1 SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
2 *
3 * -- LAPACK 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 INFO, LDA, LWORK, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
14 $ WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DSYTRD reduces a real symmetric matrix A to real symmetric
21 * tridiagonal form T by an orthogonal similarity transformation:
22 * Q**T * A * Q = T.
23 *
24 * Arguments
25 * =========
26 *
27 * UPLO (input) CHARACTER*1
28 * = 'U': Upper triangle of A is stored;
29 * = 'L': Lower triangle of A is stored.
30 *
31 * N (input) INTEGER
32 * The order of the matrix A. N >= 0.
33 *
34 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
35 * On entry, the symmetric matrix A. If UPLO = 'U', the leading
36 * N-by-N upper triangular part of A contains the upper
37 * triangular part of the matrix A, and the strictly lower
38 * triangular part of A is not referenced. If UPLO = 'L', the
39 * leading N-by-N lower triangular part of A contains the lower
40 * triangular part of the matrix A, and the strictly upper
41 * triangular part of A is not referenced.
42 * On exit, if UPLO = 'U', the diagonal and first superdiagonal
43 * of A are overwritten by the corresponding elements of the
44 * tridiagonal matrix T, and the elements above the first
45 * superdiagonal, with the array TAU, represent the orthogonal
46 * matrix Q as a product of elementary reflectors; if UPLO
47 * = 'L', the diagonal and first subdiagonal of A are over-
48 * written by the corresponding elements of the tridiagonal
49 * matrix T, and the elements below the first subdiagonal, with
50 * the array TAU, represent the orthogonal matrix Q as a product
51 * of elementary reflectors. See Further Details.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,N).
55 *
56 * D (output) DOUBLE PRECISION array, dimension (N)
57 * The diagonal elements of the tridiagonal matrix T:
58 * D(i) = A(i,i).
59 *
60 * E (output) DOUBLE PRECISION array, dimension (N-1)
61 * The off-diagonal elements of the tridiagonal matrix T:
62 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
63 *
64 * TAU (output) DOUBLE PRECISION array, dimension (N-1)
65 * The scalar factors of the elementary reflectors (see Further
66 * Details).
67 *
68 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
69 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
70 *
71 * LWORK (input) INTEGER
72 * The dimension of the array WORK. LWORK >= 1.
73 * For optimum performance LWORK >= N*NB, where NB is the
74 * optimal blocksize.
75 *
76 * If LWORK = -1, then a workspace query is assumed; the routine
77 * only calculates the optimal size of the WORK array, returns
78 * this value as the first entry of the WORK array, and no error
79 * message related to LWORK is issued by XERBLA.
80 *
81 * INFO (output) INTEGER
82 * = 0: successful exit
83 * < 0: if INFO = -i, the i-th argument had an illegal value
84 *
85 * Further Details
86 * ===============
87 *
88 * If UPLO = 'U', the matrix Q is represented as a product of elementary
89 * reflectors
90 *
91 * Q = H(n-1) . . . H(2) H(1).
92 *
93 * Each H(i) has the form
94 *
95 * H(i) = I - tau * v * v**T
96 *
97 * where tau is a real scalar, and v is a real vector with
98 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
99 * A(1:i-1,i+1), and tau in TAU(i).
100 *
101 * If UPLO = 'L', the matrix Q is represented as a product of elementary
102 * reflectors
103 *
104 * Q = H(1) H(2) . . . H(n-1).
105 *
106 * Each H(i) has the form
107 *
108 * H(i) = I - tau * v * v**T
109 *
110 * where tau is a real scalar, and v is a real vector with
111 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
112 * and tau in TAU(i).
113 *
114 * The contents of A on exit are illustrated by the following examples
115 * with n = 5:
116 *
117 * if UPLO = 'U': if UPLO = 'L':
118 *
119 * ( d e v2 v3 v4 ) ( d )
120 * ( d e v3 v4 ) ( e d )
121 * ( d e v4 ) ( v1 e d )
122 * ( d e ) ( v1 v2 e d )
123 * ( d ) ( v1 v2 v3 e d )
124 *
125 * where d and e denote diagonal and off-diagonal elements of T, and vi
126 * denotes an element of the vector defining H(i).
127 *
128 * =====================================================================
129 *
130 * .. Parameters ..
131 DOUBLE PRECISION ONE
132 PARAMETER ( ONE = 1.0D+0 )
133 * ..
134 * .. Local Scalars ..
135 LOGICAL LQUERY, UPPER
136 INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
137 $ NBMIN, NX
138 * ..
139 * .. External Subroutines ..
