1 SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
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 CHARACTER UPLO
10 INTEGER LDA, LDW, N, NB
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DLATRD reduces NB rows and columns of a real symmetric matrix A to
20 * symmetric tridiagonal form by an orthogonal similarity
21 * transformation Q**T * A * Q, and returns the matrices V and W which are
22 * needed to apply the transformation to the unreduced part of A.
23 *
24 * If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
25 * matrix, of which the upper triangle is supplied;
26 * if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
27 * matrix, of which the lower triangle is supplied.
28 *
29 * This is an auxiliary routine called by DSYTRD.
30 *
31 * Arguments
32 * =========
33 *
34 * UPLO (input) CHARACTER*1
35 * Specifies whether the upper or lower triangular part of the
36 * symmetric matrix A is stored:
37 * = 'U': Upper triangular
38 * = 'L': Lower triangular
39 *
40 * N (input) INTEGER
41 * The order of the matrix A.
42 *
43 * NB (input) INTEGER
44 * The number of rows and columns to be reduced.
45 *
46 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
47 * On entry, the symmetric matrix A. If UPLO = 'U', the leading
48 * n-by-n upper triangular part of A contains the upper
49 * triangular part of the matrix A, and the strictly lower
50 * triangular part of A is not referenced. If UPLO = 'L', the
51 * leading n-by-n lower triangular part of A contains the lower
52 * triangular part of the matrix A, and the strictly upper
53 * triangular part of A is not referenced.
54 * On exit:
55 * if UPLO = 'U', the last NB columns have been reduced to
56 * tridiagonal form, with the diagonal elements overwriting
57 * the diagonal elements of A; the elements above the diagonal
58 * with the array TAU, represent the orthogonal matrix Q as a
59 * product of elementary reflectors;
60 * if UPLO = 'L', the first NB columns have been reduced to
61 * tridiagonal form, with the diagonal elements overwriting
62 * the diagonal elements of A; the elements below the diagonal
63 * with the array TAU, represent the orthogonal matrix Q as a
64 * product of elementary reflectors.
65 * See Further Details.
66 *
67 * LDA (input) INTEGER
68 * The leading dimension of the array A. LDA >= (1,N).
69 *
70 * E (output) DOUBLE PRECISION array, dimension (N-1)
71 * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
72 * elements of the last NB columns of the reduced matrix;
73 * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
74 * the first NB columns of the reduced matrix.
75 *
76 * TAU (output) DOUBLE PRECISION array, dimension (N-1)
77 * The scalar factors of the elementary reflectors, stored in
78 * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
79 * See Further Details.
80 *
81 * W (output) DOUBLE PRECISION array, dimension (LDW,NB)
82 * The n-by-nb matrix W required to update the unreduced part
83 * of A.
84 *
85 * LDW (input) INTEGER
86 * The leading dimension of the array W. LDW >= max(1,N).
87 *
88 * Further Details
89 * ===============
90 *
91 * If UPLO = 'U', the matrix Q is represented as a product of elementary
92 * reflectors
93 *
94 * Q = H(n) H(n-1) . . . H(n-nb+1).
95 *
96 * Each H(i) has the form
97 *
98 * H(i) = I - tau * v * v**T
99 *
100 * where tau is a real scalar, and v is a real vector with
101 * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
102 * and tau in TAU(i-1).
103 *
104 * If UPLO = 'L', the matrix Q is represented as a product of elementary
105 * reflectors
106 *
107 * Q = H(1) H(2) . . . H(nb).
108 *
109 * Each H(i) has the form
110 *
111 * H(i) = I - tau * v * v**T
112 *
113 * where tau is a real scalar, and v is a real vector with
114 * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
115 * and tau in TAU(i).
116 *
117 * The elements of the vectors v together form the n-by-nb matrix V
118 * which is needed, with W, to apply the transformation to the unreduced
119 * part of the matrix, using a symmetric rank-2k update of the form:
120 * A := A - V*W**T - W*V**T.
121 *
122 * The contents of A on exit are illustrated by the following examples
123 * with n = 5 and nb = 2:
124 *
125 * if UPLO = 'U': if UPLO = 'L':
126 *
127 * ( a a a v4 v5 ) ( d )
128 * ( a a v4 v5 ) ( 1 d )
129 * ( a 1 v5 ) ( v1 1 a )
130 * ( d 1 ) ( v1 v2 a a )
131 * ( d ) ( v1 v2 a a a )
132 *
133 * where d denotes a diagonal element of the reduced matrix, a denotes
134 * an element of the original matrix that is unchanged, and vi denotes
135 * an element of the vector defining H(i).
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 DOUBLE PRECISION ZERO, ONE, HALF
141 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
142 * ..
143 * .. Local Scalars ..
144 INTEGER I, IW
145 DOUBLE PRECISION ALPHA
146 * ..
147 * .. External Subroutines ..
148 EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
149 * ..
150 * .. External Functions ..
