LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dsygs2()

 subroutine dsygs2 ( integer ITYPE, character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer INFO )

DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).

Purpose:
``` DSYGS2 reduces a real symmetric-definite generalized eigenproblem
to standard form.

If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.

B must have been previously factorized as U**T *U or L*L**T by DPOTRF.```
Parameters
 [in] ITYPE ``` ITYPE is INTEGER = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T); = 2 or 3: compute U*A*U**T or L**T *A*L.``` [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored, and how B has been factorized. = 'U': Upper triangular = 'L': Lower triangular``` [in] N ``` N is INTEGER The order of the matrices A and B. 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 INFO = 0, the transformed matrix, stored in the same format as A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] B ``` B is DOUBLE PRECISION array, dimension (LDB,N) The triangular factor from the Cholesky factorization of B, as returned by DPOTRF.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```

Definition at line 126 of file dsygs2.f.

127*
128* -- LAPACK computational routine --
129* -- LAPACK is a software package provided by Univ. of Tennessee, --
130* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
131*
132* .. Scalar Arguments ..
133 CHARACTER UPLO
134 INTEGER INFO, ITYPE, LDA, LDB, N
135* ..
136* .. Array Arguments ..
137 DOUBLE PRECISION A( LDA, * ), B( LDB, * )
138* ..
139*
140* =====================================================================
141*
142* .. Parameters ..
143 DOUBLE PRECISION ONE, HALF
144 parameter( one = 1.0d0, half = 0.5d0 )
145* ..
146* .. Local Scalars ..
147 LOGICAL UPPER
148 INTEGER K
149 DOUBLE PRECISION AKK, BKK, CT
150* ..
151* .. External Subroutines ..
152 EXTERNAL daxpy, dscal, dsyr2, dtrmv, dtrsv, xerbla
153* ..
154* .. Intrinsic Functions ..
155 INTRINSIC max
156* ..
157* .. External Functions ..
158 LOGICAL LSAME
159 EXTERNAL lsame
160* ..
161* .. Executable Statements ..
162*
163* Test the input parameters.
164*
165 info = 0
166 upper = lsame( uplo, 'U' )
167 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
168 info = -1
169 ELSE IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
170 info = -2
171 ELSE IF( n.LT.0 ) THEN
172 info = -3
173 ELSE IF( lda.LT.max( 1, n ) ) THEN
174 info = -5
175 ELSE IF( ldb.LT.max( 1, n ) ) THEN
176 info = -7
177 END IF
178 IF( info.NE.0 ) THEN
179 CALL xerbla( 'DSYGS2', -info )
180 RETURN
181 END IF
182*
183 IF( itype.EQ.1 ) THEN
184 IF( upper ) THEN
185*
186* Compute inv(U**T)*A*inv(U)
187*
188 DO 10 k = 1, n
189*
190* Update the upper triangle of A(k:n,k:n)
191*
192 akk = a( k, k )
193 bkk = b( k, k )
194 akk = akk / bkk**2
195 a( k, k ) = akk
196 IF( k.LT.n ) THEN
197 CALL dscal( n-k, one / bkk, a( k, k+1 ), lda )
198 ct = -half*akk
199 CALL daxpy( n-k, ct, b( k, k+1 ), ldb, a( k, k+1 ),
200 \$ lda )
201 CALL dsyr2( uplo, n-k, -one, a( k, k+1 ), lda,
202 \$ b( k, k+1 ), ldb, a( k+1, k+1 ), lda )
203 CALL daxpy( n-k, ct, b( k, k+1 ), ldb, a( k, k+1 ),
204 \$ lda )
205 CALL dtrsv( uplo, 'Transpose', 'Non-unit', n-k,
206 \$ b( k+1, k+1 ), ldb, a( k, k+1 ), lda )
207 END IF
208 10 CONTINUE
209 ELSE
210*
211* Compute inv(L)*A*inv(L**T)
212*
213 DO 20 k = 1, n
214*
215* Update the lower triangle of A(k:n,k:n)
216*
217 akk = a( k, k )
218 bkk = b( k, k )
219 akk = akk / bkk**2
220 a( k, k ) = akk
221 IF( k.LT.n ) THEN
222 CALL dscal( n-k, one / bkk, a( k+1, k ), 1 )
223 ct = -half*akk
224 CALL daxpy( n-k, ct, b( k+1, k ), 1, a( k+1, k ), 1 )
225 CALL dsyr2( uplo, n-k, -one, a( k+1, k ), 1,
226 \$ b( k+1, k ), 1, a( k+1, k+1 ), lda )
227 CALL daxpy( n-k, ct, b( k+1, k ), 1, a( k+1, k ), 1 )
228 CALL dtrsv( uplo, 'No transpose', 'Non-unit', n-k,
229 \$ b( k+1, k+1 ), ldb, a( k+1, k ), 1 )
230 END IF
231 20 CONTINUE
232 END IF
233 ELSE
234 IF( upper ) THEN
235*
236* Compute U*A*U**T
237*
238 DO 30 k = 1, n
239*
240* Update the upper triangle of A(1:k,1:k)
241*
242 akk = a( k, k )
243 bkk = b( k, k )
244 CALL dtrmv( uplo, 'No transpose', 'Non-unit', k-1, b,
245 \$ ldb, a( 1, k ), 1 )
246 ct = half*akk
247 CALL daxpy( k-1, ct, b( 1, k ), 1, a( 1, k ), 1 )
248 CALL dsyr2( uplo, k-1, one, a( 1, k ), 1, b( 1, k ), 1,
249 \$ a, lda )
250 CALL daxpy( k-1, ct, b( 1, k ), 1, a( 1, k ), 1 )
251 CALL dscal( k-1, bkk, a( 1, k ), 1 )
252 a( k, k ) = akk*bkk**2
253 30 CONTINUE
254 ELSE
255*
256* Compute L**T *A*L
257*
258 DO 40 k = 1, n
259*
260* Update the lower triangle of A(1:k,1:k)
261*
262 akk = a( k, k )
263 bkk = b( k, k )
264 CALL dtrmv( uplo, 'Transpose', 'Non-unit', k-1, b, ldb,
265 \$ a( k, 1 ), lda )
266 ct = half*akk
267 CALL daxpy( k-1, ct, b( k, 1 ), ldb, a( k, 1 ), lda )
268 CALL dsyr2( uplo, k-1, one, a( k, 1 ), lda, b( k, 1 ),
269 \$ ldb, a, lda )
270 CALL daxpy( k-1, ct, b( k, 1 ), ldb, a( k, 1 ), lda )
271 CALL dscal( k-1, bkk, a( k, 1 ), lda )
272 a( k, k ) = akk*bkk**2
273 40 CONTINUE
274 END IF
275 END IF
276 RETURN
277*
278* End of DSYGS2
279*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dtrsv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
DTRSV
Definition: dtrsv.f:143
subroutine dtrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
DTRMV
Definition: dtrmv.f:147
subroutine dsyr2(UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
DSYR2
Definition: dsyr2.f:147
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