LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
ssptrd.f
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1 *> \brief \b SSPTRD
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSPTRD( UPLO, N, AP, D, E, TAU, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, N
26 * ..
27 * .. Array Arguments ..
28 * REAL AP( * ), D( * ), E( * ), TAU( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SSPTRD reduces a real symmetric matrix A stored in packed form to
38 *> symmetric tridiagonal form T by an orthogonal similarity
39 *> transformation: Q**T * A * Q = T.
40 *> \endverbatim
41 *
42 * Arguments:
43 * ==========
44 *
45 *> \param[in] UPLO
46 *> \verbatim
47 *> UPLO is CHARACTER*1
48 *> = 'U': Upper triangle of A is stored;
49 *> = 'L': Lower triangle of A is stored.
50 *> \endverbatim
51 *>
52 *> \param[in] N
53 *> \verbatim
54 *> N is INTEGER
55 *> The order of the matrix A. N >= 0.
56 *> \endverbatim
57 *>
58 *> \param[in,out] AP
59 *> \verbatim
60 *> AP is REAL array, dimension (N*(N+1)/2)
61 *> On entry, the upper or lower triangle of the symmetric matrix
62 *> A, packed columnwise in a linear array. The j-th column of A
63 *> is stored in the array AP as follows:
64 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
65 *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
66 *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
67 *> of A are overwritten by the corresponding elements of the
68 *> tridiagonal matrix T, and the elements above the first
69 *> superdiagonal, with the array TAU, represent the orthogonal
70 *> matrix Q as a product of elementary reflectors; if UPLO
71 *> = 'L', the diagonal and first subdiagonal of A are over-
72 *> written by the corresponding elements of the tridiagonal
73 *> matrix T, and the elements below the first subdiagonal, with
74 *> the array TAU, represent the orthogonal matrix Q as a product
75 *> of elementary reflectors. See Further Details.
76 *> \endverbatim
77 *>
78 *> \param[out] D
79 *> \verbatim
80 *> D is REAL array, dimension (N)
81 *> The diagonal elements of the tridiagonal matrix T:
82 *> D(i) = A(i,i).
83 *> \endverbatim
84 *>
85 *> \param[out] E
86 *> \verbatim
87 *> E is REAL array, dimension (N-1)
88 *> The off-diagonal elements of the tridiagonal matrix T:
89 *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
90 *> \endverbatim
91 *>
92 *> \param[out] TAU
93 *> \verbatim
94 *> TAU is REAL array, dimension (N-1)
95 *> The scalar factors of the elementary reflectors (see Further
96 *> Details).
97 *> \endverbatim
98 *>
99 *> \param[out] INFO
100 *> \verbatim
101 *> INFO is INTEGER
102 *> = 0: successful exit
103 *> < 0: if INFO = -i, the i-th argument had an illegal value
104 *> \endverbatim
105 *
106 * Authors:
107 * ========
108 *
109 *> \author Univ. of Tennessee
110 *> \author Univ. of California Berkeley
111 *> \author Univ. of Colorado Denver
112 *> \author NAG Ltd.
113 *
114 *> \ingroup realOTHERcomputational
115 *
116 *> \par Further Details:
117 * =====================
118 *>
119 *> \verbatim
120 *>
121 *> If UPLO = 'U', the matrix Q is represented as a product of elementary
122 *> reflectors
123 *>
124 *> Q = H(n-1) . . . H(2) H(1).
125 *>
126 *> Each H(i) has the form
127 *>
128 *> H(i) = I - tau * v * v**T
129 *>
130 *> where tau is a real scalar, and v is a real vector with
131 *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
132 *> overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
133 *>
134 *> If UPLO = 'L', the matrix Q is represented as a product of elementary
135 *> reflectors
136 *>
137 *> Q = H(1) H(2) . . . H(n-1).
138 *>
139 *> Each H(i) has the form
140 *>
141 *> H(i) = I - tau * v * v**T
142 *>
143 *> where tau is a real scalar, and v is a real vector with
144 *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
145 *> overwriting A(i+2:n,i), and tau is stored in TAU(i).
146 *> \endverbatim
147 *>
148 * =====================================================================
149  SUBROUTINE ssptrd( UPLO, N, AP, D, E, TAU, INFO )
150 *
151 * -- LAPACK computational routine --
152 * -- LAPACK is a software package provided by Univ. of Tennessee, --
153 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
154 *
155 * .. Scalar Arguments ..
156  CHARACTER UPLO
157  INTEGER INFO, N
158 * ..
159 * .. Array Arguments ..
160  REAL AP( * ), D( * ), E( * ), TAU( * )
161 * ..
162 *
163 * =====================================================================
164 *
165 * .. Parameters ..
166  REAL ONE, ZERO, HALF
167  parameter( one = 1.0, zero = 0.0, half = 1.0 / 2.0 )
168 * ..
