1 SUBROUTINE ZHPTRD( 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 D( * ), E( * )
14 COMPLEX*16 AP( * ), TAU( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
21 * real symmetric tridiagonal form T by a unitary similarity
22 * transformation: Q**H * 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 * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
35 * On entry, the upper or lower triangle of the Hermitian matrix
36 * A, packed columnwise in a linear array. The j-th column of A
37 * is stored in the array AP as follows:
38 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
39 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
40 * On exit, if UPLO = 'U', the diagonal and first superdiagonal
41 * of A are overwritten by the corresponding elements of the
42 * tridiagonal matrix T, and the elements above the first
43 * superdiagonal, with the array TAU, represent the unitary
44 * matrix Q as a product of elementary reflectors; if UPLO
45 * = 'L', the diagonal and first subdiagonal of A are over-
46 * written by the corresponding elements of the tridiagonal
47 * matrix T, and the elements below the first subdiagonal, with
48 * the array TAU, represent the unitary matrix Q as a product
49 * of elementary reflectors. See Further Details.
50 *
51 * D (output) DOUBLE PRECISION array, dimension (N)
52 * The diagonal elements of the tridiagonal matrix T:
53 * D(i) = A(i,i).
54 *
55 * E (output) DOUBLE PRECISION array, dimension (N-1)
56 * The off-diagonal elements of the tridiagonal matrix T:
57 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
58 *
59 * TAU (output) COMPLEX*16 array, dimension (N-1)
60 * The scalar factors of the elementary reflectors (see Further
61 * Details).
62 *
63 * INFO (output) INTEGER
64 * = 0: successful exit
65 * < 0: if INFO = -i, the i-th argument had an illegal value
66 *
67 * Further Details
68 * ===============
69 *
70 * If UPLO = 'U', the matrix Q is represented as a product of elementary
71 * reflectors
72 *
73 * Q = H(n-1) . . . H(2) H(1).
74 *
75 * Each H(i) has the form
76 *
77 * H(i) = I - tau * v * v**H
78 *
79 * where tau is a complex scalar, and v is a complex vector with
80 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
81 * overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
82 *
83 * If UPLO = 'L', the matrix Q is represented as a product of elementary
84 * reflectors
85 *
86 * Q = H(1) H(2) . . . H(n-1).
87 *
88 * Each H(i) has the form
89 *
90 * H(i) = I - tau * v * v**H
91 *
92 * where tau is a complex scalar, and v is a complex vector with
93 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
94 * overwriting A(i+2:n,i), and tau is stored in TAU(i).
95 *
96 * =====================================================================
97 *
98 * .. Parameters ..
99 COMPLEX*16 ONE, ZERO, HALF
100 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
101 $ ZERO = ( 0.0D+0, 0.0D+0 ),
102 $ HALF = ( 0.5D+0, 0.0D+0 ) )
103 * ..
104 * .. Local Scalars ..
105 LOGICAL UPPER
106 INTEGER I, I1, I1I1, II
107 COMPLEX*16 ALPHA, TAUI
108 * ..
109 * .. External Subroutines ..
110 EXTERNAL XERBLA, ZAXPY, ZHPMV, ZHPR2, ZLARFG
111 * ..
112 * .. External Functions ..
113 LOGICAL LSAME
114 COMPLEX*16 ZDOTC
115 EXTERNAL LSAME, ZDOTC
116 * ..
117 * .. Intrinsic Functions ..
118 INTRINSIC DBLE
119 * ..
120 * .. Executable Statements ..
121 *
122 * Test the input parameters
123 *
124 INFO = 0
125 UPPER = LSAME( UPLO, 'U' )
126 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
127 INFO = -1
128 ELSE IF( N.LT.0 ) THEN
129 INFO = -2
130 END IF
131 IF( INFO.NE.0 ) THEN
132 CALL XERBLA( 'ZHPTRD', -INFO )
133 RETURN
134 END IF
135 *
136 * Quick return if possible
137 *
138 IF( N.LE.0 )
139 $ RETURN
140 *
141 IF( UPPER ) THEN
142 *
143 * Reduce the upper triangle of A.
144 * I1 is the index in AP of A(1,I+1).
