1 SUBROUTINE DSPTRD( UPLO, N, AP, 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, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DSPTRD reduces a real symmetric matrix A stored in packed form to
20 * symmetric tridiagonal form T by an orthogonal similarity
21 * transformation: Q**T * A * Q = T.
22 *
23 * Arguments
24 * =========
25 *
26 * UPLO (input) CHARACTER*1
27 * = 'U': Upper triangle of A is stored;
28 * = 'L': Lower triangle of A is stored.
29 *
30 * N (input) INTEGER
31 * The order of the matrix A. N >= 0.
32 *
33 * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
34 * On entry, the upper or lower triangle of the symmetric matrix
35 * A, packed columnwise in a linear array. The j-th column of A
36 * is stored in the array AP as follows:
37 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
38 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
39 * On exit, if UPLO = 'U', the diagonal and first superdiagonal
40 * of A are overwritten by the corresponding elements of the
41 * tridiagonal matrix T, and the elements above the first
42 * superdiagonal, with the array TAU, represent the orthogonal
43 * matrix Q as a product of elementary reflectors; if UPLO
44 * = 'L', the diagonal and first subdiagonal of A are over-
45 * written by the corresponding elements of the tridiagonal
46 * matrix T, and the elements below the first subdiagonal, with
47 * the array TAU, represent the orthogonal matrix Q as a product
48 * of elementary reflectors. See Further Details.
49 *
50 * D (output) DOUBLE PRECISION array, dimension (N)
51 * The diagonal elements of the tridiagonal matrix T:
52 * D(i) = A(i,i).
53 *
54 * E (output) DOUBLE PRECISION array, dimension (N-1)
55 * The off-diagonal elements of the tridiagonal matrix T:
56 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
57 *
58 * TAU (output) DOUBLE PRECISION array, dimension (N-1)
59 * The scalar factors of the elementary reflectors (see Further
60 * Details).
61 *
62 * INFO (output) INTEGER
63 * = 0: successful exit
64 * < 0: if INFO = -i, the i-th argument had an illegal value
65 *
66 * Further Details
67 * ===============
68 *
69 * If UPLO = 'U', the matrix Q is represented as a product of elementary
70 * reflectors
71 *
72 * Q = H(n-1) . . . H(2) H(1).
73 *
74 * Each H(i) has the form
75 *
76 * H(i) = I - tau * v * v**T
77 *
78 * where tau is a real scalar, and v is a real vector with
79 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
80 * overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
81 *
82 * If UPLO = 'L', the matrix Q is represented as a product of elementary
83 * reflectors
84 *
85 * Q = H(1) H(2) . . . H(n-1).
86 *
87 * Each H(i) has the form
88 *
89 * H(i) = I - tau * v * v**T
90 *
91 * where tau is a real scalar, and v is a real vector with
92 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
93 * overwriting A(i+2:n,i), and tau is stored in TAU(i).
94 *
95 * =====================================================================
96 *
97 * .. Parameters ..
98 DOUBLE PRECISION ONE, ZERO, HALF
99 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
100 $ HALF = 1.0D0 / 2.0D0 )
101 * ..
102 * .. Local Scalars ..
103 LOGICAL UPPER
104 INTEGER I, I1, I1I1, II
105 DOUBLE PRECISION ALPHA, TAUI
106 * ..
107 * .. External Subroutines ..
108 EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 DOUBLE PRECISION DDOT
113 EXTERNAL LSAME, DDOT
114 * ..
115 * .. Executable Statements ..
116 *
117 * Test the input parameters
118 *
119 INFO = 0
120 UPPER = LSAME( UPLO, 'U' )
121 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
122 INFO = -1
123 ELSE IF( N.LT.0 ) THEN
124 INFO = -2
125 END IF
126 IF( INFO.NE.0 ) THEN
127 CALL XERBLA( 'DSPTRD', -INFO )
128 RETURN
129 END IF
130 *
131 * Quick return if possible
132 *
133 IF( N.LE.0 )
134 $ RETURN
135 *
136 IF( UPPER ) THEN
137 *
138 * Reduce the upper triangle of A.
139 * I1 is the index in AP of A(1,I+1).