140 EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA
141 * ..
142 * .. Intrinsic Functions ..
143 INTRINSIC MAX
144 * ..
145 * .. External Functions ..
146 LOGICAL LSAME
147 INTEGER ILAENV
148 EXTERNAL LSAME, ILAENV
149 * ..
150 * .. Executable Statements ..
151 *
152 * Test the input parameters
153 *
154 INFO = 0
155 UPPER = LSAME( UPLO, 'U' )
156 LQUERY = ( LWORK.EQ.-1 )
157 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
158 INFO = -1
159 ELSE IF( N.LT.0 ) THEN
160 INFO = -2
161 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
162 INFO = -4
163 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
164 INFO = -9
165 END IF
166 *
167 IF( INFO.EQ.0 ) THEN
168 *
169 * Determine the block size.
170 *
171 NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
172 LWKOPT = N*NB
173 WORK( 1 ) = LWKOPT
174 END IF
175 *
176 IF( INFO.NE.0 ) THEN
177 CALL XERBLA( 'DSYTRD', -INFO )
178 RETURN
179 ELSE IF( LQUERY ) THEN
180 RETURN
181 END IF
182 *
183 * Quick return if possible
184 *
185 IF( N.EQ.0 ) THEN
186 WORK( 1 ) = 1
187 RETURN
188 END IF
189 *
190 NX = N
191 IWS = 1
192 IF( NB.GT.1 .AND. NB.LT.N ) THEN
193 *
194 * Determine when to cross over from blocked to unblocked code
195 * (last block is always handled by unblocked code).
196 *
197 NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
198 IF( NX.LT.N ) THEN
199 *
200 * Determine if workspace is large enough for blocked code.
201 *
202 LDWORK = N
203 IWS = LDWORK*NB
204 IF( LWORK.LT.IWS ) THEN
205 *
206 * Not enough workspace to use optimal NB: determine the
207 * minimum value of NB, and reduce NB or force use of
208 * unblocked code by setting NX = N.
209 *
210 NB = MAX( LWORK / LDWORK, 1 )
211 NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
212 IF( NB.LT.NBMIN )
213 $ NX = N
214 END IF
215 ELSE
216 NX = N
217 END IF
218 ELSE
219 NB = 1
220 END IF
221 *
222 IF( UPPER ) THEN
223 *
224 * Reduce the upper triangle of A.
225 * Columns 1:kk are handled by the unblocked method.
226 *
227 KK = N - ( ( N-NX+NB-1 ) / NB )*NB
228 DO 20 I = N - NB + 1, KK + 1, -NB
229 *
230 * Reduce columns i:i+nb-1 to tridiagonal form and form the
231 * matrix W which is needed to update the unreduced part of
232 * the matrix
233 *
234 CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
235 $ LDWORK )
236 *
237 * Update the unreduced submatrix A(1:i-1,1:i-1), using an
238 * update of the form: A := A - V*W**T - W*V**T
239 *
240 CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
241 $ LDA, WORK, LDWORK, ONE, A, LDA )
242 *
243 * Copy superdiagonal elements back into A, and diagonal
244 * elements into D
245 *
246 DO 10 J = I, I + NB - 1
247 A( J-1, J ) = E( J-1 )
248 D( J ) = A( J, J )
249 10 CONTINUE
250 20 CONTINUE
251 *
252 * Use unblocked code to reduce the last or only block
253 *
254 CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
255 ELSE
256 *
257 * Reduce the lower triangle of A
258 *
259 DO 40 I = 1, N - NX, NB
260 *
261 * Reduce columns i:i+nb-1 to tridiagonal form and form the
262 * matrix W which is needed to update the unreduced part of
263 * the matrix
264 *
265 CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
266 $ TAU( I ), WORK, LDWORK )
267 *
268 * Update the unreduced submatrix A(i+ib:n,i+ib:n), using
269 * an update of the form: A := A - V*W**T - W*V**T
270 *
271 CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
272 $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
273 $ A( I+NB, I+NB ), LDA )
274 *
275 * Copy subdiagonal elements back into A, and diagonal
276 * elements into D
277 *
278 DO 30 J = I, I + NB - 1
279 A( J+1, J ) = E( J )
280 D( J ) = A( J, J )
281 30 CONTINUE
282 40 CONTINUE
283 *
284 * Use unblocked code to reduce the last or only block
285 *
286 CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
287 $ TAU( I ), IINFO )
288 END IF
289 *
290 WORK( 1 ) = LWKOPT
291 RETURN
292 *
293 * End of DSYTRD
294 *
295 END
2 *
3 * -- LAPACK 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 INFO, LDA, LWORK, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
14 $ WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DSYTRD reduces a real symmetric matrix A to real symmetric
21 * tridiagonal form T by an orthogonal similarity transformation:
22 * Q**T * A * Q = T.
