LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
ssytd2.f
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1 *> \brief \b SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, LDA, N
26 * ..
27 * .. Array Arguments ..
28 * REAL A( LDA, * ), D( * ), E( * ), TAU( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
38 *> form T by an orthogonal similarity transformation: Q**T * A * Q = T.
39 *> \endverbatim
40 *
41 * Arguments:
42 * ==========
43 *
44 *> \param[in] UPLO
45 *> \verbatim
46 *> UPLO is CHARACTER*1
47 *> Specifies whether the upper or lower triangular part of the
48 *> symmetric matrix A is stored:
49 *> = 'U': Upper triangular
50 *> = 'L': Lower triangular
51 *> \endverbatim
52 *>
53 *> \param[in] N
54 *> \verbatim
55 *> N is INTEGER
56 *> The order of the matrix A. N >= 0.
57 *> \endverbatim
58 *>
59 *> \param[in,out] A
60 *> \verbatim
61 *> A is REAL array, dimension (LDA,N)
62 *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
63 *> n-by-n upper triangular part of A contains the upper
64 *> triangular part of the matrix A, and the strictly lower
65 *> triangular part of A is not referenced. If UPLO = 'L', the
66 *> leading n-by-n lower triangular part of A contains the lower
67 *> triangular part of the matrix A, and the strictly upper
68 *> triangular part of A is not referenced.
69 *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
70 *> of A are overwritten by the corresponding elements of the
71 *> tridiagonal matrix T, and the elements above the first
72 *> superdiagonal, with the array TAU, represent the orthogonal
73 *> matrix Q as a product of elementary reflectors; if UPLO
74 *> = 'L', the diagonal and first subdiagonal of A are over-
75 *> written by the corresponding elements of the tridiagonal
76 *> matrix T, and the elements below the first subdiagonal, with
77 *> the array TAU, represent the orthogonal matrix Q as a product
78 *> of elementary reflectors. See Further Details.
79 *> \endverbatim
80 *>
81 *> \param[in] LDA
82 *> \verbatim
83 *> LDA is INTEGER
84 *> The leading dimension of the array A. LDA >= max(1,N).
85 *> \endverbatim
86 *>
87 *> \param[out] D
88 *> \verbatim
89 *> D is REAL array, dimension (N)
90 *> The diagonal elements of the tridiagonal matrix T:
91 *> D(i) = A(i,i).
92 *> \endverbatim
93 *>
94 *> \param[out] E
95 *> \verbatim
96 *> E is REAL array, dimension (N-1)
97 *> The off-diagonal elements of the tridiagonal matrix T:
98 *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
99 *> \endverbatim
100 *>
101 *> \param[out] TAU
102 *> \verbatim
103 *> TAU is REAL array, dimension (N-1)
104 *> The scalar factors of the elementary reflectors (see Further
105 *> Details).
106 *> \endverbatim
107 *>
108 *> \param[out] INFO
109 *> \verbatim
110 *> INFO is INTEGER
111 *> = 0: successful exit
112 *> < 0: if INFO = -i, the i-th argument had an illegal value.
113 *> \endverbatim
114 *
115 * Authors:
116 * ========
117 *
118 *> \author Univ. of Tennessee
119 *> \author Univ. of California Berkeley
120 *> \author Univ. of Colorado Denver
121 *> \author NAG Ltd.
122 *
123 *> \ingroup realSYcomputational
124 *
125 *> \par Further Details:
126 * =====================
127 *>
128 *> \verbatim
129 *>
130 *> If UPLO = 'U', the matrix Q is represented as a product of elementary
131 *> reflectors
132 *>
133 *> Q = H(n-1) . . . H(2) H(1).
134 *>
135 *> Each H(i) has the form
136 *>
137 *> H(i) = I - tau * v * v**T
138 *>
139 *> where tau is a real scalar, and v is a real vector with
140 *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
141 *> A(1:i-1,i+1), and tau in TAU(i).
142 *>
143 *> If UPLO = 'L', the matrix Q is represented as a product of elementary
144 *> reflectors
145 *>
146 *> Q = H(1) H(2) . . . H(n-1).
147 *>
148 *> Each H(i) has the form
149 *>
150 *> H(i) = I - tau * v * v**T
151 *>
152 *> where tau is a real scalar, and v is a real vector with
153 *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
154 *> and tau in TAU(i).
155 *>
156 *> The contents of A on exit are illustrated by the following examples
157 *> with n = 5:
158 *>
159 *> if UPLO = 'U': if UPLO = 'L':
160 *>
161 *> ( d e v2 v3 v4 ) ( d )
162 *> ( d e v3 v4 ) ( e d )
163 *> ( d e v4 ) ( v1 e d )
164 *> ( d e ) ( v1 v2 e d )
165 *> ( d ) ( v1 v2 v3 e d )
166 *>
167 *> where d and e denote diagonal and off-diagonal elements of T, and vi
168 *> denotes an element of the vector defining H(i).
169 *> \endverbatim
170 *>
171 * =====================================================================
172  SUBROUTINE ssytd2( UPLO, N, A, LDA, D, E, TAU, INFO )
173 *
174 * -- LAPACK computational routine --
175 * -- LAPACK is a software package provided by Univ. of Tennessee, --
176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177 *
178 * .. Scalar Arguments ..