140 *
141 I1 = N*( N-1 ) / 2 + 1
142 DO 10 I = N - 1, 1, -1
143 *
144 * Generate elementary reflector H(i) = I - tau * v * v**T
145 * to annihilate A(1:i-1,i+1)
146 *
147 CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
148 E( I ) = AP( I1+I-1 )
149 *
150 IF( TAUI.NE.ZERO ) THEN
151 *
152 * Apply H(i) from both sides to A(1:i,1:i)
153 *
154 AP( I1+I-1 ) = ONE
155 *
156 * Compute y := tau * A * v storing y in TAU(1:i)
157 *
158 CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
159 $ 1 )
160 *
161 * Compute w := y - 1/2 * tau * (y**T *v) * v
162 *
163 ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
164 CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
165 *
166 * Apply the transformation as a rank-2 update:
167 * A := A - v * w**T - w * v**T
168 *
169 CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
170 *
171 AP( I1+I-1 ) = E( I )
172 END IF
173 D( I+1 ) = AP( I1+I )
174 TAU( I ) = TAUI
175 I1 = I1 - I
176 10 CONTINUE
177 D( 1 ) = AP( 1 )
178 ELSE
179 *
180 * Reduce the lower triangle of A. II is the index in AP of
181 * A(i,i) and I1I1 is the index of A(i+1,i+1).
182 *
183 II = 1
184 DO 20 I = 1, N - 1
185 I1I1 = II + N - I + 1
186 *
187 * Generate elementary reflector H(i) = I - tau * v * v**T
188 * to annihilate A(i+2:n,i)
189 *
190 CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
191 E( I ) = AP( II+1 )
192 *
193 IF( TAUI.NE.ZERO ) THEN
194 *
195 * Apply H(i) from both sides to A(i+1:n,i+1:n)
196 *
197 AP( II+1 ) = ONE
198 *
199 * Compute y := tau * A * v storing y in TAU(i:n-1)
200 *
201 CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
202 $ ZERO, TAU( I ), 1 )
203 *
204 * Compute w := y - 1/2 * tau * (y**T *v) * v
205 *
206 ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
207 $ 1 )
208 CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
209 *
210 * Apply the transformation as a rank-2 update:
211 * A := A - v * w**T - w * v**T
212 *
213 CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
214 $ AP( I1I1 ) )
215 *
216 AP( II+1 ) = E( I )
217 END IF
218 D( I ) = AP( II )
219 TAU( I ) = TAUI
220 II = I1I1
221 20 CONTINUE
222 D( N ) = AP( II )
223 END IF
224 *
225 RETURN
226 *
227 * End of DSPTRD
228 *
229 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, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * DSPTRD reduces a real symmetric matrix A stored in packed form to
20 * symmetric tridiagonal form T by an orthogonal similarity
21 * transformation: Q**T * A * Q = T.
22 *
23 * Arguments
24 * =========
25 *
26 * UPLO (input) CHARACTER*1
27 * = 'U': Upper triangle of A is stored;
28 * = 'L': Lower triangle of A is stored.
29 *
30 * N (input) INTEGER
31 * The order of the matrix A. N >= 0.
32 *
33 * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
34 * On entry, the upper or lower triangle of the symmetric matrix
35 * A, packed columnwise in a linear array. The j-th column of A
36 * is stored in the array AP as follows:
37 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
38 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
39 * On exit, if UPLO = 'U', the diagonal and first superdiagonal
40 * of A are overwritten by the corresponding elements of the
41 * tridiagonal matrix T, and the elements above the first
42 * superdiagonal, with the array TAU, represent the orthogonal
43 * matrix Q as a product of elementary reflectors; if UPLO
44 * = 'L', the diagonal and first subdiagonal of A are over-
45 * written by the corresponding elements of the tridiagonal
46 * matrix T, and the elements below the first subdiagonal, with
47 * the array TAU, represent the orthogonal matrix Q as a product
48 * of elementary reflectors. See Further Details.
49 *
50 * D (output) DOUBLE PRECISION array, dimension (N)
51 * The diagonal elements of the tridiagonal matrix T:
52 * D(i) = A(i,i).
53 *
54 * E (output) DOUBLE PRECISION array, dimension (N-1)
55 * The off-diagonal elements of the tridiagonal matrix T:
56 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
57 *
58 * TAU (output) DOUBLE PRECISION array, dimension (N-1)
59 * The scalar factors of the elementary reflectors (see Further
60 * Details).
61 *
62 * INFO (output) INTEGER
63 * = 0: successful exit
64 * < 0: if INFO = -i, the i-th argument had an illegal value
65 *
66 * Further Details
67 * ===============
68 *
69 * If UPLO = 'U', the matrix Q is represented as a product of elementary
70 * reflectors
71 *
72 * Q = H(n-1) . . . H(2) H(1).