23 *
24 * Arguments
25 * =========
26 *
27 * UPLO (input) CHARACTER*1
28 * = 'U': Upper triangle of A is stored;
29 * = 'L': Lower triangle of A is stored.
30 *
31 * N (input) INTEGER
32 * The order of the matrix A. N >= 0.
33 *
34 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
35 * On entry, the symmetric matrix A. If UPLO = 'U', the leading
36 * N-by-N upper triangular part of A contains the upper
37 * triangular part of the matrix A, and the strictly lower
38 * triangular part of A is not referenced. If UPLO = 'L', the
39 * leading N-by-N lower triangular part of A contains the lower
40 * triangular part of the matrix A, and the strictly upper
41 * triangular part of A is not referenced.
42 * On exit, if UPLO = 'U', the diagonal and first superdiagonal
43 * of A are overwritten by the corresponding elements of the
44 * tridiagonal matrix T, and the elements above the first
45 * superdiagonal, with the array TAU, represent the orthogonal
46 * matrix Q as a product of elementary reflectors; if UPLO
47 * = 'L', the diagonal and first subdiagonal of A are over-
48 * written by the corresponding elements of the tridiagonal
49 * matrix T, and the elements below the first subdiagonal, with
50 * the array TAU, represent the orthogonal matrix Q as a product
51 * of elementary reflectors. See Further Details.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,N).
55 *
56 * D (output) DOUBLE PRECISION array, dimension (N)
57 * The diagonal elements of the tridiagonal matrix T:
58 * D(i) = A(i,i).
59 *
60 * E (output) DOUBLE PRECISION array, dimension (N-1)
61 * The off-diagonal elements of the tridiagonal matrix T:
62 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
63 *
64 * TAU (output) DOUBLE PRECISION array, dimension (N-1)
65 * The scalar factors of the elementary reflectors (see Further
66 * Details).
67 *
68 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
69 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
70 *
71 * LWORK (input) INTEGER
72 * The dimension of the array WORK. LWORK >= 1.
73 * For optimum performance LWORK >= N*NB, where NB is the
74 * optimal blocksize.
75 *
76 * If LWORK = -1, then a workspace query is assumed; the routine
77 * only calculates the optimal size of the WORK array, returns
78 * this value as the first entry of the WORK array, and no error
79 * message related to LWORK is issued by XERBLA.
80 *
81 * INFO (output) INTEGER
82 * = 0: successful exit
83 * < 0: if INFO = -i, the i-th argument had an illegal value
84 *
85 * Further Details
86 * ===============
87 *
88 * If UPLO = 'U', the matrix Q is represented as a product of elementary
89 * reflectors
90 *
91 * Q = H(n-1) . . . H(2) H(1).
92 *
93 * Each H(i) has the form
94 *
95 * H(i) = I - tau * v * v**T
96 *
97 * where tau is a real scalar, and v is a real vector with
98 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
99 * A(1:i-1,i+1), and tau in TAU(i).
100 *
101 * If UPLO = 'L', the matrix Q is represented as a product of elementary
102 * reflectors
103 *
104 * Q = H(1) H(2) . . . H(n-1).
105 *
106 * Each H(i) has the form
107 *
108 * H(i) = I - tau * v * v**T
109 *
110 * where tau is a real scalar, and v is a real vector with
111 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
112 * and tau in TAU(i).
113 *
114 * The contents of A on exit are illustrated by the following examples
115 * with n = 5:
116 *
117 * if UPLO = 'U': if UPLO = 'L':
118 *
119 * ( d e v2 v3 v4 ) ( d )
120 * ( d e v3 v4 ) ( e d )
121 * ( d e v4 ) ( v1 e d )
122 * ( d e ) ( v1 v2 e d )
123 * ( d ) ( v1 v2 v3 e d )
124 *
125 * where d and e denote diagonal and off-diagonal elements of T, and vi
126 * denotes an element of the vector defining H(i).
127 *
128 * =====================================================================
129 *
130 * .. Parameters ..
131 DOUBLE PRECISION ONE
132 PARAMETER ( ONE = 1.0D+0 )
133 * ..
134 * .. Local Scalars ..
135 LOGICAL LQUERY, UPPER
136 INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
137 $ NBMIN, NX
138 * ..
139 * .. External Subroutines ..
