LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ssptrd()

 subroutine ssptrd ( character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, integer INFO )

SSPTRD

Purpose:
``` SSPTRD reduces a real symmetric matrix A stored in packed form to
symmetric tridiagonal form T by an orthogonal similarity
transformation: Q**T * A * Q = T.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] AP ``` AP is REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.``` [out] D ``` D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).``` [out] E ``` E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.``` [out] TAU ``` TAU is REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).```

Definition at line 149 of file ssptrd.f.

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 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
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
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