73 *
74 * Each H(i) has the form
75 *
76 * H(i) = I - tau * v * v**T
77 *
78 * where tau is a real scalar, and v is a real vector with
79 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
80 * overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
81 *
82 * If UPLO = 'L', the matrix Q is represented as a product of elementary
83 * reflectors
84 *
85 * Q = H(1) H(2) . . . H(n-1).
86 *
87 * Each H(i) has the form
88 *
89 * H(i) = I - tau * v * v**T
90 *
91 * where tau is a real scalar, and v is a real vector with
92 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
93 * overwriting A(i+2:n,i), and tau is stored in TAU(i).
94 *
95 * =====================================================================
96 *
97 * .. Parameters ..
98 DOUBLE PRECISION ONE, ZERO, HALF
99 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
100 $ HALF = 1.0D0 / 2.0D0 )
101 * ..
102 * .. Local Scalars ..
103 LOGICAL UPPER
104 INTEGER I, I1, I1I1, II
105 DOUBLE PRECISION ALPHA, TAUI
106 * ..
107 * .. External Subroutines ..
108 EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 DOUBLE PRECISION DDOT
113 EXTERNAL LSAME, DDOT
114 * ..
115 * .. Executable Statements ..
116 *
117 * Test the input parameters
118 *
119 INFO = 0
120 UPPER = LSAME( UPLO, 'U' )
121 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
122 INFO = -1
123 ELSE IF( N.LT.0 ) THEN
124 INFO = -2
125 END IF
126 IF( INFO.NE.0 ) THEN
127 CALL XERBLA( 'DSPTRD', -INFO )
128 RETURN
129 END IF
130 *
131 * Quick return if possible
132 *
133 IF( N.LE.0 )
134 $ RETURN
135 *
136 IF( UPPER ) THEN
137 *
138 * Reduce the upper triangle of A.
139 * I1 is the index in AP of A(1,I+1).
140 *
141 I1 = N*( N-1 ) / 2 + 1
142 DO 10 I = N - 1, 1, -1
143 *
144 * Generate elementary reflector H(i) = I - tau * v * v**T
145 * to annihilate A(1:i-1,i+1)
146 *
147 CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
148 E( I ) = AP( I1+I-1 )
149 *
150 IF( TAUI.NE.ZERO ) THEN
151 *
152 * Apply H(i) from both sides to A(1:i,1:i)
153 *
154 AP( I1+I-1 ) = ONE
155 *
156 * Compute y := tau * A * v storing y in TAU(1:i)
157 *
158 CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
159 $ 1 )
160 *
161 * Compute w := y - 1/2 * tau * (y**T *v) * v
162 *
163 ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
164 CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
165 *
166 * Apply the transformation as a rank-2 update:
167 * A := A - v * w**T - w * v**T
168 *
169 CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
170 *
171 AP( I1+I-1 ) = E( I )
172 END IF
173 D( I+1 ) = AP( I1+I )
174 TAU( I ) = TAUI
175 I1 = I1 - I
176 10 CONTINUE
177 D( 1 ) = AP( 1 )
178 ELSE
179 *
180 * Reduce the lower triangle of A. II is the index in AP of
181 * A(i,i) and I1I1 is the index of A(i+1,i+1).
182 *
183 II = 1
184 DO 20 I = 1, N - 1
185 I1I1 = II + N - I + 1
186 *
187 * Generate elementary reflector H(i) = I - tau * v * v**T
188 * to annihilate A(i+2:n,i)
189 *
190 CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
191 E( I ) = AP( II+1 )
192 *
193 IF( TAUI.NE.ZERO ) THEN
194 *
195 * Apply H(i) from both sides to A(i+1:n,i+1:n)
196 *
197 AP( II+1 ) = ONE
198 *
199 * Compute y := tau * A * v storing y in TAU(i:n-1)
200 *
201 CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
202 $ ZERO, TAU( I ), 1 )
203 *
204 * Compute w := y - 1/2 * tau * (y**T *v) * v
205 *
206 ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
207 $ 1 )
208 CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
209 *
210 * Apply the transformation as a rank-2 update:
211 * A := A - v * w**T - w * v**T
212 *
213 CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
214 $ AP( I1I1 ) )
215 *
216 AP( II+1 ) = E( I )
217 END IF
218 D( I ) = AP( II )
219 TAU( I ) = TAUI
220 II = I1I1
221 20 CONTINUE
222 D( N ) = AP( II )
223 END IF
224 *
225 RETURN
226 *
227 * End of DSPTRD
228 *
229 END