145 *
146 I1 = N*( N-1 ) / 2 + 1
147 AP( I1+N-1 ) = DBLE( AP( I1+N-1 ) )
148 DO 10 I = N - 1, 1, -1
149 *
150 * Generate elementary reflector H(i) = I - tau * v * v**H
151 * to annihilate A(1:i-1,i+1)
152 *
153 ALPHA = AP( I1+I-1 )
154 CALL ZLARFG( I, ALPHA, AP( I1 ), 1, TAUI )
155 E( I ) = ALPHA
156 *
157 IF( TAUI.NE.ZERO ) THEN
158 *
159 * Apply H(i) from both sides to A(1:i,1:i)
160 *
161 AP( I1+I-1 ) = ONE
162 *
163 * Compute y := tau * A * v storing y in TAU(1:i)
164 *
165 CALL ZHPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
166 $ 1 )
167 *
168 * Compute w := y - 1/2 * tau * (y**H *v) * v
169 *
170 ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, AP( I1 ), 1 )
171 CALL ZAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
172 *
173 * Apply the transformation as a rank-2 update:
174 * A := A - v * w**H - w * v**H
175 *
176 CALL ZHPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
177 *
178 END IF
179 AP( I1+I-1 ) = E( I )
180 D( I+1 ) = AP( I1+I )
181 TAU( I ) = TAUI
182 I1 = I1 - I
183 10 CONTINUE
184 D( 1 ) = AP( 1 )
185 ELSE
186 *
187 * Reduce the lower triangle of A. II is the index in AP of
188 * A(i,i) and I1I1 is the index of A(i+1,i+1).
189 *
190 II = 1
191 AP( 1 ) = DBLE( AP( 1 ) )
192 DO 20 I = 1, N - 1
193 I1I1 = II + N - I + 1
194 *
195 * Generate elementary reflector H(i) = I - tau * v * v**H
196 * to annihilate A(i+2:n,i)
197 *
198 ALPHA = AP( II+1 )
199 CALL ZLARFG( N-I, ALPHA, AP( II+2 ), 1, TAUI )
200 E( I ) = ALPHA
201 *
202 IF( TAUI.NE.ZERO ) THEN
203 *
204 * Apply H(i) from both sides to A(i+1:n,i+1:n)
205 *
206 AP( II+1 ) = ONE
207 *
208 * Compute y := tau * A * v storing y in TAU(i:n-1)
209 *
210 CALL ZHPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
211 $ ZERO, TAU( I ), 1 )
212 *
213 * Compute w := y - 1/2 * tau * (y**H *v) * v
214 *
215 ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, AP( II+1 ),
216 $ 1 )
217 CALL ZAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
218 *
219 * Apply the transformation as a rank-2 update:
220 * A := A - v * w**H - w * v**H
221 *
222 CALL ZHPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
223 $ AP( I1I1 ) )
224 *
225 END IF
226 AP( II+1 ) = E( I )
227 D( I ) = AP( II )
228 TAU( I ) = TAUI
229 II = I1I1
230 20 CONTINUE
231 D( N ) = AP( II )
232 END IF
233 *
234 RETURN
235 *
236 * End of ZHPTRD
237 *
238 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 D( * ), E( * )
14 COMPLEX*16 AP( * ), TAU( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
21 * real symmetric tridiagonal form T by a unitary similarity
22 * transformation: Q**H * 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 * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
35 * On entry, the upper or lower triangle of the Hermitian matrix
36 * A, packed columnwise in a linear array. The j-th column of A
37 * is stored in the array AP as follows:
38 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
39 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
40 * On exit, if UPLO = 'U', the diagonal and first superdiagonal
41 * of A are overwritten by the corresponding elements of the
42 * tridiagonal matrix T, and the elements above the first
43 * superdiagonal, with the array TAU, represent the unitary
44 * matrix Q as a product of elementary reflectors; if UPLO
45 * = 'L', the diagonal and first subdiagonal of A are over-
46 * written by the corresponding elements of the tridiagonal
47 * matrix T, and the elements below the first subdiagonal, with
48 * the array TAU, represent the unitary matrix Q as a product
49 * of elementary reflectors. See Further Details.
50 *
51 * D (output) DOUBLE PRECISION array, dimension (N)
52 * The diagonal elements of the tridiagonal matrix T:
53 * D(i) = A(i,i).
54 *
55 * E (output) DOUBLE PRECISION array, dimension (N-1)
56 * The off-diagonal elements of the tridiagonal matrix T:
57 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
58 *
59 * TAU (output) COMPLEX*16 array, dimension (N-1)
60 * The scalar factors of the elementary reflectors (see Further
61 * Details).
62 *
63 * INFO (output) INTEGER
64 * = 0: successful exit
65 * < 0: if INFO = -i, the i-th argument had an illegal value
66 *
67 * Further Details
68 * ===============
69 *
70 * If UPLO = 'U', the matrix Q is represented as a product of elementary
71 * reflectors
72 *
73 * Q = H(n-1) . . . H(2) H(1).
74 *
75 * Each H(i) has the form
76 *
77 * H(i) = I - tau * v * v**H
78 *
79 * where tau is a complex scalar, and v is a complex vector with
80 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
81 * overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
82 *
83 * If UPLO = 'L', the matrix Q is represented as a product of elementary
84 * reflectors
85 *
86 * Q = H(1) H(2) . . . H(n-1).
87 *
88 * Each H(i) has the form
89 *
90 * H(i) = I - tau * v * v**H
91 *
92 * where tau is a complex scalar, and v is a complex vector with
93 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
94 * overwriting A(i+2:n,i), and tau is stored in TAU(i).
