1 SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, 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, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
20 * form T by an orthogonal similarity transformation: Q**T * A * Q = T.
21 *
22 * Arguments
23 * =========
24 *
25 * UPLO (input) CHARACTER*1
26 * Specifies whether the upper or lower triangular part of the
27 * symmetric matrix A is stored:
28 * = 'U': Upper triangular
29 * = 'L': Lower triangular
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 * INFO (output) INTEGER
69 * = 0: successful exit
70 * < 0: if INFO = -i, the i-th argument had an illegal value.
71 *
72 * Further Details
73 * ===============
74 *
75 * If UPLO = 'U', the matrix Q is represented as a product of elementary
76 * reflectors
77 *
78 * Q = H(n-1) . . . H(2) H(1).
79 *
80 * Each H(i) has the form
81 *
82 * H(i) = I - tau * v * v**T
83 *
84 * where tau is a real scalar, and v is a real vector with
85 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
86 * A(1:i-1,i+1), and tau in TAU(i).
87 *
88 * If UPLO = 'L', the matrix Q is represented as a product of elementary
89 * reflectors
90 *
91 * Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
99 * and tau in TAU(i).
100 *
101 * The contents of A on exit are illustrated by the following examples
102 * with n = 5:
103 *
104 * if UPLO = 'U': if UPLO = 'L':
105 *
106 * ( d e v2 v3 v4 ) ( d )
107 * ( d e v3 v4 ) ( e d )
108 * ( d e v4 ) ( v1 e d )
109 * ( d e ) ( v1 v2 e d )
110 * ( d ) ( v1 v2 v3 e d )
111 *
112 * where d and e denote diagonal and off-diagonal elements of T, and vi
113 * denotes an element of the vector defining H(i).
114 *
115 * =====================================================================
116 *
117 * .. Parameters ..
118 DOUBLE PRECISION ONE, ZERO, HALF
119 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
120 $ HALF = 1.0D0 / 2.0D0 )
121 * ..
122 * .. Local Scalars ..
123 LOGICAL UPPER
124 INTEGER I
125 DOUBLE PRECISION ALPHA, TAUI
126 * ..
127 * .. External Subroutines ..
128 EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
129 * ..
130 * .. External Functions ..
131 LOGICAL LSAME
132 DOUBLE PRECISION DDOT
133 EXTERNAL LSAME, DDOT
134 * ..
135 * .. Intrinsic Functions ..
136 INTRINSIC MAX, MIN
137 * ..
138 * .. Executable Statements ..
139 *
140 * Test the input parameters
141 *
142 INFO = 0
143 UPPER = LSAME( UPLO, 'U' )
144 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
145 INFO = -1
146 ELSE IF( N.LT.0 ) THEN
147 INFO = -2
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -4
150 END IF
151 IF( INFO.NE.0 ) THEN
152 CALL XERBLA( 'DSYTD2', -INFO )
153 RETURN
154 END IF
155 *
156 * Quick return if possible
157 *
158 IF( N.LE.0 )
159 $ RETURN
160 *
161 IF( UPPER ) THEN
162 *
163 * Reduce the upper triangle of A
164 *
165 DO 10 I = N - 1, 1, -1
166 *
167 * Generate elementary reflector H(i) = I - tau * v * v**T
168 * to annihilate A(1:i-1,i+1)
169 *
170 CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
171 E( I ) = A( I, I+1 )
172 *
173 IF( TAUI.NE.ZERO ) THEN
174 *
175 * Apply H(i) from both sides to A(1:i,1:i)
176 *
177 A( I, I+1 ) = ONE
178 *
179 * Compute x := tau * A * v storing x in TAU(1:i)
180 *
181 CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
182 $ TAU, 1 )
183 *
184 * Compute w := x - 1/2 * tau * (x**T * v) * v
185 *
186 ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
187 CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
188 *
189 * Apply the transformation as a rank-2 update:
190 * A := A - v * w**T - w * v**T
191 *
192 CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
193 $ LDA )
194 *
195 A( I, I+1 ) = E( I )
196 END IF
197 D( I+1 ) = A( I+1, I+1 )
198 TAU( I ) = TAUI
199 10 CONTINUE
200 D( 1 ) = A( 1, 1 )
201 ELSE
202 *
203 * Reduce the lower triangle of A
204 *
205 DO 20 I = 1, N - 1
206 *
207 * Generate elementary reflector H(i) = I - tau * v * v**T
208 * to annihilate A(i+2:n,i)
209 *
210 CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
211 $ TAUI )
212 E( I ) = A( I+1, I )
213 *
214 IF( TAUI.NE.ZERO ) THEN
215 *
216 * Apply H(i) from both sides to A(i+1:n,i+1:n)
217 *
218 A( I+1, I ) = ONE
219 *
220 * Compute x := tau * A * v storing y in TAU(i:n-1)
221 *
222 CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
223 $ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
224 *
225 * Compute w := x - 1/2 * tau * (x**T * v) * v
226 *
227 ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ),
228 $ 1 )
229 CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
230 *
231 * Apply the transformation as a rank-2 update:
232 * A := A - v * w**T - w * v**T
233 *
234 CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
235 $ A( I+1, I+1 ), LDA )
236 *
237 A( I+1, I ) = E( I )
238 END IF
239 D( I ) = A( I, I )
240 TAU( I ) = TAUI
241 20 CONTINUE
242 D( N ) = A( N, N )
243 END IF
244 *
245 RETURN
246 *
247 * End of DSYTD2
248 *
249 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, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
20 * form T by an orthogonal similarity transformation: Q**T * A * Q = T.
