
123 * ..
124 * .. Local Scalars ..
125 INTEGER I
126 DOUBLE PRECISION EI
127 * ..
128 * .. External Subroutines ..
129 EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
130 $ DLARFG, DSCAL, DTRMM, DTRMV
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**T
148 *
149 CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
150 $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
151 *
152 * Apply I - V * T**T * V**T to this column (call it b) from the
153 * left, using the last column of T as workspace
154 *
155 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
156 * ( V2 ) ( b2 )
157 *
158 * where V1 is unit lower triangular
159 *
160 * w := V1**T * b1
161 *
162 CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
163 CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
164 $ I-1, A( K+1, 1 ),
165 $ LDA, T( 1, NB ), 1 )
166 *
167 * w := w + V2**T * b2
168 *
169 CALL DGEMV( 'Transpose', N-K-I+1, I-1,
170 $ ONE, A( K+I, 1 ),
171 $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
172 *
173 * w := T**T * w
174 *
175 CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
176 $ I-1, T, LDT,
177 $ T( 1, NB ), 1 )
178 *
179 * b2 := b2 - V2*w
180 *
181 CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
182 $ A( K+I, 1 ),
183 $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
184 *
185 * b1 := b1 - V1*w
186 *
187 CALL DTRMV( 'Lower', 'NO TRANSPOSE',
188 $ 'UNIT', I-1,
189 $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
190 CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
191 *
192 A( K+I-1, I-1 ) = EI
193 END IF
194 *
195 * Generate the elementary reflector H(I) to annihilate
196 * A(K+I+1:N,I)
197 *
198 CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
199 $ TAU( I ) )
200 EI = A( K+I, I )
201 A( K+I, I ) = ONE
202 *
203 * Compute Y(K+1:N,I)
204 *
205 CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
206 $ ONE, A( K+1, I+1 ),
207 $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
208 CALL DGEMV( 'Transpose', N-K-I+1, I-1,
209 $ ONE, A( K+I, 1 ), LDA,
210 $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
211 CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
212 $ Y( K+1, 1 ), LDY,
213 $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
214 CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
215 *
216 * Compute T(1:I,I)
217 *
218 CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
219 CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
220 $ I-1, T, LDT,
221 $ T( 1, I ), 1 )
222 T( I, I ) = TAU( I )
223 *
224 10 CONTINUE
225 A( K+NB, NB ) = EI
226 *
227 * Compute Y(1:K,1:NB)
228 *
229 CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
230 CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
231 $ 'UNIT', K, NB,
232 $ ONE, A( K+1, 1 ), LDA, Y, LDY )
233 IF( N.GT.K+NB )
234 $ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
235 $ NB, N-K-NB, ONE,
236 $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
237 $ LDY )
238 CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
239 $ 'NON-UNIT', K, NB,
240 $ ONE, T, LDT, Y, LDY )
241 *
242 RETURN
243 *
244 * End of DLAHR2
245 *
246 END