KTH framework for Nek5000 toolboxes; testing version  0.0.1
dsytd2.f
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1  SUBROUTINE dsytd2( UPLO, N, A, LDA, D, E, TAU, INFO )
2 *
3 * -- LAPACK routine (version 3.0) --
4 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
5 * Courant Institute, Argonne National Lab, and Rice University
6 * October 31, 1992
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' * 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'
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'
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'
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'*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' - w * v'
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'
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'*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' - w * v'
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
subroutine daxpy(n, da, dx, incx, dy, incy)
Definition: daxpy.f:2
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
Definition: dlarfg.f:2
subroutine dsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
Definition: dsymv.f:3
subroutine dsyr2(UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
Definition: dsyr2.f:2
subroutine dsytd2(UPLO, N, A, LDA, D, E, TAU, INFO)
Definition: dsytd2.f:2
subroutine xerbla(SRNAME, INFO)
Definition: xerbla.f:2