21 *
22 * Arguments
23 * =========
24 *
25 * UPLO (input) CHARACTER*1
26 * Specifies whether the upper or lower triangular part of the
27 * symmetric matrix A is stored:
28 * = 'U': Upper triangular
29 * = 'L': Lower triangular
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 * INFO (output) INTEGER
69 * = 0: successful exit
70 * < 0: if INFO = -i, the i-th argument had an illegal value.
71 *
72 * Further Details
73 * ===============
74 *
75 * If UPLO = 'U', the matrix Q is represented as a product of elementary
76 * reflectors
77 *
78 * Q = H(n-1) . . . H(2) H(1).
79 *
80 * Each H(i) has the form
81 *
82 * H(i) = I - tau * v * v**T
83 *
84 * where tau is a real scalar, and v is a real vector with
85 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
86 * A(1:i-1,i+1), and tau in TAU(i).
87 *
88 * If UPLO = 'L', the matrix Q is represented as a product of elementary
89 * reflectors
90 *
91 * Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
99 * and tau in TAU(i).
100 *
101 * The contents of A on exit are illustrated by the following examples
102 * with n = 5:
103 *
104 * if UPLO = 'U': if UPLO = 'L':
105 *
106 * ( d e v2 v3 v4 ) ( d )
107 * ( d e v3 v4 ) ( e d )
108 * ( d e v4 ) ( v1 e d )
109 * ( d e ) ( v1 v2 e d )
110 * ( d ) ( v1 v2 v3 e d )
111 *
112 * where d and e denote diagonal and off-diagonal elements of T, and vi
113 * denotes an element of the vector defining H(i).
114 *
115 * =====================================================================
116 *
117 * .. Parameters ..
118 DOUBLE PRECISION ONE, ZERO, HALF
119 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
120 $ HALF = 1.0D0 / 2.0D0 )
121 * ..
122 * .. Local Scalars ..
123 LOGICAL UPPER
124 INTEGER I
125 DOUBLE PRECISION ALPHA, TAUI
126 * ..
127 * .. External Subroutines ..
128 EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
129 * ..
130 * .. External Functions ..
131 LOGICAL LSAME
132 DOUBLE PRECISION DDOT
133 EXTERNAL LSAME, DDOT
134 * ..
135 * .. Intrinsic Functions ..
136 INTRINSIC MAX, MIN
137 * ..
138 * .. Executable Statements ..
139 *
140 * Test the input parameters
141 *
142 INFO = 0
143 UPPER = LSAME( UPLO, 'U' )
144 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
145 INFO = -1
146 ELSE IF( N.LT.0 ) THEN
147 INFO = -2
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -4
150 END IF
151 IF( INFO.NE.0 ) THEN
152 CALL XERBLA( 'DSYTD2', -INFO )
153 RETURN
154 END IF
155 *
156 * Quick return if possible
157 *
158 IF( N.LE.0 )
159 $ RETURN
160 *
161 IF( UPPER ) THEN
162 *
163 * Reduce the upper triangle of A
164 *
165 DO 10 I = N - 1, 1, -1
166 *
167 * Generate elementary reflector H(i) = I - tau * v * v**T
168 * to annihilate A(1:i-1,i+1)
169 *
170 CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
171 E( I ) = A( I, I+1 )
172 *
173 IF( TAUI.NE.ZERO ) THEN
174 *
175 * Apply H(i) from both sides to A(1:i,1:i)
176 *
177 A( I, I+1 ) = ONE
178 *
179 * Compute x := tau * A * v storing x in TAU(1:i)
180 *
181 CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
182 $ TAU, 1 )
183 *
184 * Compute w := x - 1/2 * tau * (x**T * v) * v
185 *
186 ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
187 CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
188 *
189 * Apply the transformation as a rank-2 update:
190 * A := A - v * w**T - w * v**T
191 *
192 CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
193 $ LDA )
194 *
195 A( I, I+1 ) = E( I )
196 END IF
197 D( I+1 ) = A( I+1, I+1 )
198 TAU( I ) = TAUI
199 10 CONTINUE
200 D( 1 ) = A( 1, 1 )
201 ELSE
202 *
203 * Reduce the lower triangle of A
204 *
205 DO 20 I = 1, N - 1
206 *
207 * Generate elementary reflector H(i) = I - tau * v * v**T
208 * to annihilate A(i+2:n,i)
209 *
210 CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
211 $ TAUI )
212 E( I ) = A( I+1, I )
213 *
214 IF( TAUI.NE.ZERO ) THEN
215 *
216 * Apply H(i) from both sides to A(i+1:n,i+1:n)
217 *
218 A( I+1, I ) = ONE
219 *
220 * Compute x := tau * A * v storing y in TAU(i:n-1)
221 *
222 CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
223 $ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
224 *
225 * Compute w := x - 1/2 * tau * (x**T * v) * v
226 *
227 ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ),
228 $ 1 )
229 CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
230 *
231 * Apply the transformation as a rank-2 update:
232 * A := A - v * w**T - w * v**T
233 *
234 CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
235 $ A( I+1, I+1 ), LDA )
236 *
237 A( I+1, I ) = E( I )
238 END IF
239 D( I ) = A( I, I )
240 TAU( I ) = TAUI
241 20 CONTINUE
242 D( N ) = A( N, N )
243 END IF
244 *
245 RETURN
246 *
247 * End of DSYTD2
248 *
249 END