151 LOGICAL LSAME
152 DOUBLE PRECISION DDOT
153 EXTERNAL LSAME, DDOT
154 * ..
155 * .. Intrinsic Functions ..
156 INTRINSIC MIN
157 * ..
158 * .. Executable Statements ..
159 *
160 * Quick return if possible
161 *
162 IF( N.LE.0 )
163 $ RETURN
164 *
165 IF( LSAME( UPLO, 'U' ) ) THEN
166 *
167 * Reduce last NB columns of upper triangle
168 *
169 DO 10 I = N, N - NB + 1, -1
170 IW = I - N + NB
171 IF( I.LT.N ) THEN
172 *
173 * Update A(1:i,i)
174 *
175 CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
176 $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
177 CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
178 $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
179 END IF
180 IF( I.GT.1 ) THEN
181 *
182 * Generate elementary reflector H(i) to annihilate
183 * A(1:i-2,i)
184 *
185 CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )
186 E( I-1 ) = A( I-1, I )
187 A( I-1, I ) = ONE
188 *
189 * Compute W(1:i-1,i)
190 *
191 CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
192 $ ZERO, W( 1, IW ), 1 )
193 IF( I.LT.N ) THEN
194 CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),
195 $ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
196 CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
197 $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
198 $ W( 1, IW ), 1 )
199 CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),
200 $ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
201 CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
202 $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
203 $ W( 1, IW ), 1 )
204 END IF
205 CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
206 ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
207 $ A( 1, I ), 1 )
208 CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
209 END IF
210 *
211 10 CONTINUE
212 ELSE
213 *
214 * Reduce first NB columns of lower triangle
215 *
216 DO 20 I = 1, NB
217 *
218 * Update A(i:n,i)
219 *
220 CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
221 $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
222 CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
223 $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
224 IF( I.LT.N ) THEN
225 *
226 * Generate elementary reflector H(i) to annihilate
227 * A(i+2:n,i)
228 *
229 CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
230 $ TAU( I ) )
231 E( I ) = A( I+1, I )
232 A( I+1, I ) = ONE
233 *
234 * Compute W(i+1:n,i)
235 *
236 CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
237 $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
238 CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,
239 $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
240 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
241 $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
242 CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,
243 $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
244 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
245 $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
246 CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
247 ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
248 $ A( I+1, I ), 1 )
249 CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
250 END IF
251 *
252 20 CONTINUE
253 END IF
254 *
255 RETURN
256 *
257 * End of DLATRD
258 *
259 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 CHARACTER UPLO
10 INTEGER LDA, LDW, N, NB
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DLATRD reduces NB rows and columns of a real symmetric matrix A to
20 * symmetric tridiagonal form by an orthogonal similarity
21 * transformation Q**T * A * Q, and returns the matrices V and W which are
22 * needed to apply the transformation to the unreduced part of A.
23 *
24 * If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
25 * matrix, of which the upper triangle is supplied;
26 * if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
27 * matrix, of which the lower triangle is supplied.
28 *
29 * This is an auxiliary routine called by DSYTRD.
30 *
31 * Arguments
32 * =========
33 *
34 * UPLO (input) CHARACTER*1
35 * Specifies whether the upper or lower triangular part of the
36 * symmetric matrix A is stored:
37 * = 'U': Upper triangular
38 * = 'L': Lower triangular
39 *
40 * N (input) INTEGER
41 * The order of the matrix A.
42 *
43 * NB (input) INTEGER
44 * The number of rows and columns to be reduced.
45 *
46 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
47 * On entry, the symmetric matrix A. If UPLO = 'U', the leading
48 * n-by-n upper triangular part of A contains the upper
49 * triangular part of the matrix A, and the strictly lower
50 * triangular part of A is not referenced. If UPLO = 'L', the
51 * leading n-by-n lower triangular part of A contains the lower
52 * triangular part of the matrix A, and the strictly upper
53 * triangular part of A is not referenced.
54 * On exit:
55 * if UPLO = 'U', the last NB columns have been reduced to
56 * tridiagonal form, with the diagonal elements overwriting
57 * the diagonal elements of A; the elements above the diagonal
58 * with the array TAU, represent the orthogonal matrix Q as a
59 * product of elementary reflectors;
60 * if UPLO = 'L', the first NB columns have been reduced to
61 * tridiagonal form, with the diagonal elements overwriting
62 * the diagonal elements of A; the elements below the diagonal
63 * with the array TAU, represent the orthogonal matrix Q as a
64 * product of elementary reflectors.
65 * See Further Details.
66 *
67 * LDA (input) INTEGER
68 * The leading dimension of the array A. LDA >= (1,N).
69 *
70 * E (output) DOUBLE PRECISION array, dimension (N-1)
71 * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
72 * elements of the last NB columns of the reduced matrix;
73 * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
74 * the first NB columns of the reduced matrix.
