LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dsytd2()

 subroutine dsytd2 ( character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAU, integer INFO )

DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).

Purpose:
``` DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
form T by an orthogonal similarity transformation: Q**T * A * Q = T.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] D ``` D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).``` [out] E ``` E is DOUBLE PRECISION 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 DOUBLE PRECISION 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
A(1:i-1,i+1), and tau 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 A(i+2:n,i),
and tau in TAU(i).

The contents of A on exit are illustrated by the following examples
with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

(  d   e   v2  v3  v4 )              (  d                  )
(      d   e   v3  v4 )              (  e   d              )
(          d   e   v4 )              (  v1  e   d          )
(              d   e  )              (  v1  v2  e   d      )
(                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).```

Definition at line 172 of file dsytd2.f.

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 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
184* ..
185*
186* =====================================================================
187*
188* .. Parameters ..
189 DOUBLE PRECISION ONE, ZERO, HALF
190 parameter( one = 1.0d0, zero = 0.0d0,
191 \$ half = 1.0d0 / 2.0d0 )
192* ..
193* .. Local Scalars ..
194 LOGICAL UPPER
195 INTEGER I
196 DOUBLE PRECISION ALPHA, TAUI
197* ..
198* .. External Subroutines ..
199 EXTERNAL daxpy, dlarfg, dsymv, dsyr2, xerbla
200* ..
201* .. External Functions ..
202 LOGICAL LSAME
203 DOUBLE PRECISION DDOT
204 EXTERNAL lsame, ddot
205* ..
206* .. Intrinsic Functions ..
207 INTRINSIC max, min
208* ..
209* .. Executable Statements ..
210*
211* Test the input parameters
212*
213 info = 0
214 upper = lsame( uplo, 'U' )
215 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
216 info = -1
217 ELSE IF( n.LT.0 ) THEN
218 info = -2
219 ELSE IF( lda.LT.max( 1, n ) ) THEN
220 info = -4
221 END IF
222 IF( info.NE.0 ) THEN
223 CALL xerbla( 'DSYTD2', -info )
224 RETURN
225 END IF
226*
227* Quick return if possible
228*
229 IF( n.LE.0 )
230 \$ RETURN
231*
232 IF( upper ) THEN
233*
234* Reduce the upper triangle of A
235*
236 DO 10 i = n - 1, 1, -1
237*
238* Generate elementary reflector H(i) = I - tau * v * v**T
239* to annihilate A(1:i-1,i+1)
240*
241 CALL dlarfg( i, a( i, i+1 ), a( 1, i+1 ), 1, taui )
242 e( i ) = a( i, i+1 )
243*
244 IF( taui.NE.zero ) THEN
245*
246* Apply H(i) from both sides to A(1:i,1:i)
247*
248 a( i, i+1 ) = one
249*
250* Compute x := tau * A * v storing x in TAU(1:i)
251*
252 CALL dsymv( uplo, i, taui, a, lda, a( 1, i+1 ), 1, zero,
253 \$ tau, 1 )
254*
255* Compute w := x - 1/2 * tau * (x**T * v) * v
256*
257 alpha = -half*taui*ddot( i, tau, 1, a( 1, i+1 ), 1 )
258 CALL daxpy( i, alpha, a( 1, i+1 ), 1, tau, 1 )
259*
260* Apply the transformation as a rank-2 update:
261* A := A - v * w**T - w * v**T
262*
263 CALL dsyr2( uplo, i, -one, a( 1, i+1 ), 1, tau, 1, a,
264 \$ lda )
265*
266 a( i, i+1 ) = e( i )
267 END IF
268 d( i+1 ) = a( i+1, i+1 )
269 tau( i ) = taui
270 10 CONTINUE
271 d( 1 ) = a( 1, 1 )
272 ELSE
273*
274* Reduce the lower triangle of A
275*
276 DO 20 i = 1, n - 1
277*
278* Generate elementary reflector H(i) = I - tau * v * v**T
279* to annihilate A(i+2:n,i)
280*
281 CALL dlarfg( n-i, a( i+1, i ), a( min( i+2, n ), i ), 1,
282 \$ taui )
283 e( i ) = a( i+1, i )
284*
285 IF( taui.NE.zero ) THEN
286*
287* Apply H(i) from both sides to A(i+1:n,i+1:n)
288*
289 a( i+1, i ) = one
290*
291* Compute x := tau * A * v storing y in TAU(i:n-1)
292*
293 CALL dsymv( uplo, n-i, taui, a( i+1, i+1 ), lda,
294 \$ a( i+1, i ), 1, zero, tau( i ), 1 )
295*
296* Compute w := x - 1/2 * tau * (x**T * v) * v
297*
298 alpha = -half*taui*ddot( n-i, tau( i ), 1, a( i+1, i ),
299 \$ 1 )
300 CALL daxpy( n-i, alpha, a( i+1, i ), 1, tau( i ), 1 )
301*
302* Apply the transformation as a rank-2 update:
303* A := A - v * w**T - w * v**T
304*
305 CALL dsyr2( uplo, n-i, -one, a( i+1, i ), 1, tau( i ), 1,
306 \$ a( i+1, i+1 ), lda )
307*
308 a( i+1, i ) = e( i )
309 END IF
310 d( i ) = a( i, i )
311 tau( i ) = taui
312 20 CONTINUE
313 d( n ) = a( n, n )
314 END IF
315*
316 RETURN
317*
318* End of DSYTD2
319*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function ddot(N, DX, INCX, DY, INCY)
DDOT
Definition: ddot.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DSYMV
Definition: dsymv.f:152
subroutine dsyr2(UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
DSYR2
Definition: dsyr2.f:147
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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