95 *
96 * =====================================================================
97 *
98 * .. Parameters ..
99 COMPLEX*16 ONE, ZERO, HALF
100 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
101 $ ZERO = ( 0.0D+0, 0.0D+0 ),
102 $ HALF = ( 0.5D+0, 0.0D+0 ) )
103 * ..
104 * .. Local Scalars ..
105 LOGICAL UPPER
106 INTEGER I, I1, I1I1, II
107 COMPLEX*16 ALPHA, TAUI
108 * ..
109 * .. External Subroutines ..
110 EXTERNAL XERBLA, ZAXPY, ZHPMV, ZHPR2, ZLARFG
111 * ..
112 * .. External Functions ..
113 LOGICAL LSAME
114 COMPLEX*16 ZDOTC
115 EXTERNAL LSAME, ZDOTC
116 * ..
117 * .. Intrinsic Functions ..
118 INTRINSIC DBLE
119 * ..
120 * .. Executable Statements ..
121 *
122 * Test the input parameters
123 *
124 INFO = 0
125 UPPER = LSAME( UPLO, 'U' )
126 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
127 INFO = -1
128 ELSE IF( N.LT.0 ) THEN
129 INFO = -2
130 END IF
131 IF( INFO.NE.0 ) THEN
132 CALL XERBLA( 'ZHPTRD', -INFO )
133 RETURN
134 END IF
135 *
136 * Quick return if possible
137 *
138 IF( N.LE.0 )
139 $ RETURN
140 *
141 IF( UPPER ) THEN
142 *
143 * Reduce the upper triangle of A.
144 * I1 is the index in AP of A(1,I+1).
145 *
146 I1 = N*( N-1 ) / 2 + 1
147 AP( I1+N-1 ) = DBLE( AP( I1+N-1 ) )
148 DO 10 I = N - 1, 1, -1
149 *
150 * Generate elementary reflector H(i) = I - tau * v * v**H
151 * to annihilate A(1:i-1,i+1)
152 *
153 ALPHA = AP( I1+I-1 )
154 CALL ZLARFG( I, ALPHA, AP( I1 ), 1, TAUI )
155 E( I ) = ALPHA
156 *
157 IF( TAUI.NE.ZERO ) THEN
158 *
159 * Apply H(i) from both sides to A(1:i,1:i)
160 *
161 AP( I1+I-1 ) = ONE
162 *
163 * Compute y := tau * A * v storing y in TAU(1:i)
164 *
165 CALL ZHPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
166 $ 1 )
167 *
168 * Compute w := y - 1/2 * tau * (y**H *v) * v
169 *
170 ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, AP( I1 ), 1 )
171 CALL ZAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
172 *
173 * Apply the transformation as a rank-2 update:
174 * A := A - v * w**H - w * v**H
175 *
176 CALL ZHPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
177 *
178 END IF
179 AP( I1+I-1 ) = E( I )
180 D( I+1 ) = AP( I1+I )
181 TAU( I ) = TAUI
182 I1 = I1 - I
183 10 CONTINUE
184 D( 1 ) = AP( 1 )
185 ELSE
186 *
187 * Reduce the lower triangle of A. II is the index in AP of
188 * A(i,i) and I1I1 is the index of A(i+1,i+1).
189 *
190 II = 1
191 AP( 1 ) = DBLE( AP( 1 ) )
192 DO 20 I = 1, N - 1
193 I1I1 = II + N - I + 1
194 *
195 * Generate elementary reflector H(i) = I - tau * v * v**H
196 * to annihilate A(i+2:n,i)
197 *
198 ALPHA = AP( II+1 )
199 CALL ZLARFG( N-I, ALPHA, AP( II+2 ), 1, TAUI )
200 E( I ) = ALPHA
201 *
202 IF( TAUI.NE.ZERO ) THEN
203 *
204 * Apply H(i) from both sides to A(i+1:n,i+1:n)
205 *
206 AP( II+1 ) = ONE
207 *
208 * Compute y := tau * A * v storing y in TAU(i:n-1)
209 *
210 CALL ZHPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
211 $ ZERO, TAU( I ), 1 )
212 *
213 * Compute w := y - 1/2 * tau * (y**H *v) * v
214 *
215 ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, AP( II+1 ),
216 $ 1 )
217 CALL ZAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
218 *
219 * Apply the transformation as a rank-2 update:
220 * A := A - v * w**H - w * v**H
221 *
222 CALL ZHPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
223 $ AP( I1I1 ) )
224 *
225 END IF
226 AP( II+1 ) = E( I )
227 D( I ) = AP( II )
228 TAU( I ) = TAUI
229 II = I1I1
230 20 CONTINUE
231 D( N ) = AP( II )
232 END IF
233 *
234 RETURN
235 *
236 * End of ZHPTRD
237 *
238 END