169 * .. Local Scalars ..
170  LOGICAL UPPER
171  INTEGER I, I1, I1I1, II
172  REAL ALPHA, TAUI
173 * ..
174 * .. External Subroutines ..
175  EXTERNAL saxpy, slarfg, sspmv, sspr2, xerbla
176 * ..
177 * .. External Functions ..
178  LOGICAL LSAME
179  REAL SDOT
180  EXTERNAL lsame, sdot
181 * ..
182 * .. Executable Statements ..
183 *
184 * Test the input parameters
185 *
186  info = 0
187  upper = lsame( uplo, 'U' )
188  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
189  info = -1
190  ELSE IF( n.LT.0 ) THEN
191  info = -2
192  END IF
193  IF( info.NE.0 ) THEN
194  CALL xerbla( 'SSPTRD', -info )
195  RETURN
196  END IF
197 *
198 * Quick return if possible
199 *
200  IF( n.LE.0 )
201  \$ RETURN
202 *
203  IF( upper ) THEN
204 *
205 * Reduce the upper triangle of A.
206 * I1 is the index in AP of A(1,I+1).
207 *
208  i1 = n*( n-1 ) / 2 + 1
209  DO 10 i = n - 1, 1, -1
210 *
211 * Generate elementary reflector H(i) = I - tau * v * v**T
212 * to annihilate A(1:i-1,i+1)
213 *
214  CALL slarfg( i, ap( i1+i-1 ), ap( i1 ), 1, taui )
215  e( i ) = ap( i1+i-1 )
216 *
217  IF( taui.NE.zero ) THEN
218 *
219 * Apply H(i) from both sides to A(1:i,1:i)
220 *
221  ap( i1+i-1 ) = one
222 *
223 * Compute y := tau * A * v storing y in TAU(1:i)
224 *
225  CALL sspmv( uplo, i, taui, ap, ap( i1 ), 1, zero, tau,
226  \$ 1 )
227 *
228 * Compute w := y - 1/2 * tau * (y**T *v) * v
229 *
230  alpha = -half*taui*sdot( i, tau, 1, ap( i1 ), 1 )
231  CALL saxpy( i, alpha, ap( i1 ), 1, tau, 1 )
232 *
233 * Apply the transformation as a rank-2 update:
234 * A := A - v * w**T - w * v**T
235 *
236  CALL sspr2( uplo, i, -one, ap( i1 ), 1, tau, 1, ap )
237 *
238  ap( i1+i-1 ) = e( i )
239  END IF
240  d( i+1 ) = ap( i1+i )
241  tau( i ) = taui
242  i1 = i1 - i
243  10 CONTINUE
244  d( 1 ) = ap( 1 )
245  ELSE
246 *
247 * Reduce the lower triangle of A. II is the index in AP of
248 * A(i,i) and I1I1 is the index of A(i+1,i+1).
249 *
250  ii = 1
251  DO 20 i = 1, n - 1
252  i1i1 = ii + n - i + 1
253 *
254 * Generate elementary reflector H(i) = I - tau * v * v**T
255 * to annihilate A(i+2:n,i)
256 *
257  CALL slarfg( n-i, ap( ii+1 ), ap( ii+2 ), 1, taui )
258  e( i ) = ap( ii+1 )
259 *
260  IF( taui.NE.zero ) THEN
261 *
262 * Apply H(i) from both sides to A(i+1:n,i+1:n)
263 *
264  ap( ii+1 ) = one
265 *
266 * Compute y := tau * A * v storing y in TAU(i:n-1)
267 *
268  CALL sspmv( uplo, n-i, taui, ap( i1i1 ), ap( ii+1 ), 1,
269  \$ zero, tau( i ), 1 )
270 *
271 * Compute w := y - 1/2 * tau * (y**T *v) * v
272 *
273  alpha = -half*taui*sdot( n-i, tau( i ), 1, ap( ii+1 ),
274  \$ 1 )
275  CALL saxpy( n-i, alpha, ap( ii+1 ), 1, tau( i ), 1 )
276 *
277 * Apply the transformation as a rank-2 update:
278 * A := A - v * w**T - w * v**T
279 *
280  CALL sspr2( uplo, n-i, -one, ap( ii+1 ), 1, tau( i ), 1,
281  \$ ap( i1i1 ) )
282 *
283  ap( ii+1 ) = e( i )
284  END IF
285  d( i ) = ap( ii )
286  tau( i ) = taui
287  ii = i1i1
288  20 CONTINUE
289  d( n ) = ap( ii )
290  END IF
291 *
292  RETURN
293 *
294 * End of SSPTRD
295 *
296  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 ssptrd(UPLO, N, AP, D, E, TAU, INFO)
SSPTRD
Definition: ssptrd.f:150
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine sspmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
SSPMV
Definition: sspmv.f:147
subroutine sspr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
SSPR2
Definition: sspr2.f:142