179  CHARACTER UPLO
180  INTEGER INFO, LDA, N
181 * ..
182 * .. Array Arguments ..
183  REAL A( LDA, * ), D( * ), E( * ), TAU( * )
184 * ..
185 *
186 * =====================================================================
187 *
188 * .. Parameters ..
189  REAL ONE, ZERO, HALF
190  parameter( one = 1.0, zero = 0.0, half = 1.0 / 2.0 )
191 * ..
192 * .. Local Scalars ..
193  LOGICAL UPPER
194  INTEGER I
195  REAL ALPHA, TAUI
196 * ..
197 * .. External Subroutines ..
198  EXTERNAL saxpy, slarfg, ssymv, ssyr2, xerbla
199 * ..
200 * .. External Functions ..
201  LOGICAL LSAME
202  REAL SDOT
203  EXTERNAL lsame, sdot
204 * ..
205 * .. Intrinsic Functions ..
206  INTRINSIC max, min
207 * ..
208 * .. Executable Statements ..
209 *
210 * Test the input parameters
211 *
212  info = 0
213  upper = lsame( uplo, 'U' )
214  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
215  info = -1
216  ELSE IF( n.LT.0 ) THEN
217  info = -2
218  ELSE IF( lda.LT.max( 1, n ) ) THEN
219  info = -4
220  END IF
221  IF( info.NE.0 ) THEN
222  CALL xerbla( 'SSYTD2', -info )
223  RETURN
224  END IF
225 *
226 * Quick return if possible
227 *
228  IF( n.LE.0 )
229  \$ RETURN
230 *
231  IF( upper ) THEN
232 *
233 * Reduce the upper triangle of A
234 *
235  DO 10 i = n - 1, 1, -1
236 *
237 * Generate elementary reflector H(i) = I - tau * v * v**T
238 * to annihilate A(1:i-1,i+1)
239 *
240  CALL slarfg( i, a( i, i+1 ), a( 1, i+1 ), 1, taui )
241  e( i ) = a( i, i+1 )
242 *
243  IF( taui.NE.zero ) THEN
244 *
245 * Apply H(i) from both sides to A(1:i,1:i)
246 *
247  a( i, i+1 ) = one
248 *
249 * Compute x := tau * A * v storing x in TAU(1:i)
250 *
251  CALL ssymv( uplo, i, taui, a, lda, a( 1, i+1 ), 1, zero,
252  \$ tau, 1 )
253 *
254 * Compute w := x - 1/2 * tau * (x**T * v) * v
255 *
256  alpha = -half*taui*sdot( i, tau, 1, a( 1, i+1 ), 1 )
257  CALL saxpy( i, alpha, a( 1, i+1 ), 1, tau, 1 )
258 *
259 * Apply the transformation as a rank-2 update:
260 * A := A - v * w**T - w * v**T
261 *
262  CALL ssyr2( uplo, i, -one, a( 1, i+1 ), 1, tau, 1, a,
263  \$ lda )
264 *
265  a( i, i+1 ) = e( i )
266  END IF
267  d( i+1 ) = a( i+1, i+1 )
268  tau( i ) = taui
269  10 CONTINUE
270  d( 1 ) = a( 1, 1 )
271  ELSE
272 *
273 * Reduce the lower triangle of A
274 *
275  DO 20 i = 1, n - 1
276 *
277 * Generate elementary reflector H(i) = I - tau * v * v**T
278 * to annihilate A(i+2:n,i)
279 *
280  CALL slarfg( n-i, a( i+1, i ), a( min( i+2, n ), i ), 1,
281  \$ taui )
282  e( i ) = a( i+1, i )
283 *
284  IF( taui.NE.zero ) THEN
285 *
286 * Apply H(i) from both sides to A(i+1:n,i+1:n)
287 *
288  a( i+1, i ) = one
289 *
290 * Compute x := tau * A * v storing y in TAU(i:n-1)
291 *
292  CALL ssymv( uplo, n-i, taui, a( i+1, i+1 ), lda,
293  \$ a( i+1, i ), 1, zero, tau( i ), 1 )
294 *
295 * Compute w := x - 1/2 * tau * (x**T * v) * v
296 *
297  alpha = -half*taui*sdot( n-i, tau( i ), 1, a( i+1, i ),
298  \$ 1 )
299  CALL saxpy( n-i, alpha, a( i+1, i ), 1, tau( i ), 1 )
300 *
301 * Apply the transformation as a rank-2 update:
302 * A := A - v * w**T - w * v**T
303 *
304  CALL ssyr2( uplo, n-i, -one, a( i+1, i ), 1, tau( i ), 1,
305  \$ a( i+1, i+1 ), lda )
306 *
307  a( i+1, i ) = e( i )
308  END IF
309  d( i ) = a( i, i )
310  tau( i ) = taui
311  20 CONTINUE
312  d( n ) = a( n, n )
313  END IF
314 *
315  RETURN
316 *
317 * End of SSYTD2
318 *
319  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine ssytd2(UPLO, N, A, LDA, D, E, TAU, INFO)
SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity tran...
Definition: ssytd2.f:173
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine ssymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SSYMV
Definition: ssymv.f:152
subroutine ssyr2(UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
SSYR2
Definition: ssyr2.f:147