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

 subroutine chetrf ( character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO )

CHETRF

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

A = U*D*U**H  or  A = L*D*L**H

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is Hermitian 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 array, dimension (LDA,N) On entry, the Hermitian 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 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.``` [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**H, 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**H, 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 176 of file chetrf.f.

177*
178* -- LAPACK computational routine --
179* -- LAPACK is a software package provided by Univ. of Tennessee, --
180* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181*
182* .. Scalar Arguments ..
183 CHARACTER UPLO
184 INTEGER INFO, LDA, LWORK, N
185* ..
186* .. Array Arguments ..
187 INTEGER IPIV( * )
188 COMPLEX A( LDA, * ), WORK( * )
189* ..
190*
191* =====================================================================
192*
193* .. Local Scalars ..
194 LOGICAL LQUERY, UPPER
195 INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
196* ..
197* .. External Functions ..
198 LOGICAL LSAME
199 INTEGER ILAENV
200 EXTERNAL lsame, ilaenv
201* ..
202* .. External Subroutines ..
203 EXTERNAL chetf2, clahef, xerbla
204* ..
205* .. Intrinsic Functions ..
206 INTRINSIC max
207* ..
208* .. Executable Statements ..
209*
210* Test the input parameters.
211*
212 info = 0
213 upper = lsame( uplo, 'U' )
214 lquery = ( lwork.EQ.-1 )
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 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
222 info = -7
223 END IF
224*
225 IF( info.EQ.0 ) THEN
226*
227* Determine the block size
228*
229 nb = ilaenv( 1, 'CHETRF', uplo, n, -1, -1, -1 )
230 lwkopt = n*nb
231 work( 1 ) = lwkopt
232 END IF
233*
234 IF( info.NE.0 ) THEN
235 CALL xerbla( 'CHETRF', -info )
236 RETURN
237 ELSE IF( lquery ) THEN
238 RETURN
239 END IF
240*
241 nbmin = 2
242 ldwork = n
243 IF( nb.GT.1 .AND. nb.LT.n ) THEN
244 iws = ldwork*nb
245 IF( lwork.LT.iws ) THEN
246 nb = max( lwork / ldwork, 1 )
247 nbmin = max( 2, ilaenv( 2, 'CHETRF', uplo, n, -1, -1, -1 ) )
248 END IF
249 ELSE
250 iws = 1
251 END IF
252 IF( nb.LT.nbmin )
253 \$ nb = n
254*
255 IF( upper ) THEN
256*
257* Factorize A as U*D*U**H using the upper triangle of A
258*
259* K is the main loop index, decreasing from N to 1 in steps of
260* KB, where KB is the number of columns factorized by CLAHEF;
261* KB is either NB or NB-1, or K for the last block
262*
263 k = n
264 10 CONTINUE
265*
266* If K < 1, exit from loop
267*
268 IF( k.LT.1 )
269 \$ GO TO 40
270*
271 IF( k.GT.nb ) THEN
272*
273* Factorize columns k-kb+1:k of A and use blocked code to
274* update columns 1:k-kb
275*
276 CALL clahef( uplo, k, nb, kb, a, lda, ipiv, work, n, iinfo )
277 ELSE
278*
279* Use unblocked code to factorize columns 1:k of A
280*
281 CALL chetf2( uplo, k, a, lda, ipiv, iinfo )
282 kb = k
283 END IF
284*
285* Set INFO on the first occurrence of a zero pivot
286*
287 IF( info.EQ.0 .AND. iinfo.GT.0 )
288 \$ info = iinfo
289*
290* Decrease K and return to the start of the main loop
291*
292 k = k - kb
293 GO TO 10
294*
295 ELSE
296*
297* Factorize A as L*D*L**H using the lower triangle of A
298*
299* K is the main loop index, increasing from 1 to N in steps of
300* KB, where KB is the number of columns factorized by CLAHEF;
301* KB is either NB or NB-1, or N-K+1 for the last block
302*
303 k = 1
304 20 CONTINUE
305*
306* If K > N, exit from loop
307*
308 IF( k.GT.n )
309 \$ GO TO 40
310*
311 IF( k.LE.n-nb ) THEN
312*
313* Factorize columns k:k+kb-1 of A and use blocked code to
314* update columns k+kb:n
315*
316 CALL clahef( uplo, n-k+1, nb, kb, a( k, k ), lda, ipiv( k ),
317 \$ work, n, iinfo )
318 ELSE
319*
320* Use unblocked code to factorize columns k:n of A
321*
322 CALL chetf2( uplo, n-k+1, a( k, k ), lda, ipiv( k ), iinfo )
323 kb = n - k + 1
324 END IF
325*
326* Set INFO on the first occurrence of a zero pivot
327*
328 IF( info.EQ.0 .AND. iinfo.GT.0 )
329 \$ info = iinfo + k - 1
330*
332*
333 DO 30 j = k, k + kb - 1
334 IF( ipiv( j ).GT.0 ) THEN
335 ipiv( j ) = ipiv( j ) + k - 1
336 ELSE
337 ipiv( j ) = ipiv( j ) - k + 1
338 END IF
339 30 CONTINUE
340*
341* Increase K and return to the start of the main loop
342*
343 k = k + kb
344 GO TO 20
345*
346 END IF
347*
348 40 CONTINUE
349 work( 1 ) = lwkopt
350 RETURN
351*
352* End of CHETRF
353*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine clahef(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
CLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kauf...
Definition: clahef.f:177
subroutine chetf2(UPLO, N, A, LDA, IPIV, INFO)
CHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (...
Definition: chetf2.f:186
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