LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ zsytrf()

 subroutine zsytrf ( character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info )

ZSYTRF

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

Purpose:
``` ZSYTRF computes the factorization of a complex symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method.  The form of the
factorization is

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.```
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] A ``` A is COMPLEX*16 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.```
Further Details:
```  If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    v    0   )   k-s
U(k) =  (   0    I    0   )   s
(   0    0    I   )   n-k
k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    0     0   )  k-1
L(k) =  (   0    I     0   )  s
(   0    v     I   )  n-k-s+1
k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).```

Definition at line 181 of file zsytrf.f.

182*
183* -- LAPACK computational routine --
184* -- LAPACK is a software package provided by Univ. of Tennessee, --
185* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186*
187* .. Scalar Arguments ..
188 CHARACTER UPLO
189 INTEGER INFO, LDA, LWORK, N
190* ..
191* .. Array Arguments ..
192 INTEGER IPIV( * )
193 COMPLEX*16 A( LDA, * ), WORK( * )
194* ..
195*
196* =====================================================================
197*
198* .. Local Scalars ..
199 LOGICAL LQUERY, UPPER
200 INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
201* ..
202* .. External Functions ..
203 LOGICAL LSAME
204 INTEGER ILAENV
205 EXTERNAL lsame, ilaenv
206* ..
207* .. External Subroutines ..
208 EXTERNAL xerbla, zlasyf, zsytf2
209* ..
210* .. Intrinsic Functions ..
211 INTRINSIC max
212* ..
213* .. Executable Statements ..
214*
215* Test the input parameters.
216*
217 info = 0
218 upper = lsame( uplo, 'U' )
219 lquery = ( lwork.EQ.-1 )
220 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
221 info = -1
222 ELSE IF( n.LT.0 ) THEN
223 info = -2
224 ELSE IF( lda.LT.max( 1, n ) ) THEN
225 info = -4
226 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
227 info = -7
228 END IF
229*
230 IF( info.EQ.0 ) THEN
231*
232* Determine the block size
233*
234 nb = ilaenv( 1, 'ZSYTRF', uplo, n, -1, -1, -1 )
235 lwkopt = max( 1, n*nb )
236 work( 1 ) = lwkopt
237 END IF
238*
239 IF( info.NE.0 ) THEN
240 CALL xerbla( 'ZSYTRF', -info )
241 RETURN
242 ELSE IF( lquery ) THEN
243 RETURN
244 END IF
245*
246 nbmin = 2
247 ldwork = n
248 IF( nb.GT.1 .AND. nb.LT.n ) THEN
249 iws = ldwork*nb
250 IF( lwork.LT.iws ) THEN
251 nb = max( lwork / ldwork, 1 )
252 nbmin = max( 2, ilaenv( 2, 'ZSYTRF', uplo, n, -1, -1, -1 ) )
253 END IF
254 ELSE
255 iws = 1
256 END IF
257 IF( nb.LT.nbmin )
258 \$ nb = n
259*
260 IF( upper ) THEN
261*
262* Factorize A as U*D*U**T using the upper triangle of A
263*
264* K is the main loop index, decreasing from N to 1 in steps of
265* KB, where KB is the number of columns factorized by ZLASYF;
266* KB is either NB or NB-1, or K for the last block
267*
268 k = n
269 10 CONTINUE
270*
271* If K < 1, exit from loop
272*
273 IF( k.LT.1 )
274 \$ GO TO 40
275*
276 IF( k.GT.nb ) THEN
277*
278* Factorize columns k-kb+1:k of A and use blocked code to
279* update columns 1:k-kb
280*
281 CALL zlasyf( uplo, k, nb, kb, a, lda, ipiv, work, n, iinfo )
282 ELSE
283*
284* Use unblocked code to factorize columns 1:k of A
285*
286 CALL zsytf2( uplo, k, a, lda, ipiv, iinfo )
287 kb = k
288 END IF
289*
290* Set INFO on the first occurrence of a zero pivot
291*
292 IF( info.EQ.0 .AND. iinfo.GT.0 )
293 \$ info = iinfo
294*
295* Decrease K and return to the start of the main loop
296*
297 k = k - kb
298 GO TO 10
299*
300 ELSE
301*
302* Factorize A as L*D*L**T using the lower triangle of A
303*
304* K is the main loop index, increasing from 1 to N in steps of
305* KB, where KB is the number of columns factorized by ZLASYF;
306* KB is either NB or NB-1, or N-K+1 for the last block
307*
308 k = 1
309 20 CONTINUE
310*
311* If K > N, exit from loop
312*
313 IF( k.GT.n )
314 \$ GO TO 40
315*
316 IF( k.LE.n-nb ) THEN
317*
318* Factorize columns k:k+kb-1 of A and use blocked code to
319* update columns k+kb:n
320*
321 CALL zlasyf( uplo, n-k+1, nb, kb, a( k, k ), lda, ipiv( k ),
322 \$ work, n, iinfo )
323 ELSE
324*
325* Use unblocked code to factorize columns k:n of A
326*
327 CALL zsytf2( uplo, n-k+1, a( k, k ), lda, ipiv( k ), iinfo )
328 kb = n - k + 1
329 END IF
330*
331* Set INFO on the first occurrence of a zero pivot
332*
333 IF( info.EQ.0 .AND. iinfo.GT.0 )
334 \$ info = iinfo + k - 1
335*
337*
338 DO 30 j = k, k + kb - 1
339 IF( ipiv( j ).GT.0 ) THEN
340 ipiv( j ) = ipiv( j ) + k - 1
341 ELSE
342 ipiv( j ) = ipiv( j ) - k + 1
343 END IF
344 30 CONTINUE
345*
346* Increase K and return to the start of the main loop
347*
348 k = k + kb
349 GO TO 20
350*
351 END IF
352*
353 40 CONTINUE
354 work( 1 ) = lwkopt
355 RETURN
356*
357* End of ZSYTRF
358*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zsytf2(uplo, n, a, lda, ipiv, info)
ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting ...
Definition zsytf2.f:191
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
subroutine zlasyf(uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
ZLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagona...
Definition zlasyf.f:177
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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