75 *
76 * TAU (output) DOUBLE PRECISION array, dimension (N-1)
77 * The scalar factors of the elementary reflectors, stored in
78 * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
79 * See Further Details.
80 *
81 * W (output) DOUBLE PRECISION array, dimension (LDW,NB)
82 * The n-by-nb matrix W required to update the unreduced part
83 * of A.
84 *
85 * LDW (input) INTEGER
86 * The leading dimension of the array W. LDW >= max(1,N).
87 *
88 * Further Details
89 * ===============
90 *
91 * If UPLO = 'U', the matrix Q is represented as a product of elementary
92 * reflectors
93 *
94 * Q = H(n) H(n-1) . . . H(n-nb+1).
95 *
96 * Each H(i) has the form
97 *
98 * H(i) = I - tau * v * v**T
99 *
100 * where tau is a real scalar, and v is a real vector with
101 * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
102 * and tau in TAU(i-1).
103 *
104 * If UPLO = 'L', the matrix Q is represented as a product of elementary
105 * reflectors
106 *
107 * Q = H(1) H(2) . . . H(nb).
108 *
109 * Each H(i) has the form
110 *
111 * H(i) = I - tau * v * v**T
112 *
113 * where tau is a real scalar, and v is a real vector with
114 * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
115 * and tau in TAU(i).
116 *
117 * The elements of the vectors v together form the n-by-nb matrix V
118 * which is needed, with W, to apply the transformation to the unreduced
119 * part of the matrix, using a symmetric rank-2k update of the form:
120 * A := A - V*W**T - W*V**T.
121 *
122 * The contents of A on exit are illustrated by the following examples
123 * with n = 5 and nb = 2:
124 *
125 * if UPLO = 'U': if UPLO = 'L':
126 *
127 * ( a a a v4 v5 ) ( d )
128 * ( a a v4 v5 ) ( 1 d )
129 * ( a 1 v5 ) ( v1 1 a )
130 * ( d 1 ) ( v1 v2 a a )
131 * ( d ) ( v1 v2 a a a )
132 *
133 * where d denotes a diagonal element of the reduced matrix, a denotes
134 * an element of the original matrix that is unchanged, and vi denotes
135 * an element of the vector defining H(i).
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 DOUBLE PRECISION ZERO, ONE, HALF
141 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
142 * ..
143 * .. Local Scalars ..
144 INTEGER I, IW
145 DOUBLE PRECISION ALPHA
146 * ..
147 * .. External Subroutines ..
148 EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
149 * ..
150 * .. External Functions ..
151 LOGICAL LSAME
152 DOUBLE PRECISION DDOT
153 EXTERNAL LSAME, DDOT
154 * ..
155 * .. Intrinsic Functions ..
156 INTRINSIC MIN
157 * ..
158 * .. Executable Statements ..
159 *
160 * Quick return if possible
161 *
162 IF( N.LE.0 )
163 $ RETURN
164 *
165 IF( LSAME( UPLO, 'U' ) ) THEN
166 *
167 * Reduce last NB columns of upper triangle
168 *
169 DO 10 I = N, N - NB + 1, -1
170 IW = I - N + NB
171 IF( I.LT.N ) THEN
172 *
173 * Update A(1:i,i)
174 *
175 CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
176 $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
177 CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
178 $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
179 END IF
180 IF( I.GT.1 ) THEN
181 *
182 * Generate elementary reflector H(i) to annihilate
183 * A(1:i-2,i)
184 *
185 CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )
186 E( I-1 ) = A( I-1, I )
187 A( I-1, I ) = ONE
188 *
189 * Compute W(1:i-1,i)
190 *
191 CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
192 $ ZERO, W( 1, IW ), 1 )
193 IF( I.LT.N ) THEN
194 CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),
195 $ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
196 CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
197 $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
198 $ W( 1, IW ), 1 )
199 CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),
200 $ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
201 CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
202 $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
203 $ W( 1, IW ), 1 )
204 END IF
205 CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
206 ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
207 $ A( 1, I ), 1 )
208 CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
209 END IF
210 *
211 10 CONTINUE
212 ELSE
213 *
214 * Reduce first NB columns of lower triangle
215 *
216 DO 20 I = 1, NB
217 *
218 * Update A(i:n,i)
219 *
220 CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
221 $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
222 CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
223 $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
224 IF( I.LT.N ) THEN
225 *
226 * Generate elementary reflector H(i) to annihilate
227 * A(i+2:n,i)
228 *
229 CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
230 $ TAU( I ) )
231 E( I ) = A( I+1, I )
232 A( I+1, I ) = ONE
233 *
234 * Compute W(i+1:n,i)
235 *
236 CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
237 $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
238 CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,
239 $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
240 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
241 $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
242 CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,
243 $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
244 CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
245 $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
246 CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
247 ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
248 $ A( I+1, I ), 1 )
249 CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
250 END IF
251 *
252 20 CONTINUE
253 END IF
254 *
255 RETURN
256 *
257 * End of DLATRD
258 *
259 END