140 EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA
141 * ..
142 * .. Intrinsic Functions ..
143 INTRINSIC MAX
144 * ..
145 * .. External Functions ..
146 LOGICAL LSAME
147 INTEGER ILAENV
148 EXTERNAL LSAME, ILAENV
149 * ..
150 * .. Executable Statements ..
151 *
152 * Test the input parameters
153 *
154 INFO = 0
155 UPPER = LSAME( UPLO, 'U' )
156 LQUERY = ( LWORK.EQ.-1 )
157 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
158 INFO = -1
159 ELSE IF( N.LT.0 ) THEN
160 INFO = -2
161 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
162 INFO = -4
163 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
164 INFO = -9
165 END IF
166 *
167 IF( INFO.EQ.0 ) THEN
168 *
169 * Determine the block size.
170 *
171 NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
172 LWKOPT = N*NB
173 WORK( 1 ) = LWKOPT
174 END IF
175 *
176 IF( INFO.NE.0 ) THEN
177 CALL XERBLA( 'DSYTRD', -INFO )
178 RETURN
179 ELSE IF( LQUERY ) THEN
180 RETURN
181 END IF
182 *
183 * Quick return if possible
184 *
185 IF( N.EQ.0 ) THEN
186 WORK( 1 ) = 1
187 RETURN
188 END IF
189 *
190 NX = N
191 IWS = 1
192 IF( NB.GT.1 .AND. NB.LT.N ) THEN
193 *
194 * Determine when to cross over from blocked to unblocked code
195 * (last block is always handled by unblocked code).
196 *
197 NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
198 IF( NX.LT.N ) THEN
199 *
200 * Determine if workspace is large enough for blocked code.
201 *
202 LDWORK = N
203 IWS = LDWORK*NB
204 IF( LWORK.LT.IWS ) THEN
205 *
206 * Not enough workspace to use optimal NB: determine the
207 * minimum value of NB, and reduce NB or force use of
208 * unblocked code by setting NX = N.
209 *
210 NB = MAX( LWORK / LDWORK, 1 )
211 NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
212 IF( NB.LT.NBMIN )
213 $ NX = N
214 END IF
215 ELSE
216 NX = N
217 END IF
218 ELSE
219 NB = 1
220 END IF
221 *
222 IF( UPPER ) THEN
223 *
224 * Reduce the upper triangle of A.
225 * Columns 1:kk are handled by the unblocked method.
226 *
227 KK = N - ( ( N-NX+NB-1 ) / NB )*NB
228 DO 20 I = N - NB + 1, KK + 1, -NB
229 *
230 * Reduce columns i:i+nb-1 to tridiagonal form and form the
231 * matrix W which is needed to update the unreduced part of
232 * the matrix
233 *
234 CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
235 $ LDWORK )
236 *
237 * Update the unreduced submatrix A(1:i-1,1:i-1), using an
238 * update of the form: A := A - V*W**T - W*V**T
239 *
240 CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
241 $ LDA, WORK, LDWORK, ONE, A, LDA )
242 *
243 * Copy superdiagonal elements back into A, and diagonal
244 * elements into D
245 *
246 DO 10 J = I, I + NB - 1
247 A( J-1, J ) = E( J-1 )
248 D( J ) = A( J, J )
249 10 CONTINUE
250 20 CONTINUE
251 *
252 * Use unblocked code to reduce the last or only block
253 *
254 CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
255 ELSE
256 *
257 * Reduce the lower triangle of A
258 *
259 DO 40 I = 1, N - NX, NB
260 *
261 * Reduce columns i:i+nb-1 to tridiagonal form and form the
262 * matrix W which is needed to update the unreduced part of
263 * the matrix
264 *
265 CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
266 $ TAU( I ), WORK, LDWORK )
267 *
268 * Update the unreduced submatrix A(i+ib:n,i+ib:n), using
269 * an update of the form: A := A - V*W**T - W*V**T
270 *
271 CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
272 $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
273 $ A( I+NB, I+NB ), LDA )
274 *
275 * Copy subdiagonal elements back into A, and diagonal
276 * elements into D
277 *
278 DO 30 J = I, I + NB - 1
279 A( J+1, J ) = E( J )
280 D( J ) = A( J, J )
281 30 CONTINUE
282 40 CONTINUE
283 *
284 * Use unblocked code to reduce the last or only block
285 *
286 CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
287 $ TAU( I ), IINFO )
288 END IF
289 *
290 WORK( 1 ) = LWKOPT
291 RETURN
292 *
293 * End of DSYTRD
294 *
295 END