1 SUBROUTINE DLAHRD( 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 DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
13 $ Y( LDY, NB )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DLAHRD 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 OBSOLETE auxiliary routine.
26 * This routine will be 'deprecated' in a future release.
27 * Please use the new routine DLAHR2 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)
55 * The scalar factors of the elementary reflectors. See Further
56 * Details.
57 *
58 * T (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 >= 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**T
80 *
81 * where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 DOUBLE PRECISION ZERO, ONE
109 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
110 * ..
111 * .. Local Scalars ..
112 INTEGER I
113 DOUBLE PRECISION EI
114 * ..
115 * .. External Subroutines ..
116 EXTERNAL DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
117 * ..
118 * .. Intrinsic Functions ..
119 INTRINSIC MIN
120 * ..
121 * .. Executable Statements ..
122 *
123 * Quick return if possible
124 *
125 IF( N.LE.1 )
126 $ RETURN
127 *
128 DO 10 I = 1, NB
129 IF( I.GT.1 ) THEN
130 *
131 * Update A(1:n,i)
132 *
133 * Compute i-th column of A - Y * V**T
134 *
135 CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
136 $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
137 *
138 * Apply I - V * T**T * V**T to this column (call it b) from the
139 * left, using the last column of T as workspace
140 *
141 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
142 * ( V2 ) ( b2 )
143 *
144 * where V1 is unit lower triangular
145 *
146 * w := V1**T * b1
147 *
148 CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
149 CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
150 $ LDA, T( 1, NB ), 1 )
151 *
152 * w := w + V2**T *b2
153 *
154 CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
155 $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
156 *
157 * w := T**T *w
158 *
159 CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
160 $ T( 1, NB ), 1 )
161 *
162 * b2 := b2 - V2*w
163 *
164 CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
165 $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
166 *
167 * b1 := b1 - V1*w
168 *
169 CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
170 $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
171 CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
172 *
173 A( K+I-1, I-1 ) = EI
174 END IF
175 *
176 * Generate the elementary reflector H(i) to annihilate
177 * A(k+i+1:n,i)
178 *
179 CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
180 $ TAU( I ) )
181 EI = A( K+I, I )
182 A( K+I, I ) = ONE
183 *
184 * Compute Y(1:n,i)
185 *
186 CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
187 $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
188 CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA,
189 $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
190 CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
191 $ ONE, Y( 1, I ), 1 )
192 CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
193 *
194 * Compute T(1:i,i)
195 *
196 CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
197 CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
198 $ T( 1, I ), 1 )
199 T( I, I ) = TAU( I )
200 *
201 10 CONTINUE
202 A( K+NB, NB ) = EI
203 *
204 RETURN
205 *
206 * End of DLAHRD
207 *
208 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 2011 --
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 * DLAHRD 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 OBSOLETE auxiliary routine.
26 * This routine will be 'deprecated' in a future release.
27 * Please use the new routine DLAHR2 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)
55 * The scalar factors of the elementary reflectors. See Further
56 * Details.
57 *
58 * T (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 >= 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**T
80 *
81 * where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 DOUBLE PRECISION ZERO, ONE
109 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
110 * ..
111 * .. Local Scalars ..
112 INTEGER I
113 DOUBLE PRECISION EI
114 * ..
115 * .. External Subroutines ..
116 EXTERNAL DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
117 * ..
118 * .. Intrinsic Functions ..
119 INTRINSIC MIN
120 * ..
121 * .. Executable Statements ..
122 *
123 * Quick return if possible
124 *
125 IF( N.LE.1 )
126 $ RETURN
127 *
128 DO 10 I = 1, NB
129 IF( I.GT.1 ) THEN
130 *
131 * Update A(1:n,i)
132 *
133 * Compute i-th column of A - Y * V**T
134 *
135 CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
136 $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
137 *
138 * Apply I - V * T**T * V**T to this column (call it b) from the
139 * left, using the last column of T as workspace
140 *
141 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
142 * ( V2 ) ( b2 )
143 *
144 * where V1 is unit lower triangular
145 *
146 * w := V1**T * b1
147 *
148 CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
149 CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
150 $ LDA, T( 1, NB ), 1 )
151 *
152 * w := w + V2**T *b2
153 *
154 CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
155 $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
156 *
157 * w := T**T *w
158 *
159 CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
160 $ T( 1, NB ), 1 )
161 *
162 * b2 := b2 - V2*w
163 *
164 CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
165 $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
166 *
167 * b1 := b1 - V1*w
168 *
169 CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
170 $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
171 CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
172 *
173 A( K+I-1, I-1 ) = EI
174 END IF
175 *
176 * Generate the elementary reflector H(i) to annihilate
177 * A(k+i+1:n,i)
178 *
179 CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
180 $ TAU( I ) )
181 EI = A( K+I, I )
182 A( K+I, I ) = ONE
183 *
184 * Compute Y(1:n,i)
185 *
186 CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
187 $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
188 CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA,
189 $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
190 CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
191 $ ONE, Y( 1, I ), 1 )
192 CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
193 *
194 * Compute T(1:i,i)
195 *
196 CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
197 CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
198 $ T( 1, I ), 1 )
199 T( I, I ) = TAU( I )
200 *
201 10 CONTINUE
202 A( K+NB, NB ) = EI
203 *
204 RETURN
205 *
206 * End of DLAHRD
207 *
208 END