LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ ssytd2()

 subroutine ssytd2 ( character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, integer INFO )

SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).

Download SSYTD2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` SSYTD2 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 REAL 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 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
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 ssytd2.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 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*
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 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
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